<p>In typical mean-variance analysis, the risk-adjusted relative value of an individual asset takes the general form</p>
<p><span class="math-container">$\frac{\mu}{\sigma^2}$</span></p>
<p>with further weighting and normalization depending on the particular optimization conditions.</p>
<p>In economics books, I have also read the suggestion that this structure is a consequence of assuming a quadratic utility function in combination with the assumption of gaussian returns. (Is that sufficient?)</p>
<p>I have seen some examples of expectation value computations, but I have not seen the general connection between utility functions, distributions, and the form of risk derived explicitly. What is the proper procedure to derive the form of risk-adjusted value from an arbitrary utility function and distribution? Or is there a good reference that shows this?</p>

CC BY-SA 4.0
2024-05-07T12:50:33.900Z
Related: https://quant.stackexchange.com/questions/79292
Richard Hardy
CC BY-SA 4.0
2024-05-07T16:10:05.180Z
@RichardHardy the word you are looking for is "cross-posted".
Giskard
CC BY-SA 4.0
2024-05-07T20:04:00.503Z
@Giskard, right. I was in a rush and got this one a bit wrong.
Richard Hardy
<utility><statistics><financial-economics><risk><variance>
What is the proper way to derive risk definitions from utility functions?
Machinus
# What is the proper way to derive risk definitions from utility functions?

(Asked by: username_0 on 2024-05-07)

In typical mean-variance analysis, the risk-adjusted relative value of an individual asset takes the general form

$\frac{\mu}{\sigma^2}$

with further weighting and normalization depending on the particular optimization conditions.

In economics books, I have also read the suggestion that this structure is a consequence of assuming a quadratic utility function in combination with the assumption of gaussian returns. (Is that sufficient?)

I have seen some examples of expectation value computations, but I have not seen the general connection between utility functions, distributions, and the form of risk derived explicitly. What is the proper procedure to derive the form of risk-adjusted value from an arbitrary utility function and distribution? Or is there a good reference that shows this?

**Tags:** utility, statistics, financial-economics, risk, variance

> **Comments:**
> **username_1**: Related: https://quant.stackexchange.com/questions/79292  
> **username_2**: @RichardHardy the word you are looking for is "cross-posted".  
> **username_1**: @username_2, right. I was in a rush and got this one a bit wrong.  

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## 0 answers

