Equity Premium Puzzle

Definition

The equity premium puzzle (Mehra and Prescott 1985) is the observation that the historical average excess return of equities over risk-free bonds (~6.9% per year in the U.S. 1889–1978) is far too large to be explained by a representative-agent model with constant relative risk aversion (CRRA), unless the coefficient of relative risk aversion R is implausibly large ( $R \gg 10$ ).

Key Ideas

Under the CRRA representative-agent model, the pricing kernel is $M_t = \beta(C_{t+1}/C_t)^{-R}$ . Matching observed consumption growth, the risk-free rate, and the equity premium in U.S. data requires $R \approx 20$ – $50$ , far outside survey estimates (~2–4).
Mehra-Prescott (1985) calibrated the puzzle using a Markov chain model for consumption/dividend growth with parameters estimated from 1889–1978 U.S. data.
The risk-free rate puzzle is the companion observation: the model-implied real risk-free rate is also too high relative to the observed ~0.8% real T-bill rate.
Proposed resolutions divide into: (i) preference modifications (habit formation, recursive utility), (ii) rare events (disaster risk), and (iii) information/learning (parameter uncertainty).

How It Works

The Mehra-Prescott Calibration

In the simplest version, log-dividend growth $\xi_{t+1} = \log(y_{t+1}/y_t)$ follows an independent and identically distributed (i.i.d.) normal distribution with known mean $\mu$ and precision $\tau$ . Given parameters $(\beta, R, \mu, \tau)$ , the equilibrium stock price and bond price in the Gordon growth model satisfy:

$S_t = y_t \cdot \frac{\beta e^{(1-R)\mu + (1-R)^2/(2\tau)}}{1 - \beta e^{(1-R)\mu + (1-R)^2/(2\tau)}}$

$B_t = \beta e^{-R\mu + R^2/(2\tau)}$

The equity premium is $\log(S_{t+1} + y_{t+1}) - \log(S_t) - \log(1/B_t)$ . Setting $\mu$ and $\tau$ to their sample estimates from 1889–1978 data, matching the observed equity premium requires $R \gg 10$ .

Parameter Uncertainty Resolution (Jobert-Platania-Rogers 2006)

When the agent treats $(\mu, \tau)$ as unknown and holds a Gamma-Gaussian conjugate prior $\pi_0(\mu, \tau)$ , stock and bond prices average over the full posterior:

$S_t = y_t \, E_t\!\left[\frac{1}{1 - \beta e^{-\nu\mu + \nu^2/(2\tau)}}\right] - y_t, \qquad \nu = R - 1$

Because $\mu$ requires ~1,550 years of data to estimate to $\pm 0.01$ at 95% confidence, the posterior over $\mu$ remains wide throughout any realistic sample. Posterior spread over $\mu$ raises the expected stock price relative to the known-parameter case, generating a higher equity premium at low $R$ . Jobert et al. (2006) show that $R \in (1, 2)$ matches the data once parameter uncertainty is properly incorporated. A convergence prefactor in the prior ensures the pricing integral is well-defined.

Habit Formation (Sundaresan 1989; Constantinides 1990)

When utility depends on a habit stock $H_t$ (a moving average of past consumption), effective risk aversion at date t equals $-C_t U_{CC}/(C_t - H_t) \cdot U_C$ , which rises sharply as $C_t \to H_t$ in recessions. The stochastic discount factor (SDF) has higher volatility in downturns, partially resolving the puzzle without high unconditional R.

Recursive Utility (Epstein-Zin 1989)

Separating the coefficient of relative risk aversion from the intertemporal elasticity of substitution (IES) allows high risk aversion without an implausibly low IES. The long-run risk model (Bansal-Yaron 2004) uses Epstein-Zin preferences with persistent predictable components in consumption growth.

Rare Disasters (Rietz 1988)

Occasional large consumption drops (wars, Great Depressions) raise the equity premium in expected-utility models without affecting average consumption growth in a censored sample. Data on disasters is by nature sparse, making these tail events hard to estimate.

Why It Matters

The puzzle is a fundamental test of the CRRA representative-agent paradigm — the workhorse model of asset pricing and macroeconomics.
Resolution determines welfare calculations, pension design, and optimal policy: if $R \approx 2$ , the welfare cost of business cycles is trivial; if $R \gg 10$ , fluctuations impose large welfare losses.
Parameter uncertainty as a resolution has direct implications for portfolio allocation: it reduces optimal equity exposure, working against the canonical "stocks for the long run" recommendation (see Estimation Risk and Asset Allocation).
The SDF puzzle — mismatch between preference-implied and option-implied pricing kernels — is related but distinct.

Open Questions

No single resolution is universally accepted. Habit formation requires specific functional forms; disaster risk requires calibrated probabilities of events with very few historical instances; parameter uncertainty is sensitive to prior specification.
Simultaneously matching the equity premium, the risk-free rate, and the return volatility ("the Sharpe ratio") constrains all proposed resolutions.
Whether the puzzle persists out-of-sample (international data, post-1978 U.S.) is debated.
Under parameter uncertainty resolutions, the implied portfolio allocation shrinks with horizon — paradoxically reversing the long-run equity premium intuition.