# Prove that the portfolio that maximizes utility lies on the efficient frontier

When maximizing mean-variance utility in a portfolio optimization framework

$$max \{R - \lambda \sigma ^2\}$$

where R is portfolio return, $$\lambda$$ is a risk aversion parameter, and $$\sigma^2$$ is portfolio volatility, how can I be sure that the result lies on the efficient frontier? I can show that $$\lambda$$ is effectively equals $$\frac{R}{\sigma^2}$$ but I don't quite see how this problem develops the efficient frontier

The efficient frontier is defined as the set of portfolios which have the highest return for a given measure of volatility, i.e. $$\{S: s \in P \; s.t. \nexists \; t \in P \; \text{where} \;R(s) < R(t) \; \text{and} \; \sigma(s)=\sigma(t) \}$$, where $$P$$ is the set of all validly constructed portfolios.
The optimisation framework $$max \{R-\lambda\sigma^2 \}$$ must therefore return the efficient frontier since by definition there does not exist a valid portfolio for a given risk or variance where $$R$$ is greater and therefore increases the objective function above that of the efficient frontier.
• Thanks. If I understand it correctly, selecting values of $\lambda$ is equivalent to selecting values of $\frac{R}{\sigma^2}$ which is effectively a point on the efficient frontier. You are asking the optimizer to find the value of $R$ and $\sigma^2$ on the point on the frontier where the slope equals $\lambda$. But what if there are multiple points on the efficient frontier where the slope equals $\lambda$ ? In that case, there isnt a unique solution. Am i thinking about this correctly? – Wadstk Oct 28 '19 at 14:05