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Why does investors having quadratic utility function mean that their optimal portfolios can be chosen by only considering mean and variance of returns i.e. imply $\mu-\sigma$ preference?

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  • $\begingroup$ Hint: the quadratic utility function exhibits increasing absolute risk aversion $\endgroup$ – user20105 Oct 18 '19 at 19:19
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If you have quadratic preferences then your utility function is: $$ U(W) = W - \lambda W^2$$ this implies your expected utility function looks like: $$ E[U(W)] = E[W - \lambda W^2] = E[W] - \lambda E[W^2]$$ $$ = E[W] - \lambda E[W^2 - E[W]^2 + E[W]^2]$$ $$ = E[W] - \lambda E[W^2 - E[W]^2] + \lambda E[E[W]^2]]$$ $$ = \mu_w - \lambda \sigma_w^2 + \lambda \mu_w^2$$

Therefore, we have established that expected utility depends only only the mean $(\mu_W)$ and variance $(\sigma^2_W)$ of wealth and the risk parameter $(\lambda)$.

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