Natural-Rate-of-Interest | Macro Paper Warehouse

Dynamics of the Long-Term Housing Yield: Evidence from Natural Experiments

Mon, 01 Jan 0001 00:00:00 +0000

Each month a fraction of UK property leases are extended by 90 years or more, creating thousands of natural experiments in which the same property’s rent and capital value are revealed simultaneously. This paper uses these lease extensions — and Massachusetts and Cambridge rent-control removals as a second identification strategy — to estimate the expected long-term housing yield (annual rent-to-price ratio) and decompose its dynamics into rent-growth expectations and discount-rate components. The central finding is that housing yield movements are dominated by discount-rate shocks: variation in required returns on housing explains the overwhelming majority of yield variance, while expected rent growth contributes less than 10 percent. Housing booms are therefore primarily driven by falling required returns, not by rational expectations of higher future rents. The yield responds to real long-term interest rates with a slope significantly below one, consistent with a non-pecuniary convenience yield on housing that is not fully displaced by interest rate changes.

Summary of a forthcoming paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.

In depth

Q1. What do the natural experiments identify?

Lease extensions reveal the market’s valuation of the same physical dwelling at two points — just before and just after the 90-year extension — with the extension itself creating a clean variation in the remaining lease term (and hence in the present value of ownership) without changing the property’s rent-generating characteristics. This design separates the rent and price components of the yield at the property level, allowing identification of discount-rate and rent-growth contributions free of compositional differences across properties.

Q2. Why do discount rates dominate yield variation?

A present-value decomposition of the housing yield into expected rent growth and the discount rate assigns more than 90 percent of variance to the discount rate component, implying that periods of low housing yields (high prices relative to rent) reflect primarily that investors demand a lower return on housing — not that they expect rents to rise faster. This result mirrors Campbell-Shiller findings for equity markets but is especially striking for housing, where naive narratives often attribute booms to expected rent appreciation.

Q3. What does the convenience yield interpretation imply?

Housing yields respond less than one-for-one to real interest rate movements — a slope well below one in the yield-rate regression — implying that housing carries a non-pecuniary convenience yield (liquidity, collateral value, direct utility of ownership) that buffers the required return on housing against interest rate changes. When real rates rise, housing yields rise by less, so price-to-rent ratios decline by less than a frictionless model would predict.

Key concepts

housing yield : the annual rent-to-price ratio on residential property; the paper’s central object, decomposed into discount-rate and rent-growth components.

discount-rate channel : the dominant source of housing yield variation in this paper; movements in investors’ required return on housing, not expected rent growth, drive the observed yield dynamics.

convenience yield : the non-pecuniary value of housing ownership (liquidity, collateral, direct utility) that drives a wedge between the housing yield and the risk-free real interest rate; explains the less-than-one slope in the yield-rate relationship.

Monetary Policy and the Drifting Natural Rate of Interest

Mon, 01 Jan 0001 00:00:00 +0000

This paper analyzes how monetary policy should respond to a long-run natural interest rate that can drift permanently — following a bounded random walk with upper bound 3 percent and lower bound 0 percent — when the zero lower bound (ZLB) on nominal interest rates is a binding constraint. The central result is that the long-run neutral rate (the real policy rate consistent with stable inflation in long-run equilibrium) should fall more than one-for-one with the long-run natural rate as the latter approaches zero, because the mere risk of future ZLB episodes — even when the economy is currently away from the ZLB — imparts a persistent downward bias on inflation expectations that can only be offset by maintaining a pre-emptive expansionary bias. Quantitatively, the model implies that the neutral rate should be zero as soon as the long-run natural rate falls to 75 basis points — well above the near-zero estimates prevailing in the late 2010s — and that the ZLB would bind one-third of the time under optimal policy when the natural rate fluctuates between 0 and 3 percent. Price level targeting with a 10-basis-point upward drift closely approximates optimal commitment policy and has the advantage of not requiring knowledge of the natural rate level.

Summary of a forthcoming paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.

In depth

Q1. What empirical fact motivates the model?

Empirical analyses of the long-run natural rate — the real interest rate prevailing over a long-run equilibrium in which nominal rigidities are absent — consistently find that it is time-varying in a manner best described by a random walk, meaning it can drift without reverting to a constant long-run level. The paper cites Holston, Laubach, and Williams (2017), Fiorentini et al. (2018), and Hamilton et al. (2016) as the main empirical references. Holston et al. (2017) place the long-run natural rate at between 0 and 1 percent in the U.S. and possibly slightly negative in the euro area as of 2016. The paper draws one central lesson: because the natural rate is time-varying and its future level is uncertain, a model with constant natural rate will give unreliable guidance for monetary policy, especially at low natural rate levels near zero.

