# 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:

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What do you mean by indifference equation? – Bob Jansen Aug 5 '13 at 18:40
Do you mean utility function? – John Aug 5 '13 at 19:47
I am afraid you need to elaborate because your "indifference curves", "utility curves" (whatever they are because you have not really defined them yet) are very off, meaning, of non standard shape. I would even argue that a trade-off between risk and return should not look like general utility curves at all. The more you increase risk the higher the chance your expected return is actually gonna be negative. Also, this should not be mean return but expected return in combination with volatility, which I assume reflects future realized portfolio variation and not historical one. – Matt Wolf Aug 6 '13 at 5:15

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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