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?

  • $\begingroup$ Hint: the quadratic utility function exhibits increasing absolute risk aversion $\endgroup$
    – Ali
    Oct 18, 2019 at 19:19

1 Answer 1


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 $(\lambda)$.

  • 1
    $\begingroup$ Line 3 doesn't distribute the negative from $-\lambda$ to the final term. The solution should be $=\mu_w-\lambda\sigma_w^2-\lambda\mu_w^2$. See for reference. $\endgroup$
    – J.R.
    Nov 5, 2021 at 14:00
  • $\begingroup$ Good catch and now fixed. $\endgroup$
    – BKay
    Nov 5, 2021 at 18:43

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