8. The Dual Puzzle

Hansen and Singleton (1982) use GMM to test the stochastic Euler equation (1.5) implied by consumption-based models. Its conditional version for any asset \(i\) is

where \(C_t,C_{t+1}\) are consumption in \(t,t+1\), \(R^i_{t,t+1}\) is asset \(i\)'s gross return from \(t\) to \(t+1\), and \(\mathcal F_t\) is the period-\(t\) information set. With CRRA utility \(u(C)=\frac{C^{1-\gamma}-1}{1-\gamma}\) plugged into (8.1):

Using an instrument vector \(\mathbf y_t\in\mathcal F_t\) from the period-\(t\) information set, turn the conditional moment into an estimable unconditional moment restriction (8.3):

Estimate \((\beta,\gamma)\) by GMM. Hansen–Singleton's finding: no matter how the relative risk aversion \(\gamma\) is calibrated, the moment restriction (8.3) is not satisfied — the consumption-based CRRA model disagrees with the data. This disagreement shows up as two puzzles at once, jointly the dual puzzle.

From (8.2), taking unconditional expectations (law of iterated expectations) gives (8.4):

Let \(c_t=\ln C_t\), \(\Delta c_{t+1}=c_{t+1}-c_t\), \(r^i_{t,t+1}=\ln R^i_{t,t+1}\). Then (8.4) becomes (8.5). Assume \((\Delta c_{t+1},r^i_{t,t+1})\) is bivariate normal (8.6), and expand with the log-normal expectation formula.

(8.12) also holds for the market portfolio \(R^m\) (8.13):

Plugging in the empirical facts \(\mathbb E[R^i]\approx1.07\), \(\mathbb E[R^f]\approx1.01\), \(\mathbb E[\Delta c]\approx0.0172\), \(\mathrm{Var}(\Delta c)\approx(0.036)^2\), \(\mathrm{Cov}(\Delta c,r^m)\approx0.003\) into (8.13):

Empirical evidence usually suggests \(\gamma\) way lower than 20, so \(\gamma\approx20\) is implausibly high — this is the equity premium puzzle.

Solve (8.8) for the risk-free rate:

Plugging in different \((\beta,\gamma)\) gives the implied gross risk-free return (Table 8.1). The \(-\ln\beta\) term rises as patience falls (smaller \(\beta\)); the \(\gamma\mathbb E[\Delta c]\) term pushes the rate up with \(\gamma\) (consumption growth → want to borrow → high rate); but \(-\frac12\gamma^2\mathrm{Var}(\Delta c)\) is a negative quadratic in \(\gamma\) (precautionary saving → lower rate).

Table 8.1: Implied gross risk-free rate \(R^f_{t,t+1}\)

When \(\gamma\) takes the high value needed for the equity premium, the implied risk-free rate is far above the observed \(\approx1.01\); to push it back to reality one needs \(\beta>1\) (a negative time preference, unreasonable). The risk-free rate puzzle.

By the Hansen–Jagannathan bound (5.17), \(\frac{\sigma_m}{\mathbb E[m]}\ge\frac{|\mathbb E[R]-R_f|}{\sigma_R}\), which traces a hyperbola (5.18) in the \((\mathbb E[m],\sigma_m)\) plane (with \(A,B,C\) as in §2.1.2). A low-risk asset sharply shrinks the feasible SDF region.

Figure 8.1: (a) the SDF volatility bound without a low-risk asset; (b) adding a low-risk asset (e.g. T-bill) sharply narrows the feasible region — an economically sensible SDF must sit inside a much tighter cone.

Further, given the risk-free gross return \(R_f\), a feasible SDF must have \(\mathbb E[m]=\frac{1}{R_f}\), the red line in Figure 8.2.

Figure 8.2: given \(R_f\), feasible SDFs lie only on the vertical red line \(\mathbb E[m]=1/R_f\), above its intersection with the hyperbola.

Revealing the dual puzzle. From the CRRA SDF (1.12) \(m_{t+1}=\beta(\frac{C_{t+1}}{C_t})^{-\gamma}\), with \(\Delta c_{t+1}\sim\mathcal N(\mathbb E[\Delta c_{t+1}],\sigma_c^2)\), log-normality gives (8.14):

And \(\mathbb E[m_{t+1}^2]=\mathbb E[m_{t+1}]^2 e^{\gamma^2\sigma_c^2}\), so \(\sigma_{m_{t+1}}=\mathbb E[m_{t+1}]\sqrt{e^{\gamma^2\sigma_c^2}-1}\), making the Sharpe-ratio bound monotonically increasing in \(\gamma\) (8.15):

Figure 8.3: the green line is the Hansen–Jagannathan volatility bound; the other three are the \((\mathbb E[m],\sigma_m)\) curves of the consumption-based SDF (\(\beta=0.95,0.97,0.99\), \(\gamma\) from 1 to 70). As \(\gamma\) grows the curve moves up-left first, then up-right.

