# Sovereign Default and the Decline in Interest Rates **Authors:** Max Miller, James D. Paron, Jessica A. Wachter **Date:** November 21, 2025 **Affiliation:** Stanford University Graduate School of Business **PDF (definitive version):** https://jamesparon.github.io/papers/sovereign-default/paper.pdf **Keywords:** Savings glut, Inflation expectations, Rare disasters, Secular stagnation **JEL codes:** E31, E43, G12 > 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 Sovereign debt yields have undergone a historic decline over the last half-century. Standard explanations, including aging populations and increases in asset demand from abroad, encounter difficulties when confronted with the full range of evidence. We propose an explanation based on a decline in inflation and default risk. We show that a model with sovereign default captures the decline in interest rates, the stability of equity valuation ratios, and the reduction in investment and output growth. Calibrations of the model post-COVID suggest that sovereign default risk may have returned. ## Full text [] The period from 1980 to 2021 marked a historic decline in sovereign yields. The prolonged combination of low interest rates and sluggish growth---especially following the Great Recession of 2009---has revived the idea of "secular stagnation," a persistent period of low investment and weak demand (Hansen 1939). (Summers 2015) and (Gordon 2015) argue for the relevance of Hansen's concept from two angles: demand side---an increase in desired savings arising from changing demographics (Eggertsson et al. 2019; Auclert et al. 2021) or growing inequality (Auclert and Rognlie 2017; Mian et al. 2021)---and supply side---stemming from a decline in the ideas and dynamism that fueled growth in the postwar era. A complementary idea is that of a global savings glut (Bernanke 2005; Caballero et al. 2008): there is too great a supply of savings, mainly from patient investors outside the United States, relative to the demand for productive investment. Yet the excess-savings explanation faces difficulties. If investors' desire to save had increased, both bonds and equities should have become more valuable, and investment should have surged. Instead, equity valuations rose only modestly and investment declined. A greater demand for savings, low interest rates, and low growth together thus present a puzzle: why did stock valuations and real investment not rise in line with bond prices? In response, (Farhi and Gourio 2018) jointly consider growth, interest rates, and stock valuations in a neoclassical growth model with rare disasters (Barro 2009; Gourio 2012). They show that a substantial increase in the risk of rare disasters is needed to reconcile the levels of interest rates and stock prices. Although (Farhi and Gourio 2018) succeed in accounting for the joint behavior of stock prices and interest rates, an explanation based on heightened fears of disaster encounters its own problems. First, such fears should be reflected in option prices and in the volatility index (VIX). Although the VIX varies over time, its average level in the first and second halves of our sample is nearly identical. Second, a rare-disaster explanation is fragile because it depends on the elasticity of intertemporal substitution (EIS). If the EIS exceeds one, rising disaster risk depresses valuations; if it is below one, it raises them---deepening the puzzle. Moreover, in a production economy, rising disaster risk and an EIS below one counterfactually imply rising investment and growth. One could instead posit a decline in disaster risk, but this only highlights the lack of evidence of an increase in disaster risk. We therefore propose a different explanation---one based on a decline in the risk of sovereign default. Greater trust in the sovereign's ability and willingness to repay debts could have driven the historic fall in interest rates observed over recent centuries (Ferguson 2018). In the modern era, this decline in perceived default risk likely manifests through reduced inflation expectations. Indeed, there is substantial evidence for a steady fall in inflation expectations from 1980--2020, coinciding with the decline in interest rates. Evidence from options markets indicates that inflation expectations became "anchored" in the twenty-first century, meaning investors no longer feared extreme inflation or deflation (Reis 2020). This evidence makes it possible to jointly explain the decline in interest rates and the stability of equity valuation ratios. Because the true real rate has not fallen as much as the nominal rate, valuation ratios rise less, and there is no need to invoke a large increase in the probability of rare disasters. If it is simply inflation expectations that have declined, why have measured real rates---nominal rates minus inflation expectations---also fallen? This apparent disconnect disappears once we account for inflation risk, the risk that the price level rises during recessions or depressions. With perfect foresight, a change in expected inflation would not affect ex post real rates. However, inflation (or the lack thereof) can surprise investors. A decline in inflation risk lowers the premium required to hold nominal securities. Interest rates will fall if this premium declines, even when measured in real terms ex post. Because inflation is a form of default, a decline in inflation risk is effectively a decline in the probability of default---one that may even influence yields on inflation-protected securities. Our first contribution is to show that a model combining rare disasters with a decline in sovereign default risk can explain both the fall in interest rates and the stability of valuations. We find direct empirical support for this interpretation. First, we compare returns on nominal and inflation-indexed bonds. In the absence of an inflation risk premium, the average real returns on nominal and inflation-linked bonds should coincide. However, in the 1980s and 1990s, inflation-adjusted returns on United Kingdom nominal bonds were almost twice as high as those on inflation-linked bonds. This premium disappeared in the twenty-first century, as our model predicts. Second, we document the decline in inflation risk using survey expectations and the changing correlation between inflation and growth. This evidence aligns with recent findings that the once-positive correlations between inflation and output growth (Campbell et al. 2020) and between bond and stock returns (Campbell et al. 2017) turned negative in the 2000s. Finally, because sovereign risk depends on institutions that have evolved substantially over the centuries, our explanation can account for the striking fact that today's low rates are unprecedented not just in the last 40 years, but in the last 400 (Schmelzing 2020). We first account for the joint behavior of interest rates, inflation expectations, and stock prices in an endowment economy. However, the puzzle extends beyond an endowment setting. Just as the joint behavior of equities and interest rates poses a challenge, so too does that of investment and interest rates. A stronger desire to save should have produced an investment boom; instead, investment fell. When embedded in a production economy, our model also explains the joint decline in investment and growth, even as interest rates fell. Our paper poses a puzzle: given the large decline in interest rates, equity valuations should have risen far more than they did. We offer a resolution: the drop in yields reflected lower sovereign default risk, not a lower real rate. This claim naturally invites questions. One concern is that default risk appeared to fall even as debt-to-GDP ratios climbed across advanced economies. In our model---where government bonds are priced as claims on future surpluses---a smaller default premium can indeed coincide with higher debt, provided investors grow more confident in the government's capacity to generate those surpluses. The United States briefly ran surpluses in the 1990s, yet has not done so since, so it is natural to ask why confidence may have risen, especially in light of the magnitude of the debt. A second question arises from the surge in inflation and interest rates after 2021. In a quantitative exercise, we show that this episode can be interpreted as an increase in sovereign default risk. Because yields rose on both nominal Treasuries and Treasury Inflation-Protected Securities (TIPS), this increase cannot be attributed solely to inflation risk. The evidence is consistent instead with investors broadening their perception of default risk to include the possibility of losses on all government liabilities, whether nominal or inflation-protected. It is too early to tell whether this episode represents a historic shift, either in the risk of sovereign default, or in the form it might take. # Summary of the Data Panel A of Figure 1 shows nominal government rates in a seven-century-long data set collected by (Schmelzing 2020). Interest rates are highly volatile, as (Jordà et al. 2019) emphasize.[^2] Periods of extreme spikes and also low rates occurred around the American Revolution, Napoleonic Wars, and World War II, reflecting a tension between an increase in risk of sovereign default and precautionary savings around disasters. The high rates in the 1970s and 1980s clearly stand out. Nonetheless, the figure shows a steady decline. Panel B shows the Bank of England lending rate starting from 1700, from the start of when the series was available. Only in the very most recent period did this rate reach a zero lower bound.
A. Sovereign yields over the long run B. Bank of England lending rate
Alt text: Two-panel figure showing long-term nominal sovereign yields. Panel A displays yields for five countries from 1311 to 2018, showing a long-run decline. Panel B shows Bank of England rates from 1694 to 2024.
Figure 2 narrows down to the last 40 years, the focus of much of the literature. The federal funds rate in the United States declined sharply from 8.4% to 0.1% in 2021 before rising to 4.5% in 2024 (panel A).[^3] On the other hand, the price-dividend ratio went from 22.9 in 1984 to 68.8 in 2021 and rose further to 65.9 in 2024. This implies that dividend yields went from 4.4% down to 1.5% in 2021---a smaller decline relative to interest rates (panel B). The last row of Figure 2 shows the decline in the investment-capital ratio (panel C) and the real GDP growth (panel D) since the 1980s. Investment as a percentage of capital stock went from an average of 7.7% to 6.8% in 2024, while real GDP growth decreased from an average of 3.5% to 2.1%.
A. Federal funds rate B. Price-dividend ratio
C. Investment-capital ratio D. Real GDP Growth
Alt text: Four-panel figure showing U.S. data from 1984 to 2024: the federal funds rate, the price-dividend ratio, the investment-capital ratio, and real GDP growth.
