Expected Return and Variance of Financial Assets

Core Idea

The expected return of an asset is the probability-weighted average of its possible returns: E[r] = Σ pᵢrᵢ. Variance measures dispersion around the mean: σ² = Σ pᵢ(rᵢ − E[r])². For a portfolio of two assets with weights w₁ and w₂, portfolio variance is σ²_p = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(r₁,r₂). This covariance term is the key insight behind diversification: assets that do not move perfectly together reduce portfolio risk below the weighted average of individual risks. The magnitude of this reduction depends entirely on the correlation between the two assets.

How It's Best Learned

Calculate expected return and variance from a probability distribution, then repeat using historical return data to see how the statistical formulas apply empirically. Compute two-asset portfolio variance at correlations of −1, 0, and +1 to see the full range of what diversification can achieve.

Common Misconceptions

• Omitting the covariance term from portfolio variance is the most common calculation error and entirely misses the point of diversification.
• Assuming returns are normally distributed is convenient but wrong — financial returns exhibit fat tails and negative skewness that the normal distribution systematically underestimates.

Explainer

You already know from probability theory that the expected value of a random variable is its probability-weighted average, and that variance measures how spread out the outcomes are around that average. Financial assets are random variables: their returns are uncertain. Applying those statistical definitions to returns gives you the expected return E[r] = Σ pᵢrᵢ and variance σ² = Σ pᵢ(rᵢ − E[r])². These are the same formulas you learned — just applied to future returns instead of abstract outcomes.

The interesting material begins when you combine assets into a portfolio. Portfolio expected return is simply the weighted average of individual expected returns: E[rₚ] = w₁E[r₁] + w₂E[r₂]. Simple and intuitive. Portfolio variance, however, is not the weighted average of individual variances. The formula is σ²_p = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(r₁,r₂). That third term — the covariance — is where all the action is. It measures whether the two assets tend to move together or in opposite directions.

The covariance term is the mathematical basis for diversification. If two assets have zero correlation, their covariance is zero, and the portfolio variance is less than the weighted average of the individual variances — you got risk reduction for free, just by combining them. If they have negative correlation, the covariance term is negative, reducing portfolio variance further. In the extreme case of correlation = −1, you can combine the assets in specific weights to achieve zero portfolio variance entirely. The intuition: when one asset zigs, the other zags, and the movements cancel out.

The practical upshot is that what matters for a portfolio is not just an asset's own variance but its covariance with everything else already in the portfolio. An asset with high individual variance but low correlation with the rest of the portfolio can actually reduce total portfolio risk when added. This is why international diversification works (equity markets across countries have historically been less than perfectly correlated) and why bonds are valuable in equity-heavy portfolios (bond and equity returns often move in opposite directions during market stress).

One important caveat: correlations between assets are not stable. During market crises, correlations across asset classes tend to spike toward +1 — the assets that were supposed to diversify each other start moving together exactly when you need diversification most. This "correlation breakdown" is one reason why realized portfolio losses during crashes are often larger than models predicted, and why variance alone understates actual risk in tail scenarios.