Q2. What is the model and what are the key equilibrium concepts?

The paper embeds a new Keynesian model in which the long-run natural rate follows a bounded random walk with upper bound 3 percent and lower bound 0 percent, calibrated to post-WWII U.S. TFP data, and studies optimal monetary policy under commitment while imposing the zero lower bound. A critical distinction separates two notions of the long-run equilibrium interest rate: the “long-run natural rate” (denoted ¯r) is the real rate that would prevail in flexible-price equilibrium, determined by fundamentals outside the central bank’s control; the “neutral rate” (r*) is the real policy rate consistent with stable inflation in the long run, which the central bank operationally targets. The two coincide in standard models with constant ¯r, but diverge in this paper because ZLB risk drives a wedge between them.

Q3. What is the main theoretical result?

Under optimal commitment, the neutral rate r should fall more than one-for-one with the long-run natural rate ¯r — that is, the central bank should maintain a negative gap (r < ¯r) that widens as ¯r falls toward zero — because permanent downward movements in ¯r make future ZLB binding episodes permanently more likely, creating a persistent downward bias on inflation expectations that requires pre-emptive accommodation even in periods when the ZLB is not currently binding.** This result contrasts with the existing literature on optimal commitment at the ZLB, which has emphasized forward guidance — the promise to maintain low rates even after the economy recovers from a ZLB episode — as the primary stabilization tool. The paper shows that forward guidance alone is not sufficient when ¯r can permanently drift lower, because each downward drift permanently raises the probability of future ZLB episodes, reducing the central bank’s scope for fulfilling future inflation promises.

Q4. What are the quantitative implications?

The model implies that the neutral rate r reaches zero when the long-run natural rate ¯r is at 75 basis points — a level that was well above the near-zero estimates of ¯r prevailing at the end of the 2010s — and that the ZLB binds one-third of the time under optimal policy when ¯r fluctuates between 0 and 3 percent.* The 75 basis-point threshold means that a central bank operating in an environment where ¯r has declined to its estimated late-2010s levels would already be constrained to a neutral rate of zero under optimal policy. The one-third ZLB frequency is higher than what would be predicted by models with constant ¯r at typical calibrations, reflecting the permanent nature of ¯r shocks and their cumulative effect on the neutral rate.

Q5. What do the adjustment dynamics look like after a negative ¯r shock?

Following a permanent reduction in ¯r, the real policy rate adjusts gradually rather than immediately — remaining temporarily above the new long-run neutral rate during the transition — implying that monetary policy is contractionary along the adjustment path and that a permanent decline in ¯r is followed by a temporary disinflation before the economy settles at the new r.* This history-dependence of optimal commitment policy means the central bank does not immediately jump to the new, lower r* after a ¯r shock; it moves gradually, making the short-run policy stance more contractionary than the long-run position. The temporary disinflation is consistent with the general principle of history-dependence of optimal policy under commitment.

Q6. What role does price level targeting play?

Price level targeting variants — particularly a rule with an optimally chosen upward drift of 10 basis points — closely approximate the economic outcomes achieved under optimal commitment policy in the model, with the practical advantage that such rules do not require the central bank to know or estimate the current level of the long-run natural rate ¯r. The Eggertsson-Woodford (2003) price level target works well in models with constant ¯r by generating positive inflation expectations in the wake of deflationary ZLB episodes. Adding a small upward drift of 10 basis points strengthens this property under a drifting ¯r, because it provides additional buffer against the downward expectations bias that permanent ¯r drift generates. Under price level targeting rules, the neutral rate reaches the ZLB as soon as ¯r falls below 1 percent.

