Expectation-driven term structure of equity and bond yields
What this paper finds — and why it matters
Overview
Research Question. What drives the joint historical dynamics of the term structure of equity yields and nominal bond yields — and can a single unified equilibrium model explain the procyclical equity yield slope, the switch in bond-stock correlation from positive to negative after the late 1990s, the maturity-declining predictability of dividend strip returns, and standard aggregate stock market puzzles?
Key Departure from Prior Literature. Existing equilibrium models (habit formation, long-run risk, disaster risk) rely on time-varying risk premia to explain asset prices. Recent survey evidence challenges this: De La O and Myers (2021) show that most aggregate stock price movements are driven by cash-flow growth expectations rather than return expectations, and Van Binsbergen et al. (2013) show that equity yields are driven mainly by dividend growth expectations. This paper constructs an equilibrium model in which equity (bond) yield variation is attributable to subjective dividend growth (GDP growth) expectations, with a constant subjective risk premium implied by CRRA utility.
Model Architecture. The representative agent has CRRA utility with risk-aversion coefficient γ = 4 and subjective discount factor β = 1.0065 (calibrated to the average 10-year equity yield). The agent departs from rational expectations by having the “belief in the law of small numbers” (Tversky and Kahneman 1971): she perceives small samples to represent their population as well as large samples, leading to subjective learning gains that differ from the rational Kalman gain. The subjective belief updating rule is a modified Kalman filter in which the likelihood is exaggerated by factor (1+θ), producing a subjective learning gain ν that exceeds the Kalman gain K when overreaction applies and falls below it when underreaction applies.
The model has three blocks of fundamentals, each decomposed into a stable and a transitory component. (1) Real GDP growth is decomposed into PCE growth (stable, with a random-walk trend state µ_g) and a volatile gap component (stationary state x_g, persistence ρ_g = 0.941). (2) Inflation is decomposed into core inflation (stable, with trend state µ_π) and a volatile gap (persistence ρ_π = 0.932). (3) Real aggregate dividend is decomposed into a long-duration dividend component dl (levered on log real GDP with leverage λ = 3) and the share of long-duration dividend ds (stationary with persistence ρ_d = 0.94). This cross-sectional decomposition uses firm-level long-term earnings growth (LTG) forecasts from IBES as a model-free equity duration measure.
Estimation. State-space parameters are estimated by maximum likelihood with the Kalman filter on data from NYSE/NASDAQ/AMEX firms (CRSP/Compustat), quarterly, from 1987Q4 to 2019Q4. Subjective learning gains are estimated by minimizing RMSE between model-implied expectations and consensus forecasts: 1-year real GDP growth and inflation from the Survey of Professional Forecasters (SPF, 1981Q3–2019Q4), and 1-year aggregate dividend growth extended from De La O and Myers (2021) to 2019Q4. Equity yield data are from Giglio et al. (2021); bond yields are end-of-quarter zero-coupon nominal yields from Gürkaynak et al. (2007).
Main Findings.
Equity Term Structure Dynamics. The model’s subjective dividend growth expectations drive equity yields. The 1-year model-implied equity yield correlates 0.68 with data; the 10-year correlates 0.79; the 10Y–1Y slope correlates 0.59 with data. Consistent with “belief in the law of small numbers,” the agent overreacts to dividend news (estimated learning gains νl_d = 0.166 and νs_d = 0.458, both below their Kalman gains, which under the level-to-growth translation implies overreaction to dividend growth news, confirmed by negative CG(2015) regression slope coefficients of −0.69 at 1Y and −0.97 at 5Y).
Procyclical Equity Yield Slope. During recessions, the average equity yield slope (10Y–1Y) in the model is −3.77%; during expansions it is +3.96%, matching the data (−5.50% in recessions, +3.93% in expansions). The sign reversal is driven primarily by the dividend-specific component of the decomposition: in recessions, short-run dividend growth expectations fall much more sharply than long-run expectations.
Bond Pricing. The model’s 1-year and 10-year nominal bond yields achieve correlations of 0.92 and 0.95 with their data counterparts, inheriting the explanatory power of Zhao (2020) for the bond market. The agent underreacts to GDP growth and inflation news (estimated learning gains well below Kalman gains, confirmed by positive CG(2015) slope coefficients of +2.08 at 1Y for GDP growth and +1.01 at 1Y for inflation).
Bond-Stock Correlation Switch. In data, 10Y bond vs. dividend strip return correlation (5Y strip) goes from +0.46 before 2000 to −0.49 after 2000. The model produces +0.14 before and −0.56 after (for the 5Y strip). Decomposing the change in bond-stock return covariance: the “inflation real effect” (correlation between expected inflation and real growth) accounts for approximately 27–31% of total changes (for 5Y to 10Y strips); the “real growth correlation” channel — stronger co-movement between real GDP and real dividend growth expectations after 2000 — accounts for approximately 89–95% of total changes. The paper identifies this real bond hedging channel as the dominant and previously unexamined driver.
