# Who Gains When Interest Rates Fall?

**Authors:** Sylvain Catherine, Max Miller, James D. Paron, Natasha Sarin
**Date:** March 20, 2026
**Affiliation:** Stanford University Graduate School of Business
**PDF (definitive version):** https://jamesparon.github.io/papers/who-gains-interest-rates/paper.pdf
**Keywords:** Interest rates, Portfolio choices, Inequality, Social Security
**JEL codes:** D31, E21, G51, H55

> Full text below is converted from the paper's LaTeX source, with mathematics in TeX math mode. The linked PDF is the definitive version.

## Abstract

We study the interest-rate sensitivity of household wealth in a realistic life-cycle model. The model predicts that middle-aged and wealthier households should hold more long-term assets, as observed in the US data. Consequently, optimal portfolio rules imply that falling interest rates increase wealth inequality, while rising rates reduce inequality, consistent with historical experience. However, these long-run shifts in wealth inequality are largely offset by changes in the valuation of human capital and Social Security benefits, mitigating the passthrough of interest-rate fluctuations to expected lifetime consumption.

## Full text

Over the past four decades, declining real interest rates have been a driving force behind the rise in asset prices, especially for long-duration assets such as equities and long-term bonds (Binsbergen 2021). Because these assets are disproportionately held by the wealthy, the resulting capital gains have contributed to the surge in wealth inequality (Greenwald et al. 2023). Still, the distributional consequences of falling interest rates must be interpreted with caution. When future rates of return are lower, the same level of wealth can fund less consumption over the life cycle. In order to finance \$100 of consumption at age 70, a 35-year-old must set aside \$71 at a 1% return, compared to only \$36 at a 3% return. Hence, falling interest rates only benefit households if their capital gains outweigh the slower rate of future wealth accumulation.

Overall, the net distributional effect of falling interest rates depends on whether households hold portfolios with the appropriate duration, such that their capital gains can offset changes in future rates of return. The appropriate portfolio depends on multiple factors, including one's age, wealth, and future labor income and retirement benefits. To take all these factors into account, we build and calibrate a life-cycle model with stochastic interest rates in which households can choose the interest-rate sensitivity of their wealth by investing in assets of different maturities.

Our analysis delivers two sets of findings. First, the model's predictions for individually optimal interest-rate sensitivity align with key stylized facts about households' portfolios in the US Survey of Consumer Finances. As in the data, the optimal interest-rate sensitivity of wealth is hump-shaped over the life cycle and increasing in both wealth and earnings. Second, the model implies that declines (increases) in real interest rates generate large and persistent rises (falls) in wealth concentration. Overall, the model accounts for roughly two-thirds of the decline in wealth inequality between the 1960s and the mid-1980s and its subsequent rise. However, the impact on welfare---as measured by lifetime utility---is much more even across households, highlighting the important role that Social Security has played in insuring households against falling interest rates.

Our first contribution is to develop a realistic portfolio choice model with stochastic real interest rates to understand how investors should think about the interest-rate sensitivity of their balance sheet. We then compare the predictions of the model to the data to evaluate how close households get to this benchmark in practice.

We find that the individually optimal interest-rate sensitivity of wealth is highest for middle-aged, high-income, and high-wealth households. These patterns reflect the interaction of several forces. Without labor income or Social Security, younger households would prefer portfolios with high interest-rate sensitivity because of their long investment horizons. Introducing labor income changes this prediction: for young workers, human capital acts like an implicit long-term bond, pushing them toward investing their wealth in short-term assets. As workers age, human capital shortens in duration and the sign of its substitution effect flips, inducing households to invest more in long-term assets. This effect fades as retirement approaches and the value of human capital converges to zero. Together, these forces generate a hump-shaped relationship between age and the optimal interest-rate sensitivity of wealth. Social Security adds another layer: because benefits behave like very long-term annuities, they reduce the need for households to hold long-duration assets. Since these benefits represent a smaller share of lifetime resources for high earners, this substitution effect is weaker at the top of the distribution, producing a positive relationship between wealth and the optimal interest-rate sensitivity.

Empirically, US household portfolios align closely with these predictions, suggesting that households approximately follow optimal rules. This alignment raises the question of how household financial behaviors replicate these portfolio rules in practice. To fully eliminate interest-rate risk, households should invest in assets with cashflows of different maturities so that, together with expected future earnings and benefits, these payments would match the timing of their desired consumption plan. If held to maturity, these assets would finance a predetermined consumption path regardless of interest-rate fluctuations, because each cashflow would arrive exactly when needed. In practice, households engage in a mix of voluntary and mandatory arrangements that mimic aspects of this abstract portfolio rule.

At the lower end of the earnings distribution, workers save primarily through contributions to Social Security, whose rate of return is unaffected by interest rates. Payroll taxes and benefits redistribute labor earnings over the life cycle to match the timing of a smoothed consumption plan, so these households do not need to actively invest in long-term assets to hedge against interest-rate changes---Social Security already provides that protection. For lower middle-income households, Social Security covers most, but not all, of retirement consumption. These households typically supplement it with private savings, most often by buying a house with a fixed-rate mortgage. This arrangement exchanges future housing services for a fixed stream of payments, locking in the cost of residential consumption. It protects households against both interest-rate risk as well as aggregate house-price risks. Social Security then finances most of their other retirement spending. Higher earners, who receive the lowest replacement rates from Social Security, rely more heavily on private retirement savings, often in stock-based accounts. Because stock values tend to rise when interest rates fall, these assets help sustain retirement consumption when rates of return fall.

Our second contribution is to study what individually optimal portfolio rules imply for trends in wealth inequality and their interpretation. As (Moll 2021) notes, interpreting these trends is difficult because the welfare implications of capital gains from discount-rate shocks are ambiguous. They depend on whether households fully hedge their interest-rate exposure (Auclert 2019).

We show that, given the historical path of interest rates, optimal portfolio rules generate large fluctuations in wealth concentration. These fluctuations account for roughly two-thirds of the decline in wealth inequality between 1960 and 1985 and its subsequent increase. Moreover, because wealthy households invest a larger share of wealth in stocks, rising stock prices---net of what would be predicted by the stock market's interest-rate exposure---further amplified these trends. On the other hand, because middle-class families invest a greater fraction of their wealth in housing, changes in national house prices had an opposing (but small) effect on wealth inequality.

These trends look markedly different under broader wealth concepts, a prediction that echoes the empirical finding that trends in inequality strongly depend on the inclusion or exclusion of accrued Social Security benefits (Catherine et al. 2025). In our model, heterogeneity in portfolios (and thus in wealth returns) is largely driven by substitution effects from off-balance-sheet assets. Therefore, once Social Security and human capital are included in wealth, much of this heterogeneity disappears. We document that this prediction of our model is consistent with the data: including Social Security in our measure of wealth evens out the interest-rate sensitivity of wealth across the earnings and wealth distributions.

An implication of this insight is that trends in wealth inequality induced by interest-rate fluctuations must be interpreted with caution. Indeed, our model generates large changes in the concentration of wealth---roughly two-thirds of historical variations since 1960---without any increase in within-cohort *welfare* inequality. This follows from the optimal portfolio decision: households target the interest-rate sensitivity of their wealth to offset the implicit exposure of their background assets, so that, ex post, their lifetime consumption is similarly exposed to interest-rate shocks.

On the other hand, changes in the interest-rate environment can affect cohorts differently for two reasons. First, because households are not infinitely risk-averse, they do not fully hedge interest-rate risk. Consequently, falling interest rates lower the lifetime consumption of younger households disproportionately more than their parents. Second, households cannot hedge against the level of interest rates when they enter the labor force. As such, we estimate that the baby boomers, who entered the labor force in 1970 before most of the rise in asset prices, saw about 10% more lifetime consumption due to the level of interest rates.

Social Security plays a central role in our analysis because it provides a powerful hedge against declines in real interest rates for lower- and middle-class workers. At the same time, Social Security faces a well-known long-run funding shortfall, which could be addressed by reducing benefits or raising the retirement age.[^6] Using the model, we show that an unanticipated, permanent benefit reduction carries a higher welfare cost in a low-rate economy because it strips away the retirement resources households rely on precisely when their other assets are bound to deliver lower returns.[^7]

#### Related literature

Our paper bridges the literatures on portfolio choice and wealth inequality.

First, we contribute to the study of households' portfolio choices. Following the seminal work of (Samuelson 1969) and (Merton 1969), this literature has focused on the allocation of wealth between different asset classes, and in particular the choice of investing in the stock-market portfolio or a short-term riskfree bond. More recent studies have expanded this class of models to incorporate real-estate investment (Cocco 2005). Our paper builds on these studies to underline the role of another risk factor: the interest-rate sensitivity of household balance sheets.

A growing empirical literature underscores the importance of household exposure to interest rates over long horizons. (Greenwald et al. 2023) document that wealthy households invest more in long-term assets, so their capital gains are larger when rates fall, which can explain much of the rise in wealth inequality since the mid 1980s. (Binsbergen 2021) shows that, between 1986 and 2021, a portfolio of 25-year US Treasuries earned annualized returns comparable to the S&P 500. (Catherine et al. 2025) further document that trends in wealth inequality are largely offset by the inclusion of accrued Social Security benefits, which increased in value as interest rates declined.

Our contribution is to derive optimal portfolio rules rather than take them from the data, as in (Greenwald et al. 2023). We build on the intuition of (Campbell and Viceira 2001) that investors can hedge interest-rate risk with long-term bonds,[^8] to study how mortality, human capital, and Social Security generate heterogeneity in the optimal hedging demand. Low-earnings and low-wealth households optimally hold portfolios with low interest-rate sensitivity because they are already implicitly invested in a long-duration asset---their future Social Security benefits. Empirically, we show that once Social Security is included, the relationship between wealth and duration documented by (Greenwald et al. 2023) largely disappears, as predicted by the model. This mechanism also explains why market-based wealth inequality closely tracks interest rates, yet remains flat when accrued Social Security benefits are included, as documented by (Catherine et al. 2025). Overall, our framework unifies empirical evidence on cross-sectional portfolio differences and trends in wealth inequality across alternative wealth concepts.

Our findings complement existing work analyzing the welfare implications of interest-rate-driven redistribution. (Gomez and Gouin-Bonenfant 2024) show that lower rates increase wealth inequality through entrepreneurship, because raising equity is cheaper and the valuation of private business is higher. (Fagereng et al. 2025) develop a sufficient-statistics approach to measure how historical changes in asset prices redistributed welfare using transaction data. (Auclert 2019) studies the implications of unhedged interest-rate exposure for monetary policy transmission. Like (Auclert 2019), we emphasize that the interest-rate exposure of wealth inclusive of non-financial income is the welfare-relevant measure. Our life-cycle approach provides a benchmark for evaluating the extent to which the interest-rate exposure of wealth is individually optimal.

In endogenizing portfolio choices, we explain an important driver of portfolio-return heterogeneity. The importance of this work is highlighted by (Moll 2021), who argues that explaining the portfolio choices that generate heterogeneous returns is essential. (Benhabib et al. 2019), (Bach et al. 2020), and (Hubmer et al. 2021) have empirically documented that the higher returns of the wealthy are crucial for explaining wealth inequality and its evolution. We provide an explanation for a large part of this heterogeneity: wealthier households should invest more in long-term assets in order to be equally hedged against interest-rate risk.

Recent work by (Fagereng et al. 2021) shows that capital gains explain most of the rise in wealth inequality because the rich "save by holding" instead of selling assets to consume. This behavior is consistent with interest-rate hedging. As an extreme example, if households hold a perfectly interest-rate-hedged portfolio to maturity, then they will be able to afford the same consumption plan regardless of discount-rate-induced changes in asset prices. As (Cochrane 2022) puts it, a family should not pay attention to these fluctuations if it intends to live off of the coupons.

Finally, this paper builds on recent studies that have focused on the ways in which progressive government programs attenuate wealth inequality (Catherine et al. 2025), income inequality (Auten and Splinter 2024), and their passthrough to lifetime consumption (Auerbach et al. 2023). Our paper shows that including transfer income in wealth yields a measure that is more relevant for the cross-sections of both portfolios and welfare. In particular, we show how the interplay between public programs like Social Security and optimal portfolio choices can generate diverging trends in wealth and consumption inequality. Our mechanism helps explain (Meyer and Sullivan 2023)'s finding that, over the past five decades, the rise in overall consumption inequality was small.

# Data and measurement

This section describes our data sources and the methodology we adopt to measure the interest-rate sensitivity of wealth. We provide more details about the data in Appendix 11.

## Definitions

#### Interest-rate sensitivity of an asset

Consider an asset with current price $P_t$ and future cashflows $\{D_{t+k}\}_{k=1}^\infty$. Let $r_{ft}$ denote the log real interest rate. We define the *interest-rate sensitivity* of the asset as the change in the asset's price in response to a decline in the interest rate, holding future cashflows and risk premia fixed: $$
 \varepsilon_r(P_t) \equiv -\frac{\partial\log P_t}{\partial r_{ft}}.
$$ Appendix 10 gives a formal definition, along with an explanation of how this concept relates to an asset's *duration*, which is defined as the value-weighted timing of its cashflows.

#### Interest-rate sensitivity of wealth

The interest-rate sensitivity of wealth of household $i$ is the value-weighted sum of each component elasticity: $$
 \varepsilon_r(\text{W}_i) = \sum_{j} \frac{\text{A}_{ji}}{\text{W}_i}\times \varepsilon_r(\text{A}_{ji}),
$$ where $\text{A}_{ji}$ denotes the value of the asset or debt $j$, $\varepsilon_r(\text{A}_{ji})$ its interest-rate elasticity, and $\text{W}_i$ the household's net worth.

## Main data sources

#### Survey of Consumer Finances

To measure the interest-rate sensitivity of assets on households' balance sheets, our primary data source is the triennial Survey of Consumer Finances (SCF), from which we use all survey years from 1989 to 2019. We use three main series: (i) detailed information on household assets and liabilities, (ii) interest rate and maturity data for household liabilities, and (iii) income data for household assets, which we use to compute valuation ratios.

#### Interest-rate sensitivity estimates

We supplement the SCF with data on Social Security wealth and its interest-rate sensitivity from (Catherine et al. 2025), duration estimates from (Greenwald et al. 2023) and Bloomberg, and the real yield curve, computed by subtracting inflation projections from the SSA annual reports from the nominal yield curve.

#### Yield curves

Historical data on the nominal yield curves are obtained from the Federal Reserve System.[^9] The zero-coupon yield curve is estimated using off-the-run Treasury coupon securities for horizons up to 30 years.

## Interest-rate process

#### Parametric assumptions

The interest-rate sensitivity of assets depends on how shocks to current interest rates will affect future interest rates. We assume that log interest rates follow a first-order autoregression: $$
 r_{f,t+1} = (1-\varphi)\bar{r}_f + \varphi r_{ft} + \sigma_r\epsilon_{r,t+1},
 
$$ where $\epsilon_r$ is a standard normal shock. When interest rates follow an AR(1) process, the interest-rate sensitivity of a zero-coupon bond paying off 1 in $k$ years is $$
 \hat{\varepsilon}_r(P_{kt}) = \frac{1-\varphi^{k}}{1-\varphi}.
 
$$ Here, the bond's maturity $k$ is also its duration. The interest-rate sensitivity is increasing in the duration; they are identical ($\hat{\varepsilon}_r(P_{kt})\to k$) if interest-rate shocks are permanent ($\varphi\to1$).

#### Calibration

Since our focus is on obtaining realistic asset-price levels and capital gains for rate-sensitive assets, we calibrate the stationary mean, persistence, and volatility to match moments of the real yield curve. Over our sample period of 1989--2019, we target (i) the slope from a linear regression of the 30-year real forward rate ($f_{30}$) on the current one-year real yield, (ii) the average 30-year real forward rate, and (iii) the unconditional volatility of the one-year real yield. These three moments provide an exactly identified system that defines each parameter in terms of data moments. Table 1 reports the data moments, their relationship to model parameters, and implied parameter estimates.

+:------------------------------------------------:+:--------------------------:+:----------:+:-:+:------------+:-----------+
| Moment conditions | | Parameter estimates |
+--------------------------------------------------+----------------------------+------------+---+-------------+------------+
| Data moment | Model equivalent | Data value | | Parameter | Value |
+--------------------------------------------------+----------------------------+------------+---+-------------+------------+
| $\text{cov}(f_{30,t},r_{ft})/\text{var}(r_{ft})$ | $\varphi^{30}$ | 0.2569 | | $\varphi$ | 0.9557 |
+--------------------------------------------------+----------------------------+------------+---+-------------+------------+
| $\bar{f}_{30,t}$ | $\bar{r}_f$ | 0.0193 | | $\bar{r}_f$ | 0.0193 |
+--------------------------------------------------+----------------------------+------------+---+-------------+------------+
| $\text{var}(r_{ft})$ | $\sigma_r^2/(1-\varphi^2)$ | 0.0167 | | $\sigma_r$ | 0.0049 |
+--------------------------------------------------+----------------------------+------------+---+-------------+------------+

: **Parameter values and moments for the riskfree rate process**

## Empirical estimates in the SCF 

We obtain the rate sensitivity of households' wealth in three steps. First, we estimate cashflow duration for each asset and liability on households' balance sheets. Second, we apply equation ) to this cashflow duration, taking the rate sensitivity of the asset to be the same as the rate sensitivity of a riskfree zero-coupon bond with the same duration. Third, we take a value-weighted sum of the assets' rate sensitivities to arrive at the overall portfolio rate sensitivity of each household.[^10]

Although we use similar cash-flow duration estimates for equities as (Greenwald et al. 2023), our mapping from duration to interest-rate sensitivity is more conservative. We allow for mean reversion in the short rate ($\varphi<1$), so a rate shock does not fully pass through to expected future rates. This attenuates the implied interest-rate sensitivity of long-term assets, implying smaller interest-rate sensitivity gaps across assets than under a near-permanent-rate-change benchmark.

Table presents the averages of our interest-rate sensitivity and duration estimates by asset group and for wealth for households in the SCF. The average balance sheet interest-rate sensitivity suggests that interest-rate risk generates substantial volatility in returns to wealth.

 

@ l\*4Y @ & IR Sensitivity & Duration &\
& & & equal-weighted & wealth-weighted\
**Assets** & & & &\
Private Business & 14.14 & 21.73 & 0.03 & 0.16\
Equity & 20.31 & 50.76 & 0.12 & 0.23\
Real Estate & 9.78 & 12.53 & 0.41 & 0.36\
Fixed Income & 3.26 & 3.44 & 0.10 & 0.14\
Vehicles & 3.17 & 3.34 & 0.19 & 0.03\
Cash and Deposits & 0.25 & 0.25 & 0.14 & 0.07\
& & & &\
**Liabilities** & & & &\
Mortgage Debt & 7.34 & 8.68 & 0.52 & 0.83\
Other Debt & 2.59 & 2.69 & 0.48 & 0.17\
& & & &\
**Networth** & & & &\
Value-weighted & 12.28 & 17.33 & &\
Equal-weighted & 8.38 & 10.24 & &\

*Note*: This table reports the average interest-rate sensitivity, duration, and portfolio share of each asset group for households in the SCF. Interest-rate sensitivity is calculated from asset duration as in equation ). To calculate average duration for each asset group, we take the duration estimate for each household's holdings in the group and then average them across households, weighting each household's contribution by its SCF sample weight and the value of its holding in the asset group. To obtain the equal-weighted portfolio shares, we take the share of each asset group within a household's portfolio and then average the shares across households using SCF sample weights. We repeat this process for the wealth-weighted portfolio shares, but in this case we weight each household by both its SCF sample weight and its networth.

To obtain these estimates, we adopt a variety of methods to compute the cashflow durations of assets and liabilities. We only provide an overview of our methodology in this section; we offer more information about our construction of all of the different components in Appendix 11.2.

#### Equity, real estate, liquid assets, and fixed income

For all equity, real estate, and liquid assets, we apply the annual group-wide duration estimates provided by (Greenwald et al. 2023). For fixed-income assets, we collect annual average duration estimates from Bloomberg for government debt, municipal bonds, mortgage-backed securities, foreign bonds, and corporate bonds and apply them to each asset within the fixed-income group accordingly.

#### Private business wealth

Private business wealth duration varies across entrepreneurs and is estimated from price-dividend ratios inferred from the SCF.[^11] In particular, we assume $$
 \text{dur}(\text{Private Business}_{ct}) = \underbrace{\left(\frac{\sum_{t,i \in c}\text{Bus. Value}_{it}}{\sum_{t,i \in c}\text{Bus. Income}_{it}}\right)}_{\text{Wealth-group-specific valuation multiple}} \times \:\:\underbrace{\left(\frac{\sum_i \text{Bus. Income}_{it}}{\sum_i(\text{Bus. Income}_{it} - \text{Wages}_{it})}\right)}_{\text{Business-income-to-dividend conversion ratio}}
 
$$ where $i$ denotes individual households and $c$ denotes groups in the wealth distribution. We do this because valuation ratios are higher at the top of the earnings distribution, since wealthy entrepreneurs tend to run different types of businesses (Schoar 2010). The first term is the ratio of business value to income for each wealth group, which can be computed directly in the SCF. The second term converts this valuation multiple into a price-dividend ratio. This is because business income includes both dividends and wages paid to the entrepreneur. Wages to business owners are not always reported in the SCF; therefore, we estimate them using information on households' level of education and age. Since this estimation procedure entails some noise, we assume all private businesses have a common conversion ratio across wealth groups and over time. For more information on this, see Appendix 11.2.4.

#### Vehicles

To determine the duration of vehicles, we compute the time left on a vehicle's life using the age of the vehicle provided in the SCF and an estimate of its maximum lifetime. We then assume a constant depreciation rate to determine its cashflow duration.

#### Defined benefit pensions

Defined benefit (DB) pension entitlements are not directly reported in the SCF at the asset level, so we impute them using total DB pension assets from the Distributional Financial Accounts and individual characteristics in the SCF. We allocate DB pension wealth across households based on (i) DB plan participation, (ii) predicted accrual rates, and (iii) income. The funded portion of these entitlements is then split across equity, fixed income, and cash deposits based on Flow of Funds pension fund asset allocations (Table L.117), and added to the corresponding asset categories in the household balance sheet. They then receive the wealth-weighted average duration of that category. For more information on how this is constructed please see Appendix 11.2.6.

#### Liabilities

For each household's liabilities in the SCF, we assume a fixed repayment schedule and estimate duration as $$
 
 \text{dur}(\text{Debt}_t) = \sum_{n=1}^N\left(\frac{P_{nt}}{\sum_{n'=1}^NP_{n't}}\right) n,
$$ where $N$ is the number of years remaining on the loan given in the SCF and $P_{nt}$ is the price of an $n$-year zero-coupon bond. The number of years remaining on the loan is either given explicitly in the SCF or can be inferred from the interest rate and loan balance outstanding. The exception to this is adjustable rate mortgages, for which we assign a maximum duration of 4---the average number of years in the data until they begin to float.

For student loans, we adjust the debt payments to account for income-driven repayment programs as in (Catherine and Yannelis 2021). To do this, we compute the present value of expected income-driven repayment payments over a 25-year horizon, where the annual payment equals 10% of income above a threshold tied to the Social Security wage index. This present value replaces the raw SCF balance in our net worth measure.

# Stylized facts 

We begin by documenting five stylized facts about the directly observable side of households' interest-rate exposure: the interest-rate sensitivity of wealth $\varepsilon_r(W)$. Although it only paints a partial picture of households' exposures to interest rates, the distribution of $\varepsilon_r(W)$ is informative about households' portfolio allocation decisions. These facts will later guide our structural analysis.

#### **Fact 1: Interest-rate sensitivity is hump-shaped over the life cycle**

The first stylized fact is that the rate sensitivity of wealth is hump-shaped over the life cycle: it is lowest for 20-year-olds, rises to a high for 40- to 45-year-olds, and steadily declines thereafter. Figure 1 decomposes this pattern, showing the relative contribution of each asset to the total portfolio rate sensitivity. The difference in portfolio interest-rate sensitivities at each age is determined by the assets households choose to hold. For example, 20- to 25-year-old households have relatively low interest-rate sensitivities because the majority of their wealth (70.4%) is invested in liquid accounts (e.g., checking and savings accounts) and vehicles.

<figure id="fig:lf_by_earnings" data-latex-placement="h">
<table>
<tbody>
<tr>
<td style="text-align: left;"><strong>A. First earnings tercile</strong></td>
<td style="text-align: left;"><strong>B. Second earnings tercile</strong></td>
<td style="text-align: left;"><strong>C. Third earnings tercile</strong></td>
</tr>
<tr>
<td style="text-align: left;"><embed src="Plots/irs_stacklines_age_all_ter1.pdf" /></td>
<td style="text-align: left;"><embed src="Plots/irs_stacklines_age_all_ter2.pdf" /></td>
<td style="text-align: left;"><embed src="Plots/irs_stacklines_age_all_ter3.pdf" /></td>
</tr>
</tbody>
</table>
<p><em>Note</em>: This figure reports the interest-rate sensitivity of wealth by age and tercile of earnings. The rate sensitivity is decomposed into the contribution of six components of wealth. From bottom to top, we calculate the sensitivity of partial portfolios, adding components step-by-step. First, we report the interest-rate sensitivity of liquid assets and fixed-income assets. We then report the rate sensitivity of a larger portfolio that also includes vehicles, and so forth. Thus, the interest-rate sensitivity of the partial portfolio inclusive of the first <span class="math inline"><em>k</em></span> components of wealth is <span class="math inline">$\hat{\varepsilon}_r(\text{Portfolio}_k) = \frac{\text{Portfolio}_{k-1}}{\text{Portfolio}_k}\hat{\varepsilon}_r(\text{Portfolio}_{k-1})+\frac{\text{Component}_{k}}{\text{Portfolio}_k}\hat{\varepsilon}_r(\text{Component}_{k})$</span>.</p>
<figcaption><strong>Interest-rate sensitivity of wealth by age</strong> </figcaption>
</figure>

As households approach midlife, the composition of assets changes and the interest-rate sensitivities of their portfolios grow. The majority of their portfolio (48.7% for 40-year-olds) is now made up of longer-term assets like equity and real estate. Moreover, leverage---in particular, mortgages and other debts---plays a more important role, increasing the rate sensitivity of the wealth portfolio by nearly 20%. The reason leverage increases the rate exposure of the household's portfolio is because the (equal-weighted) average rate sensitivity of assets is approximately 40% higher than that of debts over our sample.

As midlife turns to retirement, the rate sensitivity of wealth begins to fall. The decline in rate exposure is driven not by the asset side of the portfolio, but rather by the disappearance of leverage, which reduces the interest-rate sensitivity of the wealth portfolio. This is consistent with the conventional narrative in saving for retirement: households with a large stock of human capital take on mortgages in early adulthood to guarantee housing consumption flows in old age.

#### **Fact 2: Interest-rate sensitivity is increasing in earnings**

The second stylized fact is that high-earning households hold more rate-sensitive portfolios. Comparing the three panels of Figure 1 shows that, for a 1% decline in interest rates, those in the top earnings tercile will see approximately 4 percentage points larger capital gains than those in the bottom earnings tercile. High earners investing more in equity explain most of this difference.

#### **Fact 3: Interest-rate sensitivity is increasing in wealth**

The third stylized fact is that interest-rate sensitivity is generally increasing in wealth. This fact is shown in Figure 2, which decomposes the average rate sensitivity for households between ages 40 and 45 over the log of their wealth scaled by the Social Security Wage Index in their survey year.

<figure id="fig:duration_wealth_scf" data-latex-placement="H">
<div class="center">
<p><embed src="Plots/irs_stacklines_ln_networth_all.pdf" style="width:100.0%" /><br />
</p>
</div>
<p><em>Note</em>: This figure decomposes the interest-rate sensitivity for households in which the head of the household is between 40 and 45. The methodology is the same as in Figure <a href="#fig:lf_by_earnings" data-reference-type="ref" data-reference="fig:lf_by_earnings">1</a>, except that here the x-axis is the log of wealth scaled by the Social Security Wage Index in the survey year.</p>
<figcaption><strong>Interest-rate sensitivity of wealth at ages 40–45 by level of wealth</strong> </figcaption>
</figure>

For low-wealth households, liquid accounts, vehicles, and non-mortgage debt contribute the most to the interest-rate sensitivity of their portfolios. For middle-wealth households, real estate becomes the dominant asset, with its rate sensitivity amplified by the mortgage taken on to finance the purchase. The large, indivisible nature of a house leads lower-middle-wealth homebuyers to take out large mortgages and expose themselves to interest-rate fluctuations, which is reflected in the small bump in rate sensitivity near the lower-middle portion of the wealth distribution. As wealth increases, portfolio rate exposures increase with larger positions in highly rate-sensitive assets like publicly traded equity and private businesses.

