# Why does quadratic utility function imply $\mu-\sigma$ preference?

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?

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

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 $$(\lambda)$$.
• 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.