Mean-Variance Indifference Curves and Risk-Return Utility

Summary

This short discussion addresses why indifference curves in mean-variance portfolio theory appear convex, and how their appearance depends on the graph’s axes. The answer suggests reversing the axes to make the relationship resemble a familiar utility curve: expected portfolio return takes the role of utility, while portfolio volatility squared is penalized according to a risk-aversion parameter. Under that formulation, investors require greater expected return to accept additional risk, with the marginal compensation changing along the curve.

The source points to a derivation but does not include it, so it provides only a compact intuition rather than a formal proof or worked portfolio example. Its central teaching is that the apparent shape should be read together with the utility function and axis orientation. The explanation also uses variance as the risk measure; it does not discuss alternative risk measures, parameter estimation, or how an investor should choose the risk-aversion coefficient.

Key ideas

• Mean-variance utility rewards expected portfolio return and penalizes variance according to risk aversion.
• Indifference-curve shape depends on which quantity is placed on each graph axis.
• Reversing the axes can make the relationship resemble a familiar utility curve.
• The brief answer offers intuition but omits the referenced derivation and an empirical example.

Tags

# Why are indifference equations in mean-variance portfolio theory convex shaped


# Why are indifference equations in mean-variance portfolio theory convex shaped












As the title suggests why is the indifference equations in mean variance portfolio theory convex shaped?

Indifference Equation: https://en.wikipedia.org/wiki/Indifference_curve

A graph:

## Answer by Bob Jansen (score 1)

https://quant.stackexchange.com/a/8680

I agree with @MattWolf The graph you show is confusing and evil, it makes me feel dumb every time I look at it. So I inverted the axis.

Now we see the familiar shape of an utility curve, discussed in your previous question. It is upward sloping at a declining rate. In this case $u$ takes the place of $R_p$ and the general form of mean variance utility is $$u(R_p, \sigma_p) = R_p - \lambda \sigma^2_p$$

This derivation might be of interest.

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