#### **Fact 4: Wealth inequality follows interest rates**

Our fourth stylized fact is that, as documented by (Greenwald et al. 2023), the wealth share of the top 10% has tracked the price of real bonds over the past six decades. Figure 3 replicates their finding, approximating real bond prices by one minus the estimated 10-year forward rate and measuring wealth inequality using estimates from (Smith et al. 2023) (Panel A) and (Piketty et al. 2018) (Panel B). While this historical correlation does not, by itself, imply that interest-rate fluctuations account for most changes in wealth inequality, it suggests that the mechanisms illustrated in Figures 1 and 2 can plausibly explain an important share of these long-run trends.

<figure id="fig:wealthInqRR" data-latex-placement="H">
<div class="center">
<table>
<tbody>
<tr>
<td style="text-align: center;"><strong>A. Smith, Zidar, and Zwick (2023)</strong></td>
<td style="text-align: center;"><strong>B. Piketty, Saez, and Zucman (2018)</strong></td>
</tr>
<tr>
<td style="text-align: center;"><embed src="Plots/inequality_and_interest_rates_szz.pdf" /></td>
<td style="text-align: center;"><embed src="Plots/inequality_and_interest_rates_sz.pdf" /></td>
</tr>
</tbody>
</table>
</div>
<p><em>Note</em>: This figure presents the time series of the top 10% wealth share from <span class="citation" data-cites="SZZ2023">(Smith et al. 2023)</span> in Panel A and <span class="citation" data-cites="PSZ2018">(Piketty et al. 2018)</span> in Panel B against <span class="math inline">1 − <em>f̂</em><sub>10, <em>t</em></sub></span>, one minus our estimated 10-year real forward rate from equation (<a href="#eq:campbeel" data-reference-type="ref" data-reference="eq:campbeel">[eq:campbeel]</a>).</p>
<figcaption><strong>Wealth inequality and estimated 10-year real forward rates</strong></figcaption>
</figure>

#### **Fact 5: Social Security offsets differences in rate sensitivity**

Finally, we extend our definition of wealth to include the net present value of Social Security payments---that is, of expected benefits minus expected payroll taxes to be paid into the system. Figure 4 displays the fifth stylized fact: the inclusion of Social Security wealth strongly attenuates the relationships between interest-rate exposure and wealth (Panel A) and earnings (Panel B).

<figure id="fig:duration_wealth_with_SS" data-latex-placement="H">
<div class="center">
<table>
<tbody>
<tr>
<td style="text-align: center;"><strong>A. Wealth</strong></td>
<td style="text-align: center;"><strong>B. Earnings</strong></td>
</tr>
<tr>
<td style="text-align: center;"><embed src="Plots/irs_stacklines_ln_networth_all_ssw.pdf" /></td>
<td style="text-align: center;"><embed src="Plots/irs_stacklines_ln_wageinc_all_ssw.pdf" /></td>
</tr>
</tbody>
</table>
</div>
<p><em>Note</em>: This figure decomposes the interest-rate sensitivity for households in which the head of the household is between 40 and 45. The methodology is the same as in Figure <a href="#fig:lf_by_earnings" data-reference-type="ref" data-reference="fig:lf_by_earnings">1</a>, except that here the x-axis is the log of wealth scaled by the Social Security Wage Index in the survey year in Panel A and scaled earnings in Panel B. Estimates for the net present value of Social Security at the individual level come from the risk-adjusted valuation of <span class="citation" data-cites="CMS2025">(Catherine et al. 2025)</span>.</p>
<figcaption><strong>Interest-rate sensitivity of wealth at ages 40–45: Role of Social Security</strong> </figcaption>
</figure>

# Model 

We model household consumption and investment decisions over a life cycle divided into two stages: working age and retirement. We first describe the economic environment and then define households' consumption-saving problems, including market frictions.

## Interest rates and economic growth

Interest rates and economic growth vary together over time. We thus model them as cointegrated stochastic processes. As in our empirics, we assume that the log riskfree rate, $r_{ft}=\log R_{ft}$, follows a first-order autoregression, $$
 r_{f,t+1} = (1-\varphi)\bar{r}_f + \varphi r_{ft} + \sigma_r\epsilon_{r,t+1}.
$$ To capture the cointegrating relationship between rates and growth, we assume that the log growth rate of aggregate income (defined below) is the sum of an autoregressive process and a loading on the riskfree rate:[^12] $$

 g_{t+1} = g_{1,t+1}+ \lambda_{gr} r_{f,t+1},
$$ where $$

 g_{1,t+1} = (1-\varphi_g)\bar{g} + \varphi_g g_{1t} + \epsilon_{g,t+1}.
$$ To capture business-cycle dynamics, we assume that systematic shocks to growth follow a normal mixture distribution, $$
 \epsilon_{g,t+1} = \begin{cases}\epsilon_{g,t+1}^{-} \sim \mathcal{N}\left(\mu_g^{-}, \sigma_{g_1}^2\right) & \text { with probability } p_R, \\ 
 \epsilon_{g,t+1}^{+} \sim \mathcal{N}\left(\mu_g^{+}, \sigma_{g_1}^2\right) & \text { with probability } 1-p_R,\end{cases}
$$ where $\mu_g^{+}>\mu_g^{-}$, so $p_R$ is the probability of a recession. The shocks $\epsilon_{r,t+1}$ and $\epsilon_{g,t+1}$ are independent.

## Asset returns

Households can invest their wealth in long-term bonds, stocks, and housing. We continue to denote log returns by lowercase $r = \log R$.

#### Long-term bonds

The long-term bond is a riskless claim to one unit of real consumption in $n$ periods. Its price, denoted $P_{nt}$, satisfies the expectations hypothesis, generalized to include constant term premia. Specifically, we assume that the term premium on each $n$-period bond is some constant $\mu_n$ (with $\mu_1=0$). As we show in Appendix 12.1, these assumptions imply an explicit relation between the dynamics of long-term bond returns and short-term rate fluctuations: the log bond return equals $$
 r_{n,t+1} = r_{ft} + \mu_n - \sigma_n\epsilon_{r,t+1},
 
$$ where the sensitivity to rate shocks $\sigma_n$ is given by $$
 \sigma_n = \frac{1-\varphi^{n-1}}{1-\varphi}\sigma_r.
$$ In addition, we set $\mu_n=-\sigma_n^2/2$, so that there is no risk premium.[^13]

Recall that the interest-rate sensitivity of the bond (or any asset) is the percentage change in the price caused by an unexpected decline in the interest rate, which equals $$
 \varepsilon_r(P_{nt}) \equiv -\frac{\partial\log P_{nt}}{\partial r_{ft}} = \frac{1-\varphi^n}{1-\varphi}.
 
$$ This sensitivity is increasing in maturity $n$. This expression summarizes the effect of unexpected changes in interest rates: if the riskfree rate unexpectedly falls, then the long-term bond has an unexpectedly high return from capital gains next period. The longer is the maturity $n$, and the higher is the persistence $\varphi$ of the rate shock, the larger is this response.

#### Stock market

The stock market is a claim to public firms' cashflows. Appendix 12.2 shows that, if cashflows are cointegrated with aggregate income, the stock's log return can be written $$
 r_{s,t+1} = r_{ft} - \lambda_{sr} \epsilon_{r,t+1} + \lambda_{sg} \epsilon_{g,t+1} + \epsilon_{s,t+1},
 
$$ where the market-specific shock is $$
 \epsilon_{s,t+1} = \begin{cases}
 \epsilon_{s,t+1}^{-} \sim \mathcal{N}\left(\mu_s^{-},\sigma_s^2\right) & \text { if } \epsilon_{g,t+1}=\epsilon_{g,t+1}^{-}, \\
 \epsilon_{s,t+1}^{+} \sim \mathcal{N}\left(\mu_s^{+},\sigma_s^2\right) & \text { if } \epsilon_{g,t+1}=\epsilon_{g,t+1}^{+}.
 \end{cases}
$$ The market-specific shocks depend on the realization of a recession ($\epsilon_{g,t+1}=\epsilon_{g,t+1}^{-}$), capturing kurtosis and heteroskedasticity in returns, specifically the fact that the stock market is more likely to fall in recessions (${\mu_s^{-}} < {\mu_s^{+}}$). The loading $\lambda_{sr}$ summarizes the net effect of interest-rate shocks on returns. On the one hand, a fall in rates will increase the value of the market, because it is a long-duration asset; but on the other hand, if growth and interest rates are positively correlated ($\lambda_{gr}>0$), then this will also tend to come with bad news about cashflows.[^14] We will find that the former dominates, so $\lambda_{sr}>0$.

#### Housing

The price of one unit of housing, denoted $P_{it}$, is the product of a regional (i.e., systematic) component $P_t$ and a house-specific (i.e., idiosyncratic) component $\tilde{P}_{it}$.[^15] The regional component can be decomposed as $$
 P_t = P_{ht}P_{n_ht},
$$ where $P_{n_ht}$ is the price of an $n_h$-period bond, matching the interest-rate sensitivity (and duration) of housing; and where $$
 \log\frac{P_{h,t+1}}{P_{ht}} = \mu_h + \sigma_h\epsilon_{h,t+1}, 
 
$$ captures the drift of house prices over time, with $\epsilon_{h,t+1}$ standard normal. The idiosyncratic component $\tilde{P}_{it}$ evolves as a random walk with Gaussian innovations $\tilde{\sigma}_h\tilde{\epsilon}_{hi,t+1}$.

For an owner-occupier, the return on a house is then[^16] $$
 r_{hi,t+1} = \underbrace{r_{ft} + \mu_{n_h} - \sigma_{n_h}\epsilon_{r,t+1}}_{r_{n_h,t+1}} + \underbrace{\mu_{h} + \sigma_{h}\epsilon_{h,t+1}}_{\Delta\log P_{h,t+1}} + \underbrace{\tilde{\sigma}_h\tilde{\epsilon}_{hi,t+1}}_{\Delta\log\tilde{P}_{i,t+1}},
 
$$ which includes both the return on the long-term bond component and the sum of the regional and idiosyncratic price appreciation.

## Labor income

We model labor-income dynamics using the empirically realistic process estimated by Catherine (2022) from the data of Guvenen et al. (2022). Each household $i$ earns labor income $L_{it}$, which is the product of the aggregate wage index $\bar{L}_t$ and an idiosyncratic component $\tilde{L}_{it}$. The aggregate wage grows at the rate $$
 \log\frac{\bar {L}_{t+1}}{\bar {L}_{t}} = g_{t+1},
$$ defined in ). The idiosyncratic component equals $$
 \tilde{L}_{it}=\exp\left\{\ell(a_{it})+z_{it}+\eta_{it}\right\}.
 
$$ The deterministic component $\ell(a_{it})$ is a polynomial of age $a_{it}$; it captures the common life-cycle profile of income. The persistent component of earnings, denoted by $z_{it}$, follows a first-order autoregression $$
 z_{it} = \rho z_{i,t-1} + \zeta_{it},
$$ with innovations $\zeta_{it}$ drawn from a mixture of normal distributions $$
\zeta_{it} = 
\begin{cases} 
 \zeta_{it}^-\sim\mathcal{N}(\mu_{zt}^-,{\sigma_z^-}^2) & \text{with probability } p_z, \\
 \zeta_{it}^+\sim\mathcal{N}(\mu_{zt}^+,{\sigma_z^+}^2) & \text{with probability } 1-p_z.
\end{cases}
$$ Idiosyncratic labor-income growth tends to be negatively skewed, so the state $\zeta_{it}=\zeta_{it}^-$ is rare ($p_z<0.5$) and tends to be negative. This skewness is also cyclical (more negative in recessions), so the mean outcome co-moves with aggregate income growth[^17] $$
 \mu_{zt}^- = \bar{\mu}_z^- + \lambda_{zg}g_t,
$$ with $\lambda_{zg}>0$. This rare state is also more volatile ($\sigma_z^- \gg \sigma_z^+$), meaning that extreme income events become more likely. The initial cross-sectional distribution of the persistent component of earnings is drawn from $z_{i0}\sim\mathcal{N}(0, \sigma_{z0}^2)$.

The transitory component of idiosyncratic earnings $\eta_{it}$ is also drawn from a mixture of normal distributions, with outcomes contingent on the permanent-income shock: $$
\eta_{it} \sim 
\begin{cases}
 \mathcal{N}(\mu_\eta^-,{\sigma_\eta^-}^2) & \text{if } \zeta_{it}=\zeta_{it}^-, \\
 \mathcal{N}(\mu_\eta^+,{\sigma_\eta^+}^2) & \text{if } \zeta_{it}=\zeta_{it}^+.
\end{cases}
$$ Large transitory shocks generally coincide with large persistent shocks ($\sigma_\eta^- \gg \sigma_\eta^+$).

## Social Security and income taxes

#### Social Security

Agents pay Social Security payroll taxes $T_{it}^{\rm SS}$ on their labor income during working life, then receive benefits $B_{it}$ in retirement. We assume all workers retire at the full-retirement age $a_{t_{\rm ret}}$, the age at which they receive 100% of their scheduled benefits. The tax payments are 10.6% of all income below the Social Security wage base, which is 2.5 times the average wage: $$
T_{it}^{\rm SS} = 0.106\min\{L_{it},2.5\bar{L}_t\}.
$$ Social Security retirement benefits depend on the agent's average indexed yearly earnings (AIYE), which is an average of their indexed earnings. The indexed earnings of worker $i$ for year $t$ is: $$
L_{it}^{\rm indexed} = \min\{L_{it},2.5\bar{L}_t\}\frac{\bar{L}_{t_{60}}}{\bar{L}_t}
$$ up to retirement, where $\bar{L}_{t_{60}}$ is the wage index when the worker was 60. In words, indexed earnings are the income below the wage base at a given age, adjusted for growth in the aggregate wage index $\bar{L}_t$ up to age 60. Income earned after age 60 but before retirement at $t_{\rm ret}$ can still contribute to the worker's AIYE, but it is indexed to $t_{60}$. Total benefits are then a piecewise-linear function of the AIYE when the worker retires: $$
B_{it} = \left\{
\begin{array}{ll}
 0.9 \text{AIYE}_{it_{\rm ret}} & if \text{AIYE}_{it_{\rm ret}} <b_{1t}, \\
 0.9 b_{1t}
 + 0.32 (\text{AIYE}_{it_{\rm ret}} - b_{1t}) & if b_{1t} \leq \text{AIYE}_{it_{\rm ret}} < b_{2t}, \\
 0.9 b_{1t} + 0.32 (b_{2t} - b_{1t}) + 0.15 (\text{AIYE}_{it_{\rm ret}} - b_{2t}) & if b_{2t} \leq \text{AIYE}_{it_{\rm ret}}. \\
\end{array}
\right.
$$ The kinks in this benefit formula are determined by the "bend points" $b_{1t}$ and $b_{2t}$, which historically are about 21% and 125% of the wage index, respectively. The formula is progressive: as AIYE (lifetime income) increases, the marginal benefit declines. Note that AIYE is itself bounded above due to the wage base, so benefits have an upper bound. Benefits after the retirement year are held constant in real terms---that is, they are adjusted in nominal terms to account for CPI inflation. Before retirement, we keep track of average indexed earnings as: $$
 \text{AIYE}_{it} = \frac{1}{t-t_0+1}\sum_{s=t_0}^{t} \min\{L_{is},2.5\bar{L}_{s}\} \frac{\bar{L}_{t}}{\bar{L}_s} = \frac{\bar{L}_t}{t-t_0+1}\sum_{s=t_0}^{t} \min\{\tilde{L}_{is},2.5\} .
$$

#### Safety net

Working-age households can receive SNAP benefits ("food stamps") if their income falls sufficiently low. Under SNAP, households receive a transfer equal to 6% of the wage index minus 30% of their pre-tax earnings, if this difference is positive.

Retirees with sufficiently low Social Security benefits receive Supplemental Security Income (SSI). SSI ensures total retirement income is at least 20% of the average wage, subject to a household's wealth remaining below 5% of the average wage.

#### Income tax

Households pay taxes on income and benefits according to the income tax brackets faced by US households in 2020, adjusted for changes in the aggregate wage.[^18] Marginal tax rates are progressively increasing in idiosyncratic income $\tilde{L}_i$; we report the formula in Appendix 13.3.

## Households 

#### Objective and preferences

Household $i$ chooses goods consumption $C_{it}$, housing consumption $H_{it}$, and portfolio shares $\pi_{it}=\{\pi_{n,it},\pi_{s,it},\pi_{h,it}\}$ to maximize lifetime utility $$
V_{it} = \max_{\{C_{is},H_{is},\pi_{is}\}}\mathbb{E}_t\sum_{s=t}^{t_{\rm max}}\beta^{s-t} p_{it,s-1}\left[(1-m_{i,s-1})u(C_{is},H_{is})+m_{i,s-1}b_s(W_{is})\right],

$$ where $\beta$ is the rate of time preference, $t_{\rm max}$ corresponds to the maximum lifespan, $m_{it}$ is the age- and income-dependent mortality probability from $t$ to $t+1$, and $p_{it,s} = \prod_{u=t}^{s-1}(1-m_{iu})$ is the probability of surviving from $t$ to $s$.

The household's flow utility is given by $$
 u(C_{it},H_{it}) = \frac{(C_{it}^{1-\nu}H_{it}^\nu)^{1-\gamma}}{1-\gamma}.%(C_{it}^{1-\nu}H_{it}^\nu)^{1-\gamma}.

$$ The parameter $\nu$ governs the preference for housing relative to goods; $\gamma$ is both the coefficient of relative risk aversion and the inverse of the elasticity of intertemporal substitution.

Households also bequeath to their children an inheritance from their terminal financial wealth. In modeling utility over bequests, one must consider the fact that inheritance does not necessarily constitute a one-time transfer of liquid wealth; it might instead be a long-lived flow of consumption. Fluctuating interest rates therefore affect the value of bequests by changing the price at which future consumption can be purchased by the bequeathed wealth. Hence, we model the bequest motive as the expected utility from distributing the household's terminal wealth $W_{it}$ over the next $\bar{b}$ years. In particular, the bequest utility is $$
 b_t(W_{it}) = b(\bar{L}_t,P_{ht},W_{it},r_{ft},g_{1t}) = \max_{\{C_{is},H_{is},\pi_{is}\}}\mathbb{E}_t\sum_{s=t}^{t+\bar{b}}\beta^{s-t} u(C_{is},H_{is})

$$ subject to the budget constraints (below) and assuming that no labor income or Social Security benefits are earned over this period.

#### Budget constraints and frictions

Households cannot short-sell stocks or housing, so portfolio shares must be non-negative. If the household decides to invest a positive amount in the stock market, then it must pay both a 1% management fee and a fixed, per-period participation cost of $$
 \Phi_{s,it}= \begin{cases}c_s \bar{L}_t & \text {if } \pi_{s,it}>0, \\ 0 & \text {otherwise}.\end{cases}
$$ Households also cannot borrow to invest in the stock market, meaning the equity share $\pi_{s,it}\leq1$.

The household must also decide whether to rent or own a house. If the household chooses to rent ($\pi_{h,it}=0$), then the rental price per unit of housing equals $$
 X_t = \begin{cases}\chi P_{ht} & \text {if $\pi_{h,it}=0$ (renter),} \\ 0 & \text {if $\pi_{h,it}>0$ (homeowner),}\end{cases}
$$ for rent-to-price ratio $\chi$. A renter does not have any collateral with which to borrow, and therefore must have $W_{it}\geq0$. If instead the household chooses to own ($\pi_{h,it}>0$), then, as in (Cocco 2005), the size of the house must exceed a minimum size, given by $$
 P_t H_{it} \geq \kappa_{\min} \bar{L}_t.
$$ Homeowners are also constrained to buy a house that is not too large: $$
 P_t H_{it} \leq \kappa_{\max} W_{it}
 
$$ This constraint is equivalent to a downpayment requirement: the household must have enough wealth $W_{it}$ to cover a fraction $1/\kappa_{\max}$ of the house's value. Homeowners pay proportionate property taxes. Finally, homeowners can use their housing and equity as collateral for a mortgage. That is, if their investments in housing and equity exceed their networth, $\pi_{h,it}+\pi_{s,it}>1$, then they must finance this with a mortgage that charges interest $R_{Mit}=e^{\theta_{it}}R_{ft}$ at a premium $\theta_{it}>0$, which is an increasing function of the leverage ratio $\pi_{s,it}+\pi_{h,it}$.

Putting these constraints together, the utility maximization ) is solved subject to the dynamic budget constraint $$
 W_{i,t+1} = (W_{it}+L_{it}+B_{it}-T_{it}-C_{it}-X_tH_{it}-\Phi_{s,it})R_{Wi,t+1},
 
$$ where the return on savings equals $$\begin{multline}
 R_{Wi,t+1} = R_{ft}+\pi_{n,it}(R_{n,t+1}-R_{ft}) +\pi_{s,it}(R_{s,t+1}-R_{ft}) \\ +\pi_{h,it}(R_{h,t+1}-R_{ft}) - (\pi_{h,it}+\pi_{s,it}-1)^{+}(R_{Mit}-R_{ft})
 
\end{multline}$$ for $(\pi_{h,it}+\pi_{s,it}-1)^{+} \equiv \max(\pi_{h,it}+\pi_{s,it}-1,0)$.

## Entrepreneurs

Some households own and manage a private business instead of working in the labor market. In particular, with some probability $p_E(a_{it})$, a worker of age $a_{it}$ will start a private business and operate it through working life. If this occurs, then the entrepreneur will no longer earn labor income according to ); rather, he or she will earn idiosyncratic business income according to $$
 \tilde{L}_{it}=\exp\left\{\ell_E(a_{it})+z_{it}\right\},
 
$$ which is the sum of a polynomial of age $\ell_E(a_{it})$ and a random walk $$
 z_{i,t+1}=z_{it} + \sigma_{Ez}\epsilon_{z,t+1}.
 
$$ At the beginning of the life cycle, the initial value of $z_{it}$ is drawn from $\mathcal{N}(0,\sigma_{Ez0}^2)$.

The entrepreneur operates the business until retirement or death---whichever is first---at which point the business is sold on the market and liquidated into market wealth at a value $E_{it}$. In Appendix 12.4, we show that the value of the business can be expressed as $$
 E_{it} = L_{it}\exp\left\{\upsilon + \frac{1}{1-\kappa_E(\upsilon)\varphi_g}(g_{1t} - \bar{g}_1) - \frac{1-\lambda_{gr}}{1-\kappa_E(\upsilon)\varphi}(r_{ft} - \bar{r}_f) \right\},
 
$$ where $\upsilon$ is the mean log valuation ratio (a function of mean discount rates and cashflow growth) and $\kappa_E(\upsilon)\equiv1/(1+\exp\{-\upsilon\})$. All business valuation ratios fluctuate over time with predictable variation in interest rates $r_{ft}$ and growth $g_{1t}$. Note that the value ) also corresponds to the market value of the business at any time before retirement.[^19]

# Economic intuition 

To communicate the key intuitions of our model, we present an analytical solution to a linearized version with no income risk or bequests, which we derive in Appendix 16. We proceed in three steps. First, we discuss consumption and portfolio rules for an agent without human capital or Social Security, trading in only a long- and a short-term bond. Second, we show that the same portfolio rule applies to *total* wealth in the presence of background assets such as human capital and Social Security. And third, we explain why these results continue to hold even when we introduce other tradable assets (e.g. equity, real estate) and risk factors (e.g., growth, house prices).

## Optimal choices without labor income

#### Consumption rule

Without labor income, the linearized model implies the optimal consumption policy $$
\frac{C_{it}^*}{W_{it}} = \underbrace{\vphantom{\left(\frac{1}{\gamma}\right)}(1-\beta(1-m_{it}))}_{\textrm{time discounting}} \times \underbrace{\exp\left\{\left(1-\frac{1}{\gamma}\right)\left(\varrho_{0t}+\varrho_{rt}r_{ft}\right)\right\}}_{\textrm{income and substitution effects}}.

$$ The first term represents the positive effect of impatience and mortality on consumption. The second term represents the net of income and substitution effects from interest rates. Higher rates mean higher interest income, so that households can consume more today (the income effect). At the same time, higher rates mean agents get more consumption tomorrow in exchange for their savings (the substitution effect). The income effect dominates the substitution effect when the elasticity of intertemporal substitution (the EIS, $1/\gamma$) is less than one ($\gamma>1$). The sensitivity of consumption to interest rates depends on the coefficient[^20] $$
 \varrho_{rt} = \sum_{j=1}^\infty\varphi^{j-1}\beta^{j}p_{it,t+j},
 
$$ where $p_{it,t+j}$ is the survival probability from year $t$ to $t+j$. The coefficient $\varrho_{rt}$ is declining in age because agents with shorter horizons are less affected by persistent rate changes.

#### Portfolio rule

The optimal allocation to the $n$-period bond is:[^21] $$
\pi_{it}^* = \underbrace{\frac{1}{\gamma}\frac{\mu_n+\frac{1}{2}\sigma_n^2}{\sigma_n^2}}_{\textrm{myopic demand}} + \underbrace{\left(1-\frac{1}{\gamma}\right)\varrho_{r,t+1}
\left(\frac{1-\varphi^{n-1}}{1-\varphi}\right)^{-1}}_{\textrm{interest-rate risk hedging demand}}.

$$ The first term represents the traditional risk-return tradeoff of (Merton 1969). Our assumption of $\mu_n=-\sigma_n^2/2$ sets this term to zero. The second term is the demand from intertemporal hedging of interest-rate fluctuations, the focus of our paper. Because the long-term bond's value increases when rates unexpectedly decline, it offers protection against the deterioration of investment opportunities. This portfolio rule implies that the optimal interest-rate sensitivity of wealth equals[^22] $$
 \varepsilon_r^*(W_{i,t+1}) = \pi_{it}^*\frac{\sigma_n}{\sigma_r} = \frac{1}{\gamma}\frac{\mu_n+\frac{1}{2}\sigma_n^2}{\sigma_n\sigma_r} + \left(1-\frac{1}{\gamma}\right) \varrho_{r,t+1}.
$$ Note that this optimal sensitivity is independent of the long-term bond maturity $n$. This illustrates the fact that households can mix any combination of short- and long-term assets to target their optimal total exposure to interest-rate risk.

The sensitivity of consumption to rate shocks $\varrho_{rt}$ declines with the investor's horizon, so the hedging demand decreases in age toward zero. From equation ), we see that $\varrho_{rt}$ represents the cumulative effect of an interest-rate shock over time, weighted by the importance of each future period in the agent's lifetime utility, taking into account mortality and impatience. In other words, it results from the interplay between the agent's horizon and the persistence of interest-rate shocks. Agents with shorter horizons are less sensitive to future rate changes; therefore, for $\gamma>1$, the optimal interest-rate sensitivity of wealth declines over the life cycle.

Interestingly, for a risk-neutral agent ($\gamma=0$), the hedging demand is infinitely negative. A risk-neutral agent would like to move as much wealth as possible to the high-rate states of the world, in order to enjoy the higher compounding. This is accomplished by short-selling the long-term bond, which depreciates when rates rise, generating capital gains. Such an investor chooses infinite expected consumption (with infinite volatility). At the other extreme of infinite risk aversion ($\gamma=\infty$), the agent buys long-term bonds in order to be perfectly hedged against the deterioration of investment opportunities. Such an investor chooses zero consumption volatility at the cost of lower expected consumption. We discuss this case at length below. In the log-utility case ($\gamma=1$), these two forces perfectly offset and the portfolio rule is reduced to the myopic demand.

## Substitution effects of labor income and Social Security 

Now let us consider the effect of labor income and Social Security. Suppose that labor income $L_{it}$, taxes $T_{it}$, and benefits $B_{it}$ are deterministic. The value of human capital, $H_{it}$, is the present value of future earnings discounted using the prevailing yield curve. Social Security wealth, $S_{it}$, is the present value of future benefits net of future payroll taxes. Define total wealth $\overline{W}_{it}=W_{it}+H_{it}+S_{it}$ as the sum of wealth and these present values of background assets.

Implementing the same linearization implies that the consumption rule relative to total wealth is the same as in the no-income solution: $C_i/\overline{W}_i$ equals the right-hand side of equation ). Similarly, the optimal allocation to bonds out of total wealth is $\bar{\pi}_i=\pi_i^*$ from equation ). The optimal allocation out of wealth $W_i$ therefore takes the form $$

\pi_{it} = \pi_{it}^* - \underbrace{(\pi_{it}^H-\pi_{it}^*)\frac{H_{it}}{W_{it}}}_{\substack{\textrm{human capital}\\ \textrm{substitution effect}}}
 - \underbrace{(\pi_{it}^S-\pi_{it}^*)\frac{S_{it}}{W_{it}}}_{\substack{\textrm{Social Security}\\ \textrm{substitution effect}}},
$$ or, in terms of interest-rate sensitivities, $$
 
 \varepsilon_r(W_{i,t+1}) = \varepsilon_{r,it}^* - (\varepsilon_r(H_{i,t+1})-\varepsilon_{r,it}^*)\frac{H_{it}}{W_{it}} - (\varepsilon_r(S_{i,t+1})-\varepsilon_{r,it}^*)\frac{S_{it}}{W_{it}},
$$ such that the interest-rate sensitivity of total wealth $\varepsilon_r(\overline{W}_{i,t+1})$ remains equal to $\varepsilon_{r,it}^*$.

