Interpreting Hansen-Jagannathan Bounds on Discount Factor Volatility

Summary

The document asks how to interpret the Hansen-Jagannathan bound, which relates the volatility of a stochastic discount factor to the Sharpe ratio of excess returns. The question focuses on constructing a lower bound on discount-factor volatility when its mean is specified. It describes an approach that treats the reciprocal of the assumed mean as a hypothetical risk-free rate, then considers the maximum Sharpe ratio available by subtracting that rate from returns.

The author questions whether this constructed maximum is valid when it varies with the assumed discount-factor mean, unlike the maximum Sharpe ratio in excess-return space. The text provides no answer or derivation, so it does not resolve the concern. It identifies a conceptual issue in applying the bound and distinguishes the fixed excess-return opportunity set from a return-space calculation using a hypothetical risk-free rate. No empirical evidence or practical application is given.

Key ideas

• The Hansen-Jagannathan relation connects excess-return Sharpe ratios to discount-factor volatility.
• The question considers bounding discount-factor volatility for a specified mean value.
• A proposed construction uses the reciprocal of the mean as a hypothetical risk-free rate.
• The document asks whether a varying return-space Sharpe ratio can validly bound the excess-return Sharpe ratio, but supplies no resolution.

Tags

# Question on Cochrane's Asset Pricing Section 5.6: HJ Bounds


# Question on Cochrane's Asset Pricing Section 5.6: HJ Bounds












I'm having trouble understanding pg. 93 of Cochrane's Asset Pricing textbook.

As seen in equation 5.23,

$$\frac{\sigma(m)}{E(m)} \ge \frac{|E(R^e)|}{\sigma(R^e)}$$

the Sharpe ratio on excess returns bounds the discount factor. However, to find a lower bound on $\sigma(m)$ for a given value of $E[m]$, it seems like the author is varying the value of $E[m]$, using the value to get a hypothetical risk-free rate ($1/E[m]$), finding the maximum Sharpe ratio that can be constructed by subtracting the hypothetical risk-free rate from a return in the return space, and using that maximum Sharpe ratio to bound $\frac{\sigma(m)}{E(m)}$.

I don't get why this is a valid approach. The maximum Sharpe ratio of excess returns in the excess return space is a constant, and the maximum Sharpe ratio that can be constructed by subtracting the hypothetical risk-free rate from a return in the return space varies with the assumed value of $E[m]$. So at best it must be that the maximum Sharpe ratio that can be constructed by subtracting the hypothetical risk-free rate from a return in the return space is an upper bound for the Sharpe ratio of excess returns, for any value of $E[m]$.

Is this so? If so, how, and if not, where did I make a mistake?

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