The Return and Risk of a Financial Portfolio | Free CFA Level I Lesson

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.

Portfolio expected return is a weighted average of component expected returns with weights summing to one.

$E (R_{p}) = i = 1 \sum n w_{i} E (R_{i})$

Variance is harder because assets co-move. For two assets:

$σ_{p}^{2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} Cov (R_{1}, R_{2})$

Covariance measures co-movement in raw units and is unbounded. Correlation rescales it to a bounded range.

$ρ_{1, 2} = \frac{Cov ( R _{1} , R _{2} )}{σ _{1} σ _{2}}$

Substituting $Cov = ρ σ_{1} σ_{2}$ :

$σ_{p}^{2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} ρ_{1, 2} σ_{1} σ_{2}$

KEY: When $ρ = + 1$ , portfolio SD equals the weighted average of SDs and there is no diversification. When $ρ < + 1$ , portfolio SD is lower than the weighted average. When $ρ = - 1$ , portfolio SD can equal zero with the right weights.

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Common mistakes

Treating portfolio SD as the weighted average of SDs. SD is weighted-average only when $ρ = + 1$ . For any $ρ < + 1$ , 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. Portfolios below the GMVP are dominated.
Believing diversification requires negative correlation. Any $ρ < + 1$ produces diversification. Even $ρ = + 0.9$ 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 $2 w_{1} w_{2} ρ σ_{1} σ_{2}$ is essential
Correlation below +1 produces diversification benefit. Lower correlation means greater variance reduction
Correlation rescales covariance to $[- 1, + 1]$ , enabling cross-asset comparison
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 $σ_{2}^{2} - Cov$ : the asset whose weight you want has the OTHER asset's variance on top. For CAL vs CML, ask whether the problem invoked market equilibrium or homogeneous expectations. If yes, CML. If no, CAL.

The full lesson (about 1,816 words, 12 min read) adds 2 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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