# Endowment Economy We first turn to a standard endowment economy with a representative agent. To interpret the secular decline in interest rates, we calibrate the model separately to two sample periods, 1984--2000 and 2001--2021. We identify the year of this structural break (2001) by conducting a break test on the time series of 1-year inflation-adjusted Treasury bill yields. Details can be found in Internet Appendix . This is the same breakpoint used by (Farhi and Gourio 2018), who perform a similar analysis; it is also consistent with evidence from (Campbell et al. 2020), who find a structural break in the relationship between GDP growth and inflation in 2001. Although this approach of comparing sample averages means that certain features of the data (such as high-frequency volatility of prices and interest rates) remain outside the scope of the analysis, it allows us to consider the possibility of long-run unforeseen structural changes.[^4] Farhi and Gourio (2018) assume a neoclassical growth model. We will return to such a model in the next section, but for the analysis at hand, the extra degree of complication is not necessary. As far as prices and interest rates are concerned and in this i.i.d.-growth-rate setting, the production model and the endowment model yield the same predictions. The aggregate endowment evolves according to $$ C_{t+1} = C_te^{\mu}(1-\chi_{t+1}), $$ where $\chi_{t+1}$ represents an occurrence of rare disaster: $$ \chi_{t+1} = \begin{cases} 0 & \textrm{with probability}\quad 1-p \\ \eta& \textrm{with probability}\quad p, \end{cases} $$ for $\eta\in (0,1)$. Note that $p$ represents the probability of a disaster and $\eta$ its magnitude. We assume that the representative agent has Epstein-Zin-Weil recursive preferences (Epstein and Zin 1989; Weil 1990) with risk aversion $\gamma$, EIS $\psi$, and discount factor $\beta$. Let $W_t$ denote the wealth of the representative agent, here assumed to be the cum-dividend value of the consumption claim. Let $R_{W,t+1} \equiv W_{t+1}/(W_t - C_t)$ denote the return on wealth from time $t$ to time $t+1$. The stochastic discount factor (SDF) then equals $$ M_{t+1} \equiv \begin{cases} \beta^{\theta}\big(\frac{C_{t+1}}{C_{t}}\big)^{-\frac{\theta}{\psi}}R_{W,t+1}^{\theta-1} &\text{ if }\psi\neq1, \\ \beta\mathbb{E}_t\big[\big(\frac{C_{t+1}}{C_{t}}\big)^{1-\gamma}\big]^{-1}\big(\frac{C_{t+1}}{C_{t}}\big)^{-\gamma} &\text{ if }\psi=1, \end{cases} $$ for $\theta \equiv (1-\gamma)/(1-1/\psi)$. In this section, we assume that the aggregate stock market equals aggregate wealth (ex-dividend) and that the ex post real return on the Treasury bill equals the risk-free rate. We relax these assumptions in the following sections. In equilibrium, $R_{W,t+1}$ must satisfy $$ {\mathbb{E}_t}\left[M_{t+1}R_{W,t+1}\right] = 1. $$ Our assumptions on endowment and preferences imply a constant price-dividend ratio $(W_t-C_t)/C_t$, which we denote by $\kappa$. Standard arguments (see Internet Appendix ) then imply that $$ \kappa=\frac{\beta e^{(1-\frac{1}{\psi})\mu}\bigg[1+p((1-\eta)^{1-\gamma}-1)\bigg]^{\frac{1}{\theta}}}{1-\beta e^{(1-\frac{1}{\psi})\mu}\bigg[1+p((1-\eta)^{1-\gamma}-1)\bigg]^{\frac{1}{\theta}}}. $$ Given the return on the wealth portfolio, the Euler equation provides the return on the one-period riskless bond $$\begin{multline} \log{R_f} = -\log\beta+\frac{1}{\psi}\mu-\log(1+p((1-\eta)^{-\gamma}-1)) \\ +\bigg(\frac{\theta-1}{\theta}\bigg)\log(1+p((1-\eta)^{1-\gamma}-1)). \end{multline}$$ Equations ) and ), constitute a system of two equations in two unknowns, $p$ and $\beta$. Combining ) and ) gives the equity premium: $$\begin{eqnarray} \log \mathbb{E}_t [R_{W,t+1}] - \log{R_f} & = & \log(1-p\eta)+\log(1+p((1-\eta)^{-\gamma}-1)) \\ & & \quad \quad \quad \quad \quad \quad \quad \quad - \log(1+p((1-\eta)^{1-\gamma}-1)) \\ & \approx & p \eta ((1-\eta)^{-\gamma}-1) \end{eqnarray}$$ where the approximation is accurate for small $p$. ## Increasing disaster probability We calibrate this model using measured growth rates of real per capital consumption $\mu =0.0253$ from 1984 to 2000 and $\mu = 0.0154$ from 2001 to 2021.[^5] For comparability with Farhi and Gourio (2018), we first show results for their calibration, corresponding to $\gamma = 12$, $\psi = 2$, and a disaster size $\eta = 0.15$. We find similar results in that we match the data using a discount factor ($\beta$) of 0.969 in the early period and 0.983 in the later period, and a disaster probability ($p$) of 2.10% in the early period and 4.82% in the later period. We thus arrive at our first result: matching the combined stability of valuations and the decrease in risk-free rates requires a large increase in the disaster probability, even after accounting for decreased growth.[^6] It may seem surprising that we require such a large increase in $p$. After all, the interest rate did fall because of decreased growth and increased patience. Moreover, a decrease in growth moderates the interest rate effect on stock prices, leading to a price-dividend ratio lower than it would have been, which is precisely the problem we are trying to solve. As it happens, the reason we require such a large increase is that, like the growth rate, the increase in disaster probability has two offsetting effects which cancel out when the EIS equals unity. The EIS acts as a free parameter in this explanation. While there is no fundamental reason to believe that the EIS should differ greatly from the inverse of risk aversion (the former governs the desire of the agent to smooth across time; the latter, to smooth across states), (Farhi and Gourio 2018) follow others in the literature in assuming a high value of risk aversion and a high EIS in order to match the equity premium without other counterfactual implications. For an increase in disaster risk to be the explanation, not only must the change be large enough to overcome the offsetting effects, but it is essential to assume that the EIS is above unity. To illustrate this point, panel C of Table 1 sets the EIS to $0.5$ rather than $2$, while keeping everything else the same. Lower growth and a rising disaster probability cause valuations to increase, not decrease. Matching ) and ) with $p$ and $\beta$ still requires an increase in $p$ in the second period. However, $\beta$ must now decline, implying that investors became less patient, not more, contradicting the demand-side intuition for the decline in interest rates (Summers 2015). +:---------------------------------------------------------+:-----------:+:----------:+:----------:+ | | | Values | +----------------------------------------------------------+-------------+------------+------------+ | | Parameter | 1984--2000 | 2001--2021 | +----------------------------------------------------------+-------------+------------+------------+ | *A. Moments in the data* | | | | +----------------------------------------------------------+-------------+------------+------------+ | Price-dividend ratio | $\kappa$ | 42.35 | 50.86 | +----------------------------------------------------------+-------------+------------+------------+ | Ex ante inflation-adjusted Treasury yield | $\bar{y}_b$ | 0.0287 | -0.0053 | +----------------------------------------------------------+-------------+------------+------------+ | *B. $\gamma = 12.0$, $\textrm{EIS}= 2.0$, $\eta = 0.15$* | | | | +----------------------------------------------------------+-------------+------------+------------+ | Average consumption growth | $\mu$ | 0.0253 | 0.0154 | +----------------------------------------------------------+-------------+------------+------------+ | Discount factor | $\beta$ | 0.969 | 0.983 | +----------------------------------------------------------+-------------+------------+------------+ | Probability of disaster | $p$ | 0.0210 | 0.0482 | +----------------------------------------------------------+-------------+------------+------------+ | *C. $\gamma = 12.0$, $\textrm{EIS}= 0.5$, $\eta = 0.15$* | | | | +----------------------------------------------------------+-------------+------------+------------+ | Average consumption growth | $\mu$ | 0.0253 | 0.0154 | +----------------------------------------------------------+-------------+------------+------------+ | Discount factor | $\beta$ | 0.993 | 0.977 | +----------------------------------------------------------+-------------+------------+------------+ | Probability of disaster | $p$ | 0.0210 | 0.0482 | +----------------------------------------------------------+-------------+------------+------------+ : This table shows the parameters necessary to match the data, assuming an endowment economy with rare disasters and no inflation risk. Unless otherwise noted, we take average consumption growth from the data, and calibrate the disaster probability $p$ and the subjective discount factor $\beta$ to fit average interest rates and the price-dividend ratios in each of two sample periods. Because there is no inflation premium, the Treasury bill yield minus forecasted inflation (ex ante inflation-adjusted Treasury yield) proxies for the risk-free rate, and the wealth-consumption ratio proxies for the price-dividend ratio on the aggregate market. The table shows how $p$ and $\beta$ change depending on assumptions regarding the EIS and on growth. Treasury yields in the data and parameters in the model are annual. \ ## Did the equity premium rise? We now ask whether the equity premium did, in fact, increase. The literature studying long-term variation in the equity premium generally comes to the conclusion that the equity premium has declined during the postwar period, including from the first to the second periods that are our focus (Fama and French 2002; Lettau et al. 2008; [van Binsbergen and Koijen] 2010; Avdis and Wachter 2017). This evidence contradicts an increase in the risk of disasters. Options markets are another place to look for evidence of an increase in the equity premium (Barro and Liao 2021).[^7] Virtually any explanation for an increase in the ex ante equity premium involves an increase in risk or risk aversion. Although such a risk may not be realized in the sample, option prices incorporate the probability that market participants assign to this risk materializing. Figure 3 shows the VIX, reported by the Chicago Board Options Exchange (CBOE). The VIX is the risk-neutral expectation of quadratic volatility, which is tightly tied to the equity premium. Although the VIX is highly volatile at high frequencies, the average level of the VIX is remarkably stable between the two periods: equal to around 21 before 2001 and 20 after. It is hard to reconcile this stability with a secular increase in the equity premium.
Alt text: Line graph showing the VIX from 1986 to 2025, with a structural break around 2001 separating the two periods.
## Sovereign default risk A standard proxy for the equilibrium risk-free rate is the real return on government debt; however, this return is not necessarily riskless, as the government can default either outright or through inflation. We now price the claim to defaultable debt.[^8] A decline in default risk can explain the secular trends in risk-free rates and valuation ratios since 1980 without appealing to rising disaster risk. Suppose that, in a disaster, creditors lose a fraction $\lambda \eta$ relative to the face value of the bond. That is, a bond issued at time $t$ pays $1-L_{t+1}$ at time $t+1$, where loss $L_{t+1}=\lambda\chi_{t+1}$ represents a loss of zero if there is no disaster, and $\lambda \eta$ if a disaster should occur. If $\lambda = 1$, the loss to creditors is equal, in percentage terms, to the decrease in consumption $\eta$. If $\lambda = 0$, then the bond is risk-free. Let $Q_t$ be the price of the defaultable bond. In equilibrium, $$ Q_{t}={\mathbb{E}_t}\left[M_{t+1}(1-L_{t+1})\right]. $$ Let $R_{b,t+1} \equiv (1-L_{t+1})/Q_t$ denote the return on the defaultable bond. The expected return is $$\begin{eqnarray} \log{\mathbb{E}[R_{b,t+1}]} & = & \log{R_f} + \log(1- p\lambda \eta)+\log{(1+p( (1-\eta)^{-\gamma}-1))} \\ & & -\log{(1+p((1-\lambda\eta)(1-\eta)^{-\gamma}-1) )} \\ & \approx & \log{R_f} + p\lambda \eta ((1-\eta)^{-\gamma}-1). \end{eqnarray}$$ The term $p\lambda \eta ((1-\eta)^{-\gamma}-1)$ is the default risk premium. We have thus far been agnostic as to the means of default. Inflation offers one such means, and for simplicity, we now assume that it is the only means (we discuss this assumption further below). Let $\Pi_t$ denote the price level, so that $\Pi_{t+1}/\Pi_t$ is inflation. A capital loss of $L_{t+1}$ through default is equivalent to an inflation of $1/(1-L_{t+1})$. Our model will also allow for both expected and unexpected inflation outside of disasters. To do this, we introduce $q_t$ to allow for nondisaster expected inflation and $\sigma_\pi\epsilon_{t+1}$ to be nondisaster unexpected inflation, where $\epsilon_{t+1}$ is iid normal, and assumed, for now, to be uncorrelated with consumption growth.[^9] To summarize, the price level follows the process $$ \Pi_{t+1} = \Pi_t e^{q_t + \sigma_\pi\epsilon_{t+1}} (1-L_{t+1})^{-1}. $$ Let $Q^\$_t$ denote the nominal price of the nominal bond. In equilibrium, $$ Q^\$_t = \mathbb{E}_t \left[ M_{t+1}\frac{\Pi_t}{\Pi_{t+1}}\right]. $$ Let $y_{b,t}^\$$ denote the continuously compounded nominal yield on the nominal bond, namely, $y_{b,t}^\$ \equiv - \log Q^\$_t$. We will identify $y^\$_{b,t}$ with the yield on the 1-year Treasury. Substituting ) into ) implies the nominal yield $$\begin{multline} y^\$_{b,t} = \log{R_f} + \log{(1+p( (1-\eta)^{-\gamma}-1))} \\ -\log{(1+p((1-\lambda\eta)(1-\eta)^{-\gamma}-1) )} + q_t - \frac{1}{2}\sigma_\pi^2. \end{multline}$$ It also implies the same expected real bond return ) as in the case of outright default. To map bond yields to the data in a way that allows us to identify the default parameter $\lambda$, we consider the nominal Treasury yield less expected inflation, $$ \bar{y}_b \equiv y^\$_{b,t} - \log\mathbb{E}_t\left[\frac{\Pi_{t+1}}{\Pi_t}\right]. $$ We refer to $\bar{y}_b$ as the ex ante inflation-adjusted Treasury yield (because it uses ex ante expected inflation rather than ex post realized inflation). We will use the log of 1-year inflation forecasts from the data to measure expected inflation, equal to $$ \log\mathbb{E}_t\left[\frac{\Pi_{t+1}}{\Pi_t}\right] = q_t +\frac{1}{2}\sigma_\pi^2 + \log(1+p\lambda\eta(1-\lambda\eta)^{-1}). $$ Subtracting ) from ), we obtain $$\begin{multline} \bar{y}_b = \log{R_f} + \log{(1+p( (1-\eta)^{-\gamma}-1))} \\ -\log{(1+p((1-\lambda\eta)(1-\eta)^{-\gamma}-1) )} - \sigma_\pi^2 - \log(1+p\lambda\eta(1-\lambda\eta)^{-1}), \end{multline}$$ The difference between $\bar{y}_b$ and the log risk-free rate includes the inflation risk premium (the same as in )) and an adjustment for Jensen's inequality.