Lesson 2 · Portfolio ManagementCFA L1

CAPM and Performance Evaluation

Last lesson asked how an investor should build a portfolio. The Capital Asset Pricing Model asks the reverse: if everyone optimises, what must each asset's risk-return relationship be in equilibrium? The answer makes the market portfolio efficient and prices every asset by its beta — its exposure to undiversifiable risk. We then turn that idea into the four classic performance measures: Sharpe, M², Treynor and Jensen's alpha.

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Equilibrium & the market portfolio Capital market line The CAPM & beta Security market line The single-index model Diversification Performance measures Which measure to use

In equilibrium, demand equals supply for every risky asset and the riskfree asset has zero net supply (every lender matched by a borrower). The aggregate of all risky assets is the market portfolio — which, crucially, contains no riskfree asset.

Learning outcomes

State the CAPM assumptions and explain why the market portfolio is efficient.
Write the CAPM equation and compute and interpret beta.
Distinguish systematic from non-systematic risk and the SML from the CML.
Decompose risk with the single-index model and explain diversification.
Compute and apply the Sharpe ratio, M², Treynor ratio and Jensen's alpha.

Equilibrium and the market portfolio

Under the CAPM assumptions — every investor has mean-variance preferences over a single period; all share homogeneous beliefs about means, variances and correlations; trading is frictionless (no short-sale limits, no costs, infinitely divisible); a perfect lending/borrowing market exists at one rate; and markets clear — everyone solves the same optimal-risky-portfolio problem and so holds the risky assets in the same proportions. Therefore the market portfolio is the optimal risky portfolio.

The capital market line

The CAL joining the riskfree asset to the market portfolio M is the capital market line (CML):

Capital market line

$$ \mu=r_f+\frac{\mu_M-r_f}{\sigma_M}\,\sigma $$

Since M is efficient, so is every portfolio on the CML. A key implication: passive investing is sufficient — you need not solve the optimisation yourself; mixing the riskfree asset with a market-tracking portfolio is efficient for any investor.

The CAPM and beta

Because the opportunity set of any asset combined with M cannot cross the CML, equilibrium forces the pricing relation:

CAPM and beta

$$ E(r_i)=r_f+\beta_i\big(E(r_M)-r_f\big), \qquad \beta_i=\frac{\operatorname{Cov}(r_i,r_M)}{\operatorname{var}(r_M)}=\rho_{iM}\frac{\sigma_i}{\sigma_M} $$

Beta measures an asset's sensitivity to the broad market: positive moves with the market, negative against it. The riskfree asset has $\beta=0$ (but a zero-beta asset can still be risky); the market has $\beta=1$. Beta is estimated by regressing the asset's excess return on the market's excess return — the slope is $\beta$, and the regression $R^2$ is the fraction of total risk that is systematic. The CAPM says only systematic risk is rewarded; non-systematic (firm-specific) risk is diversifiable and earns nothing. Portfolio beta is the weighted average of constituent betas, and the CAPM holds for portfolios too.

Worked example

Portfolio beta and expected return

Google 30% (β 0.99), Apple 20% (β 1.28), S&P 500 40% (β ?), riskfree 10% (β ?), $r_f=1\%$. Find the market and riskfree betas, the portfolio beta, and the expected return if $E(r_M)=10\%$.

Show solution

The market's beta is 1, the riskfree's is 0. Portfolio $\beta_p=0.3(0.99)+0.2(1.28)+0.4(1)+0.1(0)=0.953$. Then $E(r_p)=1\%+0.953(10\%-1\%)=\textbf{9.577%}$.

β(market)=1, β(rf)=0; portfolio β=0.953; E(rp)=9.577%

The security market line

Plotting expected return against beta gives the security market line (SML); if CAPM holds, every asset and portfolio lies on it. Contrast with the CML: the SML prices every asset against its systematic risk $\beta$; the CML holds only for efficient portfolios and uses total risk $\sigma$. In practice CAPM is hard to apply — the inputs ($\beta$, $E(r_M)$, $r_f$) must be estimated with no consensus method, and the true "market portfolio" (all assets worldwide) can only be proxied by an index such as MSCI World (the S&P 500 is a fair proxy for US stocks, less so for other assets).

