The Efficient Frontier
Sweep the mix between two assets and a curve appears — only its upper limb is worth holding.
The idea
The efficient frontier identifies which mixes of risky assets are worth holding. Split money between two risky assets $A$ and $B$, with weight $w$ on $A$ and $1-w$ on $B$. The portfolio's mean return is the weighted average of the two means, but its risk is not the weighted average of the two risks: the variance combines with squared weights, $w^{2}$ and $(1-w)^{2}$, plus a covariance term. The squared weights are smaller than the weights themselves, so unless the assets move in lockstep some of their shocks cancel and the mix is less risky than the average. That cancellation is diversification.
Draw every mix as a point, with risk $\sigma$ across and mean return $\mu$ up. Sweeping $w$ from $1$ to $0$ traces a curve from $A$ to $B$, and the cancellation bends it to the left, toward lower risk. The leftmost point — the nose of the curve — is the mix of least possible risk, and it normally holds some of both assets.
Definition (Efficient frontier).
A mix of the assets is efficient when no other mix offers a higher mean return at the same or lower risk. The set of efficient mixes is the efficient frontier.
Every point below the nose is beaten by the point directly above it, which carries identical risk and earns more. The efficient mixes therefore form the upper limb of the curve, from the nose up to the higher-return asset. Choosing a point on it depends on the investor's tolerance for risk.
Ways to work on it
- Walkthrough. Trace the two-asset risk-return curve and see which of its portfolios are efficient.
- Proof. Derive the frontier of two uncorrelated assets by calculus and locate its minimum-variance portfolio.
- Practice. Compute minimum-variance weights and returns, and spot dominated portfolios.
- Hardest. Pin down the minimum-variance portfolio completely — or build a riskless mix from two perfectly anticorrelated assets.
Not sure where to start? Take the ten-question placement test.