[^10] Absent data on inflation forecasts, one could instead use the ex post inflation-adjusted yield, defined as the nominal yield ) less average log inflation in the data sample: $$ \hat{y}_b \equiv y^\$_{b,t} - \mathbb{E}_t\left[\log\frac{\Pi_{t+1}}{\Pi_t}\bigg|L_{t+1}=0\right] = y^\$_{b,t} - q_t, $$ where we have imposed the fact that there were no disasters in our data sample. The ex post and ex ante measures differ only in the Jensen's inequality terms. We now calibrate the model, keeping $p$ constant at 2.10% (the calibrated value from 1984--2000), and allowing $\lambda$ to vary. The form of Table 2 is the same as Table 1. We first show the price-dividend ratio and the ex ante inflation-adjusted Treasury yield for the two sample periods. For the ex ante inflation-adjusted, we use the yield on 1-year Treasuries and 1-year SPF inflation forecasts. Note that the model with inflation implies a different and more precise interpretation of the ex ante inflation-adjusted Treasury yield. The model counterpart is ), whereas in the previous model it was simply the risk-free rate.[^11] Similarly to the previous exercise, we fix all parameters other than consumption growth (which is taken from the data), patience $\beta$, and the inflationary default parameter $\lambda$. A higher $\lambda$ corresponds to a greater exposure, and hence a higher inflation premium. In line with the disaster literature, we consider a lower value of risk aversion $\gamma$ (equal to $5$) and a correspondingly larger disaster (a consumption decline of 30%). This calibration forms our benchmark; however, our points are qualitatively similar with higher $\gamma$ and smaller disasters. We first consider the case of an EIS equal to 2. We first note that the model is capable of matching the data, assuming a $\lambda$ such that 16.5% of the bond value is lost in disasters in the first sample whereas in the second sample the bond rises in value by 6.0%. Crucially, it does so with a smaller increase in the discount rate $\beta$. Rather than 1.4 percentage points, $\beta$ increases by 0.9 percentage points. This is because the model has a fundamentally different explanation for the decline in the interest rate, namely, the reduced inflation premium. Indeed, the inflation premium (which we can calculate using )), is 1.7 percentage points in the first half of the sample, falling to negative 62 basis points in the second half, accounting for the majority of the decline in the observed interest rate.[^12] The remaining decrease in $\bar{y}_b$ comes from a decrease in the real rate, driven by a combination of lower growth and greater patience. Panel C assumes an EIS equal to 1. Note that, unlike the previous explanation, the model does not depend on an EIS greater than 1. Parameters in panel C are similar to those in panel B, with an even smaller increase in $\beta$. When the EIS is equal to one, the decline in growth now does not affect the price-dividend ratio (although it still affects the risk-free rate). Thus, $\beta$ now must increase by a mere 0.4 percentage points, because there is no need to counteract the effect of lower growth on the price-dividend ratio and because growth also has a larger effect on the interest rate. In summary, changes in $\lambda$ help explain the decline in observed interest rates, making the assumption of a decrease in disaster premia unnecessary. This explanation is more robust, in that it does not require a knife-edge combination of hard-to-observe risk, patience, and willingness to substitute over time. +:-------------------------------------------------------+:-------------:+:----------:+:----------:+ | | | Values | +--------------------------------------------------------+---------------+------------+------------+ | | Parameter | 1984--2000 | 2001--2021 | +--------------------------------------------------------+---------------+------------+------------+ | *A. Moments in the data* | | | | +--------------------------------------------------------+---------------+------------+------------+ | Price-dividend ratio | $\kappa$ | 42.35 | 50.86 | +--------------------------------------------------------+---------------+------------+------------+ | Ex ante inflation-adjusted Treasury yield | $\bar{y}_b$ | 0.0287 | -0.0053 | +--------------------------------------------------------+---------------+------------+------------+ | *B. $\gamma = 5.0$, $\textrm{EIS}= 2.0$, $\eta = 0.3$* | | | | +--------------------------------------------------------+---------------+------------+------------+ | Average consumption growth | $\mu$ | 0.0253 | 0.0154 | +--------------------------------------------------------+---------------+------------+------------+ | Discount factor | $\beta$ | 0.972 | 0.981 | +--------------------------------------------------------+---------------+------------+------------+ | Fraction of bond value lost | $\lambda\eta$ | 0.165 | -0.060 | +--------------------------------------------------------+---------------+------------+------------+ | Model-implied riskfree rate | $r_f$ | 0.0140 | 0.0003 | +--------------------------------------------------------+---------------+------------+------------+ | *C. $\gamma = 5.0$, $\textrm{EIS}= 1.0$, $\eta = 0.3$* | | | | +--------------------------------------------------------+---------------+------------+------------+ | Average consumption growth | $\mu$ | 0.0253 | 0.0154 | +--------------------------------------------------------+---------------+------------+------------+ | Discount factor | $\beta$ | 0.977 | 0.981 | +--------------------------------------------------------+---------------+------------+------------+ | Fraction of bond value lost | $\lambda\eta$ | 0.165 | -0.060 | +--------------------------------------------------------+---------------+------------+------------+ | Model-implied riskfree rate | $r_f$ | 0.0140 | 0.0003 | +--------------------------------------------------------+---------------+------------+------------+ : This table shows the parameters necessary to match the data, assuming an endowment economy with rare disasters and inflationary default. We take average consumption growth from the data in each sample. We calibrate the discount factor $\beta$ and the decline in bond value $\lambda\eta$ to match the average price-dividend ratio and Treasury bill yield minus forecasted inflation (ex ante inflation-adjusted Treasury yield). We assume the disaster probability equals 2.10%, its benchmark value in Table 1. Parameters and yields are in annual terms. \ ## Evidence for declining inflation risk In contrast to the explanation based on disaster risk, independent evidence points to a decrease in the risk of sovereign default. We focus on the case that all sovereign default takes place through inflation. We first examine relative yields on nominal and inflation-protected bonds. Given that TIPS were first introduced in 1997 and did not trade with sufficient liquidity for prices to be reliably measured until around 2004 (Fleming and Krishnan 2004), we cannot use this information to distinguish inflation premia before and after the structural break separating our two samples.[^13] Index-linked gilts from the United Kingdom, on the other hand, have traded since the early 1980s and provide the ideal asset to examine this difference over the last four decades. We examine the ex post inflation-adjusted yield on nominal bonds relative to the yield on index-linked bonds, namely, ) in the model. Our explanation predicts that the difference is positive in the first sample period, declining to zero in the second. Panel A of Figure 4 shows the difference between the ex post inflation-adjusted yield on the 1-year United Kingdom government nominal bond and the yield on 5-year index-linked gilts. The figure shows that in the first sample, this difference was significantly positive and large---over 1.5 percentage points. In the second sample, this excess return is on average zero, consistent with our estimate of no default risk. This suggests an economically large decline in the inflation risk premium. A possible concern with this measure is that it compares bonds of different maturities. It may, for example, be that differences in these bond returns arise from changes in the term structure. Panel B addresses this concern by considering 5-year nominal bond yields minus (annualized) realized inflation over 5 years. The result is essentially identical: 5-year nominal bonds earned a premium of more than two percentage points in the 1980s and 1990s and no premium in the 2000s and 2010s. The parameter values in Table 2 imply a strikingly similar decline of 1.8 percentage points, validating our model calibration.
A. One-year nominal bond minus gilt B. Five-year nominal bond minus gilt
Alt text: Two-panel figure showing excess returns of nominal bonds over TIPS for 1-year (Panel A) and 5-year (Panel B) maturities, with data split into pre-2001 and post-2001 periods.
As described above, there are also reasons that yield-based break-even estimates of the sovereign default premium may be too low. For example, the above calculation assumed that all defaults occur through inflation. Nonzero prices on credit-default swaps suggest otherwise (Chernov et al. 2020).[^14] Even if one were to relax this assumption (and allow an additional term for outright default), there remain two reasons why the estimate may be too low: (1) inflation indexation is inexact (or, more precisely, errors may be correlated with factors investors care about, such as inflation itself), and (2) recovery rates may be lower on inflation-indexed bonds. Indeed, TIPS explicitly do not index for deflation. (Anderson and Sleath 1999) discuss errors in inflation indexation for Gilts. The government of Canada has discussed stopping the issuance of inflation-linked securities due to concerns about the size of potential payouts (Czitron 2022). (Dittmar et al. 2024) argue that the prices of TIPS reflect a greater risk of outright default than the prices of nominal Treasury bonds. This is not surprising as investors have no experience with whether TIPS would pay off should there be a shock to inflation. Another place to look for evidence of a declining inflation risk premium is in sample correlations between inflation and consumption growth. Ideally, to capture changes in $\lambda$, one would use these disaster observations to construct a correlation that is disaster specific. That is the literal interpretation of the model in the previous section. However, both expected and realized behavior during disasters are hard to observe; disaster expectations are inherently unobservable, and disasters are too few for accurate measurement regarding price behavior. Absent these ideal data moments, we look directly at realized inflation and consumption growth with the view that the line between a downturn and disaster is ultimately arbitrary. Figure 5 shows the result. The recession in the early 1990s was clearly accompanied by inflation, whereas the boom of the late 1990s was accompanied by deflation. However, the financial crisis was deflationary. The COVID recession saw deflation during the 2020 contraction, followed by high inflation.
Alt text: Line graph showing consumption growth and inflation from 1984 to 2024, with correlations reported for pre-2001 and post-2001 periods.
Survey data also provide evidence for the declining risk of sovereign default. Using data from the University of Michigan and the Survey of Professional Forecasters, Figure 6 shows a decline in inflation forecasts over four decades, leveling off in more recent years. (Reis 2020) finds an anchoring of inflation expectations using survey data. Figure 6 shows that break-even inflation measured by TIPS prices relative to comparable nominal bonds is consistent with low expected inflation in the second half. Tables 1 and 2 adjust for the expected inflation component of yields, and show that there is still a marked decline in ex ante inflation-adjusted rates. Though the evidence in Figure 6 does not directly address the puzzle in 1, it is relevant because it is reasonable to assume that higher inflation expectations would be accompanied by higher risk, and indeed our hypothesized process for inflation ) explicitly links the two.[^15] The link between expected inflation and the likelihood of disasters provides a further test of our model. Indeed, our assumed process for inflation makes predictions regarding expectational errors, assuming disasters do not occur in-sample: forecasters should overpredict inflation during the sample in which inflationary disasters occur and underpredict it (or accurately predict it) otherwise.[^16] Data on 1-year-ahead forecasts from the Survey of Professional Forecasters in fact show that in the first sample, forecasters consistently overestimate inflation, whereas in the second sample, their estimates are on average correct (Figure 7). Researchers have interpreted these expectation errors as evidence of slow learning due to highly persistent underlying processes (Farmer et al. 2024) or the strong pull of past experience (Goetzmann et al. 2022), both of which are also in the spirit of our model. Regardless of the interpretation, in the first sample, investors predicted inflation that did not occur, whereas in the second, they ceased to forecast inflation. This is consistent with a structural break such that, in the first sample, inflation exhibits positive skewness ($\lambda>0$), which vanishes in the second ($\lambda\approx 0$). A comparison of the estimates in Table 2 together with the estimates of Figure 7 indicates that about half of expectational "errors" are due to the absence of disasters in the sample.
Alt text: Line graph showing four measures of expected inflation from 1978 to 2024, including survey-based measures and TIPS break-even inflation.
Alt text: Line graph showing the difference between expected and realized inflation from 1979 to 2025, with pre-2001 and post-2001 sample means indicated.