The single-index model

The CAPM's implications are easiest to see through the statistical single-index model (SIM):

Single-index model and risk decomposition

$$ r_i-r_f=\alpha_i+\beta_i(r_M-r_f)+\varepsilon_i $$

$$ \underbrace{\operatorname{var}(r_i)}_{\text{total}}=\underbrace{\beta_i^2\operatorname{var}(r_M)}_{\text{systematic}}+\underbrace{\operatorname{var}(\varepsilon_i)}_{\text{firm-specific}} $$

where $\varepsilon_i$ is zero-mean noise, uncorrelated with the market and with other assets' noise. Taking expectations links the two: if CAPM holds and the index is a good proxy for the market, $\alpha_i=0$ for all assets. A non-zero $\alpha$ measures over- or under-performance versus the CAPM benchmark. (SIM is purely statistical; CAPM is an economic equilibrium model — but SIM with $\alpha=0$ is the CAPM case.)

Diversification

Form an $n$-asset portfolio. Its variance is $\sigma_p^2=\beta_p^2\sigma_M^2+\operatorname{var}(\varepsilon_p)$, and with equal weights the firm-specific term shrinks:

Why idiosyncratic risk vanishes

$$ \operatorname{var}(\varepsilon_p)=\frac{1}{n}\,\overline{\sigma_\varepsilon^2}\;\xrightarrow[n\to\infty]{}\;0 \qquad\Rightarrow\qquad \sigma_p^2\to\beta_p^2\sigma_M^2 $$

So in a large portfolio, individual (firm-specific) risk is diversified away — only the systematic risk, captured by $\beta_p$, remains. This is precisely why CAPM rewards beta and not total volatility.

Performance measures

Performance evaluation summarises an investment's risk-return quality in a single, preference-free number (ex-post or ex-ante). The four classics differ in which risk they charge for:

The four measures

$$ \text{Sharpe}=\frac{\mu-r_f}{\sigma}, \qquad M^2=r_f+\sigma_M\cdot\text{Sharpe} $$

$$ \text{Treynor}=\frac{\mu-r_f}{\beta}, \qquad \alpha=\mu-\big[r_f+\beta(E(r_M)-r_f)\big] $$

Sharpe and M² use total risk $\sigma$ — M² rescales the investment to the market's volatility, giving an interpretable return and the same ranking as Sharpe. Treynor uses systematic risk $\beta$ (excess return per unit of beta), and Jensen's alpha is the excess return over the CAPM-implied return ($\alpha>0$ = outperformance; the regression intercept). If CAPM holds, all alphas are zero. (Sharpe/M² mislead when excess return is negative — riskier portfolios look "less bad.")

Which measure to use

Total risk vs systematic risk

If the investment will be the investor's entire holding, total risk matters → use Sharpe or M². If it will be part of a well-diversified portfolio, only systematic risk matters → use Treynor or Jensen's alpha (and if those two disagree, prefer Treynor, a per-unit-of-risk measure). For everything except the Sharpe ratio, choose the benchmark market portfolio with care.

Worked example

Fund selection for two different investors

Fund A: 11%, σ 10%, β 0.7. Fund B: 12%, σ 25%, β 0.6. S&P 500: 9%, σ 19%, β 1. $r_f=1\%$. Compute the four measures, then advise Alice (already holds a diversified portfolio) and Bernard (this will be his only investment).

Show solution

Fund A: Sharpe $=\frac{0.11-0.01}{0.10}=1.0$; M² $=1\%+19\%(1.0)=20\%$; Treynor $=\frac{0.10}{0.7}=0.14$; $\alpha=0.11-[0.01+0.7(0.09-0.01)]=4.4\%$. Fund B: Sharpe 0.44, M² 9.36%, Treynor 0.18, $\alpha$ 6.2%. Alice (diversified) cares about systematic risk → choose B (higher Treynor and alpha). Bernard (sole holding) cares about total risk → choose A (higher Sharpe and M²). The same two funds rank oppositely depending on the context.

A: SR 1, M² 20%, Tr 0.14, α 4.4% · B: SR 0.44, M² 9.36%, Tr 0.18, α 6.2% — Alice→B, Bernard→A

Key takeaway

In CAPM equilibrium the market portfolio is efficient, so every investor can simply hold the market plus the riskfree asset (the CML). Each asset is priced by its beta — $E(r_i)=r_f+\beta_i(E(r_M)-r_f)$ — because only systematic risk survives diversification (the single-index decomposition). For evaluation, charge for the relevant risk: Sharpe/M² (total risk) for a standalone holding, Treynor/Jensen's alpha (systematic risk) for an addition to a diversified portfolio — and remember a non-zero alpha is the manager's value-add the CAPM says shouldn't exist.

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