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I want to calculate the expectation and variance in the following scenario:

$w$ is my initial wealth

With probability $0< q_i <1$ with$ i \in\{a,b,c\}$ I lose $a,b$ or $c$ repectively.

$U()$ is a concave utility function, so with probability $q_a$ my utility is $U(w-a)$

I think I can calculate the expected utility via:

$E(U) = q_a(U(w-a))+q_b(U(w-b))+q_c(U(w−c)) $

but I have no idea how to calculate the variance. Any hints on that?

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Variance? X? U? pi? Indications about what you tried? –  Did Feb 22 '13 at 14:43
    
Can you please your notation? –  Mohan Feb 22 '13 at 16:03
    
@Did: I edited my question to make it more clear –  Tom Feb 22 '13 at 16:45
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1 Answer

The formula for the expected utility $E(U)$ is correct.

Using $E(U)$ the variance can be obtained by: $\sigma^2=q_a\cdot(U(w-a)-E(U))^2+q_b\cdot(U(w-b)-E(U))^2+q_c\cdot(U(w-c)-E(U))^2$

or alternatively by: $\sigma^2=E(U^2)-[E(U)]^2$

with $E(U^2)=q_a\cdot [U(w-a)]^2+q_b\cdot[U(w-b)]^2+q_c\cdot[U(w-c)]^2$.

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