Finally, any amount of variability in the disaster probability within each period will induce a positive correlation between bond and stock returns in the first period and a negative correlation in the second. In the model of (Wachter 2013), an increase in the risk of rare disasters causes stock prices to fall. If bond payoffs are also negatively exposed to disaster risk ($\lambda>0$), then bond prices fall with when disaster risk increases, leading to a positive correlation. If, on the other hand, bonds experience deflation during a disaster ($\lambda<0$), then the correlation flips sign. Indeed, (Campbell et al. 2017) and (Campbell et al. 2020) observe a substantial shift in the bond-stock beta, going from positive to negative between the first and second sample.[^17] Thus the changing correlation in bond and stock returns also provides evidence in support of declining risk of sovereign default. To summarize, our model and calibration imply that the true risk-free rate and equity risk premium have remained relatively stable over time, a conclusion that is consistent with evidence from valuation ratios, which have remained relatively flat; and from the VIX. This is not to say that risk-free rates have not declined at all. In the calibrations that we present above, the risk-free rate decreases from the first sample to the second. This is mirrored in the data: the yield on index-linked gilts has also fallen. Our point is that this decline is substantially smaller than what is reflected in declines in inflation-adjusted nominal yields, and accounting for this explains the joint evolution of valuation ratios and bond yields. ## The government's ability to repay The previous discussion describes the puzzle posed by declining interest rates, unaccompanied by a rise in valuation. It also proposes a solution: that the decline in interest rates arises primarily from a decreased risk of sovereign default, rather than a decline in the risk-free rate. In recent work, (Jiang et al. 2024) highlight a tension between the growing debt burden in the developed world, the tendency to add to this burden by running deficits in good times and bad times, and the low interest rates that developed nations still enjoy (and which are the focus of this paper). Rather than pricing government bonds, Jiang et al. (2024), following (Cochrane 2023), price the totality of government debt as a claim to future surpluses (more precisely, their framework expresses the real value of government debt, divided by GDP, as an infinite sum of real future surpluses as a fraction of GDP). Our approach does not require a government budget constraint, and so the path of future surpluses remains in the background. However, the two frameworks are tightly connected. Internet Appendix  shows how the inflation process ) arises from a process for real government surpluses as a fraction of GDP. Then it relates the default parameter $\lambda$ to this surplus process. The greater the risk of default, the less the sovereign pays back after a disaster and inflation. Internet Appendix  establishes several results. The first is that a one-time decrease in $\lambda$ leads to an increase in the debt-to-GDP ratio. Thus, a decrease in the risk of sovereign default will imply that government borrowing as a fraction of the real economy will rise, which is in fact what occurred. The second point is that, if recent past behavior is a guide to future behavior when it comes to government surpluses, there is an increase, not a decrease, in the risk of sovereign default, and there have been economically significant pricing errors in government bonds. That is, we in effect replicate the main point of (Jiang et al. 2024). However, our approach is also consistent with a growing debt-to-GDP ratio, as long as investors are willing to accept future promises of surpluses instead of current surpluses (the recent past may not be a good guide to future behavior). This is in fact the solution claimed for the puzzle by (Cochrane 2023). Although outside the scope of this paper, our view is that it is not unrealistic for an investor to assume that the US government would pay back its debts. Outright default in the developed world is a rare event. Since the 1980s, the Federal Reserve has shown a commitment to keeping inflation low. Moreover, in the 1990s, the U.S. proved that it is possible to run surpluses. Our point in this paper is not to comment on the wisdom of running large deficits or on any element of fiscal policy. There is no bubble in our model, as in (Kocherlakota 2023), and the analysis in Section 2.6 suggests that convenience yields did not increase substantially over the 40-year period from 1980 to 2020, implying that convenience alone could not be the reason why interest rates fell despite the increase in the debt burden. Thus, our model simply implies that investors believed, during the 2000--2021 period, that the current fiscal behavior of developed countries in running large deficits will end. This belief was highly beneficial to U.S. taxpayers, at least in the short run, as it implied that interest payments were low. As Section 2.7 shows, our leading interpretation of events after 2021 marked the end of that belief. ## The role of convenience yields The above sections argue that a decline in sovereign default risk is necessary to explain the joint behavior of interest rates and equity valuation ratios. However, an alternative to sovereign default risk is a decline in the convenience yield. The convenience yield hypothesis states that certain assets can act as a substitute for cash and therefore have a utility that goes beyond their intrinsic value. A substantial literature examines the convenience yield (e.g., Krishnamurthy and Vissing-Jorgensen 2012; [van Binsbergen et al.] 2022; Fleckenstein 2024). Like a decline in sovereign default risk, a decline in the convenience yield affects the Treasury bill rate (which presumably has convenience value), while leaving equities unaffected. Specifically, suppose that the observed Treasury yield is equal to a sum of the traditional financial return and a "convenience" yield, representing the role of Treasury securities as cash. The latter is negative. An increase in the magnitude of the convenience yield could thus explain a decline in the observed Treasury yield. The literature on the convenience yield uses several proxies. Ideally, one would find a proxy in which convenience could be disentangled from risk. For our purposes, which involve finding a plausible upper bound on the effect of convenience, it suffices to examine the spread between highly rated corporate bonds and Treasury bonds (Cieslak et al. 2024). This spread widened by an average of 37 basis points between the first and second sample, indicating that an increase in the convenience yield plays a minor role in explaining the 3--4 percentage-point decline in interest rates. Replacing the Treasury bill rate series with a highly rated corporate debt series thus does not change our results very much: rather than a decline in fraction of bond value lost of 22.5 percentage points, we see a decline of 20.0 percentage points. Thus, while acknowledging that a rise in convenience may be part of the reason for the decline in Treasury rates, it is secondary to other factors driving government bond yield declines. ## Valuation ratio and interest rate dynamics within each sample Earlier in this section, we argued that reduced sovereign default risk is a primary driver of the decline in interest rates over the past 40 years and perhaps the past 400 years. There are many possible explanations for the decline in interest rates if taken in isolation. Jointly considering the price-dividend ratio, the interest rate, and the growth rate considerably narrows down the possible explanations. For example, if an increased propensity to save had driven down interest rates, the price-dividend ratio would have risen to a much greater degree than what we observe. We have argued that the answer lies in interpreting the decline in interest rates as a decline in the risk of sovereign default, and not entirely as a decline in the real rate. Although the decline in the risk of sovereign default explains the average levels across samples, we do not claim that it explains the variations within samples. For a full answer as to what drives rates and valuation ratios, we generalize the model to allow for Gaussian shocks to consumption, to expected inflation, and to unexpected inflation. We examine how much of the decline in interest rates can be accounted for by survey measures of expected inflation alone, and we incorporate the shift in the consumption-inflation correlation. Finally, we allow for time variation in dividend growth expectations that occur apart from consumption growth expectations. By definition, an increase in expected inflation represents an increase in the degree of default since all inflation is a form of default on nominal bonds. Moreover, the change in sign in the consumption-inflation correlation represents a decrease in the risk of sovereign default. However, incorporating these changes into the model will allow us to isolate the effect of rare disasters. Define $z_t\equiv\log(C_t/D_t)$, the consumption-dividend ratio, and let $\varepsilon_{t+1}$ be a four-dimensional vector of independent standard normal shocks. Assume $$\begin{eqnarray} C_{t+1} & = & C_te^{\mu+\sigma_C^\top\varepsilon_{t+1}}(1-\chi_{t+1}), \\ z_{t+1} & = & (1-\rho_z)\bar{z} + \rho_zz_t + \sigma_z^\top\varepsilon_{t+1} - \xi\log(1-\chi_{t+1}), \\ \Pi_{t+1} & = & \Pi_te^{q_t + \sigma_\pi^\top\varepsilon_{t+1}}(1-\lambda \chi_{t+1})^{-1}, \\ q_{t+1} & = & (1-\rho_q)\bar{q} + \rho_qq_t + \sigma_q^\top\varepsilon_{t+1}. \end{eqnarray}$$ Following (Longstaff and Piazzesi 2004) and (Wachter 2005), we assume that the consumption-dividend ratio is stationary and use $\sigma_z$ and $\xi$ to represent financial leverage that can cause dividends to fall by more than consumption after negative shocks. Internet Appendix  derives closed-form expressions for interest rates and valuation ratios in this economy.[^18] We use survey expectations measures to proxy for expected dividend growth (determined by $z_t$) and expected inflation (determined by $q_t$). Following (De la O and Myers 2021) and (Bordalo et al. 2024), we base dividend-growth expectations for the aggregate market on I/B/E/S earnings forecasts for individual firms. We use earnings rather than dividend forecasts because I/B/E/S only has dividend expectations for enough firms to reasonably construct a series starting in 2003. Moreover, earnings expectations, and in particular long-term earnings expectations, may better capture dividend expectations in the model in that decisions to pay a dividend or not over a given time horizon may be driven by dynamics that the simple model ) does not capture. De la O and Myers (2021) and Bordalo et al. (2024) develop methodologies for aggregating these individual-firm forecasts into a forecast for the aggregate market. Whereas De la O and Myers (2021) use 1-year ahead growth expectations, Bordalo et al. (2024) focus on the long-term growth variable (LTG). We follow the Bordalo et al. (2024) approach, extending their data, which ends in 2020. The mean of LTG is consistently above realized earnings growth.[^19] Explaining the disconnect between stated expectations and reality on the part of I/B/E/S forecasters is beyond the scope of this paper, nor would accounting for this effect change our results as it appears equally in both samples. Due to this disconnect, we de-mean LTG before backing out implied values for $z_t$. Internet Appendix  contains details on data construction; Internet Appendix  contains further details on our calibration. As in Figure 7, we use the 1-year-ahead inflation expectations from the Survey of Professional Forecasters. The log of this measure is equal to ). We match the cyclically adjusted price-earnings ratio (CAPE) from (Shiller 2000), converting the price-dividend ratio in the model to a price-earnings ratio using the payout ratio.[^20] As in our previous analysis, we allow for the change in average consumption growth between the two samples. We also allow the consumption-inflation correlation to change, as well as the discount factor $\beta$ and the default parameter $\lambda$.[^21] Other parameters remain constant between the samples. Table 3 gives the parameters that change between the samples, with further details in Internet Appendix . As Table 3 shows, this more complex model has nearly the same implications for the change in default risk as the simpler model presented in Table 2. The fraction of the value of the bond lost in a disaster goes from 13.2% to $-3.2\%$. Figure 8 shows the time series of interest rates (panel A) and valuation ratios (panel B) in the model and the data. The model succeeds in capturing the interest rate decline around 2000. It also matches some of the fluctuations in rates both before and after the structural break, but these fluctuations are more muted in the model compared to the data. As we explain below, the model has the potential to match the more recent increase in rates. Turning to the valuation ratios in panel B, the model again can match the levels both before and after the structural breaks, and has some success in matching fluctuations, although there are some fluctuations in I/B/E/S that do not appear in prices. We also perform the following counterfactual exercise: suppose that there was no change in the risk of default, leaving the decrease in growth and the increase in $\beta$ to account for the decline in interest rates. What would have been the consequence for equity valuations? Figure 8 shows that they would be impossible to ignore, rising to many multiples of the true price-earnings ratio. This is because $\beta$ needs to rise to 0.997 to match the decline in interest rates observed in the data.[^22] It is hard to imagine what force could have led to a mispricing of this magnitude. On the other hand, if the inflation premium is what changed, then one finds a much smaller real rate decline and a smaller increase in the price-dividend ratio.
A. Interest rate B. Valuation ratios
Alt text: Two-panel figure comparing model predictions to data. Panel A shows model versus actual government bond yields. Panel B shows model versus actual CAPE ratios.
