Portfolio Return & Risk

Means blend, risks partially cancel — correlation decides how much risk diversifies away.

The idea

A portfolio splits money between assets, and this topic computes the return and risk of the split. Put a fraction $w$ in asset $A$ and the rest, $1-w$, in asset $B$. Neither return is known in advance, so $R_{A}$ and $R_{B}$ are random variables, and so is the portfolio's return $R_{p} = wR_{A} + (1-w)R_{B}$. Its mean $\mu_{p}$ measures the expected reward and its standard deviation $\sigma_{p}$ measures the risk, and the two follow different rules:

$\mu_{p} = w\mu_{A} + (1-w)\mu_{B}, \qquad \sigma_{p}^{2} = w^{2}\sigma_{A}^{2} + (1-w)^{2}\sigma_{B}^{2} + 2w(1-w)\,\rho\,\sigma_{A}\sigma_{B}$

The mean is a weighted average of the two means. Linearity of expectation forces this, so how the assets move together cannot change the reward of a blend.

The variance is not a weighted average of the variances. The weights enter squared, which shrinks them, and a third term carries $\rho$, the correlation between the two returns, so that $\rho\,\sigma_{A}\sigma_{B}$ is their covariance. That term is the only place co-movement enters, and it is why a portfolio's risk depends on more than the risks of the assets in it.

Ways to work on it

• Walkthrough. Combine two assets into a portfolio and compute its expected return and risk.
• Practice. Compute a portfolio's mean and variance, and find the weight that removes all risk.
• Hardest. Find the minimum-variance weight for uncorrelated assets and the risk it achieves.

Not sure where to start? Take the ten-question placement test.