The Equity Premium Puzzle and the Riskfree Rate

This section derives the equity premium puzzle (Mehra & Prescott (1985)). Consider a representative agent solving the joint consumption and portfolio allocation problem:

where $\Rfree$ denotes the return on a perfectly riskless asset and $\Risky_{t+1}$ denotes the return on equities (the risky asset) held between periods $t$ and $t+1$, $\riskyshare_{t}$ is the share of end-of-period savings invested in the risky asset, $\Rport_{t+1}$ is the portfolio-weighted rate of return, and $y_{t+1}$ is noncapital income in period $t+1$.

As usual, the objective can be rewritten in recursive form:

The first order condition with respect to $c_{t}$ is

and the FOC with respect to $\riskyshare_{t}$ is

But the usual logic of the The Envelope Theorem and the Euler Equation theorem tells us that

so, substituting (5) into (3) and (4) we have

Now assume CRRA utility, $\uFunc(c) = c^{1-\CRRA}/(1-\CRRA)$ and divide both sides by $c_{t}^{-\CRRA}$ to get

Now recall the following two facts:

Fact 1: If $\Delta c_{t+1}/c_{t}$ is small, $c_{t+1}/c_{t} \approx 1+ \Delta \log c_{t+1}$.

Fact 2: If $z$ is small, $(1+z)^{\lambda} \approx 1 + \lambda z$.

Using these two facts, equation (7) can be approximated by

Using one more fact,

Fact 3: $\Ex [xy] = \Ex [x]\Ex [y] + \text{cov}(x,y)$

we get

or

where the last approximation holds because $\Ex_{t}[\Delta \log {c}_{t+1}]$ is small.

1The Equity Premium Puzzle¶

Because this expression must hold at all $t$, we can check it empirically by calculating empirical estimates of the two components and assuming that the sample averages correspond to the representative agent’s expectations. That is, if we have data for periods $1 \ldots n$, we assume that the unconditional expectations correspond to the sample means, $\Ex [\Risky] = (1/n) \sum_{s=1}^{n} \Risky_{s}$; $\Ex [\Delta \log c] = (1/n) \sum_{s=1}^{n} \Delta \log c_{s}$; and $\text{cov}(\Delta \log c,\Risky) = (1/n) \sum_{s=1}^{n} (\Delta \log c_{s} - \Ex [\Delta \log c])(\Risky_{s}-\Ex [\Risky])$.

The equity premium puzzle is essentially that $\text{cov}(\Delta \log c,\Risky)$ is very small (about 0.004) but $\Ex [\Risky]-\Rfree$ is about 0.08 (stocks have earned real returns of about 8 percent more than riskless assets over the historical period), which means that the only way equation (10) can hold is if $\CRRA$ is implausibly large (these values imply a value of $\CRRA=20$).

How do we know what plausible values of $\CRRA$ are? Consider the following. You must choose between a gamble in which you consume $50,000 for the rest of your life with probability 0.5 and $100,000 with probability 0.5, or consuming some amount $X$ with certainty. The coefficient of relative risk aversion determines the $X$ which would make you indifferent between consuming X or being exposed to the gamble. For example, if $\CRRA = 0$, then you have no risk aversion at all and you will be indifferent between $75,000 with certainty and the 50/50 gamble with expected value of $75,000. Here are the values of X associated with different values of $\CRRA$ (table taken from Mankiw & Zeldes (1991)).

2The Riskfree Rate Puzzle¶

Rewrite the consumption Euler equation (6) as

and note that from (7) we know that $\Ex_{t}[\beta \riskyshare_{t}(\Risky_{t+1}-\Rfree)\uFunc^{\prime}({c}_{t+1})] = 0$ so that (11) reduces to the ordinary Euler equation

Using the same “facts” and approximations as above, we get the standard approximation to the Euler equation,

The “riskfree rate puzzle” is that average consumption growth per capita has been about 1.5 percent (in the US in the postwar period) while real riskfree interest rates have been at most 1 percent. Even if we assume a time preference rate of $\timeRate=0$ (no impatience at all, e.g. $\beta=1$), the only way this equation can hold is if $\CRRA$ is a very small number (maybe even less than one). Of course, this is precisely the opposite of the conclusion of the equity premium puzzle, which implies the $\CRRA$ must be very large.

In principle, the riskfree rate puzzle might be explicable by overlapping generations models, though in practice it has proven difficult to make this mechanism work quantitatively. A precautionary saving motive can also help reduce the puzzle by adding a variance term to consumption growth, which would allow the Euler equation to hold even with low riskfree rates.