+:---------------------------------------------------+:---------------------:+:----------:+:----------:+:----------:+ | | | Values | +----------------------------------------------------+-----------------------+------------+------------+------------+ | | Parameter | 1984--2000 | 2001--2021 | 2022--2024 | +----------------------------------------------------+-----------------------+------------+------------+------------+ | *A. Moments in the data* | | | | | +----------------------------------------------------+-----------------------+------------+------------+------------+ | Price-dividend ratio | $\kappa$ | 42.35 | 50.86 | 59.65 | +----------------------------------------------------+-----------------------+------------+------------+------------+ | Ex ante inflation-adjusted Treasury yield | $\bar{y}_b$ | 0.0287 | -0.0053 | 0.0197 | +----------------------------------------------------+-----------------------+------------+------------+------------+ | *B. $\gamma = 5$, $\textrm{EIS}= 1$, $\eta = 0.3$* | | | | | +----------------------------------------------------+-----------------------+------------+------------+------------+ | Average consumption growth | $\mu$ | 0.0253 | 0.0154 | 0.0154 | +----------------------------------------------------+-----------------------+------------+------------+------------+ | Consumption-realized inflation correlation | $\text{Corr}(C, \pi)$ | -0.205 | 0.736 | 0.736 | +----------------------------------------------------+-----------------------+------------+------------+------------+ | Discount factor | $\beta$ | 0.976 | 0.981 | 0.981 | +----------------------------------------------------+-----------------------+------------+------------+------------+ | Fraction of bond value lost | $\lambda\eta$ | 0.132 | -0.032 | 0.186 | +----------------------------------------------------+-----------------------+------------+------------+------------+ : This table shows results for the generalized endowment model which includes rare disasters and inflationary default, and allows for time-variation in nondisaster inflation expectations and in expected dividend growth. We estimate processes --) using data from 1984--2024, allowing expected consumption growth $\mu$ and the correlation between shocks to consumption growth and inflation (here denoted $\text{Corr}(C, \pi)$) to change between periods. We calibrate the discount factor $\beta$ and the decline in bond value $\lambda\eta$ to match the average level of the price-dividend ratio and the ex ante inflation-adjusted Treasury yield in each sample. In both samples, $\gamma = 5$, $\eta = 0.30$, the disaster probability equals 2.10%, and the EIS equals 1. Internet Appendix  contains additional details. Parameters are in annual terms. \ Our focus has so far been on the period spanning the 1980s to December 2021. Interest rates increased substantially in 2022 and remained elevated, relative to recent years, at the time of this writing. Equations ) and ) show that four mechanisms that could play a significant role in an increase in interest rates are (1) a decline in $\beta$, (2) an increase in expected economic growth $\mu$ (or a decrease in the disaster probability $p$), (3) an increase in expected inflation $q_t$, or (4) an increase in the risk of default during disasters, $\lambda$.[^23] A decrease in patience $\beta$ implies a reversal of the long-term economic forces described above, and thus seems unlikely.[^24] Moreover, a decrease in $\beta$ would cause the stock market to crash, which did not occur. This is the mirror image of the previous analysis that shows a boom in the stock market should the entire decrease in interest rates arise from this mechanism. Likewise, there is no evidence for an increase in expected economic growth in 2022, with the Survey of Professional Forecasters anticipating 1-year-ahead GDP growth at 1.5% in 2022, nor is there evidence for a decline in rare-event risk. The evidence for an increase in expected inflation is mixed, with very little increase in the Survey of Professional Forecasters (0.6 percentage points), though a more sizable increase in the University of Michigan survey. However, even this falls far short of the increase in interest rates (3.9 percentage points) between 2021 and 2022. The remaining possibility is an increase in the risk of sovereign default. Figure 8 and Table 3 show the result of recalibrating the model to the 2022--2024 period. In accordance with the previous discussion, we keep the discount rate $\beta$ and the growth rate $\mu$ at their levels from the second sample 2000--2021. We continue to use data from the Survey of Professional Forecasters to proxy for $q_t$ and the LTG measure to proxy for $z_t$. We find that, to match the level of the interest rate in 2022, we require an increase in $\lambda$ such that nearly 18.6% of the value of a bond lost in a disaster, significantly more than the 13.2% in the first sample period.[^25] One question is why we need such a sizable increase, given that interest rates still remain lower than in the first sample (though high when compared with recent years). The reason is that we are operating with a higher value of $\beta$, which increases asset prices overall, and a lower growth, which decreases interest rates. Altering these assumptions would at best lead us to a slightly reduced $\lambda$ that would still represent a substantial increase relative to the 2001--2021 value. Further evidence in support of this mechanism comes from the reversal in sign between the correlation of bond and stock returns (Antolin-diaz 2025).[^26] Thus the 2022--2024 episode leads to the conclusion that reduction in sovereign wealth risk achieved between the first and second sample period has now been reversed.
Alt text: Line graph comparing model-implied inflation risk premium to the spread between nominal yields and TIPS yields at 5-year and 10-year maturities.
However, evidence from inflation-protected securities suggests an important addendum to this conclusion. Figure 9 shows the implied inflation premium from TIPS, together with the premium from sovereign default, from the start of the TIPS sample period. This figure shows that, other than in 2008 when nominal bonds experienced a high premium for default, this premium has fluctuated little and, more importantly, remained negative throughout the entire sample period.[^27] The data are thus consistent with our model prior to 2022. After 2022, the model and data diverge: the model suggests a sharp increase in the inflation risk premium, and while the premium did rise, the level remained at or near zero. The data appear at odds with our assumptions that (a) all default takes place through inflation, (b) TIPS are a perfect hedge, and (c) the risk of default is higher than it was. As we discussed in Section 2.4, (a) and (b) are simplifying assumptions, and not essential for us. Eliminating these assumptions only affects the interpretation of our model for inflation; the main implications, in Table 3 and Figure 8 for example, remain unchanged. # Production Economy A central part of the puzzle regarding valuation ratios and interest rates lies in the fact that economic growth and capital investment remained low despite historically high asset prices. In this section, we show how our conclusions under an endowment economy translate to a production economy, explaining investment-capital ratios. We introduce a model with a riskless inventory asset that can account for the zero lower bound on interest rates. ## No-inventory case We consider a standard production model in which capital quality can decline suddenly and unpredictably.[^28] Let $K_t$ denote the quantity of productive capital at time $t$. Given $K_t$ and constant productivity $A$, output equals $$ Y_t = A K_t. $$ Let $\delta$ denote depreciation and $X_t$ investment. Capital evolves according to $$\begin{eqnarray} \tilde{K}_{t+1} &\equiv& X_{t}+(1-\delta)K_{t} \\ K_{t+1} & \equiv & \tilde{K}_{t+1}(1-\chi_{t+1}), \end{eqnarray}$$ where $\chi_{t+1}$, defined in ), represents destruction of capital. We assume $A>1-\delta$, consistent with a growing economy. Following (Gomes et al. 2019), we refer to $\tilde{K}$ as planned capital, the quantity of capital available if the disaster does not occur. We can restate the agent's problem as a consumption-portfolio choice decision in which the agent allocates savings to capital and the risk-free bond. Let $B_t$ denote the time-$t$ dollar allocation to the risk-free asset. Define the agent's wealth at time $t$ as $$ W_t \equiv C_t + B_t + \tilde{K}_{t+1}. $$ If investment in capital grows at the stochastic rate $R_{K,t+1}$, wealth at time $t+1$ must equal $$ W_{t+1} = B_tR_{f,t+1} + \tilde{K}_{t+1}R_{K,t+1}. $$ What is $R_{K,t+1}$? Equations --) indicate that, should a disaster not occur, a single unit of capital creates $A$ units of output. A fraction $\delta$ is lost prior to the next period. Should a disaster occur, then a fraction $\chi_{t+1}$ is lost. Given the remaining capital, $A$ units of output are created and an additional fraction $\delta$ is lost. Therefore, the return on capital is $$ R_{K,t+1} = (1-\delta+A)(1-\chi_{t+1}). $$ We can rewrite the budget constraint in terms of flow variables. Equating ) with ) at time $t$ and substituting in for $R_{K,t}$ implies $$ C_t + B_t + \tilde{K}_{t+1} = B_{t-1} R_{f,t} + \tilde{K}_t (1-\delta +A) (1-\chi_t). $$ Using ) and ), then subtracting $(1-\delta)K_t$ from both sides implies $$ C_t + B_t + X_t = Y_t + B_{t-1}R_{f,t}. $$ That is, output from the capital stock plus wealth in bonds can be used toward consumption, bond purchases at time $t$, or investment in the productive asset. We can also rewrite the budget constraint in terms of the evolution of wealth. Define the share of savings invested in capital as $$\alpha_t \equiv \frac{\tilde{K}_{t+1}}{W_t-C_t}.$$ Substituting in for $B_t$ in ) from ) implies that $$ W_{t+1} = (W_t-C_t)(R_{f,t+1}+\alpha_t(R_{K,t+1}-R_{f,t+1})) $$ is an equivalent expression for the budget constraint. Under (Epstein and Zin 1989) and (Weil 1990) preferences with unit EIS. the agent chooses consumption $C_t$ and the capital portfolio share $\alpha_t$ to solve $$ \max_{C_t,\alpha_{t}}\left(C_t^{1-\beta}\left(\mathbb{E}_t\left[V(W_{t+1})^{1-\gamma}\right]\right)^{\frac{\beta}{1-\gamma}}\right), $$ subject to ). Conjecturing that $V(W_t)$ equals a constant multiplied by $W_t$, and applying the first-order condition for optimal consumption implies the standard unit EIS result $C_t/W_t=1-\beta$. In equilibrium, the bond is in zero net supply ($\alpha_t = 1$), and ) reduces to $$ C_t + X_t = Y_t = AK_t. $$ Furthermore, the conditions $\alpha = 1$ and $C_t = (1-\beta)W_t$ imply that consumption is a fixed percentage of planned capital $$ C_t = \frac{1-\beta}{\beta}\tilde{K}_{t+1} = \frac{1-\beta}{\beta}(X_t + (1-\delta) K_t), $$ where the second equality follows from the capital accumulation equation ). In Internet Appendix , we derive asset returns in this economy and show that there is an isomorphism between this economy and the endowment economy in Section 2. Indeed, equilibrium prices in the two models are identical if the parameters are such that the equilibrium consumption growth processes are the same.[^29] What is new in this production model is that investment and economic growth are endogenous. Substituting in for $C_t$ in ) gives us the equilibrium investment-capital ratio $$ \frac{X_t}{K_t} = \beta(1-\delta+A) - (1-\delta). $$ Our previous discussion implies that the risk-free rate has declined slightly, thus implying a slight increase in $\beta$. In order to match the declining investment-capital ratio (Figure 2, panel C), one needs either productivity $A$ to fall or depreciation $\delta$ to rise. ## Inventory case Suppose that, in addition to capital and a risk-free bond, the agent can put funds into inventory, namely, a risk-free storage technology with a zero net return. If we impose the condition that risk-free storage be in zero supply, then the economy reduces to that in the previous section. The innovation in this section is that the inventory asset can be in net positive supply. Why would there be a positive-supply risk-free asset? As mentioned in the Introduction, any store of value from one period to another could count as inventory, provided that it is, in fact, risk-free and can be frictionlessly interchanged between consumption and investment. Physical goods do not exactly fit this description because they cannot easily be changed into something other than what they are. Money fits this description if inflation is equal to zero. For the period 2000--2021, as discussed above, there is evidence that investors viewed money as indeed riskless in this sense. That said, our model is agnostic about whether the inventory is money or not; we simply assume that an inventory asset exists. Consider the agent's problem in Section 3.1, except here the agent can invest in a storage technology with quantity $I_t$. The agent maximizes unit-EIS recursive utility by choosing consumption and $B_t$, $I_t$, and $\tilde{K}_{t+1}$. That is, the agent solves $$ \max_{C_t, B_t, I_t, \tilde{K}_{t+1}} \left(C_t^{1-\beta}\left(\mathbb{E}_t\left[V(W_{t+1})^{1-\gamma}\right]\right)^{\frac{\beta}{1-\gamma}}\right), $$ subject to $$\begin{eqnarray} W_t & = & C_t + B_t + I_t + \tilde{K}_{t+1} \\ W_{t+1} & = & B_t R_{f,t+1} + I_t + \tilde{K}_{t+1} R_{K,t+1} \\ I_t & \geq & 0. \end{eqnarray}$$ The solution is still characterized by $V(W_t)\propto W_t$, implying the result $C_t/W_t = 1-\beta$. We now characterize the equilibrium. Let $R_f^*$ denote the equilibrium risk-free rate in the no-inventory economy Then: 1. If $R_f^*>1$, then in equilibrium $I_t = 0$, and the equilibrium is the same as in Section 3.1. 