Key concepts

long-run natural rate (¯r) : the real interest rate prevailing over a long-run equilibrium in which nominal rigidities are absent; in this paper modelled as a bounded random walk with upper bound 3 percent and lower bound 0 percent, calibrated to post-WWII TFP data.

neutral rate (r)* : the real policy rate consistent with stable inflation in the long run; distinct from ¯r in this paper because ZLB risk drives a negative gap (r* < ¯r) that widens as ¯r approaches zero.

zero lower bound (ZLB) : the constraint that nominal policy rates cannot fall below zero; in this model the reason that permanent reductions in ¯r create a persistent downward bias on inflation expectations even when the ZLB is not currently binding.

expansionary bias : the paper’s finding that optimal commitment policy should maintain r* < ¯r — a pre-emptive accommodation away from the ZLB — to offset the downward bias on inflation expectations created by the risk of future ZLB episodes.

price level targeting : a monetary policy rule in which the central bank targets the price level rather than the inflation rate; shown in this paper to approximate optimal commitment policy and to have the practical advantage of not requiring knowledge of ¯r.

Monetary Policy, Employment Shortfalls, and the Natural Rate Hypothesis

Mon, 01 Jan 0001 00:00:00 +0000

This paper examines optimal monetary policy under discretion when the loss function is asymmetric — placing greater weight on employment shortfalls than on equivalently sized employment strength. The model satisfies the natural rate hypothesis (NRH): monetary policy is neutral in the long run, so persistent accommodation of above-potential activity raises inflation expectations without permanently boosting employment. The central paradox the paper establishes is that an asymmetric shortfalls-oriented loss function, despite its stated goal of reducing shortfalls, exacerbates them: the mechanism runs through the NRH expectation-adjustment channel, which creates an inflationary bias structurally analogous to the Barro-Gordon result. Mandating a central bank objective that is more symmetric than the social loss function — a conservative-in-asymmetry design — lowers both the frequency of activity shortfalls and the inflationary bias. As a corollary, the analysis implies that monetary accommodation of labor market strength requires justifications beyond the asymmetric costs of shortfalls, such as permanent effects of strong labor markets on economic potential.

Summary of a forthcoming paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.

In depth

Q1. How does the asymmetric loss function exacerbate employment shortfalls?

The mechanism runs through the natural rate hypothesis: under a loss function that places no weight on activity above potential, the optimal policy fully accommodates positive supply shocks by allowing above-potential output, but the NRH then raises the expectational baseline, making shortfalls more frequent as the perceived natural rate adjusts upward. Because the central bank treats above-potential activity as costless, it does not resist the accumulation of above-potential output in good states; expectations of future activity then rise, effectively moving the benchmark against which shortfalls are measured, and making shortfalls a more common outcome. The asymmetric policy thus generates a self-defeating dynamic: attempts to minimize shortfalls through accommodation of strength create an expectational environment in which shortfalls are more frequent.

Q2. How does the inflationary bias emerge?

The inflationary bias is structurally analogous to the Barro-Gordon (1983) time-inconsistency result: the central bank’s asymmetric desire to reduce shortfalls leads it to ease policy more aggressively than a symmetric loss function would warrant, and this tendency transmits into persistently higher inflation through the NRH expectations-adjustment channel. The classic Barro-Gordon mechanism operates through the desire to push output above its natural rate; here the analog is the desire to push activity above the shortfalls threshold. The paper’s model is constructed so that no Barro-Gordon bias exists in the baseline symmetric case, isolating the asymmetry as the sole source of the inflationary bias.

Q3. What policy prescription follows from the analysis?

The paper recommends mandating a central bank objective that is more symmetric than the social loss function, analogous to Rogoff’s (1985) conservative-central-banker result but applied to the dimension of asymmetry rather than the level of inflation aversion. A mandate that requires the CB to weight above-potential and below-potential activity more equally than society does lowers both the frequency and depth of shortfalls and reduces inflationary bias, improving welfare relative to a CB that faithfully implements the asymmetric social preference. The paper further shows that optimal policy under this design does not accommodate fluctuations from aggregate demand shocks, implying that accommodation of labor market strength requires other justifications — such as permanent productivity effects — not the shortfalls-cost asymmetry alone.

Key concepts

shortfalls asymmetry : the specification in which the central bank’s or social loss function places greater weight on employment below its natural rate than on equivalently sized employment above it; the paper’s central object of analysis.

natural rate hypothesis (NRH) : the assumption that monetary policy is neutral in the long run — persistent monetary accommodation does not permanently raise employment above its natural rate but does raise the price level; imposes the constraint that bounds the central bank’s ability to durably lower shortfalls.

inflationary bias : the systematic tendency of a central bank operating under a shortfalls-oriented asymmetric loss function to allow above-target inflation on average; emerges in this model via the NRH expectations-adjustment channel, analogous to but distinct from the Barro-Gordon result.