• For the SDF curve to enter the feasible region, \(\gamma\) must be extremely high (e.g. \(\ge60\)) — far beyond plausible values, so the equity premium puzzle is not resolved.
• Such a high \(\gamma\) then (by Table 8.1) makes \(R^f<1\) and \(r^f\) negative, so the risk-free rate puzzle is not resolved.
• Hence the dual puzzle is not solved by the consumption-based model. The way out: can we raise the SDF's variance without raising \(\gamma\) much (so the SDF enters the feasible region and generates a high premium with a reasonable \(\gamma\))? This is exactly the idea behind the following chapters (Epstein–Zin, incomplete markets, rare events, habit).

A single representative household maximizes \(\mathbb E_0[\sum_{t=0}^\infty\beta^t u(c_t)]\), \(\beta\in(0,1)\), CRRA \(u(c)=\frac{c^{1-\gamma}-1}{1-\gamma}\). A single firm produces the consumption good; its equity is competitively traded and the market return is the firm's return. Each period the firm's output (real consumption good) and dividend payment (nominal fiat money) both equal \(y_t\), with \(y_t\) following a Markov process

with growth rates \(\lambda_j>0\) and \(y_0>0\). Equity is traded ex-dividend at the beginning of each period right after \(y_t\) is revealed.

Market clearing requires the dividend to be fully consumed each period: \(y_t=c_t\). Equity is held forever; by the CRRA SDF (1.12) and the Euler equation, the price (8.16):

The price is homogeneous of degree one in \(c\), giving the recursion (8.17); conjecture \(p^e(c,i)=w_i c\) (8.18), substitute to get a system of \(n\) linear equations (8.19), and solve for \(\{w_1,\dots,w_n\}\) as functions of \((\beta,\gamma)\):

Returns. Conditional risky net return (8.20), unconditional (8.21):

where \(\pi\) is the stationary distribution. Risk-free return. Price \(p^f_i=\beta\sum_j\phi_{ij}\lambda_j^{-\gamma}\), conditional net return (8.22), unconditional (8.23):

Mehra–Prescott (1985) reveal the dual puzzle step by step:

1. Assumptions: only two states \(\lambda_1=1+\mu+\sigma\), \(\lambda_2=1+\mu-\sigma\), with transition matrix \(\mathbb P=\begin{bmatrix}\psi&1-\psi\\1-\psi&\psi\end{bmatrix}\).
2. Calibration: using U.S. data 1889–1978, \(\mu=1.8\%\), \(\sigma=0.036\), \(\psi=0.43\).
3. Solve \(\{w_i\}\): plug \((\mu,\sigma,\psi)\) into (8.19) and solve \(\{w_i\}\) as functions of \((\beta,\gamma)\).
4. Compute predicted returns: equity from (8.20), (8.21); risk-free from (8.22), (8.23).
5. Reveal the dual puzzle: in the \((r^f,r^e)\) plane the model produces a family of points over \((\beta,\gamma)\); but the actual data point \((\hat r^f,\hat r^e)\) cannot be reproduced by any pair \((\beta,\gamma)\) — the dual puzzle.

Return volatility (≈16%) is much higher than dividend volatility (≈7%). If the price-dividend ratio is constant, \(P_t=hD_t\), then \(dP_t=h\,dD_t\), giving (8.24):

i.e. return volatility should equal dividend volatility; but empirically \(\sigma_P\approx16\%\gg\sigma_D\approx7\%\) — the volatility puzzle (Shiller).

Market returns are predictable by the (time-varying) dividend yield, while dividend growth is not. Annualized return volatility itself varies wildly over time: 60–70% in the 1930s and 2008–2009, only about 5% in the 1960s. And return volatility is negatively correlated with the price/earnings ratio: a crisis → high volatility, poor market prospects → low P/E; conversely peaceful times → low volatility, good prospects → high P/E.

By C-CAPM (1.23) returns are pinned down by covariance with consumption, and by CAPM (2.25) by covariance with the market; but size-sorted and book-to-market-sorted portfolio returns cannot be explained by covariances with the SDF or the market return. Fama–French and other multi-factor models work but lack a structural microfoundation.

U.S. stock excess returns are always higher under Democratic presidential terms than Republican terms. In a predictive regression \(R_{t\to t+s}=\alpha+\beta z_t+\varepsilon_{t+s}\), the coefficient \(\beta\) is significant for some predictors \(z_t\) (log dividend yield, log earnings yield, book-to-market ratio).

• Participation puzzle: many households do not participate in the stock market, while standard analysis (§1.4.4) says everyone should.
• Under-investment: those who do participate invest a smaller market share than theory suggests.
• Life-cycle puzzle: under a constant investment opportunity set one should hold a constant fraction (by (6.36)), regardless of age; but in reality investors increase investment in the first half of life and decrease it in the second.
• Home bias: diversification calls for investing in other companies, yet people tend to hold domestic / own-company equity.