2. If $R_f^*<1$, then $I_t>0$. Investment in inventory crowds out investment in productive capital. The argument is as follows (Internet Appendix  gives an alternative, more formal proof). First, consider the case of $R_f^*>1$, and conjecture that $R_{f,t+1} = R_f^*$ constitutes an equilibrium in --). This investor never chooses $I_t>0$ because bonds offer superior returns; on the other hand, ) implies that the agent cannot short-sell inventory. Therefore $I_t = 0$, namely, the inventory asset is irrelevant, and thus $\alpha = 1$ is still the market-clearing condition. Equilibrium quantities and returns are the same as in Section 3.1. Now assume that $R_f^*<1$. The only possible equilibrium value for $R_f$ is unity. This is because $R_f<1$ implies an arbitrage opportunity: the investor would borrow at $R_f$ and invest the proceeds in the inventory asset. If instead $R_f>1$, the reasoning in the above paragraph implies the agent holds no inventory. That means $R_f^* = R_f>1$, contradicting the assumption. Intuitively, we can find an equilibrium with inventory for the following reason: if the agent does not hold inventory ($\alpha=1$) and the risk-free rate equals $R^*_{f,t+1}<1$, then the agent will wish to hold more inventory, as it is a marginally better asset. Doing so, however, reduces the volatility of the return on the wealth portfolio and stochastic discount factor and thus increases the equilibrium risk-free rate. The agent will increase holdings of inventory until the equilibrium rate is equal to the return on inventory. Assume parameters are such that $R_f^*<1$; as the above argument shows, this is where inventory matters. We show it is also empirically relevant in that it prevails in the second sample period. Bonds are redundant, so we can assume $B_t = 0$. The requirement $R_f = 1$ replaces $\alpha = 1$ as the market-clearing condition. Given that the equilibrium takes this form, the optimization problem reduces again to ), but this time subject to $$ W_{t+1} = (W_t - C_t) ( 1 + \alpha_tr_{K,t+1}), $$ where $r_{K,t+1} = R_{K,t+1} - 1$ is the net return on capital and $\alpha$ is the share of capital in savings, as it was in Section 3.1. Note that technology and depreciation pins down $r_{K,t+1}$, see ). Internet Appendix  shows that $C_t/W_t = 1-\beta$ and the stochastic discount factor $M_{t+1} = \mathbb{E}_t[R_{W,t+1}^{1-\gamma}]^{-1} R_{W,t+1}^{-\gamma}$. The first-order condition with respect to $\alpha$ shows that $$ E_t\left[\frac{1}{\left(1+\alpha r_{K,t+1}\right)^{\gamma}}r_{K,t+1}\right] = 0, $$ which is special case of $E_t[M_{t+1}R_{W,t+1} - R_f] = 0$, with $R_f = 1$, and $R_{W,t+1} = 1+\alpha_tr_{K,t+1}$, from ). Let $r_{K,0} \equiv (1-\delta+A)-1$ and $r_{K,\eta} \equiv (1-\delta+A)(1-\eta)-1$ denote the net returns on capital in the nondisaster and disaster states, respectively. Then ) implies $$ \frac{pr_{K,\eta}}{\left(1+\alpha r_{K,\eta}\right)^{\gamma}}+\frac{(1-p)r_{K,0}}{\left(1+\alpha r_{K,0}\right)^{\gamma}}=0, $$ Solving for $\alpha$ implies $$ \alpha = \min\left\{1,-\frac{((1-p)r_{K,0})^{1/\gamma}-(-pr_{K,\eta})^{1/\gamma}}{((1-p)r_{K,0})^{1/\gamma}r_{K,\eta}-(-pr_{K,\eta})^{1/\gamma}r_{K,0}}\right\}, $$ The investor holds inventory when expected risk-adjusted capital returns are sufficiently low. It follows from $C_t/W_t = 1-\beta$ and ) that $$ \frac{C_{t+1}}{C_{t}} = \beta R_{W,t+1}= \beta (1+\alpha r_{K,t+1}) = \beta\left(\alpha (1-\delta+A)(1-\chi_{t+1})+ 1-\alpha\right). $$ Relative to the model in Section 3.1, consumption growth is less volatile because, in aggregate, agents use inventory to smooth aggregate fluctuations. It is also, on average, lower, because less is invested in the productive asset. Output growth, however, is more volatile. Consumption growth is no longer tethered to output as in Section 3.1. Still, the relation between growth in the capital stock and growth in wealth remains the same: $$ \frac{K_{t+1}}{K_t} = \frac{\tilde{K}_{t+1}}{\tilde{K}_t}\frac{1-\chi_{t+1}}{1-\chi_t} = \frac{W_t}{W_{t-1}}\frac{1-\chi_{t+1}}{1-\chi_t}. $$ Substituting in from ) then implies $$ \frac{Y_{t+1}}{Y_t} = \frac{K_{t+1}}{K_t} = \beta \left(\alpha (1-\delta+A)(1-\chi_{t+1}) + (1-\alpha)\left(\frac{1-\chi_{t+1}}{1-\chi_t}\right)\right). $$ Output growth is more volatile than consumption growth because it bears the full brunt of disasters: note that $1-\chi_{t+1}$ multiplies both the term with $\alpha$ (representing investment in the risky technology) and $1-\alpha$. By definition, the disaster applies to the entire existing capital stock. While this effect makes output growth more volatile than consumption in the present model, it does not, by itself, raise the volatility relative to the model in Section 3.1. There is, however, a second effect, represented by $1-\chi_t$ in the denominator. Coming out of a severe recession featuring capital destruction $\chi_t>0$, output growth is higher because agents invest more to get back to the optimal allocation. This raises the volatility of output growth relative to the model in Section 3.1. What are the properties of investment? Rewriting the capital accumulation equation ) so that $X_t$ is on the left-hand side, and dividing by $K_t$, implies $$\begin{eqnarray} \frac{X_t}{K_t} & = & \frac{\tilde{K}_{t+1}}{\tilde{K}_t}\frac{\tilde{K}_t}{K_t} - (1-\delta) \\ & = & \beta R_{W,t} (1-\chi_t)^{-1} - (1-\delta) \\ & = & \beta\big(\alpha (1-\delta + A) + (1-\alpha) (1-\chi_t)^{-1}\big) - (1-\delta), \end{eqnarray}$$ where we have used the fact that $\tilde{K}_{t+1}/\tilde{K}_t = \beta R_{W,t}$. After capital disasters, the agent invests at a higher rate to replenish the capital stock. Consequently, the investment-capital ratio is time-varying in this economy, despite i.i.d. shocks and a balanced growth path.
Alt text: Line graph showing comparative statics of the investment-capital ratio as a function of disaster size for scenarios with and without inventory.
A disaster in the prior period increases investment in productive capital. This is because the disaster affects capital disproportionately, and the agent must reinvest to return capital back to its precrisis level. For an illustration, see Figure 10, which shows the investment-capital ratio for $\chi_t = 0$ (no disaster) and $\chi_t = \eta$ (disaster) for various values of the disaster size.[^30] The figure also shows the optimal planned capital to wealth ratio $\alpha$. For comparison, the figure also shows quantities in the case of no inventory. Fixing other parameters, for disaster sizes of less than 25%, the gross risk-free rate is above one, implying that the economies with and without inventory are the same. As the size of the disaster increases, the equilibrium risk-free rate in the no-inventory economy falls sharply (we illustrate this in Figure 12). It becomes optimal to hold inventory and investment in productive assets falls. At that point, investment depends on the occurrence of a disaster in the prior period. The greater the size of the disaster, the greater the increase in investment. In contrast, with no inventory, the investment-capital ratio is always the same. We define the stock market as the claim to output $Y_t$ in all future periods. As ) shows, the growth rate of capital is no longer i.i.d. but depends on $\chi_t$ (note that $\chi_{t+1}$ is i.i.d. given time-$t$ information). Therefore, the price-dividend ratio on the output claim is a function of $\chi_t$ and solves $$\kappa^Y(\chi_t) = \mathbb{E}_t \left[M_{t+1}\left(1+ \kappa^Y(\chi_{t+1}) \right)\frac{Y_{t+1}}{Y_t} \right],$$ Under our distributional assumptions, $$\begin{align} \kappa^Y(0) & = \frac{\beta}{1-\beta}\left(\nu + (1-\nu)\left(\frac{1+\alpha r_{K,0}}{1+\alpha r_{K,\eta}}\right)(1-\eta)\right), \\ \kappa^Y(\eta) & = \frac{\beta}{1-\beta}\left((1-\nu) + \nu\left(\frac{1+\alpha r_{K,\eta}}{1+\alpha r_{K,0}}\right)(1-\eta)^{-1}\right), \end{align}$$ where $\nu \equiv ((1-p)(1+\alpha r_{K,0})^{1-\gamma})/\left((1-p)(1+\alpha r_{K,0})^{1-\gamma} + p(1+\alpha r_{K,\eta})^{1-\gamma}\right)$. See Internet Appendix  for details. In the case where $\alpha=1$, the price-dividend ratio is the constant $\kappa^Y=\beta/(1-\beta)$. Figure 11 shows the price-dividend ratio for various levels of the disaster size, both in the economy with inventory and in the economy without. The economy without inventory has a constant price-dividend ratio solely determined by $\beta$. When there is inventory, the price-dividend ratio rises in disasters because dividends are temporarily depressed (they are also low because of the disaster). This increase is due to the endogenous investment response, whereby inventory is liquidated after a disaster to rebuild the capital that is destroyed.
Alt text: Line graph showing comparative statics of the price-dividend ratio as a function of disaster size for scenarios with and without inventory.
In contrast with standard production models, the price-dividend ratio in the no-disaster case declines (in a comparative statics sense) as a function of the disaster size (see Figure 11). In the case without inventory, the price-dividend ratio is independent of disaster risk. Models with production that seek to match business-cycle fluctuations in investment and valuation ratios require the EIS to be greater than 1. Endowment models achieve the same effect by imposing exogenous leverage (dividends more sensitive to shocks than consumption). In this model, leverage is endogenous, and qualitatively correct price-dividend ratio dynamics could in principle occur, even with an EIS of one. The magnitude of the decline in Figure 11 suggests that the effect is small under our calibration. Figure 12 shows that the equity risk premium in this economy loses its usual dependence on disaster risk. The equity premium equals $rp\equiv\log \mathbb{E}_t[R_{Y,t+1}] - \log R_f$, where the return on the output claim is $$R_{Y,t+1} =\bigg( \frac{\kappa^Y(\chi_{t+1})+1}{\kappa^Y(\chi_{t})}\bigg)\bigg(\frac{Y_{t+1}}{Y_{t}}\bigg).$$ The blue line in the figure shows the equity premium in the model without inventory: it is highly dependent on the disaster size, as is the risk-free rate. However, the return on capital---which, in the economy with no inventory, is the equity return---is very slightly decreasing. This is a standard result in disaster-risk economies: the full discount rate on the equity claim decreases slightly with the probability of a disaster. While this might seem counterintuitive, it arises from the fact that, while the equity premium increases, the risk-free rate declines and more than offsets the effect. Also recall that the continuously compounded return in a standard i.i.d. economy can be expressed as the log dividend yield plus the log growth in cash flows. When the EIS equals one, the dividend yield does not depend on disaster risk, and so the only effect is the small effect of expected cash flows. In the economy with inventory, the return on capital is the same as in the economy without (this is defined by the production opportunities), and thus is slightly decreasing. The risk-free rate is constant, implying that the premium on capital is also slightly decreasing. The equity premium decreases slightly more in the disaster size as compared to $\log \mathbb{E}[R_K] - \log R_f$. This is because the increase in the price-dividend ratio counteracts the decline in output due to the disaster.
Alt text: Line graph showing comparative statics of risk premia and the risk-free rate as a function of disaster size for scenarios with and without inventory.