The endowments of human capital and Social Security wealth are implicit holdings of long-term assets, and thus substitute for the traded $n$-period bond. The values $\pi_i^H$ and $\pi_i^S$ represent the implicit percentage of each asset invested in the $n$-period bond. The agent adjusts the allocation out of wealth $\pi_i$ such that the duration of total wealth matches $\pi_i^*$. If, for instance, agents are endowed with a large stock of high-duration Social Security (i.e., $\pi_i^S$ and $S_i$ are large), they adjust their allocations to long-term bonds $\pi_i$ downward to offset this high interest-rate exposure.

Figure 5 illustrates the life-cycle pattern generated by this model. Early in life, most agents have little financial wealth and a large endowment of high-duration human capital. To match their ideal total-wealth rate exposure, they mostly hold assets with low rate sensitivity. As households get closer to retirement, they increase holdings of the long-term asset to offset short-term labor income and taxes, net of long-term benefits. As they progress through retirement, households reduce long-term bond holdings, in line with the declining target allocation implied by the policies above. In sum, substitution and aging effects explain the hump-shaped pattern in the data.

<figure id="fig:theory_lifecycle" data-latex-placement="H">
<div class="center">
<embed src="Plots/FigTheory1.eps" style="width:100.0%" />
</div>
<p><em>Note</em>: This figure shows a representative path of total wealth components, their interest-rate sensitivities, and their effect on wealth allocations over the life cycle. Panel A plots the average values of each component of total wealth, defined as the sum of wealth and the present values of labor income (human capital) and Social Security taxes and benefits. Panel B shows the interest-rate sensitivity of each component. Panel C illustrates the incremental effect of each component on the optimal interest-rate sensitivity. We assume <span class="math inline"><em>γ</em> = 5</span> and <span class="math inline"><em>β</em> = 0.95</span>. The life-cycle profile of wealth is approximated from the data; the present values of human capital and Social Security wealth are simulated.</p>
<figcaption><strong>Effect of labor income and Social Security on long-term asset share</strong> </figcaption>
</figure>

In addition to these effects, the progressivity of Social Security implies that households with lower earnings will hold less rate-sensitive portfolios. Figure 6 illustrates the economic intuition behind this prediction. Panel A describes the case without Social Security: all households have approximately the same savings rate, hence the wealth-income ratios and portfolio allocations show little variation within an age group. Panel B shows the two effects of Social Security. Social Security offers a higher replacement rate to low-earners, which means that they need to save less for retirement, leading to a lower wealth-to-human capital ratio at a given age: the retirement savings substitution effect. Second, because Social Security represents a greater share of their total endowment, it reduces the demand for long-term assets of low-earners disproportionately: the portfolio substitution effect. As Panel C shows, these effects combine to generate a steep positive relation between wealth and rate exposure, as in the data.

<figure id="fig:theory_wealth" data-latex-placement="H">
<div class="center">
<embed src="Plots/FigTheory2.eps" style="width:100.0%" />
</div>
<p><em>Note</em>: This figure illustrates the effect of Social Security on intra-cohort allocations to the long-term asset. Panel A plots the optimal long-term bond share as a function of the ratio of wealth <span class="math inline"><em>W</em></span> to human capital <span class="math inline"><em>H</em></span> when there is no Social Security. The round markers represent hypothetical wealth-to-human capital ratios <span class="math inline"><em>W</em>/<em>H</em></span>. Panel B shows the same relation but in the presence of Social Security. In Panel C, we re-plot the points in Panels A and B in terms of wealth only. Policy functions are drawn for age 42, <span class="math inline"><em>γ</em> = 5</span>, and <span class="math inline"><em>β</em> = 0.95</span>.</p>
<figcaption><strong>Substitution effects of Social Security within an age group</strong> </figcaption>
</figure>

## Adding other assets and sources of systematic risk 

All of the above intuition generalizes to multiple risk factors and multiple assets with different exposures to interest-rate risk. To see this, suppose that there are $J$ risky assets and $J$ risk factors---for example, the long-term bond, stock market, and housing. Letting $\varepsilon\sim \mathcal{N}(0,I)$ denote a $J\times J$ vector of independent shocks, we can write the interest-rate process as $$
r_{f,t+1} = (1-\varphi)\bar{r}_f + \varphi r_{ft} + \sigma_r^\top\varepsilon_{t+1}

$$ and write the $J$-dimensional vector of risky-asset returns as $$
r_{t+1} = r_{ft}\iota + \mu + \Sigma^\top\varepsilon_{t+1}.

$$ The only assumption we require is that these assets are non-redundant (the $J\times J$ matrix $\Sigma$ is invertible). Appendix 16.5 shows that, under this assumption, the vector of optimal portfolio weights on the risky assets (in total wealth) equals $$
\pi_{it}^* = \frac{1}{\gamma}(\Sigma^\top\Sigma)^{-1}\left(\mu+\frac{1}{2}\mathrm{diag}(\Sigma^\top\Sigma)\right) - \left(1-\frac{1}{\gamma}\right)\varrho_{r,t+1}\Sigma^{-1}\sigma_r,
$$ which, like before, is the sum of the myopic demand and the rate-hedging demand for each asset. If asset $j$ is the long-term bond, then $\pi_t^*(j)$ still equals ). The vector of interest-rate sensitivities across assets is then given by $$
\varepsilon_r = -\frac{\Sigma^\top\sigma_r}{\sigma_r^\top\sigma_r},
$$ so the interest-rate sensitivity of wealth is $$
\varepsilon_r^*(W_{i,t+1}) = {\pi_{it}^*}^\top\varepsilon_r = \underbrace{-\frac{1}{\gamma}\frac{\Lambda^\top\sigma_r}{\sigma_r^\top\sigma_r}}_{\text{risk premia}} + \underbrace{\left(1-\frac{1}{\gamma}\right)\varrho_{r,t+1}}_{\text{hedging}},

$$ where $$
\Lambda = (\Sigma^\top)^{-1}\left(\mu + \frac{1}{2}\mathrm{diag}(\Sigma^\top\Sigma)\right)
$$ is a vector of factor loadings ("Sharpe ratios"): the prices of risk for exposure to each risk factor. If $\Lambda^\top\sigma_r=0$, then there is no additional compensation for holding interest-rate risk, and ) takes the exact same value as in the two-asset model above.[^23]

Thus, adding additional systematic risk factors has no consequence on the overall optimal interest-rate sensitivity of wealth. This is essentially an application of the portfolio separation theorem, which states that, no matter the specific menu of assets, households should choose the same overall exposure to risk factors---exposure to interest-rate risk, house-price risk, etc.---which can be formed by a set of "separating portfolios" (often called "funds" in the finance literature).

## Real-life interpretation 

Thus far, we have built intuition by representing the interest-rate exposure of household balance sheets with two zero-coupon securities of different maturities. We now turn to a more concrete explanation of how households can construct real-world portfolios that achieve an interest-rate sensitivity close to the optimal portfolio rule.

#### Fully hedged consumption plan

In principle, agents can eliminate interest-rate risk by buying, at current prices, a portfolio of zero-coupon bonds that matches the difference between their consumption plan and their expected earnings in all periods, then holding this portfolio to maturity.

We can illustrate this intuition most clearly in the limiting case of an investor with infinite risk aversion and a zero EIS.[^24] In this case, the investor's desire to smooth consumption over states and time yields a constant, deterministic policy $C_{it}=\bar{C}_i$. Let $Y_i$ denote the agent's deterministic stream of income. Wealth is the present value of the excess consumption plan: $$
W_{it} = \sum_{k=1}^{t_{\rm max}}P_{kt}(\bar{C}_i-Y_{i,t+k}).
$$ The agent can secure the optimal consumption plan by buying $\bar{C}_i-Y_{i,t+k}$ of each $k$-period zero-coupon bond and consuming the coupons and income at maturity. The strategy is unaffected by capital gains and losses from interest-rate changes. As we prove in Appendix 16.6, the optimal allocation $\pi_i$ replicates exactly this buy-and-hold strategy as $\gamma\to\infty$.

#### Real-world implementation

In reality, households do not hold portfolios of zero-coupon bonds. Instead, they implement this strategy using the assets and contracts available to them. We illustrate this mechanism with three hypothetical households, each represented in our full model.

First, consider a worker at the bottom of the earnings distribution. Because of its high replacement rate, Social Security taxes and benefits execute all intertemporal transfers of income required to smooth consumption over the life cycle, and do so independently of the rate of return on private savings. Such a worker only needs to hold short-term assets.

Second, consider a middle-class worker. Because replacement rates fall with lifetime earnings, the worker needs to save privately as well. However, he can execute large intertemporal transfers at current prices by buying a house with a fixed-rate mortgage. By doing so, he effectively trades a flow of coupon payments later in life, when $C>Y$, in the form of rent-free housing, in exchange for a stream of mortgage payments earlier in life, when $C<Y$. This strategy eliminates interest-rate risk for workers whose Social Security benefits cover non-residential consumption in retirement.

Finally, consider a high-income worker. Because Social Security benefits are relatively small in the upper half of the earnings distribution, she complements this strategy with additional investments. In the United States, this complement typically takes the form of a retirement account. If these savings were invested in short-term assets, she would need to increase her contribution rate to maintain the same consumption level in retirement. However, if her account were mostly invested in long-term assets, capital gains would offset potential declines in future rates of return. As we see in Figure 1, high earners follow the long-term asset strategy by investing their "extra" wealth in stocks. From this point of view, the glide path strategy of pension funds also makes sense, as it invests retirement contributions in stocks early in the life cycle and moves towards safer assets when workers get older.

## Distributional effects of interest-rate fluctuations 

Because of substitution effects, the interest-rate exposure of wealth is heterogeneous and correlated with key household characteristics, both within and between cohorts. At the same time, the model suggests substantially less heterogeneity in optimal *total* rate exposure. In fact, all agents within a given age group are identically exposed to interest-rate risk, regardless of differences in wealth, income, or benefits. The only source of heterogeneous total rate exposure is different investment horizons. To see this, let us consider three measures of total rate exposure: the interest-rate sensitivity of total wealth, of consumption, and of lifetime utility.

As we showed above, the rate elasticity of total wealth is $\varepsilon_r(\overline{W}_{i,t+1}) = \varepsilon_{r,it}^*$, which depends only on risk aversion and the agent's investment horizon $\varrho_{rt}$. In other words, for any two agents with the same risk aversion and horizon, a rate shock will have an identical effect on total wealth, even if those agents have different long-term asset shares. Likewise, the optimal consumption rule ) implies that the response of consumption to a rate shock $\varepsilon_r(C_{it}^*)$ is also homogeneous within a cohort.

The third---and arguably most relevant---measure of total rate exposure is the interest-rate sensitivity of lifetime utility. Specifically, we calculate the rate sensitivity of a transformation of expected utility: $$
 U_{it} = ((1-\gamma)V_{it})^{1/(1-\gamma)},
 
$$ where $V_{it}$ is the expected utility maximand ). This transformation backs out a total-wealth certainty equivalent---it is the value of total wealth implied by the value function $V$ taking a power form. In our linearized model, lifetime utility has the closed-form solution $$
 U_{it} = \overline{W}_{it}(1-\beta(1-m_{it}))\exp\left\{\varrho_{0t}+\varrho_{rt}r_{ft}\right\}.
 
$$ This expression illustrates the complementarity between total wealth and interest rates: agents have high expected utility when they enjoy high wealth with high rates. The rate sensitivity of lifetime utility is therefore[^25] $$
 \varepsilon_r(U_{it}) = \underbrace{-\frac{\partial\log U_{it}}{\partial r_{ft}}}_{-\varrho_{rt}} + \underbrace{\frac{\partial\log U_{it}}{\partial\log\overline{W}_{it}}\varepsilon_r^*(\overline{W}_{it})}_{(1-1/\gamma)\varrho_{rt}} = -\frac{1}{\gamma}\varrho_{rt}.
 
$$ For any finite value of risk aversion, this elasticity is negative, meaning that a decline in rates is bad news for lifetime utility, while an increase is good news. In welfare terms, this means that, when rates fall, the deterioration of future investment opportunities outweighs the capital gains on long-term cashflow claims. Agents are willing to accept this unhedged interest-rate exposure because of the option value of compounding at a higher rate, as explained above.

Most notably, the amount of wealth, income, and benefits a household possesses is irrelevant to the rate sensitivity of utility, because the household always trades in long-term asset markets to rebalance back to this optimal exposure. Within a cohort, unexpected rate changes may redistribute wealth, but they do not redistribute welfare.

# Model calibration

We calibrate the model to macroeconomic time series and cross-sectional household data from the SCF. In this section, we exclude entrepreneurs from both the model and data. We report the calibration of parameters specific to entrepreneurs in Appendix 13.4 and the model fit including them in Appendix 13.5.

## Asset markets and aggregate growth

Table 1 reports estimates of the interest-rate process. We estimate the processes for stock returns and growth jointly via simulated method of moments, with the resulting parameter estimates presented in Table 2. In particular, we match (i) the first 4 moments of the stock-market return distribution, (ii) the first 4 moments of the GDP-growth distribution, (iii) the correlation between stock returns and GDP growth, (iv) the correlation between GDP growth and interest rates, and (v) the autocorrelation of GDP growth.[^26]

 Parameter Notation Estimate
 --------------------------------- -- ---------------- ----------------------------
 Mean growth (expansions) $\mu_g^+$ 0.0265
 Mean growth (recessions) $\mu_g^-$ $-$0.0340
 Riskfree-rate growth loading $\lambda_{gr}$ 0.1602
 Growth volatility $\sigma_{g}$ 0.0139
 Persistence of growth shocks $\varphi_g$ 0.1498
 Recession probability $p_R$ 0.164
 Stock return drift (expansions) $\mu_s^+$ 0.1096
 Stock return drift (recessions) $\mu_s^-$ $-$0.2930
 Stock volatility $\sigma_s$ 0.0832
 Stock-growth loading $\lambda_{sg}$ 0

 : Estimated parameters for growth and stock returns

Historically, economic growth has been positively exposed to interest-rate shocks ($\lambda_{gr}>0$), while market returns have been negatively exposed ($\lambda_{sr}\approx17$).[^27] While we estimate the loading of stock returns on growth $\lambda_{sg}$ to be indistinguishable from zero, stock returns and aggregate growth are still strongly correlated for two reasons. First, and most importantly, the market-specific shock $\epsilon_s$ tends to be strongly negative in recessions: $\mu_s^{-}\!<\!\!<\!\mu_s^+$. Second, shocks to interest rates feed into growth through $\lambda_{gr}>0$, and this feeds into stock returns via $\lambda_{sr}$.

For housing returns in equation ), the regional house price volatility is $\sigma_h=0.08$ and the idiosyncratic volatility is $\tilde{\sigma}_h=0.117$, as implied by estimates from (Flavin and Yamashita 2002) and (Piazzesi and Schneider 2016).[^28] For a renter, the rent-to-price ratio $\chi$ is 5.33% (Jordà et al. 2019). Given these parameters, the drift rate is set to match the historical average housing return (net of maintenance costs) of 6.1% estimated by (Jordà et al. 2019), which implies $\mu_h=-0.0203$.

## Preferences

We calibrate households' preferences to match the evolution of wealth over the life cycle and the average equity share observed in the SCF. We find that a discount factor of $\beta=0.97$ and a bequest motive equivalent to $\bar{b}=8$ years of consumption match the growth of wealth until the retirement age and its evolution thereafter. See Panel A of Figure 7.

A coefficient of relative risk aversion of $\gamma=6$ matches the average allocation of wealth to stocks. Our calibration of $\gamma$ is consistent with other studies matching the life-cycle profile of the share of wealth invested in stocks, which typically use values between 5 and 6 (Benzoni et al. 2007; Lynch and Tan 2011; Huggett and Kaplan 2016; Catherine 2022; Meeuwis 2022). Based on portfolios in Swedish administrative data, (Calvet et al. 2021) estimate an average $\gamma$ of 5.2.

We set the Cobb-Douglas housing preference parameter $\nu=0.25$, which is close to the aggregate housing expenditure share (Piazzesi and Schneider 2016).

## Other parameters 

#### Income process

We calibrate the stochastic parameters of the labor process following (Catherine 2022), but re-estimate the parameters to use the same time series of aggregate income shocks as for the estimation of parameters in Table 2 and assuming two earners. Table 3 reports our estimates and Appendix 13.2 details the estimation procedure.

+:--------+:---------+:---------------------+:---------------+:-------------+:-------------+:-:+:------------------+:------------------+
| | Persistent shocks | | Transitory shocks |
+---------+----------+----------------------+----------------+--------------+--------------+---+-------------------+-------------------+
| $p_{z}$ | $\rho_z$ | $\overline{\mu^-_z}$ | $\lambda_{zg}$ | $\sigma^-_z$ | $\sigma^+_z$ | | $\sigma^-_{\eta}$ | $\sigma^+_{\eta}$ |
+---------+----------+----------------------+----------------+--------------+--------------+---+-------------------+-------------------+
| .224 | .993 | -.026 | 5.219 | .352 | .059 | | .414 | .124 |
+---------+----------+----------------------+----------------+--------------+--------------+---+-------------------+-------------------+

: **Estimated parameters: Idiosyncratic labor-income shocks**

This table reports parameter estimates for the idiosyncratic income process, conditional on the group-level aggregate shock. The process is estimated as in (Catherine 2022) by targeting the time-series of cross-sectional moments of individual income log growth rates in the SSA administrative data from (Guvenen et al. 2014).

#### Initial wealth

Households enter working life with 0.1$\times$ the national wage index in networth, the equivalent of \$5,400 in 2019.

#### Frictions and constraints

The maximum house size is $\kappa_{\max}=4$ times current wealth, consistent with a minimum down-payment requirement of 25%. The minimum house size is $\kappa_{\min}=2$ times the average wage, corresponding to approximately \$110,000 per adult in 2019. The mortgage premium $\theta_{it}$ is estimated by regressing mortgage premia, adjusted for the tax benefits of interest deductibility, on leverage (assets-to-networth) ratios.[^29] Homeowners pay a property tax rate of 1.4% on the value of their house.

Equity investors pay a 1 percentage-point management fee and a 1.2 percentage-point capital gains tax.[^30] The equity participation cost is $c_s=0.05$ times the average wage (roughly \$270 per year), based on estimates from (Catherine 2022).

#### Mortality

We model mortality as a function of age and past lifetime earnings: $$
 m_{it} = \min \left\{\delta\left(\text{AIYE}_{it}/\bar{L}_{t}\right) \times \overline{m}_{it},1\right\},
$$ where $\delta(\cdot)$ is an adjustment coefficient which only depends on the average indexed earnings of the agent up to time $t$ and $\overline{m}_{it}$ is the average mortality rate by age, which we calibrate as the average across genders from the 2019 Social Security actuarial life tables. While $\delta\left(\text{AIYE}_{it}/\bar{L}_{t}\right)$ does not depend on age, the agent's life expectancy changes as he or she moves up and down the wage ladder. An advantage of this method is that the agent's life expectancy is less volatile than if it were a function of persistent income $z_{it}$. We calibrate the value of $\delta\left(\text{AIYE}_{it}/\bar{L}_{t}\right)$ at each point of the numerical grid of the $\text{AIYE}_{it}/\bar{L}_{t}$ state variable such that, given our labor-income process, we obtain the same life expectancy differential across percentiles of $\text{AIYE}_{it}/\bar{L}_{t}$ at age 40 as those reported by percentiles of earnings in (Chetty et al. 2016).

## Validation

We calibrate the preference parameters $\{\gamma,\beta,\bar{b}\}$ to match life-cycle profiles of networth and portfolio composition. Figure 7 compares model-implied allocations to their empirical counterparts. The model reproduces the hump-shaped patterns of wealth accumulation, equity holdings, and conditional house values, and it generates a homeownership rate that rises early in life and then stabilizes. Households in the model transition into homeownership slightly sooner than in the data, which mechanically leads them to build mortgage debt faster and repay it earlier. A natural explanation is that the model abstracts from the richer heterogeneity in housing markets and family dynamics that delays home purchases in practice. Earlier homeownership also implies that leverage rises more quickly and peaks earlier in the model; a more gradual renting-to-own transition would flatten the life-cycle leverage profile accordingly.

<figure id="fig:validation_lifecycle" data-latex-placement="H">
<table>
<tbody>
<tr>
<td style="text-align: left;">A. Wealth</td>
<td style="text-align: left;">B. Home ownership rate</td>
<td style="text-align: left;">C. Net Debt</td>
</tr>
</tbody>
</table>
<embed src="Plots/FigValidation1a_hbus.eps" />
<table>
<tbody>
<tr>
<td style="text-align: left;">D. Equity</td>
<td style="text-align: left;">E. Housing wealth</td>
</tr>
</tbody>
</table>
<p><embed src="Plots/FigValidation1b_hbus.eps" /><br />
<br />
<span><em>Note</em>: This figure reports wealth (Panel A), homeownership rates (Panel B), net debt (Panel C), wealth invested in stocks (Panel D) and housing (Panel E) by age, in the model and in the data. Equity wealth, debt, and wealth are measured in units of the national wage index. Net debt is defined as housing wealth minus wealth. Entrepreneurs are excluded.</span></p>
<figcaption><strong>Wealth and portfolios over the life cycle</strong></figcaption>
</figure>

Figures 8 and 9 report average equity and housing holdings, respectively, by income and wealth for households aged 40--45. The model reproduces the empirical pattern that richer and higher-income households allocate more to both equities and housing---not only in levels, but also as a share of wealth. A well-known challenge for life-cycle models is to rationalize the very low stock-market participation and equity shares among young and low-income households. In our framework, this margin is primarily driven by countercyclical labor-income risk (Catherine 2022), which raises the effective risk exposure of these households in downturns and dampens their incentives to hold equities.

<figure id="fig:validation_equity" data-latex-placement="H">
<table>
<tbody>
<tr>
<td style="text-align: left;">A. By income level</td>
<td style="text-align: left;">B. By wealth level</td>
</tr>
</tbody>
</table>
<p><embed src="Plots/FigValidation2a_hbus.eps" /><br />
<br />
<span><em>Note</em>: This figure reports the amount of wealth invested in stocks in the model and in the data, for households between age 40 and 45, by level of income and wealth. Equity wealth, income, and wealth are measured in units of the national wage index. Entrepreneurs are excluded. Appendix Figure <a href="#fig:validation_equity_full" data-reference-type="ref" data-reference="fig:validation_equity_full">21</a> shows the same results with entrepreneurs.</span></p>
<figcaption><strong>Equity holdings at age 40–45</strong></figcaption>
</figure>

<figure id="fig:validation_housing" data-latex-placement="H">
<table>
<tbody>
<tr>
<td style="text-align: left;">A. By income level</td>
<td style="text-align: left;">B. By wealth level</td>
</tr>
</tbody>
</table>
<p><embed src="Plots/FigValidation2b_hbus.eps" /><br />
<br />
<span><em>Note</em>: This figure reports the amount of wealth invested in housing in the model and in the data, for households between age 40 and 45, by level of income and wealth. Housing wealth, income, and wealth are measured in units of the national wage index. Entrepreneurs are excluded. Appendix Figure <a href="#fig:validation_housing_full" data-reference-type="ref" data-reference="fig:validation_housing_full">22</a> shows the same results with entrepreneurs.</span></p>
<figcaption><strong>Housing wealth at age 40–45</strong></figcaption>
</figure>

# Matching the stylized facts 

Although the model is calibrated to match other dimensions of household balance sheets, our primary interest remains the interest-rate sensitivity of wealth. We therefore now examine the extent to which the model reproduces the stylized facts documented in Section 2.

## Cross-section of interest-rate sensitivity 

Panel A of Figure 10 reports the evolution of the interest-rate sensitivity of wealth over the life cycle, in the data and in the model. In both the data and the model, the interest-rate sensitivity of wealth increases over the first part of working life and declines afterwards. The increase is explained by the substitution effect of human capital and Social Security early in life, as explained in Section 4.2. The introduction of labor income risk in the model reduces the agent's private valuation of human capital, which explains why the predicted hump-shaped relationship with age is less pronounced than in Figure 5.

In the model, the interest-rate sensitivity exhibits an additional hump around age 30, which coincides with the peak in leverage shown in Panel C of Figure 7 but is not present in the data. As discussed in the previous section, our model abstracts from heterogeneity in housing markets and family dynamics that, in practice, disperses the timing of homeownership. This dispersion would smooth and shift this early-life hump to later ages, bringing the simulated profile closer to the empirical pattern.

<figure id="fig:IRSCrossSection" data-latex-placement="H">
<table>
<tbody>
<tr>
<td style="text-align: left;">A. By age</td>
<td style="text-align: left;">B. By income at age 40–45</td>
<td style="text-align: left;">C. By wealth at age 40–45</td>
</tr>
</tbody>
</table>
<p><embed src="Plots/FigValidation3_hbus.eps" /> <span><em>Note</em>: This figure reports the interest-rate sensitivity of wealth in the model and in the data. Panel A reports the interest-rate sensitivity of wealth by age group. Panel B and C report it within households aged 40–45, by income and wealth groups, respectively. Income and wealth are measured in units of the national wage index. Entrepreneurs are excluded. Appendix Figure <a href="#fig:IRSCrossSection_full" data-reference-type="ref" data-reference="fig:IRSCrossSection_full">23</a> shows the same results with entrepreneurs.</span></p>
<figcaption>Interest-rate sensitivity in the cross-section</figcaption>
</figure>

During retirement, the decline in the agent's investment horizon becomes the dominant force and reduces the need to hedge against falling interest rates. As a result, the long-term asset share falls. This decline is moderated by the bequest motive, which effectively increases the investment horizon of the agent beyond his own life expectancy.

Panel B reports the relationship between the interest-rate sensitivity of wealth and income between age 40--45. In the model, high earners invest more in the long-term asset because Social Security covers a smaller share of their retirement consumption and, to a lesser extent, because they have higher life expectancy.

Panel C shows that the model also produces a positive relationship between the long-term asset share and wealth within an age group. This is partly explained by the fact that wealthier households tend to be high earners and that human capital and Social Security represent smaller fractions of their total wealth, and thus have weaker substitution effects.

In the data, we observe a hump around the third decile of the wealth distribution, which could reflect the need for these households to borrow to buy houses and vehicles. In the model, optimal rate sensitivity also displays a hump around the wealth threshold at which households transition into homeownership. At this point, the curvature of the value function with respect to wealth is different and households find it desirable to increase the volatility of their wealth due to the option value of becoming homeowners in the next period. This mechanism makes high-variance assets relatively more attractive around this threshold.[^31]

## Household interest-rate exposure

We now study the rate sensitivities of two measures that are more relevant for welfare: wealth inclusive of Social Security and expected lifetime utility. As in the linearized model, we find that there is less heterogeneity in these measures (especially in utility), suggesting that the recent rise in wealth inequality has not necessarily come with a rise in welfare inequality.

To calculate wealth inclusive of Social Security, we capitalize the expected benefits and taxes into a present value. To measure welfare, we calculate the transformed expected utility $U$ defined in ). Because $U$ is a function of both wealth $W$ and rates $r_f$, this elasticity can be approximated, to a first order, as $$
 \varepsilon_r(U) \approx \underbrace{-\frac{\partial\log U}{\partial r_f}}_{\substack{\textrm{change in investment}\\ \textrm{opportunities}}} + \;\; \underbrace{\vphantom{\frac{\partial\log U}{\partial r_f}}\frac{\partial\log U}{\partial \log W}\varepsilon_r(W)}_{\textrm{capital gains}}\biggr. .
$$ When rates decline, expected utility decreases because investment opportunities are worse, but also increases because of capital gains in financial wealth. If $\varepsilon_r(U)$ is negative, as we find, then a decline in interest rates decreases welfare.

<figure id="fig:IRSCrossSection2" data-latex-placement="H">
<table>
<tbody>
<tr>
<td style="text-align: left;">A. By age</td>
<td style="text-align: left;">B. By income at age 40–45</td>
<td style="text-align: left;">C. By wealth at age 40–45</td>
</tr>
</tbody>
</table>
<p><embed src="Plots/FigIRS_hbus.eps" /> <em>Note</em>: This figure reports the interest-rate sensitivity of wealth, wealth inclusive of Social Security, and expected utility over the life cycle. Entrepreneurs are excluded. Appendix Figure <a href="#fig:IRSCrossSection2_full" data-reference-type="ref" data-reference="fig:IRSCrossSection2_full">24</a> shows the same results with entrepreneurs.</p>
<figcaption><strong>Interest-rate sensitivities for different measures of wealth</strong> </figcaption>
</figure>

Panel A of Figure 11 shows the average paths of these sensitivities over the life cycle. Adding Social Security wealth increases the average sensitivity for the young, consistent with the fact that it is a very long-term asset. The interest-rate sensitivity of expected utility is much flatter over the life cycle. With the exception of households in their twenties, rate declines reduce welfare on net, but this negative effect attenuates gradually with age as the remaining investment horizon shortens. Very low-wealth households, especially recent labor-market entrants in their twenties, are also less able to attain their preferred interest-rate exposure because they cannot short long-duration assets.

