Comparing Exponential and Mean-Variance Risk-Sensitive Objectives

Summary

The question compares two optimization objectives: minimizing the logarithm of the expected exponential of a random outcome, and minimizing its expectation subject to a variance bound. It asks whether the exponential criterion is preferable because its expansion reflects moments beyond the mean, and whether this makes it generally more useful as a risk-sensitive cost.

The response resists calling the first objective better, describing it as rigid. It suggests that if a decision-maker wants to account for higher moments, it may be clearer to specify the moments directly rather than implicitly incorporating an unrestricted range of them through the exponential criterion. The exchange offers a qualitative preference, not a formal comparison or conditions under which either objective performs well. It does not define the application, discuss parameter choices, or establish how either formulation behaves for particular distributions.

Key ideas

• The first objective uses the logarithm of an exponential moment, while the second constrains variance.
• The question notes that the exponential criterion can reflect higher moments.
• The response describes that criterion as rigid rather than universally superior.
• Explicitly including selected higher moments is suggested as an alternative to the exponential objective.
• No formal comparison or application-specific guidance is provided.

Tags

# Which is the better risk sensitive measure?


# Which is the better risk sensitive measure?












Consider the two following optimization problem

1) $$ \min_{\theta} \ln E_{\theta}[ e^{X}]$$

2) $$ \min_{\theta} E_{\theta}[ X]$$ with the constraint $$ Var_{\theta}[X] <c$$

Is it true that the first one is a better measure for risk-sensitivity cost as it takes other moments (which is visible from the Taylor series expansion) also into account ?

In general, can we say anything about the usefulness of the first cost compared to any other risk-sensitive cost ?

## Answer by James (score 1)

https://quant.stackexchange.com/a/14715

I wouldn't say 1) is better because it's very rigid. If you want to include some higher moments (most likely you won't need more than 4th order), it's better to do it explicitly rather than to stuff the moments of all orders into the criterion.

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