We calibrate the model to match the ex ante inflation-adjusted Treasury yield, price-dividend ratio, and GDP growth in the U.S., as in the sections above. Calibrating to match these data requires solving a system of three equations in three unknowns, where the unknowns are the parameters $\beta$, $\lambda$, and $A$ and the three equations are the price-dividend ratio ), the ex ante inflation-adjusted bond yield ), and $$ \frac{Y_{t+1}(0)}{Y_t(0)} =\beta \left(\alpha (1-\delta+A) + (1-\alpha)\right) $$ which is GDP growth when the disaster does not occur. Indeed, each of the moments to which we calibrate parameters is the value in the no-disaster state ($\chi_t=0$), consistent with the fact that we do not observe any disasters in our sample. We then solve for the values of the parameters that equate the data moments with their corresponding model moments. Table 4 displays the results from calibrating the production model. The model explains the data moments with a quantitatively reasonable calibration of $\beta$, $\lambda$, and $A$. The slight increase in $\beta$ matches the modest rise in the price-dividend ratio; lower capital productivity $A$ matches the lower growth in the second sample.[^31] Inflationary default risk $\lambda\eta$ falls, in line with the estimates in Section 2.[^32] A striking result of Table 4 is the model's ability to match real investment. It turns out that matching the growth rate and asset prices is sufficient, in this model, to capture both the levels and the decline in the investment-capital ratio. Specifically, while the average in the data is 7.7% from 1984 to 2000, it is 7.4% in the model. This falls to an average of 6.9% in the data from 2001--2021. The model matches this decline, producing an average of 6.3%. As in Section 2.7, a $\beta$ sufficiently high to explain interest rates on its own implies an investment boom (see equation , or the without-inventory version, equation ). In fact the opposite occurred, offering further evidence that a force other than a savings glut must have led to the decline in interest rates. Table 4 also calibrates a production economy without inventory, allowing us to discern the impact of allowing this additional investment opportunity. In the model with inventory, reduced GDP growth occurs in part because of investment in unproductive capital. To match the growth rates, capital productivity must fall by less than in the model without. Moreover, given the same discount factor $\beta$, the unconstrained risk-free rate also falls by less because the economy is less risky. This implies a greater change in the fraction lost in default that is needed to explain the observed decline in interest rates. Overall, while inventory acts as both an insurance policy and as a drag on growth, its effects, which stem from the risk-free rate hitting the zero lower bound, are relatively small. +:------------------------------------------+:--------------------:+:----------------:+:----------------:+ | | | Values | +-------------------------------------------+----------------------+------------------+------------------+ | | Parameter | 1984--2000 | 2001--2021 | +-------------------------------------------+----------------------+------------------+------------------+ | *A. Moments in the data* | | | | +-------------------------------------------+----------------------+------------------+------------------+ | Price-dividend ratio | $\kappa$ | 42.35 | 50.86 | +-------------------------------------------+----------------------+------------------+------------------+ | Ex ante inflation-adjusted Treasury yield | $\bar{y}_b$ | 0.0287 | -0.0053 | +-------------------------------------------+----------------------+------------------+------------------+ | US GDP growth | $\frac{\Delta Y}{Y}$ | 0.0241 | 0.0133 | +-------------------------------------------+----------------------+------------------+------------------+ | Investment-capital ratio | $\frac{X}{K}$ | 0.0773 | 0.0689 | +-------------------------------------------+----------------------+------------------+------------------+ | *B. Calibration, $\gamma = 5$, $\textrm{EIS}= 1$, $\eta = 0.30$* | +-------------------------------------------+----------------------+------------------+------------------+ | Discount factor | $\beta$ | 0.977 | 0.981 | +-------------------------------------------+----------------------+------------------+------------------+ | Fraction of bond value lost | $\lambda\eta$ | 0.178 | -0.059 | +-------------------------------------------+----------------------+------------------+------------------+ | Capital productivity | $A$ | 0.0986 | 0.0839 | +-------------------------------------------+----------------------+------------------+------------------+ | Risky capital share | $\alpha$ | 1.000 | 0.981 | +-------------------------------------------+----------------------+------------------+------------------+ | Investment-capital ratio | $\frac{X}{K}$ | 0.0744 | 0.0634 | +-------------------------------------------+----------------------+------------------+------------------+ | Unconstrained riskfree rate | $r_f^*$ | 0.0130 | -0.0012 | +-------------------------------------------+----------------------+------------------+------------------+ | *C. Without inventory* | +-------------------------------------------+----------------------+------------------+------------------+ | Discount factor | $\beta$ | 0.977 | 0.981 | +-------------------------------------------+----------------------+------------------+------------------+ | Fraction of bond value lost | $\lambda\eta$ | 0.178 | -0.037 | +-------------------------------------------+----------------------+------------------+------------------+ | Capital productivity | $A$ | 0.0986 | 0.0833 | +-------------------------------------------+----------------------+------------------+------------------+ | Risky capital share | $\alpha$ | 1.000 | 1.000 | +-------------------------------------------+----------------------+------------------+------------------+ | Investment-capital ratio | $\frac{X}{K}$ | 0.0744 | 0.0634 | +-------------------------------------------+----------------------+------------------+------------------+ | Unconstrained riskfree rate | $r_f^*$ | 0.0130 | -0.0018 | +-------------------------------------------+----------------------+------------------+------------------+ : This table shows the parameters necessary to match the data, assuming a production economy with rare disasters and inflationary default. We take average GDP growth and average consumption growth from the data in each sample. We calibrate the discount factor $\beta$ and the decline in bond value $\lambda\eta$ to match the average price-dividend ratio and Treasury bill yield minus forecasted inflation (ex ante inflation-adjusted Treasury yield). The model is solved with risk aversion $\gamma = 5$ and EIS $= 1$. Consumption declines $30\%$ in a disaster ($\eta = 0.30$), the probability of disaster $p=2.10\%$, and depreciation $\delta=0.05$. Panel A assumes the existence of inventory, a risk-free production opportunity, whereas Panel B is a standard production model. \
Alt text: Line graph showing MZM and M2 money supply relative to the capital stock from 1984 to 2024.
If inventory is indeed money, then this should show up in the data as an increase in the money supply relative to the capital stock. To assess this prediction of the inventory model, we plot this ratio in Figure 13. We report two definitions of the money supply, both of which have risen relative to the capital stock in the twenty-first century. The first is the common M2 measure. The second, which we argue is a better measure of inventory, is the more inclusive "zero-maturity money" measure (MZM) from the St. Louis Fed, which takes M2, removes small illiquid time deposits, and adds institutional money market mutual funds (whereas M2 only includes retail money market funds). The rise in MZM relative to capital over the past two decades has been sizable, in line with our model's prediction of an increasing share of wealth in money-like inventory assets. # Conclusion Declining interest rates are not merely a feature of the 1980--2021 period, they are a feature of the last several centuries. They also pose a joint puzzle: why have persistently low interest rates not been accompanied by higher valuation ratios and investment rates? This article offers an explanation rooted in the decline of perceived sovereign default risk. We show that when investors require a smaller premium for default, yields can fall substantially without an equivalent decline in the true real rate. This mechanism accounts for declining interest rates, and stable valuation ratios. It is supported by reduced inflation risk premia, and a reversal in the correlations among inflation, output growth, and asset returns. Our analysis, however, abstracts from the deeper sources of changing default risk. The model is silent on why investors' confidence in sovereign repayment---and, equivalently, their expectations of inflationary default---have improved. These expectations appear to have evolved along both short- and long-term dimensions: institutionally, through credible monetary and fiscal policy over recent decades, and historically, through the slow consolidation of sovereign legitimacy. Understanding these forces lies beyond the scope of this paper but is essential to a full account of the historic decline in yields. Our results suggest several areas for future work. First, our model is based on investors' lower expectations of default, as well as lower default during disasters. Qualitatively, these can account for higher debt-to-GDP ratios. However, the increase in debt-to-GDP ratios across advanced economies has been so pronounced that it raises questions about whether default risk truly has declined relative to the 1980s or 1990s. Second, we match intraperiod dynamics using expectations data. These appear to be insufficient: a model incorporating time-varying disaster risk could explain additional facts, including the changing bond--stock correlations and the sharp decline in rates during 2008. Further work could also investigate why inflation-protected and nominal yields sometimes move together---most notably after 2021---when an inflation-based channel of default risk predicts divergence between them. These questions point toward a richer understanding of how perceptions of sovereign and macroeconomic risk jointly shape asset prices over time. The replication code and data are available in the Harvard Dataverse at Anderson, Nicola, and John Sleath. 1999. *New Estimates of the UK Real and Nominal Yield Curves*. Bank of England. Ang, Andrew, Geert Bekaert, and Min Wei. 2008. "The Term Structure of Real Rates and Expected Inflation." *Journal of Finance* 63 (2): 797--849. . Antolin-diaz, Juan. 2025. "How Did Government Bonds Become Safe?" Unpublished manuscript. Auclert, Adrien, Hannes Malmberg, Frédéric Martenet, and Matthew Rognlie. 2021. "Demographics, Wealth, and Global Imbalances in the Twenty-First Century." Unpublished manuscript. Auclert, Adrien, and Matthew Rognlie. 2017. "Aggregate Demand and the Top 1 Percent." *American Economic Review: Papers & Proceedings* 107 (5): 588--92. Avdis, Efstathios, and Jessica A. Wachter. 2017. "Maximum Likelihood Estimation of the Equity Premium." *Journal of Financial Economics* 125 (3): 589--609. . Barro, Robert J. 2009. "Rare Disasters, Asset Prices, and Welfare Costs." *American Economic Review* 99 (1): 243--64. Barro, Robert J., and Gordon Y. Liao. 2021. "Rare Disaster Probability and Options Pricing." *Journal of Financial Economics* 139 (3): 750--69. Bekaert, Geert, and Xiaozheng Sandra Wang. 2010. "Inflation and the Inflation Risk Premium." *Economic Policy* 25 (64): 757--806. . Bernanke, Ben. 2005. "The Global Savings Glut and the u.s. Current Account Deficit." Unpublished manuscript. Bianchi, Francesco, Martin Lettau, and Sydney C. Ludvigson. 2022. "Monetary Policy and Asset Valuation." *Journal of Finance* 77 (2): 967--1017. Bomfim, Antulio N. 2022. "Credit Default Swaps." Unpublished manuscript. Bordalo, Pedro, Nicola Gennaioli, Rafael La Porta, and Andrei Shleifer. 2024. "Belief Overreaction and Stock Market Puzzles." Doi: 10.1086/727713. *Journal of Political Economy* 132 (5): 1450--84. . Caballero, R. J., E. Farhi, and P. O. Gourinchas. 2008. *An Equilibrium Model of Global Imbalances and Low Interest Rates*. Campbell, John Y., Carolin Pflueger, and Luis M. Viceira. 2020. "Macroeconomic Drivers of Bond and Equity Risks." Doi: 10.1086/707766. *Journal of Political Economy* 128 (8): 3148--85. . Campbell, John Y., Adi Sunderam, and Luis M. Viceira. 2017. "Inflation Bets or Deflation Hedges? The Changing Risks of Nominal Bonds." *Critical Finance Review* 6: 263--301. Chernov, Mikhail, Lukas Schmid, and Andres Schneider. 2020. "A Macrofinance View of U.S. Sovereign CDS Premiums." *Journal of Finance* 75 (5): 2809--44. Cieslak, A., W. Li, and C. E. Pflueger. 2024. *Inflation and Treasury Convenience*. Cieslak, Anna, and Annette Vissing-Jorgensen. 2021. "[The Economics of the Fed Put]." *Review of Financial Studies* 34 (9): 4045--89. Cochrane, John H. 2023. *The Fiscal Theory of the Price Level*. Princeton University Press. Czitron, Tom. 2022. *Canada Plans to Bring an End to an Inflation Hedge Tool. Here Are Some Alternatives Investors Can Turn to.* *The Globe and Mail*, November 22. De la O, Ricardo, and Sean Myers. 2021. "Subjective Cash Flow and Discount Rate Expectations." *The Journal of Finance* 76 (3): 1339--87. Dittmar, Robert, Alex Hsu, Guillaume Roussellet, and Peter Simasek. 