Dividend Strip Return Predictability. The price-dividend ratio predicts annual market excess returns with R² of 10.3% (data) vs. 9.0% (model). Strip return predictability is downward-sloping by maturity: in data, the R² is 20.2% for 5-year strips and 14.5% for 10-year strips; the model generates 14.2% and 10.4% respectively. This is decomposed into three sources: bond return predictability (small contribution), dividend forecast error predictability (dominant for short maturities), and forecast revision predictability (negative contribution that offsets). The downward slope occurs because current news has smaller impact on long-term dividend expectations.
Aggregate Market Puzzles. The model-implied log dividend-price ratio correlates 0.86 with data, with AR(1) coefficient 0.96 (data: 0.95). Model-implied average market return is 9% (data: 8%); annualized return volatility 12% (data: 16%). The model replicates the switch of the bond-stock aggregate return correlation from +0.13 before 2000 to −0.46 after 2000 (data: +0.39 to −0.64).
Scope Conditions. Results apply to U.S. equity and bond markets over 1987Q4–2019Q4 (with bond learning using data back to 1959Q1). The model assumes a representative agent with CRRA utility and constant subjective risk premium. It is silent on the term structure of expected returns in the statistical sense (which requires identification of latent states under the physical measure). The aggregate market results require a reduced-form specification for stochastic equity duration H_t linked to the value-weighted LTG average.
Q&A
Q1: What is the core psychological mechanism generating subjective beliefs, and how does it differ from the diagnostic expectations approach?
The agent has the “belief in the law of small numbers” (Tversky and Kahneman 1971): she treats small samples as equally representative of their population as large samples. Formally, this is embedded by exaggerating the likelihood in the Bayesian update: p(x_t|I_t) ∝ p(y_t|x_t)^{1+θ} × p(x_t|I_{t-1}), where θ captures the magnitude of cognitive bias. The resulting subjective learning gain ν = (1+θ)P̃ / [(1+θ)P̃ + σ²_ε] can exceed the Kalman gain K when θ is large (overreaction) or fall below it when θ is small (underreaction). This differs from diagnostic expectations (Bordalo et al. 2019, 2020a,b), which are based on the representativeness heuristic; the paper notes the two notions of news are highly correlated in simulation (Table IA.2) and that both can imply overreaction.
Q2: Why does the model generate overreaction to dividend growth news even though the dividend-level learning gains are smaller than the Kalman gains?
The model separates dividend learning into level and growth. Section 2.2 derives that underreaction to dividend level news (νl_d < Kl_d, νs_d < Ks_d, estimated values 0.166 and 0.458 against Kalman gains 0.19 and 0.49 respectively) translates into overreaction to dividend growth news. This is confirmed by the CG(2015) rationality test: regressing forecast errors on lagged forecast revisions yields slope coefficients of −0.69 (1Y) and −0.97 (5Y) for real dividend growth, both statistically significant (t-statistics −3.63 and −3.22). In contrast, the same test yields positive slope coefficients for GDP growth (2.08 at 1Y) and inflation (1.01 at 1Y), confirming underreaction for these series.
Q3: How well does the model match subjective dividend growth expectations in the survey data?
The model-implied 1-year subjective dividend growth forecast is estimated by minimizing RMSE against the consensus dividend growth forecast series (extended from De La O and Myers 2021 to 2019Q4, with a replication correlation of 0.92 over the overlapping sample). The unconditional correlation between model-implied and data 1-year forecasts is 0.80. Although only 1-year forecasts are used in estimation, the model also achieves a correlation of 0.80 for 2-year forecasts, providing an out-of-sample validation.
Q4: What explains the higher volatility of short-term equity yields relative to long-term equity yields?
Short-term subjective dividend growth expectations are more volatile because the agent’s short-run expectation mean-reverts toward the less volatile long-run (levered) GDP growth expectation. In the model’s two-component dividend structure, the transitory dividend-share component xd has persistence ρ_d = 0.94 and its effect on equity yields decays as maturity increases (via the factor (1−ρ^n_d)/n). Similarly, the effect of the transitory GDP growth state x_g decays with maturity. Long-term equity yields are thus anchored by the slower-moving trend components µ_g and µ_d. In the data from Giglio et al. (2021), 1-year yields have a standard deviation of 8.89% annualized vs. 2.70% for 10-year yields; the model generates 8.22% and 1.89% respectively.
Q5: What is the quantitative importance of the “real growth correlation” channel vs. the “inflation real effect” channel in explaining the bond-stock correlation switch?