The Zero-Beta Interest Rate

Mon, 01 Jan 0001 00:00:00 +0000

Layer 1: Overview

This paper proposes and measures the zero-beta rate — the expected return on a portfolio of stocks with zero market beta, constructed to be orthogonal to the SDF innovations spanned by standard factors — as the correct intertemporal price of consumption, and argues that safe interest rates (Treasury bill yields) are not. Using 130 stock portfolios (81 sorted on combinations of beta, size, value, investment, and profitability; 49 industry portfolios) and GMM estimation with five macro instruments (T-bill yield, inflation, term spread, excess bond premium, and the U6 unemployment rate) over January 1973 to December 2020, the paper estimates the zero-beta rate to average 8.3% per year in real terms with a standard deviation of 9.3%, producing a spread of roughly 7.6% per year over the expected real Treasury bill yield. The paper then shows that this zero-beta rate fits the aggregate consumption Euler equation remarkably well: the macro instruments that best predict the real return of the zero-beta portfolio are nearly proportional to those that predict real consumption growth, a non-mechanical result that survives when the sample is restricted to exclude COVID. Statistical Euler equation tests (Stock-Wright [2000] weak-instrument-robust GMM) reject the Euler equation for all IES values when applied to the Treasury bill, fail to reject it for any IES value when applied to the volatile market return (weak identification), but fail to reject it only for IES below 0.5 (risk aversion above 2) when applied to the zero-beta rate — providing identification from the intermediate predictability of the zero-beta portfolio. Monetary policy shock regressions using Romer-Romer and Nakamura-Steinsson shocks further show that an unexpected monetary tightening raises the real Treasury bill yield but lowers the real zero-beta rate, consistent with the Euler equation’s prediction that the intertemporal price should fall when expected consumption growth falls. Finally, the high level and volatility of the zero-beta rate implies that the entire variation of the price-dividend ratio of a consumption claim can be attributed to variation in the zero-beta rate without requiring time-varying equity risk premia — resolving the equity premium puzzle and Campbell’s [1991] excess volatility puzzle simultaneously, at the cost of an unexplained convenience spread on safe assets.

In depth

Q1. What is the zero-beta rate and how does the paper construct it?

The zero-beta rate is the expected return on a portfolio of stocks that is constructed to have zero covariance with all included asset-pricing factors; in a model where the SDF innovations are spanned by those factors, the expected return of this portfolio equals the intertemporal marginal rate of substitution — the correct price for rearranging consumption over time. The paper follows a three-step procedure. First, it estimates the betas of 130 CRSP stock portfolios with respect to seven factors (Fama-French 5-factor model augmented with a bond excess return factor and a default spread factor) using the Ledoit-Wolf (2017) shrinkage estimator for the factor covariance matrix, to mitigate the rank problem arising from 130 portfolios and 574 monthly observations. Second, it uses the betas to construct the minimum-variance zero-beta portfolio — the portfolio that minimizes return variance subject to having zero exposure to each factor, exploiting all 130 portfolios. Third, it regresses the return of this portfolio on five macro instruments (T-bill yield, lagged inflation, term spread, excess bond premium [EBP], and U6 unemployment) using GMM, in an exactly-identified system in which the same instruments used to predict the portfolio return are used as moment conditions. The fitted value of this regression is the zero-beta rate — the predictable component of the zero-beta portfolio return.

The key econometric property is that the GMM procedure simultaneously estimates the factor loadings and the predictive regression in a way that accounts for the estimation error in both. The standard errors for the predictive coefficients (γ) account for the fact that the betas used to construct the zero-beta portfolio are themselves estimated. Comparatively, an infeasible OLS regression predicting the zero-beta portfolio return has nearly identical point estimates and only slightly smaller standard errors, showing that the main estimation uncertainty comes from the predictive regression rather than from the beta estimation step.

Q2. What are the key properties of the estimated zero-beta rate?

The estimated zero-beta rate has three properties that distinguish it from safe rates and make it a plausible intertemporal price: it is predictable by macro instruments (with all predictors except the unemployment rate individually significant at 5%), it is high and volatile (8.3% annually on average, std dev 9.3%, compared to a low and stable Treasury bill yield), and it co-moves in the direction economic theory predicts with macro conditions. The spread between the zero-beta rate and the expected real Treasury bill yield averages roughly 7.6% per year. The average real zero-beta rate is similar to the average real return of the CRSP market index (which averages 11.8% annualized nominal, or about 8.1% real), consistent with earlier estimates of the average zero-beta return by Hong and Sraer (2016) and Bali et al. (2017). The standard deviation of the zero-beta portfolio’s excess return over its expected value is about 2.7% per month (9.4% annualized), substantially below the standard deviation of the market return, which is why the paper can reject predictability for the zero-beta portfolio even though predicting market returns is notoriously difficult.

