The Hansen–Jagannathan Bound and Limits on Sharpe Ratios
Summary
The Hansen–Jagannathan bound links the volatility of a stochastic discount factor (SDF) to the Sharpe ratio of excess returns it prices. The pricing condition is that the expected product of the SDF and an excess return equals zero. This implies that the SDF’s standard deviation divided by its mean must be at least as large as the absolute Sharpe ratio.
The bound constrains which discount factors can price a given set of returns, and which returns are compatible with a given discount factor. The document illustrates the implication using estimates for the US market and risk-free rate: the SDF must be quite volatile relative to its mean. In a basic consumption-based asset-pricing model, that requirement points to very high risk aversion or unusually volatile consumption growth. The example is an economic interpretation, not a standalone test; the bound itself does not identify which assumption or model component explains the required SDF volatility.
Key ideas
- The bound requires the SDF’s coefficient of variation to be at least the absolute Sharpe ratio of priced excess returns.
- The pricing condition for excess returns is that their product with the SDF has zero expected value.
- The bound restricts both feasible discount factors and returns consistent with a specified discount factor.
- The document connects the implied SDF volatility to the equity premium puzzle and consumption-based model assumptions.
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# Intuitive explanation of the Hansen-Jagannathan bound
# 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?
## Answer by pbr142 (score 10, accepted)
https://quant.stackexchange.com/a/10538
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.