Modern Portfolio Theory and the CAPM | Free FRM Part I Lesson

A pension fund holds two equity managers. Manager A returned 14% with 22% volatility; Manager B returned 11% with 12% volatility. Headline performance favors A. Risk-adjusted performance reverses that ordering. The exam tests whether you can convert raw returns into the right risk-adjusted measure for the question's framing.

Markowitz's framework treats portfolio choice as a constrained optimization. You pick weights to minimize variance for each target expected return. Plot the optimal portfolios on a (return, standard-deviation) chart and you trace out the efficient frontier. Anything below the frontier is suboptimal: you can earn more return for the same risk by moving up to the frontier.

Two ideas drive the math. Expected portfolio return is the weighted average of asset returns:

$E [R_{p}] = i = 1 \sum N w_{i} E [R_{i}]$

But portfolio variance is not a weighted average. Covariances matter:

$σ_{p}^{2} = i \sum j \sum w_{i} w_{j} σ_{ij}$

When two assets are imperfectly correlated, combining them reduces variance below either asset's individual variance. That is the diversification benefit. It is mechanical, not magical.

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Common mistakes

Treating CML as a pricing line for individual assets. The CML prices only efficient portfolios; an individual stock can sit far to the right of the CML even when correctly priced. Use the SML for individual assets. Trap: a question asks "is XYZ stock fairly priced?" and shows it below the CML.
Computing portfolio variance as a weighted average. $σ_{p}^{2}$ is NOT $\sum w_{i} σ_{i}^{2}$ . Covariances dominate. With two assets at 20% σ each and ρ = 0, an equal-weight portfolio has σ ≈ 14.1%, not 20%. Trap: 20% appears as choice B and catches candidates who skip the covariance term.
Using Sharpe for portfolios with asymmetric returns. Funds with strong positive skew (option-selling, trend-following with stop-losses) get penalized by Sharpe for their upside volatility. Sortino is the right metric. Trap: the question says "the manager's strategy generates positive skew", so Sortino is the answer, not Sharpe.

Bottom line

Efficient frontier = portfolios that minimize variance at each level of expected return (Markowitz quadratic optimization); diversification works through covariance, not return averaging.
CAPM: $E [R_{i}] = R_{f} + β_{i} (E [R_{m}] - R_{f})$ . Single-factor model pricing systematic risk only; when $β = 1$ the asset's expected return equals the market's.
CML plots efficient portfolios in (return, total-risk) space; SML plots all assets in (return, beta) space (different x-axis, different population).
Beta = $Cov (R_{i}, R_{m}) / σ_{m}^{2} = ρ_{i, m} (σ_{i} / σ_{m})$ ; measures market sensitivity, not total risk, and portfolio beta is the weighted average.

Exam shortcut

When a question gives you total return and asks for risk-adjusted performance, the denominator decides the metric: σ → Sharpe, β → Treynor, tracking error → IR, downside σ → Sortino. Skip the formulas, match the denominator, and the answer pops out. Memory aid: "Sharpe sees Sigma, Treynor takes Two-Greek-letters [β], Information needs Index [benchmark]." And remember: CML for portfolios, SML for assets: different x-axis, different population.

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Learning objectives

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