In terms of time-series patterns: (1) the zero-beta rate increases more than one-for-one with the Treasury bill yield (consistent with Treasury bills having money-like qualities per Nagel [2016]); (2) it is decreasing in lagged inflation; (3) it falls when a recession is likely — specifically, the unemployment rate, term spread, and EBP collectively predict the zero-beta rate so that it is particularly low when U6 unemployment is low, the yield curve is inverted, and the EBP is high, conditions associated with elevated recession risk (Kiley [2022]).

Q3. How does the zero-beta rate fit the aggregate consumption Euler equation?

The central empirical finding is that the macro instruments that best predict the real return of the zero-beta portfolio are nearly proportional to the macro instruments that best predict real consumption growth — a non-mechanical result because the two regressions are estimated entirely separately, on different data series, with no consumption data used to construct the zero-beta rate. In the linearized Euler equation, the real zero-beta rate should predict real consumption growth in proportion to 1/σ (the IES). If the prediction coefficients on the instruments for the zero-beta rate are γ₀, and the corresponding coefficients for consumption growth are γc, then the vector Δ = γ₀ − σ × γc should be close to zero. Graphically, the expected real zero-beta rate and expected real consumption growth track each other closely over the full sample (Figure 1 in the paper), while the expected real Treasury bill return bears essentially no resemblance to expected consumption growth. With five instruments (L=5), the result is not mechanical: one would need L=1 to always find a σ that makes the result hold. The result is even stronger when the sample ends in December 2019, excluding the COVID episode, because the COVID consumption collapse introduces four-to-seventeen standard deviation consumption growth realizations that attenuate the fit.

Robustness: ridge-penalized estimation (using cross-validation to minimize out-of-sample squared forecast error) substantially attenuates both expected consumption growth and the zero-beta rate toward zero, but they remain approximately proportional. The result holds across alternative specifications (different factor models, different instruments) tested in Appendix Section G.

Q4. What do the statistical Euler equation tests find?

Using Stock-Wright (2000) weak-instrument-robust GMM tests of the non-linear consumption Euler equation, the paper finds: (a) the T-bill return fails the Euler equation test for all IES values (rejected); (b) the market return fails to reject the Euler equation for almost any IES values (weak-instruments problem, consistent with Yogo [2004]); and (c) the zero-beta rate fails to reject the Euler equation for IES values below 0.5 (σ above 2, i.e. risk aversion above 2 under CRRA), and rejects for IES above 0.5. The procedure tests the instrumented non-linear Euler equation: for a conjectured value of σ, it estimates δ (the discount factor) from the unconditional Euler moment, then tests the instrumented Euler moments for each of the five instruments. The test statistic is chi-square with 5 degrees of freedom; the confidence set is the set of σ values that cannot be rejected. The zero-beta portfolio’s intermediate predictability (between the easily-predicted T-bill and the hard-to-predict market return) provides the identification that gives this test meaningful power. The paper obtains its preferred IES estimate of approximately 0.2 (σ ≈ 5) by noting that scaling the zero-beta rate down by a factor of five makes it match expected consumption growth most closely.

The test faces a boundary problem: when σ is very large (above 10), the April 2020 consumption collapse creates a very large SDF realization that dwarfs all others, making the variance-covariance matrix nearly singular and the test uninformative. For this reason, the analysis is restricted to σ ≤ 10.

Q5. What happens to the zero-beta rate after a monetary policy shock?

Using the Romer-Romer (2004) and Nakamura-Steinsson (2018) identified monetary policy shocks, the paper finds that a surprise tightening raises the nominal and real Treasury bill yield but lowers the real zero-beta rate — opposite to what the standard Euler equation with the Treasury bill predicts. Both shocks are normalized to a 100-basis-point increase in the federal funds rate on impact. The estimated effect on the real Treasury bill yield is an immediate increase of roughly the same magnitude (slightly more transitory for the Romer-Romer shock). In contrast, the real zero-beta rate falls following the shock, and the effect is larger and more persistent for the Nakamura-Steinsson shock.