2024. "Default Risk and the Pricing of u.s. Sovereign Bonds." *Journal of Finance* Forthcoming. Eggertsson, Gauti B., Neil R. Mehrotra, and Jacob A. Robbins. 2019. "A Model of Secular Stagnation: Theory and Quantitative Evaluation." *American Economic Journal: Macroeconomics* 11 (1): 1--48. Epstein, Larry, and Stan Zin. 1989. "Substitution, Risk Aversion and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework." *Econometrica* 57: 937--69. Fama, Eugene F., and Kenneth R. French. 2002. "The Equity Premium." *Journal of Finance* 57 (2): 637--59. . Farhi, Emmanuel, and François Gourio. 2018. "[Accounting for Macro-Finance Trends: Market Power, Intangibles, and Risk Premia]." *Brookings Papers on Economic Activity*, 147--250. Farmer, Leland, Emi Nakamura, and Jón Steinsson. 2024. "Learning about the Long Run." *Journal of Political Economy* 132 (10): 3334--77. Favero, Carlo A, Alessandro Melone, and Andrea Tamoni. 2024. "Monetary Policy and Bond Prices with Drifting Equilibrium Rates." *Journal of Financial and Quantitative Analysis* 59 (2): 626--51. Ferguson, Niall. 2018. *The Ascent of Money*. Second. The Penguin Press. Fleckenstein, Francis A., Matthias And Longstaff. 2024. "Treasury Richness." *The Journal of Finance* 79 (4): 2797--844. https://doi.org/. Fleming, Michael J., and Neel Krishnan. 2004. *[The Microstructure of the TIPS Market]*. Gabaix, Xavier. 2011. "Disasterization: A Simple Way to Fix the Asset Pricing Properties of Macroeconomic Models." *American Economic Review: Papers & Proceedings* 101 (3): 406--9. . Goetzmann, William N., Akiko Watanabe, and Masahiro Watanabe. 2022. "Evidence on Retrieved Context: How History Matters." In *SSRN Electronic Journal*. . Gomes, João F, Marco Grotteria, and Jessica A Wachter. 2019. "Cyclical Dispersion in Expected Defaults." *Review of Financial Studies* 32 (4): 1275--308. . Gordon, Robert J. 2015. "Secular Stagnation: A Supply-Side View." *American Economic Review: Papers & Proceedings* 105 (5): 54--59. Gourio, François. 2012. "Disaster Risk and Business Cycles." *American Economic Review* 102 (6): 2734--66. Greenwald, D. L., M. Lettau, and S. C. Ludvigson. 2024. *How the Wealth Was Won: Factor Shares as Market Fundamentals*. Greenwald, Daniel L., Matteo Leombroni, Hanno Lustig, and Stijn Van Nieuwerburgh. 2022. "Financial and Total Wealth Inequality with Declining Interest Rates." Unpublished manuscript. Hansen, Alvin H. 1939. "[Economic Progress and Declining Population Growth]." *American Economic Review* 29 (1): 1--15. Haubrich, Joseph, George Pennacchi, and Peter Ritchken. 2012. "Inflation Expectations, Real Rates, and Risk Premia: Evidence from Inflation Swaps." *The Review of Financial Studies* 25 (5): 1588--629. Hillenbrand, Sebastian. 2024. "The Fed and the Secular Decline in Interest Rates." *Review of Financial Studies* Forthcoming. Jiang, Zhengyang, Hanno Lustig, Stijn Van Nieuwerburgh, and Mindy Z. Xiaolan. 2024. "The U.S. Public Debt Valuation Puzzle." *Econometrica* 92 (4): 1309--47. Jordà, Òscar, Katharina Knoll, Dmitry Kuvshinov, Moritz Schularick, and Alan M Taylor. 2019. "[The Rate of Return on Everything, 1870--2015]." *Quarterly Journal of Economics* 134 (3): 1225--98. Kocherlakota, Narayana R. 2023. "Infinite Debt Rollover in Stochastic Economies." *Econometrica* 91 (5): 1629--58. Krishnamurthy, Arvind, and Annette Vissing-Jorgensen. 2012. "The Aggregate Demand for Treasury Debt." *Journal of Political Economy* 120 (2). Lettau, Martin, Sydney C. Ludvigson, and Jessica A. Wachter. 2008. "The Declining Equity Premium: What Role Does Macroeconomic Risk Play?" *Review of Financial Studies* 21: 1653--87. Longstaff, Francis A., and Monika Piazzesi. 2004. "Corporate Earnings and the Equity Premium." *Journal of Financial Economics* 74 (1). Mian, Atif, Ludwig Straub, and Amir Sufi. 2021. "Indebted Demand." *The Quarterly Journal of Economics* 136 (4): 2243--307. Reis, Ricardo. 2020. "The Anchoring of Long-Run Inflation Expectations Today." Unpublished manuscript. Schmelzing, Paul. 2020. "Eight Centuries of Global Real Interest Rates, $R-G$, and the 'Suprasecular' Decline, 1311--2018." *Bank of England Staff Working Paper No. 845*, 1--110. Shiller, Robert J. 2000. *Irrational Exuberance*. Princeton University Press. Summers, Lawrence H. 2015. "Demand Side Secular Stagnation." *American Economic Review: Papers & Proceedings* 105 (5): 60--65. [van Binsbergen, Jules H.] 2020. "Duration-Based Stock Valuation." Unpublished manuscript. [van Binsbergen, Jules H., William F. Diamond, and Marco Grotteria]. 2022. "Risk-Free Interest Rates." *Journal of Financial Economics* 143 (1): 1--29. [van Binsbergen, Jules H., and Ralph S. J. Koijen]. 2010. "[Likelihood-Based Estimation of Exactly-Solved Present-Value Models]." *Journal of Finance* 65 (4): 1439--71. Wachter, Jessica A. 2005. "Solving Models with External Habit." *Finance Research Letters* 4 (2). Wachter, Jessica A. 2013. "Can Time-Varying Risk of Rare Disasters Explain Aggregate Stock Market Volatility?" *The Journal of Finance* 68 (3): 987--1035. Weil, Philippe. 1990. "Nonexpected Utility in Macroeconomics." *Quarterly Journal of Economics* 105: 29--42. [^1]: We thank Cristina Arellano, Jules van Binsbergen, Sylvain Catherine, John Cochrane, François Gourio, Christian Julliard, Narayana Kocherlakota, Karen Lewis, Sydney Ludvigson, Sean Myers, Aleksei Oskolkov, Carolin Pflueger, Ludwig Straub, Andrea Tamoni, Stijn Van Nieuwerburgh, Jonathan Wachter, and seminar participants at the NBER Summer Institute Capital Markets Meeting, the Conference on Macroeconomics and Monetary Policy at the San Francisco Federal Reserve, the Western Finance Association Meeting, the NBER Summer Institute Asset Pricing Meeting, the Workshop on Methods and Applications for DSGE models, Columbia University, Tel Aviv University, the Wharton School, and the University of Southern California for helpful comments. Send correspondence to Jessica A. Wachter, jwachter@wharton.upenn.edu. [^2]: (Jordà et al. 2019) note that prior observations of a real rate of zero are not unusual. However, these are observations after subtracting ex post realized inflation, not ex ante inflation-adjusted yields. Although both returns are zero from an investor's perspective, one was a realization of zero because of high inflation, whereas the other is an expected value of zero. [^3]: In our quantitative model, we will focus on the 1-year nominal yield. Notably, the literature studying monetary policy shocks and real interest rates finds that the secular decline in short-term rates has also shown up in long-term yields (Bianchi et al. 2022; Hillenbrand 2024). For example, the 30-year yield fell from 11.5% to 1.9% in 2021 before rising to 4.8% in 2024, similar to the federal funds rate. [^4]: We will consider within-sample variation in Section 2.7. [^5]: Throughout this paper, all figures, with the exception of Figure 1; Figure 4, panel B (which uses data 5 years ahead); and Figure 13 (where one series was discontinued), report data up until the end of 2024. In this section, Section 2.3, and Section 3, the model is calibrated to two sample periods 1984--2000 and 2001--2021. Section 2.7 contains a detailed analysis of the 2022--2024 period. [^6]: ([van Binsbergen] 2020) states the puzzle as follows: given the decrease in interest rates and the duration of the stock market, one would have expected a much larger capital gain if the risk premium were to remain constant. [^7]: If interest rates contain a premium for default risk, as we argue in subsequent sections, then the equity premium as measured by the ex post sample average of returns on equities minus bond yields will be biased downwards by this premium. If equity returns rise relative to bond returns, this could seem like an increase in the equity risk premium, but in fact only be a decline in sovereign default risk. [^8]: See Internet Appendix  for more detailed derivations. [^9]: We relax this assumption below. [^10]: Specifically, the Jensen's term is $\bar{y}_b-\log\mathbb{E}_t[R_{b,t+1}]= -\log\big(\mathbb{E}_t[\Pi_{t+1}/\Pi_t]^{-1}\mathbb{E}_t[\Pi_t/\Pi_{t+1}]\big)$, which is strictly negative for $\lambda\eta\neq0$ or $\sigma_\pi\neq0$. [^11]: Under the inflation process ) and imposing the fact that there were no disasters in the data sample ($L_t=0$), we can recover the variance of nondisaster inflation shocks from the difference between inflation realizations and forecasts: $\text{var}(\log(\Pi_{t+1}/\Pi_t) - \log\mathbb{E}_t[\Pi_{t+1}/\Pi_t]|L_{t+1}=0) = \sigma_\pi^2$. [^12]: Not surprisingly, this is a larger decline than that estimated using models assuming stationarity (see (Favero et al. 2024) for a discussion of the role of the stationarity assumption in interest rate modeling). For instance, (Haubrich et al. 2012) assume stationarity and Gaussian shocks, implying that the inflation risk premium depends directly on the variance of inflation. They find an average premium of 0.4%. (Greenwald et al. 2022; Greenwald et al. 2024) find similar results. (Ang et al. 2008)---more similar to our results---that the 5-year inflation risk premium has declined over time from a peak of approximately 2% in the early 1980s to around 0.15% after the 2001 recession. Similarly, (Bekaert and Wang 2010) find recent estimates of the inflation to be close to zero. Our model differs from these in that we allow for both a structural break and rare inflation events, implying not only that true volatility is difficult to capture in-sample, but also that the premium does not depend solely on this volatility. That said, our estimation implies that, if averaged across the samples, the premium is 0.5%, almost exactly what is implied by these prior estimates. [^13]: We discuss the TIPS series further below and in Section 2.7. [^14]: Some of the premium for default may reflect the probability of a temporary halt in payment (technical default), whereas the premium ) assumes missed payments are not made at a later date (Bomfim 2022). [^15]: This is also why direct observations of yields over the long term in Figure 1 is relevant. [^16]: In a sample in which disasters do not occur, the difference between forecasted and the measured ex post average inflation will be $\log\mathbb{E}_t[\Pi_{t+1}/\Pi_t]-\log\mathbb{E}_t[\Pi_{t+1}/\Pi_t|L_{t+1}=0] = - \log(1-p\lambda\eta(1-\lambda\eta)^{-1}) \approx p\lambda\eta(1-\lambda\eta)^{-1}$, which, according to our theory, is positive in the first sample but close to zero in the second. [^17]: Relatedly, (Cieslak and Vissing-Jorgensen 2021) show that the so-called "Fed Put"---the tendency of the Fed to reduce rates (increasing bond prices) when the stock market falls---began only in the late 1990s. [^18]: For simplicity, these equations do not reflect the unforeseen structural breaks, for example, in $\mu$ and $\lambda$. [^19]: LTG fluctuates around 13%, far higher than average consumption growth, and indeed higher than average realized log real earnings growth---around 4% in annual data from (Shiller 2000). [^20]: This number is 53% in the first sample and 52% in the second. Given that these numbers are so close, setting them as a single value does not noticeably change our results. [^21]: Note that the correlations in the table are between shocks to consumption and inflation (i.e., after filtering out expectations), whereas the correlations in Figure 5 were between the raw time series. [^22]: A $\beta$ of 0.997 is consistent with an interest rate of $-0.003$, whereas we estimate the Treasury yield less expected inflation over 2001--2021 to be $-0.005$. Given our other parameters, that value is inconsistent with a finite price-dividend ratio. [^23]: A reversal of the correlation of consumption and inflation outside of disasters does not play a sufficiently large role economically, given realistically low risk aversion, to be a consideration. [^24]: A caveat is that, to the extent $\beta$ reflects the mix of investors accessing U.S. markets, it is possible that this mix could fluctuate if, say, sovereign wealth of foreign nations such as China, for political reasons, partially withdraws from U.S. markets. [^25]: The model matches the slight decline in interest rates from 2022-2024 nearly perfectly based on the decrease in inflation expectations. [^26]: As Section 2.4 explains, any amount of variability in the disaster probability within each sample would cause stock and bond returns to be positively correlated if $\lambda>0$ and negatively correlated with $\lambda<0$. The change in signs of the bond-stock correlation matches the change in our $\lambda$ parameter. Data suggest that the sign of this correlation, which switched from positive to negative between the first and second period, has subsequently become positive in the 2022--2024 period. [^27]: The 2008 episode can be thought of as a temporary increase in the risk of a disaster, which would have led to declining equity valuations, plummeting risk-free rates, and a higher yield on inflation-protected securities given our estimate of $\lambda$. [^28]: See (Barro 2009), (Gabaix 2011), and (Gourio 2012). [^29]: This occurs when $\beta^{-1}e^{\mu} = (1-\delta+A)$. One can verify this by comparing ) and ). [^30]: A higher disaster size has the same effect in the model as a higher disaster probability. What matters for these mechanism is total disaster risk. [^31]: We could, equivalently, keep $A$ constant and estimate an increase in $\delta$; they are isomorphic for explaining the growth decline. [^32]: The estimates of $\lambda$ are slightly different than in Table 2 because we calibrate to average GDP growth instead of consumption growth.