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?

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

1 Answer 1


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: $$ 0 = \mathbb{E}[mR^e] $$

Economically, the HJ bound is therefore a restriction on the set of possible discount factors that can price a given set of (excess) returns and, at the same time, a restriction on the set of returns that can be observed for a given discount factor.

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.

  • $\begingroup$ Could you please clarify how you exactly you calculated 𝔼[m]=0.99. It's because 1/(1+rf) = 1/(1+0.01) = 1/(1.01)=0.99, right ? $\endgroup$ Commented Jun 4, 2014 at 20:41
  • $\begingroup$ If 𝔼[m]=0.99 then σ(m) ≥ 0.99*0.5 <=> σ(m) ≥ 0.495 or not? How is the variance of the SDF related to risk aversion? $\endgroup$ Commented Jun 4, 2014 at 20:47

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