# Intuitive explanation of the Hansen-Jagannathan bound

The Hansen-Jagannathan bound states that the maximum Sharpe ratio of a portfolio can't exceed the ratio of the standard deviation of a stochastic discount factor to its mean. I more or less understand the meaning of all three concepts: standard deviation, mean and stochastic discount factor but I'm at loss about how they relate here. So:

What is the intuition behind this result and why is this useful?

• This question is part of this weeks topic challenge, see meta.quant.stackexchange.com/q/1369/35?cb=1 for more info. Nov 2, 2013 at 12:04

The HJ bounds state that $$\frac{\sigma(m)}{\mathbb{E}[m]} \geq \frac{|\mathbb{E}[R^e]|}{\sigma(R^e)}$$ where $R^e$ is the excess return of an asset or portfolio, $\sigma$ denotes standard deviation, $\mathbb{E}$ denotes expectation w.r.t. the statistical measure, and $m$ is a stochastic discount factor (or state-price density/kernel, etc.) that prices the return: $$0 = \mathbb{E}[mR^e]$$
Chapter 21 of John Cochrane's book on asset pricing contains a nice discussion of what the HJ bounds tell us about the Equity Premium Puzzle (the following is more or less on-to-one from the chapter) Empirically, the HJ bound implies that the SDF has to be very volatility with a mean near one. This fact has been used a lot in the investigation of the equity premium puzzle. The Sharpe ratio of the US market is about 0.5 (8% return with 16% volatility). The average risk-free rate is 1%, so $\mathbb{E}[m] = 0.99$ (Assuming there is a risk-free rate $R^f = 1+r^f$, we have $\mathbb{E}[m] = \frac{1}{R^f}$). Therefore, $\sigma(m) \geq 0.5$ on an annual basis. Following the basic consumption model, this either implies very extreme risk aversion or consumption growth volatility.