Panels B and C report the distribution of these sensitivities within a middle-aged cohort, as functions of income and wealth. First, when Social Security is taken into account, the wealth of the rich and of high earners is no longer more sensitive to interest rates. This explains why, when Social Security is accounted for and discounted using the market yield curve, wealth inequality has not increased since 1989 (Catherine et al. 2025).

Panels B and C are the model's counterparts to Figure 4. The data and the model convey the same message: once Social Security is accounted for, rich households no longer hold more interest-rate-sensitive assets. As in the data, the interest-rate sensitivity of wealth, inclusive of Social Security, is around 15, though a bit higher for low-wealth households in the data.

Second, within a cohort, expected utility is uniformly elastic to interest rates across the earnings and wealth distributions, save for a very minor effect from income-driven mortality differences. As equation ) predicts, once human capital and Social Security wealth are accounted for, all households within a cohort are equally exposed to rate fluctuations.

# Trends in wealth inequality and welfare

So far, we have demonstrated that, under a reasonable calibration, the interest-rate risk hedging demand for long-term assets can explain differences in the interest-rate sensitivity of wealth over the life cycle, and across the earnings and wealth distribution. Consequently, in the model, interest-rate fluctuations will, through our mechanism, redistribute wealth across the population without significant implications for the distribution of lifetime consumption (welfare).

In this section, we use an overlapping-generations version of our life-cycle model to study the historical distributions of wealth and welfare. First, we assess the extent to which interest-rate fluctuations explain the long-run fluctuations in wealth inequality. Second, we quantify the historical implications of these fluctuations for welfare.

## Intuition: From rate shocks to inequality

Wealth of a household $i$ evolves according to: $$
 \frac{W_{i,t+1}}{W_{it}} = \underbrace{\left(1-\frac{C_{it}-Y_{it}}{W_{it}}\right)}_{\textrm{savings}}\underbrace{\vphantom{\left(\frac{C_{it}}{W_{it}}\right)} R_{Wi,t+1}}_{\textrm{portfolio}},
 
$$ where $Y_{it}$ summarizes disposable income. We can decompose changes in inequality over time by taking logs of ) and then computing cross-sectional variances. Doing so yields the change in wealth dispersion from one period to the next:[^32] $$\begin{multline}
 \mathrm{var}_I(w_{i,t+1}) - \mathrm{var}_I(w_{it}) = \mathrm{var}_I(s_{it}) + \mathrm{var}_I(r_{wi,t+1}) \\
 + 2\mathrm{cov}_I(w_{it},s_{it}) + 2\mathrm{cov}_I(w_{it},r_{wi,t+1}) + 2\mathrm{cov}_I(s_{it},r_{wi,t+1}).
 
\end{multline}$$ The first two channels through which wealth inequality may increase are the direct effects of heterogeneous savings rates $s$ and realized portfolio returns $r_w$. The remaining three channels are captured by the covariance terms. Inequality increases if (i) the wealthy tend to save more, (ii) the wealthy experience higher returns, and (iii) households with higher savings rates experience higher returns. Our model reveals why these covariance channels are all positive when interest rates fall.

Consider how wealth inequality within a cohort responds to a negative interest-rate shock. First, inequality increases because Social Security induces differential savings rates: low-income, low-wealth households with higher replacement rates will save less into financial wealth. This savings substitution effect of Social Security results in a dispersion in savings rates ($\mathrm{var}_I(s_{it})>0$) and a positive wealth-savings correlation ($\mathrm{cov}_I(w_{it},s_{it})>0$). Second, Social Security gives rise to changes in inequality via its impact on portfolio choices. As Figure 6 illustrates, the substitution effects of Social Security create a positive correlation between a household's wealth and its interest-rate exposure. Thus, households within a cohort may experience different wealth returns, and the direction of reallocation will depend on the direction of the interest-rate shock. All unexpected rate changes result in heterogeneous returns ($\mathrm{var}_I(r_{wi,t+1})>0)$. A negative rate shock will result in disproportionately high returns for the wealthy $(\mathrm{cov}_I(w_{it},r_{wi,t+1})>0)$, increasing inequality. Finally, since the wealthy save more, the savings-return covariance $\mathrm{cov}_I(s_{it},r_{wi,t+1})$ is also positive given a rate decline, amplifying the increase in inequality.

The same redistribution will not, in general, occur for welfare (lifetime utility) or for broader concepts of wealth that include human capital and Social Security.[^33] This is because background assets offset interest-rate shocks; indeed, they are precisely the reason that households choose differential savings rates and portfolios to begin with.

## Overlapping-generations simulation

To quantify the role of interest-rate risk hedging on wealth inequality trends, we set up an overlapping-generations (OLG) version of our life-cycle model. Specifically, we simulate the lives of cohorts born since 1880, feeding in the history of macroeconomic shocks over that period.[^34] The time series of interest rates is described in Appendix 13.1.1; time series for growth, stocks, and house prices are as in the model calibration. We describe the OLG simulation in more detail in Appendix 15.

The only substantive assumption we must make for the OLG model is how to distribute bequests across generations. In the data, the size of inheritance is correlated with the income and wealth of the recipient. To capture this, we assume that the persistent component of income $z_{it}$ is correlated across generations, via its initial value, and calibrate this correlation to the persistence documented in (Chetty et al. 2014). Inheritance flows are unanticipated.

## Implications for wealth inequality

We use our model to quantify the effect of each shock on the top 10% wealth share. To measure the effect of each shock, we simulate the model multiple times, starting from a counterfactual with no aggregate shocks and then sequentially adding one shock from the data at a time. We start with a specification that includes data on interest rates and growth. The next specifications add stock-market shocks, followed by national house-price shocks. The only exception is that we do not add the growth shock to stock returns in the first specification, as to make clear the impact of stock returns in the data on wealth inequality in the second specification.

Figure 12 plots the model-implied inequality series. We consider the model both without entrepreneurs (Panel A) and with entrepreneurs (Panel B). The latter case is essential for explaining the level of top wealth shares in the data. The results of the simulation suggest that the majority of post-war variation in top wealth shares is explained by declining interest rates. Consider Panel A. From 1950--1985, the top 10% share declines by over 4 percentage points in the specification where only interest-rate shocks are included. The result is nearly identical when adding data on stock returns and house prices. Similarly, the subsequent rise in top wealth shares is nearly 3 percentage points and almost entirely driven by declining interest rates.

The specification with entrepreneurs looks quite similar, but explains a greater share of the overall fluctuations in the top wealth shares since 1950. Here, interest-rate fluctuations implied 8.5 percentage points (pp) of the decline in top wealth shares from 1950--1985 and 3.5 pp of the subsequent rise. Adding stocks and housing amplifies the trend slightly, with the model generating a 10-pp decline and a 6-pp rise in top wealth shares pre- and post-1985.

How does this compare to the data? From Figure 3, we see that estimates of the top 10% share fell by about 7--10 pp prior to 1985, then rose by about 8--10 pp thereafter. Consequently, the interest-rate risk hedging channel explains roughly two-thirds of the long-run fluctuations in wealth inequality over this period.

<figure id="fig:model_wealthinequality" data-latex-placement="H">
<div class="center">
<table>
<tbody>
<tr>
<td style="text-align: left;">A. Without entrepreneurs</td>
<td style="text-align: left;">B. With entrepreneurs</td>
</tr>
<tr>
<td style="text-align: left;"><embed src="Plots/figure_12a_olg_top_10_share.eps" /></td>
<td style="text-align: left;"><embed src="Plots/figure_12b_olg_top_10_share.eps" /></td>
</tr>
</tbody>
</table>
</div>
<p><em>Notes:</em> This figure shows the evolution of the top 10% wealth share in our model using historical asset returns. Panel A reports the top 10% share excluding business owners; Panel B includes them. The “interest-rates only” specification includes the historical time series for economic growth and interest rates, along with shocks to interest rates <span class="math inline"><em>ϵ</em><sub><em>r</em></sub></span>. The other stochastic components of returns <span class="math inline">{<em>ϵ</em><sub><em>g</em></sub>, <em>ϵ</em><sub><em>s</em></sub>, <em>ϵ</em><sub><em>h</em></sub>}</span> are drawn at random. Relative to this specification, the “stock market returns” specification adds historical data on the remaining shocks that affect equity returns <span class="math inline">{<em>ϵ</em><sub><em>s</em></sub>, <em>ϵ</em><sub><em>g</em></sub>}</span>. The “house prices” specification combines aggregate national house price shocks with idiosyncratic shocks drawn from a random distribution. House price indices using repeated sales began in 1975, so we begin in that year. Appendix <a href="#section:olgsimulationAppendix" data-reference-type="ref" data-reference="section:olgsimulationAppendix">15</a> details how we infer the historical values of these shocks.</p>
<figcaption><strong>Evolution of the top 10% wealth share</strong></figcaption>
</figure>

Most of the long-run fluctuations in wealth inequality are explained by interest-rate fluctuations. After accounting for the effect of interest rates on stock and house prices, these asset classes prove to be relatively less important for explaining inequality. This is consistent with the finding of (Greenwald et al. 2023), who conduct a similar exercise, taking portfolio choices as given from the data, and find that the fall in real rates since the 1980s explains 75% of the rise in wealth inequality. In our study, portfolios are endogenous, meaning that our model provides an explanation for the effect documented in their paper.

The patterns we document would also allow us to explain a substantial fraction of the rise in income inequality documented in prior work (Auten and Splinter 2024; Piketty et al. 2018). A major proportion of rising top incomes comes from capital gains on the investments of the wealthy. As such, the large increase in capital income from realized gains---driven by falling interest rates---also helps explain rising income inequality over the last 4 decades.

## Implications for welfare

Are the wealthy actually better off from the interest-rate-induced increase in wealth inequality? As predicted in Section 4.5, increases in top wealth shares stemming from falling rates do not translate into increased welfare inequality. To be precise, falling rates have no distributional effect on welfare within cohorts and a small amount of welfare redistribution to older cohorts with shorter investment horizons. To illustrate this point, Figure 13 reports the evolution of welfare inequality within cohorts in the model, using the certainty equivalent measure of lifetime utility from equation ). Panel A reports welfare inequality in the model without entrepreneurs, and Panel B reports it with entrepreneurs.[^35] As predicted by theory, welfare inequality remains largely unchanged.[^36] This remains true even accounting for multiple assets, risk factors, portfolio constraints, and frictions.

<figure id="fig:model_welfareinequality" data-latex-placement="H">
<div class="center">
<table>
<tbody>
<tr>
<td style="text-align: left;">A. Without entrepreneurs</td>
<td style="text-align: left;">B. With entrepreneurs</td>
</tr>
<tr>
<td style="text-align: left;"><embed src="Plots/figure_13a_olg_top_10_welfare_share.eps" /></td>
<td style="text-align: left;"><embed src="Plots/figure_13b_olg_top_10_welfare_share.eps" /></td>
</tr>
</tbody>
</table>
</div>
<p><em>Notes:</em> This figure shows the evolution of the top 10% “welfare” share in our model. We take the certainty equivalent of future utility as our measure of welfare given by equation (<a href="#eq:transformedutility" data-reference-type="ref" data-reference="eq:transformedutility">[eq:transformedutility]</a>). The top 10% is computed as simulated individuals in the top 10% of the welfare distribution within an age-year. Panel A reports the top 10% share excluding business owners; Panel B includes them. The “interest-rates only” specification includes the historical time series for economic growth and interest rates, along with shocks to interest rates <span class="math inline"><em>ϵ</em><sub><em>r</em></sub></span>. The other stochastic components of returns <span class="math inline">{<em>ϵ</em><sub><em>g</em></sub>, <em>ϵ</em><sub><em>s</em></sub>, <em>ϵ</em><sub><em>h</em></sub>}</span> are drawn at random. Relative to this specification, the “stock market returns” specification adds historical data on the remaining shocks that affect equity returns <span class="math inline">{<em>ϵ</em><sub><em>s</em></sub>, <em>ϵ</em><sub><em>g</em></sub>}</span>. The “house prices” specification combines aggregate national house price shocks with idiosyncratic shocks drawn from a random distribution. House price indices using repeated sales began in 1975, so we begin in that year. Appendix <a href="#section:olgsimulationAppendix" data-reference-type="ref" data-reference="section:olgsimulationAppendix">15</a> details how we infer the historical values of these shocks.</p>
<figcaption><strong>Evolution of within-cohort top 10% welfare</strong></figcaption>
</figure>

While the wealthy within a cohort may be no better off from interest-rate shocks, there is some noteworthy variation in welfare across cohorts. This occurs as a result of market incompleteness across generations: agents in the model cannot hedge interest-rate shocks that occur prior to their birth. This means that, all else equal, cohorts born in low interest-rate environments have lower lifetime utility than cohorts born in high interest-rate environments.

Figure 14 illustrates this point by showing average wealth and lifetime utility for five generations. Both values are shown relative to an average of simulations with a random time series of interest-rate shocks, so these should be interpreted as percentage deviations from the historical norm. Examining Panel A, it is clear that the large increase in interest rates from 1960--1985 reduced wealth for every generation that lived through it. However, those generations that entered the labor force closest to the high-rate period of the 1980s (e.g., the baby boomers) had the benefit of saving at substantially higher rates, leading to higher lifetime wealth than those other cohorts.

Panel B, which instead looks at lifetime utility, reveals two things. First, those cohorts which were fortunate to be born at a time of high interest rates and subsequently enjoy capital gains did experience above-average lifetime welfare. This is, again, because agents cannot fully hedge interest-rate shocks across generations. Second, comparing Panel B to Panel A, fluctuations in welfare across cohorts are substantially smaller than fluctuations in wealth.

<figure id="fig:model_cohortinequality" data-latex-placement="H">
<div class="center">
<table>
<tbody>
<tr>
<td style="text-align: left;">A. Wealth</td>
<td style="text-align: left;">B. Welfare</td>
</tr>
<tr>
<td style="text-align: left;"><embed src="Plots/figure_14a_year_normalized_wealth.eps" /></td>
<td style="text-align: left;"><embed src="Plots/figure_14b_year_normalized_welfare.eps" /></td>
</tr>
</tbody>
</table>
</div>
<p><em>Note</em>: This figure reports lifetime utility for the five most recent generations in the simulation. Lifetime utility is calculated as <span class="math inline">$\bar{U}_{it} = \mathbb{E}_t \left[ (1-\gamma) \sum_{s=t+a_{\rm min}-a}^{t + a_{\rm max}-a}\beta^{s-t} u(C_{is},H_{is})\right]^{\frac{1}{1-\gamma}}$</span> where <span class="math inline"><em>u</em></span> is utility under optimal policies. This is a weighted average of realized consumption and expected consumption over the lifecycle, without the bequest motive. After agents die in the simulation, they are assigned their last living value of <span class="math inline"><em>U</em><sub><em>i</em></sub></span> for the remainder of the simulation. The specification presented is then the average of <span class="math inline"><em>U</em><sub><em>i</em><em>t</em></sub></span> with historical interest-rate shocks and growth, divided by the hypothetical <span class="math inline"><em>U</em><sub><em>i</em><em>t</em></sub></span> without interest-rate shocks. All other shocks are drawn randomly.</p>
<figcaption><strong>Cross-generation welfare inequality</strong></figcaption>
</figure>

# Implications for Social Security reform 

The US Social Security trust fund is expected to run out within the decade, necessitating either higher taxes or lower benefits (Congressional Budget Office 2023). These funding concerns have been amplified by rising interest rates, which increase the government's cost of servicing its debt. We use our model to assess the welfare implications of an unanticipated 25% reduction in promised benefits, the amount needed to close the funding gap. An increase in the Full Retirement Age (FRA) would be economically similar.[^37]

To implement this policy counterfactual in our model, we simulate multiple economies, each with its own series of macroeconomic shocks. For each simulated household, we compute the percentage change in welfare, defined in equation ), under a 25% permanent reduction in Social Security benefits, holding wealth and other state variables fixed. This is the sense in which the reform is unanticipated: if households foresaw the cut, they would have adjusted their portfolios and savings early in life. Then, for each economy, we bin cohorts into deciles of pre-reform welfare, and compute the average welfare change within each decile. This allows us to measure the distributional impact of an unexpected cut of Social Security benefits.

Figure 15 plots the average welfare loss for each welfare decile at age 62, as a function of the interest-rate levels at the time of the benefit cut. For a given interest rate, welfare losses are largest for households with the least total wealth, reflecting their greater reliance on Social Security to finance retirement consumption. The main exception is the bottom decile ("D1"), for which losses are muted because the social safety net (SSI) partially offsets the benefit reduction for sufficiently low Social Security payments.

<figure id="fig:model_benefitscut" data-latex-placement="H">
<div class="center">
<embed src="Plots/FigBenefitsCut_Lines.eps" style="width:100.0%" />
</div>
<p><span><em>Note</em>: This figure reports the average change in welfare, defined in equation (<a href="#eq:transformedutility" data-reference-type="ref" data-reference="eq:transformedutility">[eq:transformedutility]</a>), of 62-year-olds in response to an unanticipated, permanent 25% reduction in Social Security benefits. Each line represents the average within a decile of welfare, with D1 denoting the bottom 10% of welfare and D10 the top 10%. Entrepreneurs are not included. See the main text for more details of the simulation.</span></p>
<figcaption><strong>Welfare effect of a 25% benefit cut at 62 years old</strong></figcaption>
</figure>

Given the essential role that Social Security plays in providing interest-rate insurance for low-income households, we consider the incidence of the benefit cut under different levels of the real interest rate. Across all deciles, welfare losses are larger when benefits are cut in a low-rate economy. In a low-rate environment, the value of all assets is higher, but Social Security wealth rises disproportionately because it is a long-duration claim. As a result, Social Security wealth represents a larger share of households' total wealth when interest rates are low. A fixed proportional benefit cut therefore removes a larger fraction of total wealth---and of the resources used to finance retirement consumption---precisely in low-rate states. In this sense, cutting benefits when rates are low withdraws insurance payouts at the time when they are most valuable, implying that, holding the size of the reform fixed, closing the funding gap is least costly when real interest rates are high.

# Conclusion 

We study household exposure to interest-rate risk, beginning from the empirical observation that middle-aged and richer households invest more in long-term assets that appreciate in value when rates fall. This interest-rate sensitivity must be interpreted in light of households' life-cycle problem, so we develop a life-cycle model that incorporates the roles that human capital and Social Security play in households' portfolio choices.

The data and the model produce strikingly close patterns: the optimal interest-rate sensitivity of wealth is hump-shaped over the life cycle and, within cohorts, increases with wealth and earnings. This is driven by the heterogeneous role that background assets play: human capital displaces the need for long-term holdings for the young, and Social Security has the same effect for the low- and middle-income households on which its impact is most pronounced.

A key implication of these findings is that, while interest-rate fluctuations have generated substantial fluctuations in wealth inequality, they have had a much smaller effect on welfare inequality. This follows from our portfolio-choice findings: because households in the data are holding portfolios that are close to optimal, their consumption is similarly exposed to interest-rate changes. The dramatic changes in wealth inequality statistics should therefore be interpreted with care.

In comparing optimal portfolio rules with the data, we can conclude that households seem to be targeting close-to-optimal interest-rate exposures over their life cycles. Still, we cannot conclude from our findings that households would continue to do this if the economic environment were to change. For example, we cannot say whether households adjust their portfolios optimally in response to Social Security or whether Social Security is well-designed to correct investment mistakes that would arise in its absence. Determining the direction of this causality is necessary for evaluating long-run policy counterfactuals and is therefore an essential step for future research.

**INTERNET APPENDIX**\

# Interest-rate sensitivity: Definitions and derivations 

## Interest-rate sensitivity of an asset

Consider a claim to a stream of cashflows $\{D_{t+k}:k\in\{1,2,\dots\}\}$. Let $P_t$ denote the price of the asset and $\{R_{t+k}:k\in\{1,2,\dots\}\}$ its future one-period returns (i.e., $R_{t+k}=(P_{t+k}+D_{t+k})/P_{t+k-1}$).

The *interest-rate sensitivity* of an asset is defined as the elasticity of its price with respect to the interest rate, $$
\varepsilon_r(P_t) \equiv -\frac{\partial\log P_t}{\partial r_{ft}},
$$ holding fixed the distributions of future cashflows and future returns in excess of the riskfree rate.

Following this definition, let us derive an expression for interest-rate sensitivity under the assumption that interest rates follow an AR(1). It will be easiest to do this if we divide the asset into separate claims to the cashflows at each horizon $k$ (the "strips"). Letting $P_{kt}$ denote the price of the claim to the single cashflow $D_{t+k}$, we have $$
P_t = \sum_{k=1}^\infty P_{kt}.
$$ The interest-rate sensitivity of the asset is then the value-weighted average of the sensitivities of each strip: $$
\varepsilon_r(P_t) = \sum_{k=1}^\infty\frac{P_{kt}}{P_t}\varepsilon_r(P_{kt}).
$$ To derive these elasticities, note that, from the definition of a return, $$
P_{kt} =\frac{P_{k-1,t+1}}{R_{k,t+1}} = \left(\prod_{k'=1}^kR_{k',t+1+k-k'}^{-1}\right)D_{t+k}.
$$ Thus, defining the excess return $\hat{r}_{k',t+k}\equiv\log(R_{k',t+k}/R_{f,t+k-1})$ and taking logs then expectations, $$
\log P_{kt} = -\sum_{k'=1}^k(\mathbb{E}_t[r_{f,t+k-k'}] + \mathbb{E}_t[\hat{r}_{k',t+1+k-k'}]) + \mathbb{E}_t[\log D_{t+k}].
$$ Now, using the fact that $$
\mathbb{E}_t[r_{f,t+k}] = \bar{r}_f + \varphi^k(r_{ft} - \bar{r}_f),
$$ we have that $$
\varepsilon_r(P_{kt})=-\frac{\partial\log P_{kt}}{\partial r_{ft}} = -\sum_{k'=1}^k\left(\varphi^{k-k'} + \frac{\partial\mathbb{E}_t[\hat{r}_{k',t+1+k-k'}]}{\partial r_{ft}}\right) + \frac{\partial\mathbb{E}_t[\log D_{t+k}]}{\partial r_{ft}} = \frac{1-\varphi^k}{1-\varphi}.
$$ The last equality follows from the assumption that expected excess returns and cashflow growth are held constant, so $\partial\mathbb{E}_t[\hat{r}_{t+k'}]/\partial r_{ft}=\partial\mathbb{E}_t[D_{t+k}]/\partial r_{ft}=0$. Substituting this into the expression above, we then have that the asset's interest-rate sensitivity is $$
\varepsilon_r(P_t) = \sum_{k=1}^\infty\frac{P_{kt}}{P_t}\frac{1-\varphi^k}{1-\varphi}.
$$

## Relating interest-rate sensitivity to duration

The (Macaulay) *duration* of an asset is defined as the value-weighted timing of its cashflows: $$
\text{dur}(P_t) \equiv \sum_{k=1}^\infty\frac{P_{kt}}{P_t}k,
$$ where the strip prices $P_{kt}$ are defined above.

One can see from this expression that $$
\lim_{\varphi\to1}\varepsilon_r(P_t) = \text{dur}(P_t),
$$ meaning that the duration is the same as the response to a permanent change in interest rates. Our empirical measure of interest-rate sensitivity is derived from a first-order approximation of $\varepsilon_r$ around the duration. To see this, note that, to a first-order, $$
\varphi^k \approx \varphi^{\text{dur}(P_t)} + \varphi^{\text{dur}(P_t)} \log\varphi(k-{\text{dur}(P_t)}).
$$ Substituting this approximation back into the expression for $\varepsilon_r$ gives $$
\varepsilon_r(P_t) = \frac{1-\sum_{k=1}^\infty\frac{P_{kt}}{P_t}\varphi^k}{1-\varphi} \approx \frac{1-\varphi^{\text{dur}(P_t)}}{1-\varphi}.
$$ Thus, high duration implies high interest-rate sensitivity.

## Interest-rate sensitivity of wealth

Just as the rate sensitivity of an asset is the value-weighted average of the sensitivities of its strips, the rate sensitivity of wealth is the value-weighted average of the sensitivities of its assets. Suppose a household has wealth $$
W_t = \sum_j A_{jt} - \sum_k D_{kt}.
$$ In the case of positive net worth $W_t>0$, it is clear that $$\begin{align}
\varepsilon_r(W_t) &= -\sum_j\frac{A_{jt}}{W_t}\frac{\partial\log A_{jt}}{\partial r_{ft}} + \sum_k\frac{D_{kt}}{W_t}\frac{\partial\log D_{kt}}{\partial r_{ft}} \\
&= \sum_j\frac{A_{jt}}{W_t}\varepsilon_r(A_{jt}) - \sum_k\frac{D_{kt}}{W_t}\varepsilon_r(D_{kt}).
\end{align}$$

The definition of rate sensitivity above applies only to positive-wealth households. This is not an issue in our empirics, as we only include these households in our computations. Still, it is worth discussing how one would measure rate sensitivity for a household with negative wealth. A simple generalization that handles this case is $$
\varepsilon_r(W_t) \equiv -\frac{\partial W_t}{\partial r_{ft}}\frac{1}{|W_t|}.
$$ For $W_t>0$, one can see that this reverts to the original definition (by the chain rule). For a household with negative networth, $$

\varepsilon_r(W_t) = \sum_j\frac{A_{jt}}{|W_t|}\varepsilon_r(A_{jt}) - \sum_k\frac{D_{kt}}{|W_t|}\varepsilon_r(D_{kt}).
$$ The absolute value in the denominator is necessary for getting the sign of the elasticity right. For example, suppose a household has \$100 of assets and \$200 of debt, both with the same rate sensitivity (say, 2). If rates fall by 1%, networth should fall by roughly \$2, from $-$\$100 to about $-$\$102.[^38] In this case, ) implies a rate sensitivity of wealth of $-$2, reflecting the fact that a decline in rates has a negative effect on wealth.

# Data appendix 

This appendix documents the measurement choices underlying our empirical interest-rate sensitivity estimates. It details how we construct cashflow durations for each SCF balance-sheet component---including private business wealth, vehicles, and the full set of household liabilities---and how we translate these durations into interest-rate sensitivities. It also describes the construction of Social Security wealth sensitivity and presents complementary evidence for European household portfolios.

## Survey of Consumer Finances

Data on household portfolios come from the Survey of Consumer Finances (SCF). We construct networth as: $$\begin{multline*}
\mathtt{networth\_d} = \mathtt{cash\_dep}+\mathtt{equity}+\mathtt{fixed\_inc}+\mathtt{real\_estate} \\ +\mathtt{bus}+\mathtt{vehic}-\mathtt{mortgage\_dbt}- \mathtt{vehic\_dbt} -\mathtt{edu\_dbt}-\mathtt{other\_dbt},
\end{multline*}$$ where each of the constituent variables are defined as:

- $\mathtt{cash\_dep}$: value of cash deposits defined as liquid accounts ($\mathtt{liq}$) which are the sum of all checking, savings, and money market accounts, call accounts at brokerages, and prepaid cards, added to certificates of deposit ($\mathtt{cds}$).

- $\mathtt{equity}$: value of all financial assets invested in stock, which include directly held stock, stock mutual funds, and the portion of any combination mutual funds, annuities, trusts, IRA/Keogh accounts, and other retirement accounts invested in stock.

- $\mathtt{fixed\_inc}$: value of all other remaining financial assets ($\mathtt{fixed\_inc} = \mathtt{fin} - \mathtt{cash\_dep}-\mathtt{equity}$). The largest component of this asset category is bonds held outright, in mutual funds, and in retirement accounts.

- $\mathtt{real\_estate}$: value of the primary residence ($\mathtt{houses}$) plus the value of other residential real estate ($\mathtt{oresre}$) and net equity in nonresidential real estate ($\mathtt{nnresre}$).

- $\mathtt{bus}$: reported market value of private business interest.

- $\mathtt{vehic}$: prevailing retail value for all vehicles owned by household.

- $\mathtt{mortgage\_dbt}$: housing debt from mortgages, home equity loans, and home equity lines of credit ($\mathtt{mrthel}$) plus debt for other residential property ($\mathtt{resdbt}$).

- $\mathtt{vehic\_dbt}$: debt from vehicle loans ($\mathtt{veh\_inst}$)

- $\mathtt{edu\_dbt}$: debt from student loans

- $\mathtt{other\_dbt}$: other debt, including other lines of credit plus credit card balance ($\mathtt{ccbal}$) plus installment loans less education loans and vehicle loans ($\mathtt{other\_dbt} = \mathtt{othloc}+\mathtt{ccbal}+\mathtt{install}- \mathtt{edn\_inst}-\mathtt{veh\_inst}$).

In addition to portfolio data, we also use data on household wage income ($\mathtt{wageinc}$) which we combine with data on the number of people in the household and the Social Security wage index to create a per capita wage measure that is comparable over time. We count married and widowed respondents as two-adult households and all others as single-adult households.

## Duration component calculations 

### Duration of equity

The duration of $\mathtt{equity}$ is obtained using annual estimates for the duration of the aggregate stock market from (Greenwald et al. 2023), Figure D2 of the September 2023 working paper version. These estimates are applied uniformly to all individuals in the SCF by survey year.

