Portfolio expected return and variance

Weights: the starting point

Every portfolio calculation starts with weights. A holding's weight is its market value divided by the total market value of the portfolio. Because every currency unit sits in some holding, the weights add up to 1 (100%).

Weights are based on money invested, not on the number of assets. A portfolio with EUR 900,000 in shares and EUR 100,000 in bonds is 90/10, even though it holds two assets.

The exam often gives you amounts instead of weights. Convert first, then use the formulas below.

Expected return: a simple weighted average

The expected return of a portfolio is the weighted average of the expected returns of its holdings. If you expect 9% from equities and 4% from bonds and hold 60/40, you expect 0.6 × 9% + 0.4 × 4% = 7.0% from the mix.

This works because the portfolio return in any outcome is just the weighted sum of the holdings' returns, and the expected value of a weighted sum is the weighted sum of expected values.

Note what is missing: correlation, covariance and standard deviation play no role in expected return. Two portfolios with the same weights and the same asset expected returns have the same expected return, however differently the assets move together.

Variance: risk needs co-movement

Portfolio variance is the expected value of the squared deviation of the portfolio return from its expected return, $\sigma^2(R_p) = E\{[R_p - E(R_p)]^2\}$. It is a forward-looking measure of how widely the portfolio return may spread around its mean.

Unlike expected return, variance is not a weighted average of the holdings' variances. When you square a sum of weighted returns, cross terms appear. Those cross terms are covariances: they measure whether two returns tend to be above or below their means at the same time.

For two assets the result has three pieces. The first two, $w_1^2\sigma_1^2$ and $w_2^2\sigma_2^2$, are each asset's own risk scaled by the squared weight. The third, $2w_1w_2\text{Cov}(R_1,R_2)$, is the co-movement term. It appears twice (hence the 2) because asset 1 pairs with asset 2 and asset 2 pairs with asset 1.

If the covariance is given as a correlation, convert it first: $\text{Cov}(R_1,R_2) = \rho_{12}\sigma_1\sigma_2$. Then take the square root of the variance to get the standard deviation, $\sigma_p$.

Mixing systems is the most common source of wrong answers. If a covariance matrix is given in %², use $\sigma$ in percent (15, not 0.15). Both systems give the same final standard deviation.

Interpreting the result

Expected return tells you the centre of the distribution of portfolio returns; standard deviation tells you its typical spread. Together they are the two numbers modern portfolio theory uses to compare portfolios: reward and risk.

Because the covariance term can be small or even negative, portfolio standard deviation is normally below the weighted average of the holdings' standard deviations. It equals that average only when the assets are perfectly positively correlated. The shortfall below the weighted average is a measure of the diversification benefit, which the later lessons explore.

Special case: adding a risk-free asset. A risk-free asset (such as a government bill held to maturity) has a return known in advance, so its standard deviation is 0 and its covariance with any risky asset is 0. In the two-asset formula, both the risk-free term and the covariance term drop out, leaving $\sigma_p^2 = w^2\sigma^2$ for the risky holding.

Taking the square root gives $\sigma_p = w\,\sigma$: portfolio risk scales in a straight line with the weight in the risky asset. Put 60% in a risky fund with $\sigma$ = 15% and the rest in the risk-free asset, and $\sigma_p$ = 0.6 × 15% = 9%. Multiply by the weight itself, not by its square root, and do not ignore the risk-free slice.