For the switch in bond-stock return correlation (using the 10-year nominal bond and various maturity dividend strips), the decomposition in Table 4 shows that the “real growth correlation” channel accounts for 89.1% (5Y strip), 92.1% (7Y strip), and 94.8% (10Y strip) of total bond-stock covariance changes, while the “inflation real effect” (correlation between expected inflation and expected real growth) accounts for 27.3%, 29.3%, and 31.1% respectively. The “volatility of shocks to expected inflation and real growth” makes a negative contribution (−16.4%, −21.4%, −25.9%), mostly attributable to more volatile beliefs during the 2008 global financial crisis. The real growth correlation channel reflects that after 2000, real bonds provide a better hedge to aggregate real dividend risks because real GDP growth expectations and real dividend growth expectations became more positively correlated.
Q6: Does the same real growth correlation story hold for the “Fed model” (bond-stock yield correlation)?
Yes, but with a quantitatively different balance. For yield correlations (Table 5), the “real growth correlation” channel accounts for 72.4%–80.1% of bond-stock yield covariance changes (5Y to 10Y strip), while the “inflation real effect” now accounts for 41.2%–43.9%. The inflation real effect is proportionally larger for yield levels because persistent expected inflation correlates strongly with the level of expected real GDP growth — even though inflation expectations do not move fast enough at high frequency to explain return correlation, they co-move strongly with expected growth at low frequency.
Q7: How does the model generate a downward-sloping term structure of return predictability?
The strip excess return is decomposed into three components (Equation 44): maturity-matched bond excess return (Bond), dividend forecast error within the holding period (FE), and forecast revision regarding dividend growth after the holding period (FR). For short maturities, bond predictability contributes little (R² ≈ 6.7% for 5Y strip), while FE predictability (R² ≈ 31.5%) and FR predictability (R² ≈ 35.6%) dominate. As maturity increases, the current news has smaller impact on long-term dividend expectations, reducing the predictability of FE (R² ≈ 26.6% for 10Y) and FR (R² ≈ 26.5% for 10Y). Taken together, total model-implied strip R² declines from 14.2% (5Y) to 10.4% (10Y), matching the data pattern (20.2% to 14.5%). The paper identifies forecast revision predictability as a new channel not previously documented.
Q8: Why do forecast errors and forecast revisions have opposite signs in the predictability regressions?
Bad news (high equity yields, i.e., low current stock prices) triggers excessively pessimistic subjective dividend growth expectations because the agent overreacts to dividend news. These overly pessimistic forecasts tend to be disappointed in the future — actual dividend realizations exceed the forecast — producing positive subsequent forecast errors (FE is positively predicted by high yields, with R² ≈ 31.5% for 5Y strips). However, as dividend levels mean-revert, higher subsequent realizations cause the agent to revise down the forecast for dividend growth thereafter, leading to negative forecast revisions (FR is negatively predicted by high yields, with R² ≈ 35.6% for 5Y strips, opposite sign from FE). The net effect on return predictability is thus a combination of positive (FE) and negative (FR) contributions.
Q9: How does the model handle the aggregate market dividend-price ratio and its persistence?
The aggregate stock price is modeled as the sum of dividend strip prices up to a stochastic horizon H_t, which is parameterized as a linear function of the value-weighted average of LTG forecasts: H_t = a + b·LTG_t. Parameters a and b are estimated by minimizing RMSE between model-implied and data log dividend-price ratio. The model-implied ratio achieves a correlation of 0.86 with data, an AR(1) coefficient of 0.96 (data: 0.95), and an annualized volatility of 26% (data: 30%). The time-variation is driven entirely by strip yield variations and exogenous LTG movements.
Q10: Is the overreaction to dividend news and underreaction to GDP/inflation news consistent in a single framework?
Yes. The model’s subjective learning framework (based on “belief in the law of small numbers”) generates both over- and underreaction depending on the estimated subjective learning gain relative to the Kalman gain. For GDP growth and inflation, the learning gains (ν*_g = 0.012, νgap_g = 0.065; ν*_π = 0.049, νgap_π = 0.228) are below their Kalman gains (0.29 and 0.67 for GDP components; 0.67 and 0.48 for inflation components), producing underreaction. The paper hypothesizes this is related to the Fed’s dual mandate: agents rationally assign lower weight to GDP and inflation shocks expecting the Fed will stabilize them. For dividend growth, a level-to-growth translation converts level underreaction into growth overreaction.
Q11: What are the robustness checks, and what do they show?