To allow equity duration to vary across the wealth distribution, we apply a mean-preserving heterogeneity adjustment analogous to the one described for private business wealth in Section 11.2.4. We split households into six wealth groups $g$---bottom 50%, 50--90%, 90--99%, 99--99.9%, 99.9--99.99%, top 0.01%---and compute the equity-value-weighted price-dividend ratio for each group using equity holdings and dividend income reported in the SCF. Specifically, let $E_g = \sum_{i \in g} E_i$ and $D_g = \sum_{i \in g} D_i$ denote aggregate equity holdings and dividend income in wealth group $g$, where both are deflated by the Social Security wage index. The group-level price-dividend ratio is $m_g^{eq} = E_g / D_g$, and the equity-weighted population average is $\bar{m}^{eq} = \sum_g E_g \, m_g^{eq} / \sum_g E_g$. The mean-preserving adjustment factor for group $g$ is $$\alpha_g^{eq} = \frac{m_g^{eq}}{\bar{m}^{eq}},$$ so that the adjusted equity duration for households in group $g$ is $\text{dur}_{g}^{eq} = \alpha_g^{eq} \times \text{dur}^{eq,\text{agg}}$, where $\text{dur}^{eq,\text{agg}}$ is the aggregate duration estimate from (Greenwald et al. 2023). By construction, $\sum_g E_g \, \alpha_g^{eq} / \sum_g E_g = 1$.

### Duration of fixed income

Data on the Macaulay duration of government bonds, municipal bonds, corporate bonds, and mortgage backed securities come from Bloomberg where the series used are:

- US gov/credit: $\mathtt{LUGCTRUU}$

- US Treasury: $\mathtt{LUATTRUU}$

- Government-related: $\mathtt{LD08TRUU}$

- US aggregate: $\mathtt{LBUSTRUU}$

- Municipal bond: $\mathtt{LMBTTR}$

- Corporate: $\mathtt{LUACTRUU}$

- US MBS: $\mathtt{LUMSTRUU}$

- Global aggregate: $\mathtt{LEGATRUU}$

For holdings of US government bonds ($\mathtt{govtbnd} + \mathtt{gbmutf} + \mathtt{savbnd}$), we use the market-value weighted average Macaulay duration of the US gov/credit, US Treasury, and government-related bond categories. For holdings of tax-free and municipal bonds ($\mathtt{notxbnd} + \mathtt{tfbmutf}$), mortgage-backed securities ($\mathtt{mortbnd}$), corporate bonds ($\mathtt{corpbnd}$), and foreign bonds $(\mathtt{forbnd})$, we use the Macaulay duration of municipal bonds, corporate bonds, US MBS, and the global aggregate, respectively. For all other fixed income assets that we do not have duration measures for, we assign 5.64, which is the average fixed income wealth-weighted duration in the SCF assets for which we have data.

Similar to equity and private business, we also apply a mean-preserving heterogeneity adjustment to bond durations across wealth groups. For each of the same six wealth groups $g$, let $B_g = \sum_{i \in g} B_i$ and $I_g = \sum_{i \in g} I_i$ denote aggregate bond holdings and interest income (both deflated by the Social Security wage index). The group-level price-to-interest ratio is $m_g^{bond} = B_g / I_g$, and the bond-weighted population average is $\bar{m}^{bond} = \sum_g B_g \, m_g^{bond} / \sum_g B_g$. The mean-preserving adjustment factor for group $g$ is $$\alpha_g^{bond} = \frac{m_g^{bond}}{\bar{m}^{bond}},$$ so that the adjusted bond duration for households in group $g$ is $\text{dur}_{g}^{bond} = \alpha_g^{bond} \times \text{dur}^{bond,\text{agg}}$, where $\text{dur}^{bond,\text{agg}}$ is the Bloomberg duration estimate for the relevant bond type. By construction, $\sum_g B_g \, \alpha_g^{bond} / \sum_g B_g = 1$. This allows bond duration to vary across the wealth distribution in proportion to valuation multiples implied by reported interest income.

### Duration of real estate

The duration of $\mathtt{real\_estate}$ is obtained using the annual estimates of the duration of aggregate real estate from (Greenwald et al. 2023), Figure D2 of the September 2023 working paper version. These estimates are applied uniformly to all individuals in the SCF by survey year.

### Duration of private business wealth 

The duration of private business wealth is computed for each household as the value of household businesses, $\mathtt{bus}$, divided by the annual cashflows from those equity holdings. However, the annual cashflows from those equity holdings are not reported in the SCF, the major issue being that cashflows from private businesses partially contain implicit or explicit labor income for the entrepreneur. As such, we must estimate or difference out this labor income. We do this in four ways, depending on the household's role in the business and what is reported.

1. For households whose main respondent has an active management role in either of the household's potential actively managed businesses, reports being self-employed, and reports not receiving a salary, we estimate their predicted wage.

 - The predicted wage is estimated via ordinary least squares on all SCF respondents $j$ where the household's wage income is the dependent variable, and the independent variables are a third-degree polynomial in age interacted with dummies for each $\text{Race}\times\text{Education}\times\text{Gender}$ group.

2. For households whose main respondent has an active management role in either of the household's potential actively managed businesses and reports being self-employed and receiving a salary or reports being employed by someone else, we subtract the maximum of their predicted wage and reported wage from $\mathtt{busefarminc}$.

3. We repeat steps 1) and 2) for spouses who have an active management role in either of the household's potential actively managed businesses.

4. All other households with positive private business wealth who do not meet the criteria for a wage subtraction are given cashflows equal to $\mathtt{busefarminc}$.

We then aggregate $\mathtt{bus}$ and the estimated annual cashflows within each survey year and divide them to obtain an annual time series of valuation ratios.

Next, to allow our aggregate estimates of private business duration to vary over the wealth distribution, we perform a mean-preserving adjustment to these aggregate duration estimates. First, we split the population into different networth groups defined by whether they are in the bottom 50%, 50--90%, 90--99%, 99--99.9%, 99.9--99.99% or the top 0.01% of the wealth distribution. We then take the business wealth ($\mathtt{bus}$) divided by total income from businesses ($\mathtt{busefarminc}$) for each household, and take the business wealth-weighted average for each networth group. Provided that cashflows from equity are proportional to labor income, this provides a proxy for duration for each group. These price-total income ratios are then divided by the business wealth-weighted average for the aggregate population to obtain a mean-preserving adjustment which is applied to the annual aggregate private business duration estimates. The duration is then given by $$
 \text{dur}(\text{Private Business}_{ct}) = \underbrace{\left(\frac{\sum_{t,i \in c}\text{Bus. Value}_{it}}{\sum_{t,i \in c}\text{Bus. Income}_{it}}\right)}_{\text{Wealth-group-specific valuation multiple}} \times \:\:\underbrace{\left(\frac{\sum_i \text{Bus. Income}_{it}}{\sum_i(\text{Bus. Income}_{it} - \text{Wages}_{it})}\right)}_{\text{Business-income-to-dividend conversion ratio}}
$$

### Duration of vehicles

The $\mathtt{vehic}$ category in the SCF contains detailed information on up to 4 automobiles, up to 2 non-automobile vehicles, and an aggregation of additional automobiles and non-automobile vehicles owned by the household. For the primary automobiles of the household, we attribute an expected lifetime of 8 years for 1989 and 12 years for 2019, linearly interpolating in intermediate years. We calculate the time left on an automobile's life as the model year plus the expected age minus the survey year. We assume a fixed depreciation rate to 0 over the car's remaining years, and calculate the duration using ). We attribute a duration of one to vehicles whose age exceeds their expected lifetime.

For the aggregation of additional automobiles owned, we attribute a duration equal to the average of the duration of the first four automobiles owned by the household. For all non-automobile vehicles owned by the household, we ascribe a duration of 6 years.

### Defined benefit pension wealth 

We add defined benefit (DB) pensions to the analysis in three steps: (1) computing the funded portion of aggregate DB pension entitlements, (2) allocating these entitlements to individual SCF households, and (3) assigning the resulting pension wealth to asset categories with corresponding interest-rate sensitivities.

First, we take aggregate DB pension entitlements from the Federal Reserve's Distributional Financial Accounts (DFA), which reports total DB pension net worth by wealth group: bottom 50%, 50--90%, 90--99%, 99--99.9%, and top 0.1%. To isolate the funded portion, we compute a funding ratio from the Flow of Funds (Table L.117) as the ratio of pension fund assets (series FL592000075.A) to total pension entitlements (series FL594190045.A) for each year from 1945 to 2024. We apply this funding ratio to the DFA entitlements within each wealth group: $$\text{Funded DB}_{ct} = \text{DFA DB}_{ct} \times \frac{\text{Pension fund assets}_t}{\text{Pension entitlements}_t},$$ where $c$ denotes the wealth group. Including only the funded portion is consistent with (Saez and Zucman 2016), who also restrict attention to funded DB pension wealth in their analysis.

Second, we allocate these aggregate funded DB pension amounts to individual households in the Survey of Consumer Finances (SCF) using a weighting scheme based on four factors. For each household, we construct an allocation weight: $$w_{it} = \text{accrual}_{it} \times \text{mortality}_{it} \times \text{income}_{it} \times \mathbf{1}[\text{has DB}_{it}],$$ where:

- $\mathbf{1}[\text{has DB}_{it}]$ equals one if the respondent or spouse reports having a DB pension plan from a current or previous job (`dbplancj==1` or `dbplant==1`), and zero otherwise;

- $\text{accrual}_{it}$ is a predicted fraction of lifetime pension benefits already accrued, obtained from a regression of the ratio of accrued Social Security benefits to total expected benefits on income centile, net worth centile, age, and sex indicators, estimated on the subsample with age $\leq 65$. For households over age 65 we set $\text{accrual}_{it} = 1$; for those under 20 we set it to 0. All other predicted values are capped between 0 and 1;

- $\text{mortality}_{it} = \min\left(1,\; e_{x,t} / e_{65,t}\right)$, where $e_{x,t}$ is remaining life expectancy at age $x$ in year $t$ from the Human Mortality Database, computed separately by sex. This adjusts for the fact that younger households have longer expected payout horizons;

- $\text{income}_{it} = \min\left(\max\left(\text{wageinc}_{it} / \overline{\text{wageinc}}_{ct},\; 0\right),\; 5\right)$, where $\overline{\text{wageinc}}_{ct}$ is the mean wage income within wealth group $c$ in year $t$. This captures the correlation between pension generosity and career earnings.

Within each wealth group $c$ and year $t$, we allocate the aggregate funded DB pension amount proportionally based on survey-weighted individual weights: $$\text{DB}_{it} = \frac{w_{it}}{\sum_{j \in c} w_{jt} \cdot \omega_j} \times \text{Funded DB}_{ct},$$ where $\omega_j$ denotes the SCF survey weight for household $j$.

Third, we allocate each household's DB pension wealth to three asset categories---cash deposits, fixed income, and equity---using the asset composition of private pension funds from Table L.117 of the Flow of Funds, which we compute for each year. Specifically, cash deposits include checkable and time/savings deposits; fixed income includes Treasury securities, agency securities, corporate and foreign bonds, and mortgages; and equity includes corporate equities and mutual fund shares. Miscellaneous assets are excluded from the denominator.

These DB pension amounts are added to the corresponding categories in the household's balance sheet. For the purpose of computing interest-rate sensitivities, DB pension equity receives the aggregate stock market interest-rate sensitivity and DB pension fixed income receives the aggregate US bond interest-rate sensitivity, reflecting the diversified nature of pension fund portfolios. The interest-rate sensitivity of the combined category (e.g., public equity) is then a value-weighted average of the household's direct holdings and the DB pension component.

### Duration of debts 

For the debt categories, $\mathtt{mortgage\_dbt}$, $\mathtt{vehic\_dbt}$, and $\mathtt{other\_dbt}$, we break each up into their component loans as described in the SCF extract and calculate the duration of each loan separately. For each loan, we assume a fixed payment schedule, and thus its duration can be calculated using equation ), where $N$ is the maturity of the loan and $y_{nt}$ is the riskfree spot rate at horizon $n$ in year $t$.[^39]

Under our fixed payment assumption, the only metric we need for each loan is its time remaining. Since different loan component variables contain different amounts of information in the raw SCF, we calculate the time remaining differently depending on the available information for each component loan group: primary component loans, aggregated additional loans, and lines of credit. The primary component loans of each debt category contain information on loan origination, balance, payments, and interest rates. For these loans, we calculate the number of years remaining on the loan payments using the reported origination year, length of loan at origination, and survey year. For respondents with a positive loan balance who have missing responses for loan length or a negative calculated time remaining, we impute time remaining with balance ($B$), initial amount ($L$), interest rate ($R$), and year of origination ($p$) using the equation $$\begin{align*}
 T = \frac{\log(R^p-B/L) - \log(1 - B/L)}{\log R} - p.
\end{align*}$$ The aggregated additional loans group contains loan variables that capture an aggregation of loans that the respondents hold in addition to the primary ones in each debt category. These loans include data on only loan balance and payments ($X$). Using the average interest rates for primary loans in the same debt category, we calculate time remaining as $$\begin{align*}
 T = - \frac{\log(1- B(R-1)/X)}{\log R}.
\end{align*}$$

The third group of component loans is the lines of credit. The line of credit variables contain information on loan balance, typical payments, and interest rates. With these data points, we calculate time remaining according to the same formula used for the aggregated additional loans group. Finally, there is an aggregated additional lines of credit variable, which we assign a duration equal to the average of the duration of the other lines of credit.

We replace the duration of loans with a predicted time remaining under one year with a duration of one and give the median duration to respondents with a positive loan amount but insufficient information to calculate time remaining on the loan.

We also adjust student loan balances and duration to account for income-driven repayment (IDR) following (Catherine and Yannelis 2021). Under IDR, borrowers pay 10% of earnings above 1.5 times the Social Security wage index (scaled by 0.225) annually for up to 25 years. We discount these payments at the riskfree rate to obtain the present value and corresponding cashflow duration.

Finally, since adjustable rate mortgages (ARMs) have no interest-rate exposure after they float, we treat them differently. In particular, we cap their duration at 4---the average number of years until the interest rate begins to float in the SCF. Approximately 12% of mortgages and 19% of investment property loans are adjustable rate in the data.

### Interest-rate sensitivity of Social Security wealth

The interest-rate sensitivity of Social Security wealth comes from the methodology used in (Catherine et al. 2025). We generate their baseline risk-adjusted Social Security wealth under the net present value wealth concept and the Treasury yield curve. We then generate Social Security wealth under an identical specification where the log forward rate at horizon $h$ in survey year $t$ is given by $\tilde{f}_{h,t} =f_{h,t} + \varphi^h 0.01$ where $f_{h,t}$ is the unshocked log forward rate and $\varphi$ is our calibrated persistence for the interest rate process shown in Table 1. This is the same thing as applying a one-period shock of 0.01 to the log riskfree rate under the process in equation ).

## Evidence from Europe

This section presents the interest-rate sensitivity (IRS) of household portfolios in the European Union (EU) using data from the Household Finance and Consumption Survey from the European Central Bank. We use all four available waves (2010, 2014, 2017, and 2021) and include all participating countries. As with the SCF, we use all five multiply-imputed datasets per wave.

To obtain the IRS of assets, we assume each asset held by EU households has the same IRS as those held by their American counterparts. For the 2021 HFCS wave, we use 2019 US duration estimates, the most recent available. For other financial assets (`othfin`) and voluntary pension and whole-life insurance holdings (`pensions`), which are not broken down by asset type in the HFCS, we allocate 50% to equity and 50% to fixed income.

To obtain the IRS of debts, we follow the same procedure as in Appendix 11.2.7, with two differences. First, adjustable-rate mortgages are assigned a duration of one year (reflecting the typical annual reset frequency in European mortgage markets, compared to four years in the US). Second, for other non-mortgage loans where individual duration cannot be computed, we assign the household's own value-weighted average duration across its mortgage and property loans.

We winsorize the net worth duration at the 1st and 99th percentiles to reduce the influence of outliers. The results are presented in Figure 16. Just as in the US, the interest-rate sensitivity of household portfolios is increasing with income and wealth and hump-shaped over the lifecycle.

<figure id="fig:eu_vs_usa" data-latex-placement="H">
<div class="center">
<table>
<tbody>
<tr>
<td style="text-align: center;"><strong>A. IRS by wage tercile</strong></td>
<td style="text-align: center;"><strong>B. IRS by wealth (40–45 y.o.)</strong></td>
</tr>
<tr>
<td style="text-align: center;"><embed src="Plots/ecb_irs_wage.pdf" /></td>
<td style="text-align: center;"><embed src="Plots/ecb_irs_weath.pdf" /></td>
</tr>
</tbody>
</table>
</div>
<figcaption><strong>Interest-rate sensitivity in the European Union</strong></figcaption>
</figure>

While the broad trends are similar, the relationship between income and wealth and interest-rate sensitivity is less stark in the EU, particularly at the top of the distribution. A potential reason for this is illustrated by Figure 17, which shows that countries in the EU give more generous benefits to higher earners. Indeed, in the US, essentially no one receives more than 60% of average income, whereas in the EU it is not uncommon for an individual to receive more than 100% of the national average wage. This is what our model predicts: countries that replace more income for high earners should have a flatter relationship between income, wealth, and interest-rate sensitivity.

<figure id="fig:eu_replacement" data-latex-placement="H">
<embed src="Plots/ecb_vs_usa_replacement_rate.pdf" />
<figcaption><strong>Generosity of public pensions: US vs EU</strong></figcaption>
</figure>

# Model derivations 

## Long-term bond returns 

This section explains how the riskfree rate dynamics ) imply the $n$-period bond returns ). Since it has no intermediate cashflows, the bond's return from $t$ to $t+1$ is $$
R_{n,t+1}= \frac{P_{n-1,t+1}}{P_{nt}} \equiv \frac{e^{-(n-1)y_{n-1,t+1}}}{e^{-ny_{nt}}},

$$ where the yield $y_{nt}$, under the expectations hypothesis, is given by $$
 y_{nt} \equiv \frac{1}{n} \log\left(\frac{1}{P_{nt}}\right)= \frac{1}{n}\sum_{j=1}^n(\mathbb{E}_tr_{f,t+j-1} + \mu_j).
 
$$ Moreover, note that ) iterates backward to the expression $$
r_{f,t+j} = (1-\varphi^j)\bar{r}_f + \varphi^jr_{ft} + \sum_{k=1}^j\varphi^{j-k}\sigma_r\epsilon_{r,t+k}.

$$ Substituting the riskfree rates ) into the yield expression ) and evaluating expectations implies $$
y_{nt} = \bar{r}_f+ \frac{1}{n}\frac{1-\varphi^n}{1-\varphi}(r_{ft}-\bar{r}_f) + \frac{1}{n}\sum_{j=1}^n\mu_j.

$$ Taking logs of ) and substituting ) into the yield then implies the log return $$\begin{align}
r_{n,t+1} &= ny_{nt}-(n-1)y_{n-1,t+1} \\%\log P_{t+1}^{(n-1)} - \log P_t^{(n)} \\
&= \bar{r}_f+ \frac{1-\varphi^n}{1-\varphi}(r_{ft}-\bar{r}_f) - \frac{1-\varphi^{n-1}}{1-\varphi}(r_{f,t+1}-\bar{r}_f) + \mu_n \\
&= \bar{r}_f+\frac{1-\varphi^n}{1-\varphi}(r_{ft}-\bar{r}_f) - \frac{\varphi-\varphi^n}{1-\varphi}(r_{ft}-\bar{r}_f) - \frac{1-\varphi^{n-1}}{1-\varphi}\sigma_r\epsilon_{r,t+1} + \mu_n \\
&= r_{ft} + \mu_n - \frac{1-\varphi^{n-1}}{1-\varphi}\sigma_r\epsilon_{r,t+1},
\end{align}$$ the stated expression ).

## Stock returns 

Let $D_t$ denote the dividends of the market and $V_t$ its value. We make two assumptions. First, expected log returns equal the riskfree rate plus a constant risk premium: $$
 \mathbb{E}_t[r_{s,t+1}] = r_{ft} + \mu_s.
$$ Second, since dividends are cointegrated with aggregate income, we follow (Longstaff and Piazzesi 2004) and define the log dividend-income ratio $dl_t\equiv\log(D_t/L_t)$ by the autoregression $$
 dl_{t+1} = (1-\varphi_d)\overline{dl} + \varphi_ddl_t + \epsilon_{d,t+1},
$$ for some shock $\epsilon_{d,t+1}$ with $\mathbb{E}_t[\epsilon_{d,t+j}]=0$. This implies the log dividend growth rate $$
 g_{D,t+1} = g_{t+1} + \Delta dl_{t+1} = \lambda_{gr}r_{f,t+1} + g_{1,t+1} - (1-\varphi_d)(dl_t-\overline{dl}) + \epsilon_{d,t+1}.
$$

Next, we linearize the stock return under these assumptions. By definition, the return on the stock market is $$
 R_{s,t+1} = \frac{D_{t+1}+V_{t+1}}{V_t} = \frac{D_{t+1}}{D_t}\left(\frac{1+V_{t+1}/D_{t+1}}{V_t/D_t}\right).
$$ Now take logs and linearize to get the Campbell-Shiller decomposition. Letting $g_{D,t+1} = \log(D_{t+1}/D_t)$ and $vd_t=\log(V_t/D_t)$ represent the log of dividend growth and of the price-dividend ratio, respectively, we have $$
 r_{s,t+1} \approx g_{D,t+1} + \kappa_0 + \kappa_1vd_{t+1} - vd_t,
$$ where the linearizing constants are $$
 \kappa_1 \equiv \frac{1}{1+\exp\{-\overline{vd}\}}\quad\textrm{and}\quad \kappa_0 \equiv -\log\kappa_1 - \kappa_1\overline{vd},
$$ for $\overline{vd}=\mathbb{E}[vd_t]$. Solving the linearized equation forward for the price-dividend ratio implies $$
 vd_t = g_{D,t+1} - r_{s,t+1} + \kappa_0 + \kappa_1vd_{t+1} = \kappa_0 + \sum_{j=0}^\infty \kappa_1^j(g_{D,t+1+j} - r_{s,t+1+j}).
$$ which holds both in realization and in expectation.

We know the expected stock return $\mathbb{E}_t[r_{s,t+1}]$ by assumption. Using the linearization, the unexpected component of the stock return equals $$
 r_{s,t+1} - \mathbb{E}_t[r_{s,t+1}] = g_{D,t+1}-\mathbb{E}_t[g_{D,t+1}] + \kappa_1(vd_{t+1}-\mathbb{E}_t[vd_{t+1}]).
$$ Because risk premia are constant, the unexpected change in the price-dividend ratio equals: $$
 \kappa_1(vd_{t+1}-\mathbb{E}_t[vd_{t+1}]) = \sum_{j=1}^\infty \kappa_1^j(\mathbb{E}_{t+1}[g_{D,t+1+j}] - \mathbb{E}_t[g_{D,t+1+j}]) - \sum_{j=1}^\infty \kappa_1^j(\mathbb{E}_{t+1}[r_{f,t+j}] - \mathbb{E}_t[r_{f,t+j}]),
$$ and hence $$
 r_{s,t+1} - \mathbb{E}_t[r_{s,t+1}] = \sum_{j=0}^\infty \kappa_1^j(\mathbb{E}_{t+1}[g_{D,t+1+j}] - \mathbb{E}_t[g_{D,t+1+j}]) - \sum_{j=1}^\infty \kappa_1^j(\mathbb{E}_{t+1}[r_{f,t+j}] - \mathbb{E}_t[r_{f,t+j}]).
$$ Note first that $$
 \mathbb{E}_{t+1}[r_{f,t+j}] - \mathbb{E}_t[r_{f,t+j}] = \varphi^{j-1}\sigma_r\epsilon_{r,t+1},
$$ and therefore that $$
 \sum_{j=1}^\infty \kappa_1^j(\mathbb{E}_{t+1}[r_{f,t+1+j}] - \mathbb{E}_t[r_{f,t+1+j}]) = \frac{\kappa_1}{1-\kappa_1\varphi}\sigma_r\epsilon_{r,t+1}.
$$ Next, note that $$\begin{multline}
 \mathbb{E}_{t+1}[g_{D,t+1+j}] - \mathbb{E}_t[g_{D,t+1+j}] = \lambda_{gr}(\mathbb{E}_{t+1}[r_{f,t+1+j}] - \mathbb{E}_t[r_{f,t+1+j}]) + (\mathbb{E}_{t+1}[g_{1,t+1+j}] - \mathbb{E}_t[g_{1,t+1+j}]) \\
- (1-\varphi_d)(\mathbb{E}_{t+1}[dl_{t+j}] - \mathbb{E}_t[dl_{t+j}]) + \mathbb{I}_{j=0}\epsilon_{d,t+1}.
\end{multline}$$ The riskfree rate component equals $$
 \lambda_{gr}(\mathbb{E}_{t+1}[r_{f,t+1+j}] - \mathbb{E}_t[r_{f,t+1+j}]) = \lambda_{gr}\varphi^j(r_{f,t+1} - \mathbb{E}_t[r_{f,t+1}]) = \lambda_{gr}\varphi^j\sigma_r\epsilon_{r,t+1}.
$$ Similarly, $$
 \mathbb{E}_{t+1}[g_{1,t+1+j}] - \mathbb{E}_t[g_{1,t+1+j}] = \varphi_g^j(g_{1,t+1} - \mathbb{E}_t[g_{1,t+1}]) = \varphi_g^j\epsilon_{g,t+1},
$$ and $$
 (1-\varphi_d)(\mathbb{E}_{t+1}[dl_{t+j}] - \mathbb{E}_t[dl_{t+j}]) = (1-\varphi_d)\varphi_d^{j-1}\epsilon_{d,t+1}.
$$ Consequently, $$
 \sum_{j=0}^\infty \kappa_1^j(\mathbb{E}_{t+1}[g_{D,t+1+j}] - \mathbb{E}_t[g_{D,t+1+j}]) = \frac{\lambda_{gr}\sigma_r}{1-\kappa_1\varphi}\epsilon_{r,t+1} + \frac{1}{1-\kappa_1\varphi_g}\epsilon_{g,t+1} + \frac{1-\kappa_1}{1-\kappa_1\varphi_d}\epsilon_{d,t+1}.%+ \left(1-\frac{(1-\varphi_d)\kappa_1}{1-\kappa_1\varphi_d}\right)\epsilon_{d,t+1}.
$$ Putting all of this together, the unexpected stock return equals $$
 r_{s,t+1} - \mathbb{E}_t[r_{s,t+1}] = \frac{\lambda_{gr} - \kappa_1}{1-\kappa_1\varphi}\sigma_r\epsilon_{r,t+1} + \frac{1}{1-\kappa_1\varphi_g}\epsilon_{g,t+1} + \frac{1-\kappa_1}{1-\kappa_1\varphi_d}\epsilon_{d,t+1}.
$$ Substituting in the expected return, we have an expression for return.

Finally, to get to the expression ) in the main text, we assume that the shock $\epsilon_d$ can be decomposed such that $$
 \frac{1-\kappa_1}{1-\kappa_1\varphi_d}\epsilon_{d,t+1} = \lambda_{dg}\epsilon_{g,t+1} + \epsilon_{s,t+1}
$$ where $\epsilon_g$ is the growth shock and $\epsilon_s$ is the market-specific shock. For the loading $\lambda_{sr}$ on interest-rate shocks in ), we use the fact that, under this linearization, the interest-rate sensitivity of the stock equals $$
 \frac{\kappa_1}{1-\kappa_1\varphi} = \frac{\sigma_{n_s}}{\sigma_r} = \frac{1-\varphi^{n_s-1}}{1-\varphi},
 
$$ where $n_s$ is the duration of the stock market. This implies $$
 \kappa_1 = \frac{\sigma_{n_s}/\sigma_r}{1+\varphi\sigma_{n_s}/\sigma_r} = \frac{1-\varphi^{n_s-1}}{1-\varphi^{n_s}}.
 
$$ We then use this value to compute the loading on the interest rate, $$
 \lambda_{sr} = \frac{\kappa_1-\lambda_{gr}}{1-\kappa_1\varphi}.
 
$$ Together, this gives the stock return ).

## Housing returns 

To derive housing returns, we need to make assumptions about depreciation and maintenance of the housing stock, as well as transaction costs from buying and selling housing. First, we assume that the housing stock depreciates each period and owners offset this by investing a proportionate maintenance cost. Second, we assume that homeowners incur proportionate transaction costs, representing the cost of moving. In the model, it is costless for agents to change house size, and thus households move houses every period and pay this transaction cost each period.

