The Return and Risk of a Financial Portfolio

A two-asset portfolio's risk is not the weighted average of the assets' risks. It is less, sometimes dramatically less, and that gap is the foundation of modern portfolio theory.

Put 60% of your money in a stock fund and 40% in a bond fund. Your expected return is simply 60% of one return plus 40% of the other. You calculate portfolio expected return as a weighted average of component expected returns, with weights summing to one.

$$E(R_p) = \sum_{i=1}^{n} w_i E(R_i)$$

Risk does not average that way. On days the stock fund drops, the bond fund may hold steady, so the two partly cancel. You calculate portfolio variance with each asset's own variance plus a cross term that captures how the two move together. For two assets:

$$\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \text{Cov}(R_1, R_2)$$

The weights are squared because variance is measured in squared returns. The cross term appears twice because asset 1 moves with asset 2 and asset 2 moves with asset 1.

Common mistakes

• Treating portfolio SD as the weighted average of SDs. SD is weighted-average only when $\rho = +1$. For any $\rho < +1$, portfolio SD is lower. Trap: computing 0.60 × 20% + 0.40 × 12% = 16.80% for the 60/40 mix above instead of 14.20%.
• Confusing the minimum-variance frontier with the efficient frontier. The minimum-variance frontier is the entire left edge. The efficient frontier is only the portion at or above the GMVP. Trap: calling 100% B (6%, 12%) efficient when the GMVP offers 6.72% at 11.45%.
• Believing diversification requires negative correlation. Any $\rho < +1$ produces diversification. Even $\rho = +0.9$ reduces variance below the weighted average. Trap: "the two assets are positively correlated, so no diversification benefit exists."

Bottom line

• Portfolio expected return is the weighted average of asset returns. Portfolio variance is NOT, the cross term $2 w_1 w_2 \rho \sigma_1 \sigma_2$ is essential
• Covariance is the probability-weighted product of deviations; correlation rescales it to $[-1, +1]$ by dividing by $\sigma_1 \sigma_2$
• Correlation below +1 produces diversification benefit. Lower correlation means greater variance reduction
• Minimum-variance frontier = lowest variance for each return level. Efficient frontier = upper half above the global minimum-variance portfolio (GMVP)

Exam shortcut

For two-asset variance, memorize the three-term formula and confirm you doubled the covariance cross term. For the GMVP weight in asset 1, remember the numerator is $\sigma_2^2 - \text{Cov}$: the asset whose weight you want has the OTHER asset's variance on top. For the optimal risky weight, write $y^* = \text{risk premium} / (A\sigma^2)$ in decimals, then check the answer choices for the 2x trap from a stray one half.

The full lesson (about 3,889 words, 26 min read) adds 10 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• return and risk of a financial portfolio

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