Deriving the Hansen–Jagannathan Bound from Correlation Limits

Summary

The note clarifies a step in the Hansen–Jagannathan bound derivation for asset pricing. The starting equation expresses an asset’s excess return divided by its volatility as the negative of its correlation with the stochastic discount factor, multiplied by the discount factor’s volatility relative to its mean.

Since a correlation lies between negative one and one, its absolute value cannot exceed one. Taking absolute values in the equation therefore bounds the absolute risk-adjusted excess return by the discount factor’s volatility-to-mean ratio. The response explains this directly through the correlation identity; it does not provide a separate Cauchy–Schwarz derivation. The explanation is concise and applies to the stated equation and assumptions, without discussing broader interpretations or extensions of the bound.

Key ideas

• The equation writes a risk-adjusted excess return as a correlation times a volatility-to-mean ratio.
• The absolute value of a correlation is at most one.
• Taking absolute values therefore gives the stated upper bound.
• The answer explains the algebraic step directly rather than deriving the bound from Cauchy–Schwarz.

Tags

# Hansen-Jagannathan bounds derivation: last step is not clear


# Hansen-Jagannathan bounds derivation: last step is not clear












Pennachi's "Asset Pricing" chapter 4 derives:

$$ \frac{E[R_{i}-R_{f}]}{\sigma_{R_{i}}}=-\rho_{m_{01},R_{i}}\frac{\sigma_{m_{01}}}{E[m_{01}]} $$

Then, he states that the fact that $-1\leq \rho_{m_{01},R_{i}} \leq 1$ implies that: $$ \left | \frac{E[R_{i}-R_{f}]}{\sigma_{R_{i}}} \right | \leq \frac{\sigma_{m_{01}}}{E[m_{01}]} $$

This last step is not clear to me, could you please explain how it follows? Wikipedia says that it follows from Cauchy–Schwarz inequality, but I cannot figure out how.

P.S. There is a question about H-J bounds already, but there is an intuitive explanation, and I couldn't find an answer over there.

## Answer by Bob Jansen (score 1, accepted)

https://quant.stackexchange.com/a/11275

I believe that last sentence on Wikipedia isn't about the last step you're showing. So let's ignore that.

Basically in the first equation $a=rb$ and we have $|r| \leq 1$ so certainly $b>a$ as we have in the second equation. Taking absolutes just drops the signs.

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