This result is consistent with the consumption Euler equation applied to the zero-beta rate, because the monetary shock also lowers expected consumption growth (well-established in the impulse-response literature). The decomposition in Appendix Section E shows the mechanism: a higher Treasury bill yield raises the zero-beta rate, but the monetary tightening also flattens the yield curve and widens credit spreads (raises the EBP); since both the term spread and EBP are predictors of the zero-beta rate with large coefficients, the indirect effects through these variables dominate and lower the zero-beta rate overall. This finding resolves a tension in structural macro models: while the standard New Keynesian model uses the Euler equation with the safe rate (as in Smets and Wouters [2003, 2007]) and requires habits or wedges to match the hump-shaped consumption response to monetary shocks, the Euler equation with the zero-beta rate is satisfied without additional mechanisms. A stylized three-period New Keynesian model in Appendix Section F shows a monetary tightening can simultaneously raise the safe rate and lower the zero-beta rate through endogenous changes in the convenience spread.

Q6. Can the zero-beta rate explain valuation ratio variation without time-varying risk premia?

The paper shows that the zero-beta rate is sufficiently high, volatile, and persistent to generate the observed variation in the price-dividend ratio of a consumption claim under the assumption of a constant and small equity risk premium, reproducing the finding of Campbell (1991) that discount rates must vary — but attributing the variation to the zero-beta rate rather than to time-varying excess returns. The Campbell-Shiller decomposition of the log price-dividend ratio implies that the expected price-dividend ratio must predict either future real zero-beta rates, future excess returns, or a combination. Under the hypothesis of constant expected excess returns, the price-dividend ratio variation is driven entirely by zero-beta rate variation. In a VAR that includes the five macro instruments plus the CAPE ratio, the implied standard deviation of expected log price-dividend ratio of a consumption claim is approximately 28% — comparable to the 27% standard deviation in the Campbell-Cochrane (1999) model calibration, which achieves this variation through habit formation and time-varying risk premia. The equivalent calculation using the Treasury bill yield rather than the zero-beta rate produces a standard deviation of only 9%, consistent with Campbell’s (1991) original finding that the risk-free rate variation is insufficient.

The paper interprets this as a resolution of both the equity premium puzzle (the average zero-beta return is approximately equal to the average market return, implying a roughly zero equity premium over the zero-beta rate) and the excess volatility puzzle (the zero-beta rate generates sufficient variation in the discount rate). The trade-off is that the unexplained spread between the zero-beta rate and the Treasury bill yield — averaging 7.6% annually — is instead left as a “convenience puzzle.” The paper argues this reframing is progress because the type of models required to explain a large convenience spread on safe assets (frictions, segmented markets, money-like demand for liquid assets) are quite different from those designed to explain large time-varying risk premia.

Key Concepts

zero-beta rate: the expected return on the minimum-variance portfolio of stocks that has zero covariance with each of the included asset-pricing factors; in a model where the SDF innovations are spanned by those factors, equals the conditional expectation of the intertemporal marginal rate of substitution, making it the correct intertemporal price of consumption; measured in this paper at 8.3% annually on average using GMM on 130 CRSP stock portfolios, January 1973–December 2020.
convenience spread: the gap between the zero-beta rate and the expected real Treasury bill yield, averaging approximately 7.6% per year in this paper’s estimates; interpreted as the non-pecuniary value that holders of safe assets (Treasury bills and equivalents) receive from liquidity, collateral, and money-like services — not an expected excess return relative to consumption risk but a departure from the risk-return tradeoff for investors who value safety and liquidity independently of consumption hedging.
SDF-orthogonal equity portfolio: the minimum-variance portfolio constructed by the paper to have zero covariance with all seven asset-pricing factors (Fama-French 5 plus bond and default factors); the portfolio whose expected return equals the zero-beta rate because, by construction, no factor risk premium enters its expected return; estimated using the Ledoit-Wolf (2017) shrinkage estimator applied to 130 stock portfolios to address the rank problem in beta estimation.
Euler equation failure with safe rates: the empirical finding that the expected real Treasury bill return does not co-move with expected real consumption growth — the standard failure documented by Hansen-Singleton (1983), Dunn-Singleton (1986), and Yogo (2004) — which the paper reinterprets not as a structural failure of the representative agent model but as a consequence of using the wrong interest rate; the same Euler equation holds when applied to the zero-beta rate, which the paper argues is the correct intertemporal price.