These assumptions manifest themselves in the housing return, which equals $$
 R_{hi,t+1} = \frac{P_{n_h-1,t+1}}{P_{n_ht}}\frac{P_{h,t+1}}{P_{ht}}\frac{\tilde{P}_{i,t+1}}{\tilde{P}_{it}},
$$ as follows. The bond component of the return accounts for the fact that the house depreciates over the period, so its duration falls from $n_h$ to $n_h-1$, as would a long-term bond. The drift $\mu_h$ in ) includes not only common price appreciation, but also nets out the costs to the homeowner of transacting and of maintaining the property. The investment in maintenance offsets the depreciation, and in doing so returns the bond portion of the house back to its original duration $n_h$ for the new set of homebuyers. And finally, because the house is sold after this return is realized, the idiosyncratic house price shock only accumulates over the one period, at which point the homeowner sells the house and buys a new one at the regional price $P_t$.

## Private business valuation 

Here we derive the valuation of private business at exit ). Both in our model and in empirical measures of the value of the business, the relevant valuation is under the discount rate and cashflow dynamics after the company is sold. For discount rates, we assume that the business is sold to diversified investors---this will be the case, for instance, if the company goes public. Because the new owners will be diversified, they will not demand compensation for idiosyncratic risk, so the relevant risk premium will be the same as that of the market. Future cashflows $L_i$ can follow a general process on which we impose only one fairly minor assumption: growth in the persistent part of the idiosyncratic component of income is not predictable from current information. Formally, for all $j\geq0$, $\mathbb{E}_t[\tilde{L}_{i,t+1+j}/\tilde{L}_{i,t+j}]=\mathbb{E}[\tilde{L}_{i,t+1+j}/\tilde{L}_{i,t+j}]$. Note that this assumption allows for the possibility that the cashflow growth dynamics could change after exit. The only purpose of this assumption is to simplify the model; it affects fluctuations in valuation over time, but not the average level of valuations, as we will see from what follows.

Let $E_{it}$ denote the present value of future earnings $\{L_{i,t+j}\}$. Let $e_{it}$ and $l_{i,t+j}$ denote logs of these variables and $el_{it} \equiv e_{it}-l_{it}$ the current valuation ratio. In what follows, we will suppress $i$ subscripts. Applying the same Campbell-Shiller linearization as we did for the stock market in Appendix 12.2, we have that the one-period log return on the business is $$
 r_{e,t+1} \approx g_{L_i,t+1} + \kappa_{E0} + \kappa_Eel_{t+1} - el_t,
$$ where $$
 \kappa_E \equiv \frac{1}{1+\exp\{-\overline{el}\}}\quad\textrm{and}\quad \kappa_{E0} \equiv -\log\kappa_E - \kappa_E\overline{el},
$$ and $\overline{el}\equiv\mathbb{E}[el_t]$ is the unconditional mean valuation ratio. Iterating forward and taking expectations, we have $$
 el_t = \kappa_{E0} + \sum_{j=0}^\infty \kappa_E^j(\mathbb{E}_t[g_{l,t+1+j}] - \mathbb{E}_t[r_{e,t+1+j}]),
$$ or, in deviations from the mean, $$
 el_t - \overline{el} = \sum_{j=0}^\infty \kappa_E^j(\mathbb{E}_t[g_{l,t+1+j}] - \bar{g}_l) - \sum_{j=0}^\infty \kappa_E^j(\mathbb{E}_t[r_{e,t+1+j}] - \bar{r}_e).
$$ Because the firm is sold to diversified investors, it has the same expected return as the stock market, so $$
 \mathbb{E}_t[r_{e,t+1+j}] = \mathbb{E}_t[r_{s,t+1+j}] = \mathbb{E}_t[r_{f,t+j}]+\mu_s = (1-\varphi^j)\bar{r}_f + \varphi^jr_{ft} + \mu_s.
$$ This implies $$
 \sum_{j=0}^\infty \kappa_E^j(\mathbb{E}_t[r_{e,t+1+j}] - \bar{r}_e) = \frac{1}{1-\kappa_E\varphi}(r_{ft} - \bar{r}_f). %= \sum_{j=0}^\infty (\kappa_E\varphi)^j(r_{ft} - \bar{r}_f)
$$ The cashflow growth component can be further divided into aggregate wage index growth and idiosyncratic growth. The aggregate growth component is $$
 \sum_{j=0}^\infty \kappa_E^j(\mathbb{E}_t[g_{1,t+1+j}] - \bar{g}_1) + \frac{\lambda_{gr}}{1-\kappa_E\varphi}(r_{ft} - \bar{r}_f) = \frac{1}{1-\kappa_E\varphi_g}(g_{1t} - \bar{g}_1) + \frac{\lambda_{gr}}{1-\kappa_E\varphi}(r_{ft} - \bar{r}_f).
$$ And we have assumed that the idiosyncratic component is not predictable, so $$
 \sum_{j=0}^\infty \kappa_E^j(\mathbb{E}_t[g_{\tilde{l},t+1+j}] - \mathbb{E}[g_{\tilde{l},t+1+j}]) = 0.
$$ Consequently, $$
 el_t = \overline{el} + \frac{1}{1-\kappa_E\varphi_g}(g_{1t} - \bar{g}_1) - \frac{1-\lambda_{gr}}{1-\kappa_E\varphi}(r_{ft} - \bar{r}_f).
$$ Taking the exponential of this and relabelling $\overline{el}$ as the parameter $\upsilon$, we get the expression ).

# Model calibration 

## Macroeconomic variables 

### Time series of riskfree interest rates 

To obtain a time series of the short-term real interest rate, we use a methodology similar to that of (Beeler and Campbell 2012). Using the yield on the 10-year nominal Treasury bond $y_{10}$ and annual inflation rate $\pi$ from Global Financial Data, we estimate the annual regression $$
 y_{10,t}-\pi_{t,t+1}=\beta_{0}+\beta_{1}y_{10,t}+\beta_{2}\pi_{t-1,t}+\epsilon_{t+1}
 
$$ on the post-war period. The fitted values are then taken as our estimate of the expected riskfree rate 10-years from time $t$, $\hat{f}_{10,t}$. From this, equation ) yields the time-$t$ riskfree rate: $$
 r_{ft}=\varphi^{-10}(\hat{f}_{10,t}-(1-\varphi^{10})\bar{r}_f).
$$ We use this methodology for two main reasons. First, by using long-term rates to back out short-term rates, we smooth much of the short-term variation in measured short-term real rates that are potentially outside of our model. Second, this methodology allows us to extend our real rate series further into the past, allowing for a longer simulation prior to our period of interest. This procedure yields a time series of annual realizations of real rates $\{r_{ft}\}$ and shocks $\{\epsilon_{rt}\}$ from 1789 to 2020. The post-war time series of these rates are shown in Figure 18.

<figure id="fig:rf_ts" data-latex-placement="H">
<div class="center">
<embed src="Plots/rf_ts_post_war.pdf" style="width:100.0%" />
</div>
<p><em>Note</em>: This figure presents the time series of short-term riskfree rates as estimated by equation (<a href="#eq:campbeel" data-reference-type="ref" data-reference="eq:campbeel">[eq:campbeel]</a>) and transformed by equation (<a href="#eq:rf_trans" data-reference-type="ref" data-reference="eq:rf_trans">[eq:rf_trans]</a>).</p>
<figcaption><strong>Time series of riskfree rates, Post-war sample</strong></figcaption>
</figure>

### Economic growth and stock-market returns

We jointly estimate the dynamics of aggregate growth ) and stock returns ) via the simulated method of moments (SMM). The parameter vector to be estimated is[^40] $$

\Theta \equiv \begin{bmatrix} \bar{g} & \varphi_g & \mu_g^- & \sigma_{g} & p_R & \lambda_{gr} & \mu_s^- & \mu_s^+ & \sigma_s \end{bmatrix}.
$$ We estimate the probability of a recession $p_R$ directly from the data as the frequency of NBER recessions. To identify the remaining 8 parameters, we use the first four moments (i.e., mean, variance, skewness, and kurtosis) of log stock returns and of log GDP growth, the autocorrelation of GDP growth, the correlation of stock returns and GDP growth, and the correlation of riskfree rates with five-year moving average growth. We use the correlation between lagged returns and GDP growth as a moment because stock markets are forward looking. This shows up in the data---as noted in (Fama 1990; Schwert 1990; Catherine et al. 2024)---by stock markets returns predicting future real activity more strongly than they predict current real activity. The higher-order moments (skewness and kurtosis) are particularly useful for identifying parameters of the normal mixtures. The correlation of rates with a moving average of the growth rate, as opposed to the growth rate itself, provides identification for $\lambda_{gr}$ by isolating the low-frequency covariance.

The SMM procedure then estimates $\Theta$ as follows. Letting $m(\Theta)$ denote the vector of simulated moments implied by the model and $\hat{m}$ the corresponding vector of data moments, our estimated parameters minimize the squared (percentage) errors: $$
\hat{\Theta} = \arg\min_{\Theta} \left\{\left(\frac{\hat{m} - m(\Theta)}{\hat{m}}\right)^\top\left(\frac{\hat{m} - m(\Theta)}{\hat{m}}\right)\right\},
$$ where division of vectors is an element-wise operation. For the data moments, we use historical data from a number of sources. We construct the growth series using per capita real GDP growth as a proxy for aggregate wage dynamics. The primary data source is the Global Macro Database (GMD), from which we extract US real GDP per capita from 1900 to 2021. Stock-market return data cover 1871--2021 and come from Shiller. We use the log total return on the S&P 500. Table 2 above reports the estimated parameters. Table 4 reports the moments in the data and model.

 Moment Description Data Model
 --------------------------------------- ------------------------------------------ -------- --------
 $\mathbb{E}[r_{st}]$ Mean stock return (1900-2021) 0.066 0.0666
 $\sigma(r_{st})$ Standard deviation of stock returns 0.187 0.188
 $\text{Skew}(r_{st})$ Skewness of stock returns -0.831 -0.892
 $\text{Kurt}(r_{st})$ Kurtosis of stock returns 3.61 3.5
 $\mathbb{E}[g_t]$ Mean wage growth (1960-2021) 0.0196 0.0196
 $\sigma(g_t)$ Standard deviation of wage growth 0.0205 0.0209
 $\text{Skew}(g_t)$ Skewness of wage growth -0.629 -0.676
 $\text{Kurt}(g_t)$ Kurtosis of wage growth 3.42 3.34
 $\text{Corr}(g_t, g_{t-1})$ Autocorrelation of wage growth 0.164 0.164
 $\text{Corr}(g_t, r_{st})$ Contemporaneous correlation 0.541 0.56
 $\text{Corr}(r_{f,t}, \bar{g}_{5,t})$ Risk-free rate and 5-year average growth 0.211 0.238
 $p_R$ Recession probability 0.164 0.164

 : Moments for estimation of growth and stock returns

## Idiosyncratic income process 

Our calibration of the idiosyncratic income component of earnings follows (Catherine 2022) closely, with two differences. In this study, the parameters of the income process are estimated by targeting moments of the distribution of log earnings growth from 1978 to 2011, feeding the historical mean log earnings growth of each of these years into the simulation. The targeted moments include the standard deviation of log earnings growth at the 1-- and 5--year horizons, and the skewness at the 1--, 3--, and 5--year horizons. These moments are computed using Social Security administrative data and reported in (Guvenen et al. 2014).

We estimate the process using the same procedure as (Catherine 2022), described in detail in Appendix B.1 of his study, with two differences. First, for years prior to 1978, we feed into the simulation the historical time series of log changes in real per capita GDP growth, which we use to calibrate the process for economic growth as in Section 13.1. For the period 1978--2011, we feed into the simulation the historical mean log earnings growth rate reported in (Guvenen et al. 2014), but scale this variable such that it exhibits the same volatility over that period as per capita GDP growth, which is roughly 1.5 times lower. Our goal is to obtain comparable variability in idiosyncratic income risk over the business cycle in the model and in the data, while leveraging the longer time span of the per capita GDP growth series relative to the moments reported in (Guvenen et al. 2014).

The second difference is that we account for the diversification arising from two earners in the typical American household. As (Guvenen et al. 2014) only report moments for individual earners, we proceed in two steps to estimate the income process for a two-earner household. First, we estimate all parameters for a single-earner household by targeting the moments reported in (Guvenen et al. 2014). The estimated parameters are reported in the 'Single' row of Table 5 and are very close to those in (Catherine 2022), except for the parameter $\lambda_{zg}$, which governs the relationship between GDP growth and the skewness of idiosyncratic income shocks. This parameter is roughly 1.5 times larger than in (Catherine 2022), reflecting the fact that the GDP growth time series is itself about 1.5 times less volatile than the mean log earnings growth series used in this study.

Second, we use the estimated parameters to generate new moments for two-earner households. To do so, we simulate individuals using our estimated income process and randomly match them to create two-earner households. We then compute new log income growth moments from these households, assuming that the highest earner in each household receives 70% of the combined income. Finally, we rerun the estimation procedure targeting these new moments. The estimated parameters are reported in the 'Two earners' row of Table 5.

+:------------+:--------+:---------+:---------------------+:---------------+:-------------+:-------------+:-:+:------------------+:------------------+
| | | | | Transitory |
+-------------+---------+--------------------------------------------------------------------------------+---+---------------------------------------+
| | | Persistent shocks | | shocks |
+-------------+---------+----------+----------------------+----------------+--------------+--------------+---+-------------------+-------------------+
| | $p_{z}$ | $\rho_z$ | $\overline{\mu^-_z}$ | $\lambda_{zg}$ | $\sigma^-_z$ | $\sigma^+_z$ | | $\sigma^-_{\eta}$ | $\sigma^+_{\eta}$ |
+-------------+---------+----------+----------------------+----------------+--------------+--------------+---+-------------------+-------------------+
| One earner | .145 | .971 | -.085 | 6.542 | .644 | .012 | | .758 | .097 |
+-------------+---------+----------+----------------------+----------------+--------------+--------------+---+-------------------+-------------------+
| Two earners | .224 | .993 | -.026 | 5.219 | .352 | .059 | | .414 | .124 |
+-------------+---------+----------+----------------------+----------------+--------------+--------------+---+-------------------+-------------------+

: **Estimated parameters: Idiosyncratic labor-income shocks**

This table reports parameter estimates for the idiosyncratic income process, conditional on the group-level aggregate shock. The single-earner process is estimated as in (Catherine 2022) by targeting the time-series of cross-sectional moments of individual income log growth rates in the SSA administrative data from (Guvenen et al. 2014). To estimate the double-earner process, we first simulate data for couples using the single-earner process and recompute the target moments using the sum of two incomes. We assume a 30% gap between the highest and lowest earner and no assortative matching.

Figure 19 presents the values and fit of targeted and simulated moments from the estimation of the idiosyncratic labor income shocks. Panels on the left report targeted and simulated moments for single-earner households, where targeted moments are from (Guvenen et al. 2014). Panels on the right report targeted and simulated moments for households with two earners.

<figure id="fig:idiofitness" data-latex-placement="H">
<div class="center">
<p></p>
<table>
<tbody>
<tr>
<td style="text-align: center;"><strong>A. Standard deviation</strong></td>
</tr>
<tr>
<td style="text-align: center;"></td>
</tr>
<tr>
<td style="text-align: center;"><embed src="Plots/IdiosyncraticstdFit.eps" style="width:90.0%" /></td>
</tr>
<tr>
<td style="text-align: center;"></td>
</tr>
<tr>
<td style="text-align: center;"><strong>B. 1-Year Skewness</strong></td>
</tr>
<tr>
<td style="text-align: center;"></td>
</tr>
<tr>
<td style="text-align: center;"><embed src="Plots/Idiosyncratic1yrSknewssFit.eps" style="width:90.0%" /></td>
</tr>
<tr>
<td style="text-align: center;"></td>
</tr>
<tr>
<td style="text-align: center;"><strong>C. 3-Year Skewness</strong></td>
</tr>
<tr>
<td style="text-align: center;"></td>
</tr>
<tr>
<td style="text-align: center;"><embed src="Plots/Idiosyncratic3yrSknewssFit.eps" style="width:90.0%" /></td>
</tr>
<tr>
<td style="text-align: center;"></td>
</tr>
<tr>
<td style="text-align: center;"><strong>D. 5-Year Skewness</strong></td>
</tr>
<tr>
<td style="text-align: center;"></td>
</tr>
<tr>
<td style="text-align: center;"><embed src="Plots/Idiosyncratic5yrSknewssFit.eps" style="width:90.0%" /></td>
</tr>
<tr>
<td style="text-align: center;"></td>
</tr>
</tbody>
</table>
</div>
<figcaption><strong>Fitness of idiosyncratic labor income risk model</strong> </figcaption>
</figure>

## Income tax rates 

Households pay income tax on labor income and Social Security benefits. In the model, households face the following marginal tax rates: $$
\textrm{Marginal Tax Rate}_{it} = \left\{
\begin{array}{lcl}
 0.10 & & if \tilde{L}_{it} < 0.18, \\
 0.12 & & if 0.18 \leq \tilde{L}_{it} < 0.72, \\
 0.22 & & if 0.72 \leq \tilde{L}_{it} < 1.54, \\
 0.24 & & if 1.54 \leq \tilde{L}_{it} < 2.94, \\
 0.32 & & if 2.94 \leq \tilde{L}_{it} < 3.73, \\
 0.35 & & if 3.73 \leq \tilde{L}_{it} < 9.32, \\
 0.37 & & if \tilde{L}_{it} \geq 9.32.
\end{array}
\right.
$$ The bendpoints in this formula are the limits of the 2020 tax brackets divided by the wage index.

## Entrepreneurs 

We calibrate the private business income ) and wealth ) processes to data from the SCF. We need to calibrate the deterministic component of income $\ell_E(a_{it})$, the initial volatility $\sigma_{Ez0}$ and time-series volatility $\sigma_{Ez}$ of the permanent component $z_{it}$, the mean valuation ratio $\upsilon$, and the probability of becoming an entrepreneur by age $p_E(a_{it})$. In the SCF, we consider a household to be an entrepreneur if the main respondent actively manages a private business and the household's business and farm income exceeds its wage income. We restrict our sample to households before retirement (i.e., aged 18 to 60).

We choose the mean log valuation ratio $\upsilon$ to match the average log aggregate valuation ratio: $$
 \upsilon = \mathbb{E}\left[\log\frac{\sum_{i=1}^{N_t}E_{it}}{\sum_{i=1}^{N_t}(L_{it} - L_{it}^{\rm wage})}\right] = 3.461,
$$ where $L_{it}^{\rm wage}$ is the predicted wage paid to the entrepreneur, as computed in the empirics. Recall that this aggregate ratio was how we computed the duration and interest-rate sensitivity of private business in Section 11, so this calibration is consistent with those moments as well. To estimate the deterministic life-cycle profile of business income, we fit a cubic polynomial in age to the moments $$
 \ell_E(a_{it}) = \mathbb{E}\left[\log\frac{L_{it}}{\bar{L}_t}\biggl|a_{it}\right],
$$ which implies an estimated profile $$
 \ell_E(a_{it}) = -6.118 + 0.334 \times a_{it} -0.0545 \times \frac{a_{it}^2}{10} +0.00256 \times \frac{a_{it}^3}{100}.
$$ To estimate the variances of $z_{it}$, we use the fact that $$
 \text{var}(z_{it}|a_{it}) = \text{var}\left(\log\frac{L_{it}}{\bar{L}_t}\biggl|a_{it}\right).
$$ Consequently, the time-series volatility can be computed from the change in variance of business income over the life cycle, $$
 \sigma_{Ez}^2 = \text{var}\left(\log\frac{L_{it}}{\bar{L}_t}\biggl|a_{it}\right) - \text{var}\left(\log\frac{L_{it}}{\bar{L}_t}\biggl|a_{it}-1\right),
$$ and the initial volatility can be computed as the level at the initial age, 20, $$
 \sigma_{Ez0}^2 = \text{var}\left(\log\frac{L_{it}}{\bar{L}_t}\biggl|a_{it}=20\right).
$$ We estimate these by running a linear regression of the variances $\text{var}(\log(E_{it}/\bar{L}_t)|a_{it})$ on age $a_{it}$, and find a constant $\sigma_{Ez0}^2=2.283$ and a slope $\sigma_{Ez}^2=0.0607$. Finally, we compute the entry probability $p_E(a_{it})$ as the change in the share of households who are entrepreneurs from one age to the next.[^41] Specifically, we first compute the share of households at each age $a_{it}$ who are entrepreneurs (call this $s_E(a_{it})$), then fit a cubic polynomial to this share, and then take the change in this fitted line as the entry probability, from the fact that $$
 p_E(a_{it}) = \frac{s_E(a_{it})-s_E(a_{it}-1)}{1-s_E(a_{it}-1)}.
$$

## Model fit with entrepreneurs 

<figure id="fig:validation_lifecycle_full" data-latex-placement="H">
<table>
<tbody>
<tr>
<td style="text-align: left;">A. Wealth</td>
<td style="text-align: left;">B. Home ownership rate</td>
<td style="text-align: left;">C. Net Debt</td>
</tr>
</tbody>
</table>
<embed src="Plots/FigValidation1a_full.eps" />
<table>
<tbody>
<tr>
<td style="text-align: left;">D. Equity</td>
<td style="text-align: left;">E. Housing wealth</td>
</tr>
</tbody>
</table>
<p><embed src="Plots/FigValidation1b_full.eps" /><br />
<br />
<span><em>Note</em>: This figure reports wealth (Panel A), home ownership rates (Panel B), net debt (Panel C), wealth invested in stocks (Panel D) and housing (Panel E) by age, in the model and in the data. Equity wealth, debt and wealth are measured in units of the national wage index. Net debt is defined as housing wealth minus wealth.</span></p>
<figcaption><strong>Wealth and portfolios over the life cycle</strong></figcaption>
</figure>

<figure id="fig:validation_equity_full" data-latex-placement="H">
<table>
<tbody>
<tr>
<td style="text-align: left;">A. By income level</td>
<td style="text-align: left;">B. By wealth level</td>
</tr>
</tbody>
</table>
<p><embed src="Plots/FigValidation2a_full.eps" /><br />
<br />
<span><em>Note</em>: This figure reports the amount of wealth invested in stocks in the model and in the data, for households between age 40 and 45, by level of income and wealth. Equity wealth, income and wealth are measured in units of the national wage index.</span></p>
<figcaption><strong>Equity holdings at age 40–45</strong></figcaption>
</figure>

<figure id="fig:validation_housing_full" data-latex-placement="H">
<table>
<tbody>
<tr>
<td style="text-align: left;">A. By income level</td>
<td style="text-align: left;">B. By wealth level</td>
</tr>
</tbody>
</table>
<p><embed src="Plots/FigValidation2b_full.eps" /><br />
<br />
<span><em>Note</em>: This figure reports the amount of wealth invested in housing in the model and in the data, for households between age 40 and 45, by level of income and wealth. Housing wealth, income and wealth are measured in units of the national wage index.</span></p>
<figcaption><strong>Housing wealth at age 40–45</strong></figcaption>
</figure>

<figure id="fig:IRSCrossSection_full" data-latex-placement="H">
<table>
<tbody>
<tr>
<td style="text-align: left;">A. By age</td>
<td style="text-align: left;">B. By income at age 40–45</td>
<td style="text-align: left;">C. By wealth at age 40–45</td>
</tr>
</tbody>
</table>
<p><embed src="Plots/FigValidation3_full.eps" /> <span><em>Note</em>: This figure reports the interest-rate sensitivity of wealth in the model and in the data. Panel A reports the interest-rate sensitivity of wealth by age group. Panel B and C report it within households aged 40–45, by income and wealth groups, respectively. Income and wealth are measured in units of the national wage index.</span></p>
<figcaption>Interest-rate sensitivity in the cross-section</figcaption>
</figure>

<figure id="fig:IRSCrossSection2_full" data-latex-placement="H">
<table>
<tbody>
<tr>
<td style="text-align: left;">A. By age</td>
<td style="text-align: left;">B. By income at age 40–45</td>
<td style="text-align: left;">C. By wealth at age 40–45</td>
</tr>
</tbody>
</table>
<p><embed src="Plots/FigIRS_full.eps" /> <em>Note</em>: This figure reports the interest-rate sensitivity of wealth, wealth inclusive of Social Security, and expected utility over the life cycle.</p>
<figcaption><strong>Interest-rate sensitivities for different measures of wealth</strong> </figcaption>
</figure>

# Numerical appendix

This appendix section details how we solve the model numerically. To render the model solvable, we exploit two key properties of the model that allow us to rescale the problem, eliminating state variables; and to transform the wealth return, simplifying the portfolio choice problem.

## Rescaling the model 

We eliminate two state variables, the wage index $\bar{L}_t$ and the house price $P_{ht}$, by rescaling the problem as follows. Henceforth, let $\hat{Z}_t$ denote wage-indexed quantities $Z_t/\bar{L}_t$ for any variable $Z_t$. The utility function equals $$
 u(C_{it},H_{it}) = \bar{L}_t^{1-\gamma}P_{ht}^{-\nu(1-\gamma)}u(\hat{C}_{it},\hat{P}_{ht}H_{it}).
$$ From this, it follows (and we will prove) that there exists a function $\hat{V}$ such that the value function equals[^42] $$
 V_{it} = V(\bar{L}_t,P_{ht},W_{it},r_{ft},g_{1t},z_{it},a_{it},\text{AIYE}_{it}) = \bar{L}_t^{1-\gamma}P_{ht}^{-\nu(1-\gamma)}\hat{V}(\hat{W}_{it},r_{ft},g_{1t},z_{it},a_{it},\hat{\text{AIYE}}_{it}),
$$ provided that the agent chooses the rescaled policies $\{\hat{C}_{it},\hat{P}_{ht}H_{it}\}$ instead of $\{C_{it},H_{it}\}$. Henceforth, we will use the notation $\hat{H}_{it}\equiv\hat{P}_{ht}H_{it}$ to emphasize that this is a single choice variable.

Before proving the claim, note that all constraints can be rescaled accordingly. Wealth accumulates according to $$
 \hat{W}_{i,t+1} = \frac{\bar{L}_t}{\bar{L}_{t+1}}(\hat{W}_{it} + \tilde{L}_{it}+\hat{B}_{it}-\hat{T}_{it}-\hat{C}_{it}-\chi P_{n_ht}\hat{H}_{it}\mathbb{I}_{\{\pi_{h,it}=0\}}-c_s)R_{Wi,t+1},
$$ which is independent of $\bar{L}_t$ and $P_{ht}$. The minimum and maximum house size constraints are $$
 P_{n_ht}\hat{H}_{it} \geq \kappa_{\min} \quad\text{and}\quad \pi_{h,it} \leq \kappa_{\max}.
$$ The scaled value of a private business $\hat{E}_{it}$ is also independent of $\bar{L}_t$. Thus, the constraints can all be written in terms of controls $\{\hat{C}_{it},\hat{H}_{it}\}$ and without any dependence on $\bar{L}_t$ and $P_{ht}$.

We can now prove the claim by (backward) induction. First, consider the bequest ). In the terminal period (i.e., $\bar{b}$ periods after death), the claim is true, as $$
 V_{it} = \bar{L}_t^{1-\gamma}P_{ht}^{-\nu(1-\gamma)}\max_{\{\hat{C}_{it},\hat{P}_{ht}H_{it}\}}u(\hat{C}_{it},\hat{P}_{ht}H_{it}) = \bar{L}_t^{1-\gamma}P_{ht}^{-\nu(1-\gamma)}\hat{V}_{it}.
$$ Now suppose that it is true at any time $t+1$ within the time span after death. This implies $$\begin{align}
 V_{it} &= \max_{\{C_{it},H_{it},\pi_{it}\}}\left\{u(C_{it},H_{it}) + \beta\mathbb{E}_t\left[\bar{L}_{t+1}^{1-\gamma}P_{h,t+1}^{-\nu(1-\gamma)}\hat{V}_{i,t+1}\right]\right\} \\
 &= \bar{L}_t^{1-\gamma}P_{ht}^{-\nu(1-\gamma)}\max_{\{\hat{C}_{it},\hat{H}_{it},\pi_{it}\}}\left\{u(\hat{C}_{it},\hat{H}_{it}) + \beta\mathbb{E}_t\left[\left(\frac{\bar{L}_{t+1}}{\bar{L}_t}\right)^{1-\gamma}\left(\frac{P_{h,t+1}}{P_{ht}}\right)^{-\nu(1-\gamma)}\hat{V}_{i,t+1}\right]\right\} \\
 &= \bar{L}_t^{1-\gamma}P_{ht}^{-\nu(1-\gamma)}\hat{V}_{it}.
\end{align}$$ The second equality changes the choice variables to $\{\hat{C}_{it},\hat{H}_{it},\pi_{it}\}$, which is possible because the rescaled utility function and budget constraints only feature these choice variables. The third equality follows from the fact that the maximand has no dependence on $\bar{L}_t$ and $P_{ht}$. It follows that the initial bequest utility equals $$
 b(\bar{L}_t,P_{ht},W_{it},r_{ft},g_{1t}) = \bar{L}_t^{1-\gamma}P_{ht}^{-\nu(1-\gamma)}\hat{b}(\hat{W}_{it},r_{ft},g_{1t}).
$$ This proves the claim for the bequest.

