What is the significance of the Hansen-Jagannathan bound?

Question

The Wikipedia article on the Hansen-Jagannathan bound is short, giving only the following:

Hansen–Jagannathan bound is a theorem in financial economics that says that the ratio of the standard deviation of a stochastic discount factor to its mean exceeds the Sharpe Ratio attained by any portfolio. This result is an application of the Cauchy–Schwarz inequality.

An important detail that is missing is a discussion of the theorem's significance. The Hansen-Jagannathan (H-J) bound gives us a type of mean-variance frontier. But what is importance/significance (in theoretical and empirical work) of the H-J bound over the classic mean-variance efficient frontier? (The two are different but related. What's the main insight from the H-J bound?)

Answer 1

Fundamental theorem of asset pricing tells us that if there is no arbitrage, there must exist a positive random variable $M$ (also called stochastic discount factor) such that for any return $R$, we must have $1 = E[M R]$. Thus to understand asset prices, in cross-section as well as across time, we need to understand which factors affect $M$.

Now, the most basic Hansen-Jagannathan bound can be stated as

$$ \left| \frac{E[R^e]}{\sigma[R^e]} \right| \leq \frac{\sigma[M]}{E[M]} $$

where $R^e$ is an excess return. The main contribution is that it allows us to say something about moments of $M$ (right-hand side), which is unobservable, in terms of moments of returns (left-hand side), which can be (in principle) observed. Specifically, given the observed Sharpe ratio (say, around 0.4), the bound tells us that the SDF must be at least just as volatile.

The problem is that standard consumption-based models for $M$ have problems matching this volatility. For example, your basic Lucas-tree economy with CRRA utility leads to $M = \beta \exp(-\gamma \Delta c)$, where $\Delta c$ is log consumption growth. Since $E[M] \approx 1$, we have $\sigma[M] \approx \gamma \sigma[\Delta c]$. If $\sigma[\Delta c]$ is let's say around 1% yearly, we need to set $\gamma > 40$, an unrealistically high value of risk aversion.