The paper checks three alternative equity duration measures: those from Dechow et al. (2004), Weber (2018), and Gonçalves (2021b), as well as the book-to-market ratio following Lettau and Wachter (2007). Table IA.1 shows that replacing LTG with these measures still produces model-implied equity yields that replicate key data moments with high time-series correlations. Changing the cross-sectional breakpoint for long-duration dividends from the median LTG to the 40th or 60th percentile leaves results similar. The paper also presents an Internet Appendix extension in which the agent has ambiguity about real GDP and dividend growth (model misspecification fear), yielding equity yields and returns even closer to data.
Q12: What is the paper’s contribution to the bond market relative to Zhao (2020)?
The bond pricing block closely follows Zhao (2020), inheriting its explanatory power for bond market stylized facts. The model’s 1-year and 10-year nominal bond yields achieve correlations of 0.92 and 0.95 with data, respectively. The new contribution is the joint model covering both equity and bond markets simultaneously, enabling the decomposition of bond-stock covariance and the identification of the real growth correlation as the dominant driver of the bond-stock correlation switch — a channel not addressed by Zhao (2020), which focused on bond market puzzles alone.
Key Concepts
Equity Yield (Dividend Strip Yield). Defined as ey^(n)_t = (1/n)(d$_t − p^(n)_t), where p^(n)_t is the log price of the n-period dividend strip (a claim to the nominal dividend n periods ahead) and d$_t is the log nominal aggregate dividend. It decomposes into the bond yield, a subjective dividend growth component, and a (constant) risk premium component.
Belief in the Law of Small Numbers. A cognitive bias (Tversky and Kahneman 1971) in which the agent perceives small samples to represent their population as well as large samples. Modeled by exaggerating the likelihood in Bayesian updating: p(x_t|I_t) ∝ p(y_t|x_t)^{1+θ} × p(x_t|I_{t-1}). This generates a subjective learning gain ν that can exceed the Kalman gain (overreaction) or fall below it (underreaction) depending on θ and the signal-to-noise ratio.
Subjective Learning Gain. The coefficient ν in the subjective Kalman filter update ẽ_t x_t = ρẽ_{t-1}x_{t-1} + ν(y_t − ρẽ_{t-1}x_{t-1}). It equals (1+θ)P̃ / [(1+θ)P̃ + σ²_ε], where P̃ is the subjective predictive variance. When ν > K (the rational Kalman gain), the agent overreacts to news; when ν < K, the agent underreacts.
Long-Duration Dividend Component. The portion of aggregate real dividend (dl_t) attributable to “long-duration” firms — those with above-median analyst LTG forecasts in CRSP/Compustat/IBES data. Levered on log real GDP with leverage parameter λ = 3, it carries aggregate risk. The complementary short-duration dividend share ds_t is stationary and carries no aggregate risk. The decomposition allows the model to exploit cross-sectional cash-flow duration information when learning about future aggregate dividend growth.
Real Growth Correlation Channel. A bond-stock covariance component defined as Cov(RGDP^(N), RDIV^(n)), where RGDP^(N) is the real GDP growth expectation component of 10-year nominal bond returns and RDIV^(n) is the real dividend growth expectation component of n-period strip returns. This channel captures whether real bonds hedge aggregate real dividend risks. The paper shows this channel accounts for approximately 89–95% of the post-2000 bond-stock covariance change for dividend strips.
Inflation Real Effect. The covariance component Cov(INFL^(N)_B, RGDP^(n) + RDIV^(n)), defined as the correlation between shocks to expected inflation (embedded in nominal bond returns) and shocks to expected real growth (in strip returns). In the paper’s framework this is distinct from the standard inflation risk premium story, as it concerns the correlation between subjective beliefs rather than realized covariances under the physical measure.
Forecast Error (FE) and Forecast Revision (FR) Predictability. Two of three components of realized strip excess return (Equation 44). FE = ∆d${t+1:t+h} − ẽ_t∆d${t+1:t+h} is the realized dividend growth forecast error within the holding period; FR = (ẽ_{t+h} − ẽ_t)∆d$_{t+h+1:t+n} is the forecast revision for dividend growth beyond the holding period. Because the agent overreacts to dividend news, bad news triggers overly pessimistic forecasts (positive subsequent FE) and, as dividends mean-revert, downward forecast revisions (negative FR). These two have opposite signs in predictive regressions, generating the downward-sloping term structure of return predictability.
Fed Model. The empirical positive correlation between equity yields (real) and nominal bond yield levels. The paper shows that this yield-level correlation switched from strongly positive (≈ 0.85 before 2000) to significantly negative (≈ −0.60 to −0.62 after 2000) for 5Y–10Y dividend strips, and that the same real growth correlation and inflation real effect decomposition applies, albeit with the inflation real effect proportionally larger (≈ 40%) for yield levels than for returns (≈ 30%) because persistent inflation expectations co-move with the level of expected real GDP growth.