The rest of the proof follows the same inductive reasoning. We know, upon death, the value function rescales as claimed (since the value function is the bequest utility). Now, for the induction step, suppose the value function conditional on living to $t+1$ rescales as claimed. Then $$\begin{align}
 V_{it} &= \max_{\{C_{it},H_{it},\pi_{it}\}}\left\{u(C_{it},H_{it}) + \beta\mathbb{E}_t\left[\bar{L}_{t+1}^{1-\gamma}P_{h,t+1}^{-\nu(1-\gamma)}\left((1-m_{it})\hat{V}_{i,t+1} + m_{it}\hat{b}_{i,t+1}\right)\right]\right\} \\
 &= \bar{L}_t^{1-\gamma}P_{ht}^{-\nu(1-\gamma)}\max_{\{\hat{C}_{it},\hat{H}_{it},\pi_{it}\}}\biggl\{u(\hat{C}_{it},\hat{H}_{it}) \\
 &\qquad\qquad\qquad\qquad\qquad + \beta\mathbb{E}_t\left[\left(\frac{\bar{L}_{t+1}}{\bar{L}_t}\right)^{1-\gamma}\left(\frac{P_{h,t+1}}{P_{ht}}\right)^{-\nu(1-\gamma)}\left((1-m_{it})\hat{V}_{i,t+1} + m_{it}\hat{b}_{i,t+1}\right)\right]\biggr\} \\
 &= \bar{L}_t^{1-\gamma}P_{ht}^{-\nu(1-\gamma)}\hat{V}_{it}.
\end{align}$$ The equalities follow identical reasoning as above. Most importantly, $\hat{V}_{it}$ in the last equation is independent of $\bar{L}_t$ and $P_{ht}$, proving the conjecture.

## Transforming the portfolio problem

In choosing a portfolio, the agent must select optimal weights $\{\pi_{n,it},\pi_{s,it},\pi_{h,it}\}$ on the risky assets. Numerically, this problem becomes intractable because all three assets are exposed to interest-rate shocks. We circumvent this tractability problem by re-expressing the problem in terms of exposures to separate asset-market shocks: the agent instead chooses $\{\varepsilon_{r,it},\pi_{s,it},\pi_{h,it}\}$, where $\varepsilon_{r,it}$ is the interest-rate sensitivity of the total portfolio. This transformation is also ideal because the long-term bond share $\pi_n$ is itself not that meaningful, as it arbitrarily depends on our choice of the bond duration $n$; in contrast, $\varepsilon_r$ is invariant to $n$.

To implement this change of choice variables, we need to rewrite the wealth return ) in a way that replaces the long-term bond share $\pi_{n,it}$ with the interest-rate sensitivity of the total portfolio $\varepsilon_{r,it}$. To do this, let us start from the result of Appendix 16.5 that, to a second order, the log wealth return (suppressing $i$ subscripts) equals $$
 r_{w,t+1} = r_{ft} + \pi_t^\top(r_{t+1}-r_{ft}\iota) + \frac{1}{2}\left(\pi_t^\top\textrm{diag}(\Sigma\Sigma^\top) - \pi_t^\top\Sigma\Sigma^\top\pi_t\right) - (\pi_{h,t}+\pi_{s,t}-1)^{+}\theta,
 
$$ where $\pi_t$ is the vector of portfolio weights, $r_{t+1}$ is the vector of risky returns, $\iota$ is a vector of ones, $\Sigma\Sigma^\top$ is the covariance matrix of returns with respect to the vector of shocks, and $\theta$ is shorthand for the mortgage premium.

To transform the wealth return, first note that we can rewrite the log return on any asset $j\in\{n,s,h\}$ as $$
 r_{j,t+1} = \log R_{j,t+1} = r_{ft} + \mu_j - \sigma_{jr}\epsilon_{r,t+1} + \sigma_{j,-r}^\top\epsilon_{-r,t+1},
$$ where $\epsilon_{-r}$ is defined as the (standardized) vector of shocks orthogonal to $\epsilon_r$, $$
 \epsilon_{-r,t+1}\equiv[\epsilon_{g,t+1},\epsilon_{s,t+1},\epsilon_{h,t+1},\tilde{\epsilon}_{h,t+1}]^\top,
$$ and $\sigma_{j,-r}$ the corresponding vector of loadings. In this notation, the loadings on the rate shock are $\{\sigma_{nr},\sigma_{sr},\sigma_{hr}\} = \{\sigma_n,\lambda_{sr},\sigma_{n_h}\}$. Stacking these returns in vector form, we have $$
 r_{t+1} = r_{ft}\iota + \mu - \Sigma_r\epsilon_{r,t+1} + \Sigma_{-r}\epsilon_{-r,t+1},
$$ meaning that the wealth return is $$\begin{multline}
 r_{w,t+1} = r_{ft} + \pi_t^\top(\mu - \Sigma_r\epsilon_{r,t+1} + \Sigma_{-r}\epsilon_{-r,t+1}) + \frac{1}{2}\bigl(\pi_t^\top\textrm{diag}(\Sigma_r\Sigma_r^\top)+\pi_t^\top\textrm{diag}(\Sigma_{-r}\Sigma_{-r}^\top) \\ 
 - \pi_t^\top\Sigma_r\Sigma_r^\top\pi_t - \pi_t^\top\Sigma_{-r}\Sigma_{-r}^\top\pi_t\bigr) - (\pi_{h,t}+\pi_{s,t}-1)^{+}\theta.
\end{multline}$$ Next, let us substitute out $\pi_{n,t}$ using the fact that the interest-rate sensitivity of the portfolio equals $$
 \varepsilon_{r,t} = \pi_t^\top\Sigma_r.
$$ Substituting this into the wealth return, we have $$\begin{multline}
 r_{w,t+1} = r_{ft} + \pi_t^\top\mu - \varepsilon_{r,t}\epsilon_{r,t+1} + \pi_t^\top\Sigma_{-r}\epsilon_{-r,t+1} + \frac{1}{2}\bigl(\pi_t^\top\textrm{diag}(\Sigma_r\Sigma_r^\top)+\pi_t^\top\textrm{diag}(\Sigma_{-r}\Sigma_{-r}^\top) \\ 
 - \varepsilon_{r,t}^2 - \pi_t^\top\Sigma_{-r}\Sigma_{-r}^\top\pi_t\bigr) - (\pi_{h,t}+\pi_{s,t}-1)^{+}\theta.
 
\end{multline}$$ Because the first row of $\Sigma_{-r}$ is zero (i.e., $r_{n,t+1}$ has no loadings on $\epsilon_{-r,t+1}$), and because the first element of $\mu+\textrm{diag}(\Sigma_r\Sigma_r^\top)/2$ equals zero (i.e., $\mu_n=-\sigma_n^2/2$), this rewritten wealth return has no dependence on $\pi_{n,t}$. We have completely replaced it with $\varepsilon_{r,t}$. Indeed, ) can be written as $$
 r_{w,t+1} = r_{ft} + {\pi_t^*}^\top(\mu_t^* - \Sigma_t^*\epsilon_{t+1}) + \frac{1}{2}\bigl({\pi_t^*}^\top\textrm{diag}(\Sigma_t^*{\Sigma_t^*}^\top) - {\pi_t^*}^\top\Sigma_t^*{\Sigma_t^*}^\top\pi_t^*\bigr) - (\pi_{h,t}+\pi_{s,t}-1)^{+}\theta
 
$$ where[^43] $$
 \pi_t^* \equiv \begin{bmatrix} 1 \\ \pi_s \\ \pi_h \end{bmatrix}, \quad
 \mu_t^* \equiv \begin{bmatrix} -\varepsilon_{r,t}^2/2 \\ \mu_s + \lambda_{sr}^2/2 \\ \mu_h \end{bmatrix}, \quad\textrm{and}\quad
 \Sigma_t^* \equiv \begin{bmatrix} -\varepsilon_{r,t} & 0 & 0 & 0 & 0 \\ 0 & \lambda_{sg} & 1 & 0 & 0 \\ 0 & 0 & 0 & \sigma_h & \tilde{\sigma}_h \end{bmatrix}.
$$ This transformed expression then implies an equivalent expression for the wealth return: $$\begin{multline}
 R_{Wi,t+1} = R_{ft}+(e^{-\varepsilon_{r,it}^2/2-\varepsilon_{r,it}\epsilon_{r,t+1}}-1)R_{ft} +\pi_{s,it}(e^{r_{ft} + \mu_s + \lambda_{sr}^2/2 + \lambda_{sg}\epsilon_{g,t+1}+\epsilon_{s,t+1}}-R_{ft}) \\ +\pi_{h,it}(e^{r_{ft} + \mu_h + \sigma_h\epsilon_{h,t+1} + \tilde{\sigma}_h\tilde{\epsilon}_{hi,t+1}}-R_{ft}) - (\pi_{h,it}+\pi_{s,it}-1)^{+}(R_{Mit}-R_{ft})
 
\end{multline}$$ (To see this, simply apply the same second-order approximation to ) as we did to get ), and the result will be )). Thus, the optimal allocation can equivalently be stated as a choice over $\{\varepsilon_{r,t},\pi_{s,t},\pi_{h,t}\}$, and these choices map to an optimal long-term bond allocation equal to $$
 \pi_{n,t} = \frac{\varepsilon_{r,t} - \pi_{s,t}\lambda_{sr} - \pi_{h,t}\sigma_{n_h}}{\sigma_n}.
$$ This allocation is unique given the maturity $n$ of the bond.

# Simulation of the overlapping-generations model 

## Aggregate shocks

The overlapping-generations (OLG) has 4 aggregate shocks that must be mapped to data: these are shocks to (1) riskfree rates $\epsilon_{r}$, wage growth $\epsilon_{g}$, stock returns $\epsilon_{s}$, and house prices $\epsilon_{h}$. Shocks to the riskfree rate come from the procedure outlined in Appendix 13.1.1. This procedure also provides us with the level of real interest rates $r_f$. The series of wage income growth rates $g$ are proxied by the rate of growth of per capita real GDP, just as in Appendix 13.1. We obtain $g_1$ using the relation $$
g_{1t} = g_t - \lambda_{rg} r_{ft}.
$$

Stock returns come from the S&P500 and are added directly to the simulation after subtracting out the portion driven by shocks to interest rates---which is akin to subtracting $\sigma_{n_s} \epsilon_r$ from the log stock return. This means that we add both $\epsilon_{g}$ and $\epsilon_{s}$ into the simulation simultaneously. Similarly, aggregate house prices come from the Case-Shiller index and are added after differencing out the portion of the capital gain attributable to interest rate shocks $\sigma_{n_h} \epsilon_r$. For the idiosyncratic portion of house price shocks, we assign each simulated individual a shock drawn from normal distribution with variance $\tilde{\sigma}_h^2$.

When not using data in the simulation, we randomly assign agents aggregate shocks from 50 randomly generated time series. When computing top wealth shares, we do so within a group of shocks and then take the average top 10% share among the 50 groups.

## Inheritance and intergenerational income persistence

To model intergenerational income persistence, we assume a correlation between the levels of persistent income of parents and their children at labor market entry. Specifically, we model initial persistent income as an AR(1) process: $$
 z_{i,t_0} = \rho_{z_0} z^{\text{parent}_i}_{t_0-30} + \sigma_{z_0} \epsilon_{z_0},
$$ where $z_{i,t_0}$ denotes the child's initial value of $z$ upon entering the labor force, and $z^{\text{parent}_i}_{t_0-30}$ is the corresponding value for the parent 30 years earlier. We calibrate the parameters $\rho_{z_0}$ and $\sigma_{z_0}$ using two moment conditions. First, the variance of $z_{i,t_0}$ must equal $\sigma^2_{\alpha}$, a parameter chosen to match the observed level of income inequality. Second, we set $\rho_{z_0}$ to reproduce the empirical slope of the relationship between the child's and the parent's income ranks at age 30, as documented in (Chetty et al. 2014). Given the rest of our labor income process, these conditions imply $\rho_{z_0}=0.4564$ and $\sigma_{z_0}=0.9787$. Figure 25 plots the child's income rank against the parent's income rank, providing the model counterpart to Figure II in (Chetty et al. 2014).

<figure id="fig:skillpersistence" data-latex-placement="H">
<div class="center">
<embed src="Plots/SkillPersistence.eps" style="width:50.0%" />
</div>
<figcaption><strong>Child income rank vs parent income rank in the model</strong> </figcaption>
</figure>

# Derivation of the linearized model 

This section lays out the details of the linearization and analytical solutions presented in Section 4. The approach follows that of (Campbell and Viceira 2001), except that we add finite lives, intertemporal income, and multiple assets. To fully understand the economics, we first solve for policies in the general case of recursive utility (i.e., disentangling risk aversion and the EIS), then reduce to the time-additive case in the main text. For the remainder of this appendix section, we will suppress $i$ indices and state approximate (i.e., linearized) equalities as exact.

## Linearized conditions 

Suppose that there is no intertemporal income or housing, so the budget constraint ) simplifies to[^44] $$
W_{t+1} = (W_t-C_t)R_{W,t+1}.

$$ The first-order condition for a recursive-utility agent takes the familiar form $$
1 = \mathbb{E}_t\left[(\beta(1-m_t))^\theta\left(\frac{C_{t+1}}{C_t}\right)^{-\theta/\psi}R_{W,t+1}^{\theta-1}R_{j,t+1}\right],

$$ where $\beta(1-m_t)$ is mortality-adjusted patience, $\psi$ is the EIS, $\theta=(1-\gamma)/(1-1/\psi)$, and $R_j\in\{R_f,R_n,R_W\}$. The analytical solution follows from linearizing this budget constraint and first-order condition.

Let lowercase letters denote logs and the $\Delta$ operator denote first differences. Scaling the budget constraint ) by financial wealth $W_t$, taking logs, and linearizing $\log\left(1-e^{c_t-w_t}\right)$ around $c_t - w_t = \log( 1 - \beta(1- m_t))$ implies $$
\Delta w_{t+1} = \kappa_w(m_t)+\left(1-\frac{1}{\rho_c(m_t)}\right)(c_t-w_t) + r_{w,t+1},

$$ where $\rho_c(m_t)=\beta(1-m_t)$ and $\kappa_w(m_t)=\log{\rho_c(m_t)}+(1-\rho_c(m_t))\log{(1-\rho_c(m_t))}/\rho_c(m_t)$.[^45] (Notice that, as $m_t\to1$, $c_t\to w_t$; agents who will die almost surely consume everything.) We can also get the linearized approximation to the log wealth return $$
r_{w,t+1} = r_{ft} + \pi_t(r_{n,t+1}-r_{ft}) + \frac{1}{2}\pi_t(1-\pi_t)\mathrm{var}_t(r_{n,t+1}).

$$ This expression is a discretization of the exact continuous-time law of motion. Finally, log-linearize the Euler equation ) up to a second order: $$\begin{multline}
0 = \theta\log(\beta(1-m_t)) + \mathbb{E}_t\left[- \frac{\theta}{\psi}\Delta c_{t+1}+(\theta-1)r_{w,t+1}+r_{j,t+1}\right]\\+\frac{1}{2}\mathrm{var}_t\left(-\frac{\theta}{\psi}\Delta c_{t+1}+(\theta-1)r_{w,t+1}+r_{j,t+1}\right).

\end{multline}$$

Substituting in $r_j=r_n$ and then $r_j = r_f$ and subtracting the two equations implies the risk premium on the long-term bond $$
\mathbb{E}_t[r_{n,t+1}-r_{ft}]+\frac{1}{2}\mathrm{var}_t(r_{n,t+1}) = \frac{\theta}{\psi}\mathrm{cov}_t(r_{n,t+1},\Delta c_{t+1}) + (1-\theta)\mathrm{cov}_t(r_{n,t+1},r_{w,t+1}).

$$ Using the decomposition $$
\Delta c_{t+1} = (c_{t+1}-w_{t+1})-(c_t-w_t) +\Delta w_{t+1}

$$ and the expression for $\Delta w_{t+1}$ from the linearized budget constraint ), we can rewrite $$
\mathrm{cov}_t(r_{n,t+1},\Delta c_{t+1}) = \mathrm{cov}_t(r_{n,t+1}, c_{t+1}-w_{t+1}) + \mathrm{cov}_t(r_{n,t+1},r_{w,t+1}).

$$ Substituting this and the fact that $$
\mathrm{cov}_t(r_{n,t+1},r_{w,t+1}) = \pi_t\mathrm{var}_t(r_{n,t+1})

$$ into ) and using $\theta/\psi+1-\theta=\gamma$ implies the solution $$
\pi_t = \frac{1}{\gamma}\frac{\mathbb{E}_t[r_{n,t+1}-r_{ft}]+\frac{1}{2}\mathrm{var}_t(r_{n,t+1})}{\mathrm{var}_t(r_{n,t+1})} - \frac{1-1/\gamma}{1-\psi}\frac{\mathrm{cov}_t(r_{n,t+1},c_{t+1}-w_{t+1})}{\mathrm{var}_t(r_{n,t+1})}.

$$ As explained in the main text, the first term is the myopic risk-return portfolio; the second is intertemporal hedging of interest-rate risk.

Another fact that will become useful is that the first-order condition ) for wealth returns ($r_j=r_w$) simplifies to $$
\mathbb{E}_t[\Delta c_{t+1}] = \psi\log(\beta(1-m_t)) + \psi\mathbb{E}_t\left[r_{w,t+1}\right]+\frac{1}{2}\frac{\theta}{\psi}\mathrm{var}_t\left(\Delta c_{t+1}-\psi r_{w,t+1}\right).

$$ Using fact ) and the decomposition of $\Delta c$ from ), the variance term can be rewritten as $$
\begin{aligned}
\mathrm{var}_t\left(\Delta c_{t+1}-\psi r_{w,t+1}\right) &= \mathrm{var}_t\left(c_{t+1}-w_{t+1}+(1-\psi) r_{w,t+1}\right) \\
&= \mathrm{var}_t\left(c_{t+1}-w_{t+1}\right) + (1-\psi)^2\pi_t^2\mathrm{var}_t\left(r_{n,t+1}\right) \\
&\qquad + (1-\psi)\pi_t\mathrm{cov}_t\left(r_{n,t+1},c_{t+1}-w_{t+1}\right).
\end{aligned}

$$ We will use these expressions to solve for the equilibrium consumption-wealth ratio.

## Optimal policies in the linearized model 

We will now solve for the optimal consumption and portfolio choices using the conditions derived above. Conjecture that the optimal consumption-wealth ratio takes the form $$
c_t-w_t = \log(1-\beta(1-m_t))+(1-\psi)(\varrho_{0t}+\varrho_{rt}r_{ft}),

$$ for some functions $\varrho_{0t}=\varrho_0(\{m_s\}_{s\geq t})$ and $\varrho_{rt}=\varrho_r(\{m_s\}_{s\geq t})$ of the future mortality probabilities. Increasing utility implies the boundary conditions $\lim_{m\to1}(1-\psi)\varrho_0(m)=0$ and $\lim_{m\to1}\varrho_r(m)=0$. This conjecture implies that $$\begin{align*}
(1-\psi)^{-1}\mathrm{cov}_t(r_{n,t+1},c_{t+1}-w_{t+1}) &=\varrho_{r,t+1}\mathrm{cov}_t(r_{n,t+1},r_{f,t+1}) \\
&= -\varrho_{r,t+1}\sigma_n\sigma_r.
\end{align*}$$ Substituting this expression into ), we obtain $$\begin{align*}
\pi_t &= \frac{1}{\gamma}\frac{\mu_n+\frac{1}{2}\sigma_n^2}{\sigma_n^2} + \left(1-\frac{1}{\gamma}\right)\varrho_{r,t+1}
\frac{\sigma_r}{\sigma_n} \\
&= a_0+a_r\varrho_{r,t+1},
\end{align*}$$ which, combined with ), is our expression for the optimal share in the $n$-period bond ).

To solve for $\varrho_0$ and $\varrho_r$, notice that substituting this solution for $\pi$ into the expectation of our log-linearized wealth return ) implies $$
\begin{aligned}
\mathbb{E}_t[r_{w,t+1}] &= r_{ft} + \pi_t\mu_n + \pi_t(1-\pi_t)\sigma_n^2 \\
&= r_{ft} + \left(a_0\mu_n+(a_0-a_0^2)\sigma_n^2\right) + \left(a_r\mu_n+(a_r-2a_0a_r)\sigma_n^2\right)\varrho_{r,t+1} - a_r^2\sigma_n^2\varrho_{r,t+1}^2 \\
&= r_{ft} + d_0 + d_1\varrho_{r,t+1} - d_2\varrho_{r,t+1}^2.
\end{aligned}

$$ It also implies $$\begin{align*}
\mathrm{var}_t\left(c_{t+1}-w_{t+1}\right) &= (1-\psi)^2\varrho_{r,t+1}^2\sigma_r^2, \\
(1-\psi)^2\pi_t^2\mathrm{var}_t\left(r_{n,t+1}\right) &= (1-\psi)^2(a_0^2+2a_0a_r\varrho_{r,t+1}+a_r^2\varrho_{r,t+1}^2)\sigma_n^2, \\
(1-\psi)\pi_t\mathrm{cov}_t\left(r_{n,t+1},c_{t+1}-w_{t+1}\right) &= (1-\psi)^2(a_0+a_r\varrho_{r,t+1})(-\varrho_{r,t+1}\sigma_n\sigma_r).
\end{align*}$$ Substituting these three equations into ), we have $$
\mathrm{var}_t\left(\Delta c_{t+1}-\psi r_{w,t+1}\right) = (1-\psi)^2(g_0+g_1\varrho_{r,t+1}+g_2\varrho_{r,t+1}^2)

$$ for constants $g_j$. Finally, substituting our log-linearized budget constraint ) into the decomposition ) and applying our conjecture ), we have $$
\begin{aligned}
\mathbb{E}_t[\Delta c_{t+1}] &= \mathbb{E}_t[c_{t+1}-w_{t+1}]-\rho_c(m_t)^{-1}(c_t-w_t) + \kappa_w(m_t) + \mathbb{E}_t[r_{w,t+1}] \\
&= (1-\psi)(\varrho_{0,t+1}+\varrho_{r,t+1}((1-\varphi)\bar{r}_f+\varphi r_{ft})) \\
&\qquad -\rho_c(m_t)^{-1}(1-\psi)(\varrho_{0t}+\varrho_{rt}r_{ft}) + \log\rho_c(m_t) + \log\frac{1-\rho_c(m_{t+1})}{1-\rho_c(m_t)} + \mathbb{E}_t[r_{w,t+1}].

\end{aligned}
$$ Substituting ), ), and ) into the Euler equation for wealth returns ), then collecting coefficients on $r_{ft}$, implies the difference equation $$\varphi \varrho_{r,t+1} = \rho_c(m_t)^{-1}\varrho_{rt}-1.$$ Now iterate forward and use the boundary condition $\lim_{t\to\infty}\varrho_{rt}=0$: $$\begin{align*}
\varrho_{rt} &= \rho_c(m_t)(\varphi \varrho_{r,t+1}+1) \\
&= \rho_c(m_t) + \varphi\rho_c(m_t)\rho_c(m_{t+1}) + \varphi^2\rho_c(m_t)\rho_c(m_{t+1})\rho_c(m_{t+2}) + \dots \\
&= \beta(1-m_t)\left(1+\sum_{j=1}^\infty\varphi^j\beta^j\prod_{k=1}^j(1-m_{t+k})\right) \\
&= \sum_{j=1}^\infty\varphi^{j-1}\beta^{j}p_{t,t+j}
\end{align*}$$ The lower are the probabilities of future survival $p_{t,t+j}$, the less relevant are fluctuations in the interest rate to consumption and portfolio choices. For reference, note that, for infinitely lived agents ($m_t=0$ for all $t$), this converges to $\varrho_r=\rho_c/(1-\varphi\rho_c)$, the result from (Campbell and Viceira 2001).

Collecting the remaining constant terms implies a difference equation for $\varrho_{0t}$: $$\varrho_{0,t+1} = \rho_c(m_t)^{-1}\varrho_{0t} - q_{0t}$$ for the deterministic constant $$\begin{multline*}
q_{0t} \equiv \varrho_{r,t+1}(1-\varphi)\bar{r}_f + \log\rho_c(m_t) + (1-\psi)^{-1}(\log(1-\rho_c(m_{t+1}))-\log(1-\rho_c(m_t))) \\
+ d_0 + d_1\varrho_{r,t+1} - d_2\varrho_{r,t+1}^2 + \frac{1}{2}(\gamma-1)(g_0+g_1\varrho_{r,t+1}+g_2\varrho_{r,t+1}^2).
\end{multline*}$$ We can similarly iterate this expression forward with terminal condition $(1-\psi)\varrho_0\to0$ to arrive at a solution: $$\begin{align*}
\varrho_{0t} &= \rho_c(m_t)(q_{0t} + \varrho_{0,t+1}) \\
&= \rho_c(m_t)q_{0t} + \rho_c(m_t)\rho_c(m_{t+1})q_{0,t+1} + \rho_c(m_t)\rho_c(m_{t+1})\rho_c(m_{t+2})q_{0,t+2} + \dots \\
&= \sum_{j=1}^\infty\beta^jp_{t,t+j}q_{0,t+j-1}.
\end{align*}$$ Note that $\lim_{m\to1}\rho(m)q_{0t} = 0$, so $\varrho_{0t}$ is finite and converges to zero as $m\to1$. This verifies the conjecture.

## Adding labor income and Social Security 

We now introduce a deterministic stream of labor income $L$ and, in turn, Social Security taxes $T$ and benefits $B$. The present values of labor income (human capital) $H$ and Social Security wealth $S$ are as stated in the main text.

As we did with the wealth return above, let us linearize the returns on human capital and Social Security wealth using a continuous-time approximation. For human capital, the log return is $$
r_{H,t+1} = r_{ft} + \mu_{Ht} + \underbrace{\left(\sum_{j=1}^{t_{\rm ret}-t}\omega_{jt}^H\left(\frac{\sigma_j}{\sigma_n}\right)\right)}_{\pi_t^H}(r_{n,t+1}-r_{ft})
$$ where $$\omega_{jt}^H = \frac{p_{t,t+j}P_{jt}L_{t+j}}{\sum_{j'=1}^{t_{\rm ret}-t}p_{t,t+j'}P_{j't}L_{t+j'}} = \frac{p_{t,t+j}P_{jt}L_{t+j}}{H_t}$$ is the value weight of the $j$th labor-payment, and therefore $\pi^H$ is a value-weighted rate-sensitivity adjustment. More specifically, $\pi^H$ represents the percent holdings of $n$-period bonds implicit in the human capital asset. To see this, note that the rate sensitivity implied by $\pi^H$ is $$\pi_t^H\frac{\sigma_n}{\sigma_r} = \sum_{j=1}^{t_{\rm ret}-t}\omega_{jt}^H\left(\frac{\sigma_j}{\sigma_r}\right) = \sum_{j=1}^{t_{\rm ret}-t}\omega_{jt}^H\varepsilon_r(P_{j,t+1}) = \varepsilon_r(H_{t+1}).$$ In words, a portfolio with $\pi^H$ percent allocated to the $n$-period bond has the exact same interest-rate sensitivity as the human capital asset.

Identical logic leads us to conclude that the log return on Social Security is $$r_{S,t+1} = r_{ft} + \mu_{St} + \underbrace{\left(\sum_{j=1}^\infty\omega_{jt}^S\left(\frac{\sigma_j}{\sigma_n}\right)\right)}_{\pi_t^S}(r_{n,t+1}-r_{ft}),$$ where the value weights take the form $$\omega_{jt}^S = \omega_{jt}^B - \omega_{jt}^T = \frac{p_{t,t+j}P_{jt}(B_{t+j}-T_{t+j})}{S_t},$$ the difference between the benefits claim and the tax liability.

Now, as in the main text, define total wealth as $$
\overline{W}_t = W_t + (L_t+H_t) + (B_t-T_t+S_t).

$$ (Recall that $H$ and $S$ do not include their contemporaneous "dividends," so we must add them back in this expression.) Grossing up at the rates of return on these assets implies $$
\overline{W}_{t+1} = (W_t+L_t+B_t-T_t-C_t)R_{W,t+1} + H_tR_{H,t+1} + S_tR_{S,t+1}.

$$ Multiplying and dividing by $\overline{W}_t-C_t$, we have the dynamic budget constraint $$\overline{W}_{t+1} = (\overline{W}_t-C_t)R_{\overline{W},t+1}.$$ where the return on total wealth is $$\begin{align*}
R_{\overline{W},t+1} &= \left(\frac{W_t+L_t+B_t-T_t-C_t}{\overline{W}_t-C_t}\right)R_{W,t+1} + \left(\frac{H_t}{\overline{W}_t-C_t}\right)R_{H,t+1} + \left(\frac{S_t}{\overline{W}_t-C_t}\right)R_{S,t+1} \\
&= \alpha_{Wt}R_{W,t+1} + \alpha_{Ht}R_{H,t+1} + \alpha_{St}R_{S,t+1},
\end{align*}$$ and the return on financial wealth $R_W$ is as it was in the original problem.

Using the same linearization technique as before, the log total-wealth return can be approximated as $$r_{\overline{w},t+1} = r_{ft} + \bar{\mu}_t + \bar{\pi}_t(r_{n,t+1}-r_{ft}) + \frac{1}{2}\bar{\pi}_t(1-\bar{\pi}_t)\sigma_n^2,$$ where $$\bar{\mu}_t = \alpha_{Ht}\mu_{Ht} + \alpha_{St}\mu_{St}$$ is a value-weighted drift term from the intertemporal endowments, and $$
\bar{\pi}_t = \alpha_{Wt}\pi_t + \alpha_{Ht}\pi_t^H + \alpha_{St}\pi_t^S

$$ is the value-weighted average of positions in the long-term bond---that is, the percentage of total wealth invested in the bond. Other than the presence of $\bar{\mu}$, this budget constraint is identical in form to that of the problem with no labor income or Social Security. Following the same steps from before, we conclude that $$\bar{\pi}_t = \pi_t^*,$$ where $\pi_t^*$ is the optimal solution without intertemporal income. Substituting this into ) and solving for $\pi_t$, we see that the optimal allocation to the asset from financial wealth is $$\pi_t = \pi_t^* + \left(\frac{H_t}{W_t+L_t+B_t-T_t-C_t}\right)(\pi_t^*-\pi_t^H) + \left(\frac{S_t}{W_t+L_t+B_t-T_t-C_t}\right)(\pi_t^*-\pi_t^S).$$ In the main text, we slightly simplify notation by redefining wealth $W_t$ to include the contemporaneous income and consumption flows (thus far, we have assumed that it *excludes* these components). Doing this gives us the final expression ).

## Solution to the value function 

This section solves for the value function under optimal policies in closed form. Under recursive preferences, the transformed value function ) is implicitly defined by the aggregator $$U_t = \left[(1-\beta(1-m_t))C_t^{1-1/\psi} + \beta(1-m_t)\mathbb{E}_t\left[U_{t+1}^{1-\gamma}\right]^{\frac{1-1/\psi}{1-\gamma}}\right]^{\frac{1}{1-1/\psi}}.$$ By Euler's Theorem, $$U_t = \frac{\partial U_t}{\partial C_t}C_t + \mathbb{E}_t\left[\frac{\partial U_t}{\partial U_{t+1}}U_{t+1}\right],$$ where $$\frac{\partial U_t}{\partial C_t} = (1-\beta(1-m_t))\left(\frac{U_t}{C_t}\right)^{1/\psi}$$ and $$\frac{\partial U_t}{\partial U_{t+1}} = \beta(1-m_t)U_t^{1/\psi}\mathbb{E}_t\left[U_{t+1}^{1-\gamma}\right]^{\frac{\gamma-1/\psi}{1-\gamma}}U_{t+1}^{-\gamma}.$$ Noting that the stochastic discount factor $$M_{t+1} = \beta(1-m_t)\left(\frac{C_{t+1}}{C_t}\right)^{-1/\psi}\left[\frac{U_{t+1}}{\mathbb{E}_t\left[U_{t+1}^{1-\gamma}\right]^{1/(1-\gamma)}}\right]^{-(\gamma-1/\psi)},$$ and that $$\frac{(\partial U_t/\partial U_{t+1})(\partial U_{t+1}/\partial C_{t+1})}{\partial U_t/\partial C_t} = M_{t+1},$$ we have that $$\frac{U_t}{\partial U_t/\partial C_t} = C_t + \mathbb{E}_t\left[M_{t+1}\frac{U_{t+1}}{\partial U_{t+1}/\partial C_{t+1}}\right].$$ Iterating this recursion forward yields $$\frac{U_t}{\partial U_t/\partial C_t} = \sum_{j=0}^\infty\mathbb{E}_t\left[M_{t,t+j}C_{t+j}\right] = \overline{W}_t,$$ since total wealth is the present value of consumption. Substituting the expression for $\partial U_t/\partial C_t$ and noting that consumption is at an optimum ($C_t=C_t^*$), we get the solution ).

## Adding multiple assets 

Consider the setting with $J$ non-redundant risky assets in Section 4.3, summarized by ) and ). In this case, the linearized wealth return becomes $$
r_{w,t+1} = r_{ft} + \pi_t^\top(r_{t+1}-r_{ft}\iota) + \frac{1}{2}\left(\pi_t^\top\textrm{diag}(\Sigma^\top\Sigma) - \pi_t^\top\Sigma^\top\Sigma\pi_t\right).
$$ The log-linearized Euler equations are the same for each asset $j$, so by the exact same logic, we have the vector of optimal portfolio weights $$
\pi_t^* = \frac{1}{\gamma}(\Sigma^\top\Sigma)^{-1}\left(\mathbb{E}_t[r_{t+1}-r_{ft}\iota]+\frac{1}{2}\mathrm{diag}(\Sigma^\top\Sigma)\right) - \left(1-\frac{1}{\gamma}\right)(\Sigma^\top\Sigma)^{-1}\frac{\mathrm{cov}_t(r_{t+1},c_{t+1}-w_{t+1})}{1-\psi}.
$$ Also by the same logic, the consumption-wealth ratio $c_t-w_t$ continues to take the same form, namely affine in $r_{ft}$. Consequently, substituting in the expected log returns and the fact that $$
\mathrm{cov}_t(r_{t+1},r_{f,t+1}) = \Sigma^\top\sigma_r,
$$ we get the optimal portfolio rule $\pi_t^*$ and the resulting interest-rate sensitivity of wealth $\varepsilon_r^*(W_{t+1})$ stated in the main text.

## Optimal consumption plan in the limit 

This section derives the optimal consumption-investment strategy in the limit as risk aversion $\gamma$ approaches infinity and the EIS $\psi$ approaches zero. We first show that, in the limit as $\gamma\to\infty$ and $\psi\to0$, the optimal consumption plan is constant and financed by a perpetuity. We then prove that both of these conditions (i.e., both $\gamma\to\infty$ and $\psi\to0$) are necessary for this to be true---that is, it is insufficient to just have $\gamma\to\infty$ and not $\psi\to0$ or vice versa.[^46]

First, let us consider what happens when $\gamma=1/\psi\to\infty$. The first-order condition when $\gamma=1/\psi$ is $$
1 = \mathbb{E}_t\left[\beta(1-m_t)\left(\frac{C_{t+1}}{C_t}\right)^{-\gamma}R_{j,t+1}\right].
$$ Conjecture that the optimal consumption policy is a deterministic constant $C_t=\bar{C}_t$. Substituting this conjecture into the first-order condition implies the recursion $$
\bar{C}_t = \left(\beta(1-m_t)\mathbb{E}_t[R_{j,t+1}]\right)^{-1/\gamma}\bar{C}_{t+1}.
$$ Now taking the limit as $\gamma\to\infty$ implies that $\bar{C}_t=\bar{C}_{t+1}=\bar{C}$, meaning that consumption is indeed deterministic and in fact time-invariant.

Next, we show that a constant consumption plan is financed by the optimal portfolio strategy $\pi_t^*$, and that $\gamma=1/\psi\to\infty$ is necessary, not just sufficient.[^47] In order for the consumption plan to be constant, we need two conditions to be satisfied. First, expected consumption growth must be invariant to the level of interest rates; and second, unexpected consumption growth must be invariant to shocks to interest rates. That is, we require $$
\frac{d\mathbb{E}_t[\Delta c_{t+1}]}{dr_{ft}} = 0 \quad\text{and}\quad \frac{d(\Delta c_{t+1})}{d\epsilon_{r,t+1}} = 0.
$$ Using the optimal policy for $c_t-w_t$, consumption growth is $$
\Delta c_{t+1} = \Delta w_{t+1} + (1-\psi)(\varrho_{0,t+1} - \varrho_{0t}) + (1-\psi)(\varrho_{r,t+1}r_{f,t+1} - \varrho_{rt}r_{ft}),
$$ with $$
\Delta w_{t+1} = \kappa_w(m_t)+\left(1-\frac{1}{\rho_c(m_t)}\right)(c_t-w_t) + r_{w,t+1}.
$$ The unexpected component of this is $$
\Delta c_{t+1} - \mathbb{E}_t[\Delta c_{t+1}] = -\pi_t\sigma_n\epsilon_{r,t+1} + (1-\psi)\varrho_{r,t+1}\sigma_r\epsilon_{r,t+1}.
$$ Substituting in the optimal portfolio and differentiating, this implies $$
\frac{d(\Delta c_{t+1})}{d\epsilon_{r,t+1}} = \left(\frac{1}{\gamma}-\psi\right)\varrho_{r,t+1}\sigma_r.

$$ For expected consumption growth, differentiate ) with respect to $r_{ft}$ to see that $$
\frac{d\mathbb{E}_t[\Delta c_{t+1}]}{dr_{ft}} = (1-\psi)(\varphi\varrho_{r,t+1} - \rho_c(m_t)^{-1}\varrho_{rt}) + 1.
$$ Recalling that $$\varphi\varrho_{r,t+1} = \rho_c(m_t)^{-1}\varrho_{rt} -1,$$ this simplifies to $$
\frac{d\mathbb{E}_t[\Delta c_{t+1}]}{dr_{ft}} = \psi.

$$ The results then follow from ) and ). The consumption plan is invariant to interest rates if and only if $\psi=0$ and $\gamma=1/\psi\to\infty$. Under these preferences, the trading strategy implemented by the portfolio rule $\pi_t^*$ finances a constant consumption plan (and hence replicates a buy-and-hold perpetuity). Moreover, the only way for a constant consumption plan to be consistent with the optimal policies is to have both $\gamma\to\infty$ and $\psi\to0$ simultaneously. They are therefore both necessary and sufficient.

Aguiar, Mark, Benjamin Moll, and Florian Scheuer. 2025. "Putting the 'Finance' into 'Public Finance': A Theory of Capital Gains Taxation." Unpublished manuscript.

Ang, Andrew, Geert Bekaert, and Min Wei. 2008. "The Term Structure of Real Rates and Expected Inflation." *Journal of Finance* 63 (2): 797--849. <https://doi.org/10.1111/j.1540-6261.2008.01325.x>.

Auclert, Adrien. 2019. "[Monetary Policy and the Redistribution Channel]." *American Economic Review* 109 (6).

Auerbach, Alan J., Laurence J. Kotlikoff, and Darryl Koehler. 2023. "US Inequality and Fiscal Progressivity: An Intragenerational Accounting." *Journal of Political Economy* 131 (5): 1249--93. <https://doi.org/10.1086/722394>.

Auten, Gerald, and David Splinter. 2024. "[Income Inequality in the United States: Using Tax Data to Measure Long-term Trends]." *Journal of Political Economy* 132 (7): 2179--530.

Bach, Laurent, Laurent E. Calvet, and Paolo Sodini. 2020. "Rich Pickings? Risk, Return, and Skill in Household Wealth." *American Economic Review* 110 (9): 2703--47. <https://doi.org/10.1257/aer.20170666>.

Bansal, Ravi, and Amir Yaron. 2004. "Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles." *Journal of Finance* 59 (4): 1481--509. <https://doi.org/10.1111/j.1540-6261.2004.00670.x>.

Beeler, Jason, and John Y. Campbell. 2012. "[The Long-Run Risks Model and Aggregate Asset Prices: An Empirical Assessment]." *Critical Finance Review* 1 (1).

Benhabib, Jess, Alberto Bisin, and Mi Luo. 2019. "Wealth Distribution and Social Mobility in the US: A Quantitative Approach." *American Economic Review* 109 (5): 1623--47. <https://doi.org/10.1257/aer.20151684>.

Benzoni, Luca, Pierre Collin-Dufresne, and Robert S. Goldstein. 2007. "[Portfolio Choice over the Life-Cycle when the Stock and Labor Markets Are Cointegrated]." *Journal of Finance* 62 (5): 2123--67.

Binsbergen, Jules van. 2021. "Duration-Based Stock Valuation." Unpublished manuscript.

Calvet, Laurent E, John Y Campbell, Francisco Gomes, and Paolo Sodini. 2021. *The Cross-Section of Household Preferences*. Working Paper. Working paper series, no. 28788 (May).

Campbell, John Y., and Luis M. Viceira. 2001. "Who Should Buy Long-Term Bonds?" *American Economic Review* 91 (1): 99--127. <https://doi.org/10.1257/aer.91.1.99>.

Catherine, Sylvain. 2022. "Countercyclical Labor Income Risk and Portfolio Choices over the Life Cycle." *The Review of Financial Studies*, ahead of print. <https://doi.org/10.1093/rfs/hhab136>.

Catherine, Sylvain, Max Miller, and Natasha Sarin. 2025. "Social Security and Trends in Wealth Inequality." *The Journal of Finance* 80 (3): 1497--531. https://doi.org/<https://doi.org/10.1111/jofi.13440>.

Catherine, Sylvain, Paolo Sodini, and Yapei Zhang. 2024. "Countercyclical Income Risk and Portfolio Choices: Evidence from Sweden." *Journal of Finance* 79 (3): 1755--88.

Catherine, Sylvain, and Constantine Yannelis. 2021. "The Distributional Effects of Student Loan Forgiveness." Unpublished manuscript.

Chetty, Raj, Nathaniel Hendren, Patrick Kline, and Emmanuel Saez. 2014. "Where Is the Land of Opportunity? The Geography of Intergenerational Mobility in the United States \*." *The Quarterly Journal of Economics* 129 (4): 1553--623. <https://doi.org/10.1093/qje/qju022>.

Chetty, Raj, Michael Stepner, Sarah Abraham, et al. 2016. "[The association between income and life expectancy in the United States, 2001-2014]." *Journal of the American Medical Association* 315 (16): 1750--66.

Cocco, João F. 2005. "Portfolio Choice in the Presence of Housing." *The Review of Financial Studies* 18 (2): 535--67. <http://www.jstor.org/stable/3598045>.

Cochrane, John H. 2022. "[Portfolios for Long-Term Investors\*]." *Review of Finance* 26 (1): 1--42. <https://doi.org/10.1093/rof/rfab038>.

Congressional Budget Office. 2023. "CBO's 2023 Long-Term Projections for Social Security." Unpublished manuscript.

Fagereng, Andreas, Matthieu Gomez, Émilien Gouin-Bonenfant, Martin Blomhoff Holm, Benjamin Moll, and Gisle Natvik. 2025. "Asset-Price Redistribution." *Journal of Political Economy*.

Fagereng, Andreas, Martin Blomhoff Holm, Benjamin Moll, and Gisle Natvik. 2021. *Saving Behavior Across the Wealth Distribution: The Importance of Capital Gains*.

Fama, Eugene F. 1990. "Stock Returns, Expected Returns, and Real Activity." *The Journal of Finance* 45 (4): 1089--108.

Flavin, Marjorie, and Takashi Yamashita. 2002. "Owner-Occupied Housing and the Composition of the Household Portfolio." *American Economic Review* 92 (1): 345--62. <https://doi.org/10.1257/000282802760015775>.

Gomez, Matthieu, and Émilien Gouin-Bonenfant. 2024. "Wealth Inequality in a Low Rate Environment." *Econometrica* 92 (1): 201--46. https://doi.org/<https://doi.org/10.3982/ECTA19092>.

Gordon, Myron J., and Eli Shapiro. 1956. "Capital Equipment Analysis: The Required Rate of Profit." *Management Science* 3 (1): 102--10. <http://www.jstor.org/stable/2627177>.

Gormsen, Niels Joachim, and Eben Lazarus. 2025. *Interest Rates and Equity Valuations*.

Greenwald, Daniel, Matteo Leombroni, Hanno N. Lustig, and Stijn van Nieuwerburgh. 2023. "Financial and Total Wealth Inequality with Declining Interest Rates." Unpublished manuscript.

Guvenen, Fatih, Fatih Karahan, Serdar Ozkan, and Jae Song. 2022. "[What Do Data on Millions of U.S. Workers Reveal About Life-Cycle Earnings Risk?]" *Econometrica* 89 (5): 2303--39.

Guvenen, Fatih, Serdar Ozkan, and Jae Song. 2014. "[The Nature of Countercyclical Income Risk]." *Journal of Political Economy* 122 (3): 621--60.

Hubmer, Joachim, Per Krusell, and Anthony A. Smith. 2021. "Sources of US Wealth Inequality: Past, Present, and Future." *NBER Macroeconomics Annual* 35: 391--455. <https://doi.org/10.1086/712332>.

Huggett, Mark, and Greg Kaplan. 2016. "How Large Is the Stock Component of Human Capital?" *Review of Economic Dynamics* 22: 21--51. https://doi.org/<https://doi.org/10.1016/j.red.2016.06.002>.

Jordà, Òscar, Katharina Knoll, Dmitry Kuvshinov, Moritz Schularick, and Alan M Taylor. 2019. "The Rate of Return on Everything, 1870--2015\*." *The Quarterly Journal of Economics* 134 (3): 1225--98. <https://doi.org/10.1093/qje/qjz012>.

Longstaff, Francis A., and Monika Piazzesi. 2004. "Corporate Earnings and the Equity Premium." *Journal of Financial Economics* 74 (3): 401--21. https://doi.org/<https://doi.org/10.1016/j.jfineco.2003.12.001>.

Lynch, Anthony W., and Sinan Tan. 2011. "Labor Income Dynamics at Business-Cycle Frequencies: Implications for Portfolio Choice." *Journal of Financial Economics* 101 (2): 333--59.

Meeuwis, Maarten. 2022. "[Wealth Fluctuations and Risk Preferences: Evidence from U.S. Investor Portfolios]." *Working Paper*.

Merton, Robert C. 1969. "[Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case]." *The Review of Economics and Statistics* 51 (3): 247--57. <https://ideas.repec.org/a/tpr/restat/v51y1969i3p247-57.html>.

Meyer, Bruce D., and James X. Sullivan. 2023. "Consumption and Income Inequality in the United States Since the 1960s." *Journal of Political Economy* 131 (2): 247--84. <https://doi.org/10.1086/721702>.

Moll, Benjamin. 2021. "Comment on "Sources of US Wealth Inequality: Past, Present, and Future"." *NBER Macroeconomics Annual* 35: 468--79. <https://doi.org/10.1086/712334>.

Piazzesi, Monika, and Martin Schneider. 2016. *Housing and Macroeconomics*. Edited by John B. Taylor and Harald Uhlig. Vol. 2. Handbook of Macroeconomics. Elsevier. https://doi.org/<https://doi.org/10.1016/bs.hesmac.2016.06.003>.

Piazzesi, Monika, Martin Schneider, and Selale Tuzel. 2007. "Housing, Consumption and Asset Pricing." *Journal of Financial Economics* 83 (3): 531--69. https://doi.org/<https://doi.org/10.1016/j.jfineco.2006.01.006>.

Piketty, Thomas, Emmanuel Saez, and Gabriel Zucman. 2018. "[Distributional National Accounts: Methods and Estimates for the United States]." *The Quarterly Journal of Economics* 133 (2): 553--609.

Saez, Emmanuel, and Gabriel Zucman. 2016. "[Wealth Inequality in the United States since 1913: Evidence from Capitalized Income Tax Data]." *The Quarterly Journal of Economics* 131 (2): 519--78.

Samuelson, Paul A. 1969. "Lifetime Portfolio Selection by Dynamic Stochastic Programming." *The Review of Economics and Statistics* 51 (3): 239--46. <http://www.jstor.org/stable/1926559>.

Schoar, Antoinette. 2010. "[The Divide between Subsistence and Transformational Entrepreneurship]." *Innovation Policy and the Economy* 10 (1): 57--81.

Schwert, G. William. 1990. "Stock Returns and Real Activity: A Century of Evidence." *The Journal of Finance* 45 (4): 1237--57.

Smith, Matthew, Owen Zidar, and Eric Zwick. 2023. "[Top Wealth in America: New Estimates under Heterogeneous Returns]." *Quarterly Journal of Economics* 138 (1): 515--73.

Wachter, Jessica A. 2013. "Can Time-Varying Risk of Rare Disasters Explain Aggregate Stock Market Volatility?" *Journal of Finance* 68 (3): 987--1035. <https://doi.org/10.1111/jofi.12023>.

[^1]: University of Pennsylvania, Wharton. Email: <scath@wharton.upenn.edu>.

[^2]: Harvard Business School. Email: <mamiller@hbs.edu>.

[^3]: Stanford University, Graduate School of Business. Email: <jparon@stanford.edu>.

[^4]: Yale Law School. Email: <natasha.sarin@yale.edu>.

[^5]: This paper previously circulated under the title "Interest-rate risk and household portfolios." We thank seminar participants at the University of Maryland, Dartmouth Tuck, Berkeley Haas, Columbia University, the London School of Economics, the University of Southern California, the University of Texas at Austin, the University of Washington Seattle, Stanford GSB, MIT Sloan, HBS, Northwestern Kellogg, Yale SOM, UCLA, and Brown for helpful comments.

[^6]: Raising the retirement age is often economically close to a benefit cut, since households can typically claim earlier or later already, but only by accepting actuarial adjustments (bonus/malus) that change lifetime benefits.

[^7]: Another relevant policy is capital gains taxation under stochastic discount rates; see (Aguiar et al. 2025).

[^8]: (Campbell and Viceira 2001) study the case of an infinitely lived agent without background assets.

[^9]: <https://www.federalreserve.gov/data/nominal-yield-curve.htm>

[^10]: We formally derive the second and third steps in Appendix 10.

[^11]: We use the price-dividend ratio here because it is equal to the duration when discount rates and cashflow growth are constant (Binsbergen 2021; Gordon and Shapiro 1956). That is, for constant cashflow growth rate $g$, riskfree rate $r_f$, and risk premium $\mu$, the price-dividend ratio is $P_t/D_t=\int_0^\infty e^{-(r_f+\mu-g)n}dn = 1/(r_f+\mu-g)$ and the duration is $\int_0^\infty\frac{e^{-(r_f+\mu)n}D_{t+n}}{P_t}ndn = 1/(r_f+\mu-g)$.

[^12]: In an asset pricing model, one would typically think of interest rates as depending on growth, not the other way around. Econometrically, though, these are equivalent representations and therefore without loss of generality.

[^13]: This is not inconsistent with an upward-sloping term structure for nominal bonds, as nominal bonds have a risk premium due to inflation risk. In contrast, the term premium on inflation-indexed bonds (TIPS) is likely close to zero: (Ang et al. 2008) find that nearly all the slope of the nominal term structure is explained by inflation risk, whereas the real term structure is on average flat. Indeed, many leading asset pricing models predict a flat or downward-sloping real term structure (Wachter 2013; Bansal and Yaron 2004). As such, this assumption accords well with prior work.

[^14]: This implies the imperfect passthrough of interest rates to equities emphasized by (Gormsen and Lazarus 2025).

[^15]: (Piazzesi et al. 2007) show that about half of the volatility of housing returns in the data is idiosyncratic.

[^16]: In Appendix 12.3, we derive this return and discuss how it accounts for maintenance and transaction costs.

[^17]: The mean of the other state, $\mu_{zt}^+$, is then such that average shocks to $z_{it}$ are zero: $0 = p_z\mu_{zt}^- + (1-p_z)\mu_{zt}^+$.

[^18]: Households also receive a standard deduction of \$13,850 in 2020 dollars scaled with the aggregate wage.

[^19]: In our data, business owners are asked to report estimates of the value of their businesses if they were to sell them, so these values indeed represent estimates of public market values.

[^20]: We omit the subscript on $\varrho_{rit}$ to simplify notation and to emphasize that it is homogeneous within a cohort.

[^21]: As we verify in Appendix 16.2, ) holds even if we separate the coefficient of relative risk aversion from the elasticity of intertemporal substitution. Thus, the portfolio share is indeed governed by risk aversion, not the EIS.

[^22]: Notice that the agent targets the interest-rate sensitivity of next period's wealth, $W_{i,t+1}$, because that is what will respond to the interest-rate shock $\epsilon_{r,t+1}$ next period. This distinction disappears as the time interval becomes small.

[^23]: There still may be risk premia on every individual asset when $\Lambda^\top\sigma_r=0$, but these premia come from exposure to the other $J-1$ risk factors. For a long-term nominal bond, for example, this would include an inflation risk premium.

[^24]: See Appendix 16.6 for a derivation and more detailed technical discussion of this case. In that appendix, we prove that this case requires both infinite risk aversion and a zero EIS. Intuitively, in order for an agent to desire perfect hedging, he or she must be unwilling to substitute consumption across both states (risk aversion) and time (EIS).

[^25]: This expression remains unchanged when separating risk aversion from the EIS.

[^26]: We match the moments from the stock return distribution of the S&P 500 from 1900--2021. All other moments are calculated using data from 1960--2021. Appendix 13.1 provides additional explanation of the estimation procedure and Table 4 reports the closeness of fit to the moments we target.

[^27]: Recall that $\lambda_{sr}$ is an endogenous outcome of interest-rate shocks to growth in ) and the empirical interest-rate sensitivity of stocks in Table . We derive this expression explicitly in Appendix 12.2.

[^28]: (Flavin and Yamashita 2002) report a total house-price volatility of 0.14, which implies an idiosyncratic volatility $\tilde{\sigma}_h$ such that $\tilde{\sigma}_h^2+\sigma_h^2=0.14^2$.

[^29]: In particular, for leverage ratio $\pi_{s,it}+\pi_{h,it}$, $\theta_{it} \approx 0.020 + 0.010\times(\pi_{s,it}+\pi_{h,it}) - 0.001\times(\pi_{s,it}+\pi_{h,it})^2$.

[^30]: This is implied by a 15% capital gains tax rate on an 8 percentage-point expected return.

[^31]: Consider a function $V=\max\left\{V_\text{rent},V_\text{own}\right\}$. Even if $V_\text{rent}$ and $V_\text{own}$ are both concave in $W$, $V$ is locally convex where $V_\text{own}=V_\text{rent}$. Panel A in Figure 4 of (Catherine 2022) shows a similar effect in favor of equities over treasuries around the homeownership threshold, meaning this effect increases the demand for higher volatility assets in general.

[^32]: Lowercase letters denote logs; $s\equiv\log(1-(C-Y)/W)$ denotes the log savings rate. We abstract from entry and exit, which keep the wealth distribution stationary on a balanced-growth path (i.e., absent aggregate shocks).

[^33]: We say "in general" because this will only be precisely true insofar as households are not constrained in making consumption and portfolio choices. We account for the presence of such constraints in our model.

[^34]: We begin feeding data shocks to growth post-World War II and house prices after 1975. This is because macroeconomic growth was substantially more volatile in the US prior to WWII and house price indices were not based on repeated sales until 1975. Moreover, to ensure that we arrive at a stationary distribution of wealth and welfare, we begin the simulation in 1523 with random shocks until the data begin.

[^35]: Counterintuitively, adding entrepreneurs has a much more modest effect on welfare inequality than on wealth inequality. There are several reasons why the welfare effect is smaller. An important one is that, unlike workers, entrepreneurs' wealth includes the capitalized value of future earnings. Capitalizing those future earnings raises measured wealth, and thus the top wealth share, but does not increase lifetime consumption.

[^36]: If anything, welfare inequality trends go in the opposite direction. This is because, in the full model, high earners have higher life expectancy and are therefore slightly more exposed to interest-rate risk, as predicted in equation ).

[^37]: Workers can generally claim their benefits earlier or later than the FRA already. However, benefits are permanently adjusted for the claiming age: claiming before the FRA triggers an *actuarial reduction* in monthly benefits, while delaying claiming beyond the FRA increases benefits through *Delayed Retirement Credits* (DRCs). Raising the FRA therefore reduces benefits for a given claiming age (or, equivalently, requires delaying claiming to avoid a reduction), making it close to a benefit cut in terms of lifetime retirement resources.

[^38]: Under a rate sensitivity of 2, assets will rise from \$100 to \$102 and debt will rise from \$200 to \$204.

[^39]: For credit card balances---which do not have this information---we assign a duration of 1 year.

[^40]: Recall that $\lambda_{sr}$ is given by ) and that $\mu_g^+$ sets $p_R\mu_g^- + (1-p_R)\mu_g^+=0$, so these are implicitly estimated here.

[^41]: We do not model exit, so the change in the share of entrepreneurs must all be attributed to entry in the model.

[^42]: For simplicity, we will consider the case of a worker who never becomes an entrepreneur, and the same logic will apply when we add entrepreneurship.

[^43]: The third element of $\mu_t^*$ is $\mu_h+\mu_{n_h}+\sigma_{n_h}^2/2$ and we have substituted in $\mu_{n_h}=-\sigma_{n_h}^2/2$.

[^44]: It would be fine to assume rented housing, in which case consumption here represents total consumption of both goods and housing. We abstract from this for simplicity.

[^45]: In infinite-horizon models like that of (Campbell and Viceira 2001), one typically chooses $\rho_c=1-\exp\{\mathbb{E}[c_t-w_t]\}$, which reduces to $\rho_c=\beta$ for an EIS of 1. Here, to capture the effect of aging, we linearize instead around the unit-EIS solution, which is exact in our model.

[^46]: This result was first shown by (Campbell and Viceira 2001) for an infinitely lived agent with no background assets. An alternative (and closely related) proof can be found in their appendix.

[^47]: Because optimal policies $\{C_t^*,\pi_t^*\}$ were derived under linearized first-order conditions, this result will only be exactly true under those conditions.