∫

∑

∂

∞

√

∏

Home/Financial Mathematics/Course 5

Course 5

Core Theory

6-7 hours

Capital Asset Pricing Model

Market equilibrium, systematic risk, and the Security Market Line

Learning Objectives

•Derive the CAPM equation from market equilibrium conditions
•Understand beta as a measure of systematic risk and calculate it
•Apply CAPM to cost of equity estimation and capital budgeting
•Evaluate portfolio performance using Jensen's alpha and Treynor ratio
•Understand CAPM's assumptions, limitations, and empirical evidence

1. CAPM Assumptions and Market Equilibrium

The Capital Asset Pricing Model, developed by Sharpe (1964), Lintner (1965), and Mossin (1966), describes asset pricing in equilibrium when all investors follow mean-variance portfolio theory.

Definition 1.1: CAPM Assumptions

Investors are mean-variance optimizers: Care only about expected return and variance
Homogeneous expectations: All investors have identical beliefs about returns, variances, covariances
Single period: Investors optimize over a single period
Perfect capital markets: No taxes, no transaction costs, divisible assets
Price takers: Investors cannot influence prices
Risk-free borrowing/lending: All can borrow and lend at risk-free rate $r_f$

Theorem 1.1: Market Portfolio Optimality

Under CAPM assumptions, in equilibrium:

All investors hold the same risky portfolio (the market portfolio $M$ )
The market portfolio contains all risky assets weighted by their market capitalization
The market portfolio is the tangency portfolio from Portfolio Theory

Proof of Market Portfolio Optimality (Sketch):

Step 1: By the separation theorem, all investors combine the risk-free asset with the same tangency portfolio $T$ .

Step 2: In equilibrium, markets must clear - supply equals demand for each asset.

Step 3: If everyone holds portfolio $T$ , then aggregate demand for each asset must equal its supply. Therefore, $T$ must be the market-cap weighted portfolio of all assets.

Step 4: This market portfolio $M$ is mean-variance efficient - it lies on the efficient frontier.

∎

Remark 1.1: Implications of Market Portfolio

The market portfolio is:

Fully diversified (contains all risky assets)
Observable (in principle) - often proxied by broad stock indices like S&P 500
The optimal risky portfolio for all investors regardless of risk aversion
Mean-variance efficient by construction

Individual risk preferences only affect allocation between $M$ and the risk-free asset.

2. The CAPM Equation and Beta

Definition 2.1: Beta

The beta of asset $i$ with respect to the market portfolio $M$ is:

\beta_i = \frac{\text{Cov}(R_i, R_M)}{\text{Var}(R_M)} = \frac{\sigma_{iM}}{\sigma_M^2}

Beta measures the asset's sensitivity to market movements - how much $R_i$ changes when $R_M$ changes.

Theorem 2.1: CAPM / Security Market Line

In equilibrium, the expected return on any asset $i$ satisfies:

\mathbb{E}[R_i] = r_f + \beta_i (\mathbb{E}[R_M] - r_f)

where:

$r_f$ = risk-free rate
$\mathbb{E}[R_M]$ = expected return on market portfolio
$\mathbb{E}[R_M] - r_f$ = market risk premium
$\beta_i(\mathbb{E}[R_M] - r_f)$ = risk premium for asset $i$

Proof of CAPM Equation:

Method 1: Using CML and Covariance

Consider a portfolio $P$ with weight $\alpha$ in asset $i$ and $(1-\alpha)$ in market portfolio $M$ :

R_P = \alpha R_i + (1-\alpha)R_M

\mathbb{E}[R_P] = \alpha \mathbb{E}[R_i] + (1-\alpha)\mathbb{E}[R_M]

\sigma_P^2 = \alpha^2 \sigma_i^2 + (1-\alpha)^2\sigma_M^2 + 2\alpha(1-\alpha)\sigma_{iM}

At $\alpha = 0$ (pure market portfolio), the marginal rate of substitution between risk and return must equal the CML slope:

\frac{d\mathbb{E}[R_P]/d\alpha}{d\sigma_P/d\alpha}\bigg|_{\alpha=0} = \frac{\mathbb{E}[R_M] - r_f}{\sigma_M}

Computing derivatives at $\alpha = 0$ :

\frac{d\mathbb{E}[R_P]}{d\alpha}\bigg|_{\alpha=0} = \mathbb{E}[R_i] - \mathbb{E}[R_M]

\frac{d\sigma_P}{d\alpha}\bigg|_{\alpha=0} = \frac{\sigma_{iM} - \sigma_M^2}{\sigma_M}

Therefore:

\frac{\mathbb{E}[R_i] - \mathbb{E}[R_M]}{(\sigma_{iM} - \sigma_M^2)/\sigma_M} = \frac{\mathbb{E}[R_M] - r_f}{\sigma_M}

\mathbb{E}[R_i] - \mathbb{E}[R_M] = \frac{\sigma_{iM} - \sigma_M^2}{\sigma_M^2}(\mathbb{E}[R_M] - r_f)

\mathbb{E}[R_i] = r_f + \frac{\sigma_{iM}}{\sigma_M^2}(\mathbb{E}[R_M] - r_f) = r_f + \beta_i(\mathbb{E}[R_M] - r_f)

∎

Proposition 2.1: Properties of Beta

Market beta: $\beta_M = 1$ by definition
Risk-free beta: $\beta_{rf} = 0$ (no covariance with market)
Portfolio beta: $\beta_P = \sum_i w_i \beta_i$ (weighted average)
High beta: $eta > 1$ means more volatile than market (aggressive)
Low beta: $0 < \beta < 1$ means less volatile than market (defensive)

Example 2.1: Calculating Beta from Data

Given monthly returns data, estimate beta using regression:

R_i - r_f = \alpha_i + \beta_i(R_M - r_f) + \epsilon_i

Ordinary least squares (OLS) gives:

\hat{\beta}_i = \frac{\text{Cov}(R_i - r_f, R_M - r_f)}{\text{Var}(R_M - r_f)} \approx \frac{\text{Cov}(R_i, R_M)}{\text{Var}(R_M)}

Example: If stock has correlation 0.6 with market, $\sigma_i = 30\%$ , $\sigma_M = 20\%$ :

\beta = \rho_{iM} \frac{\sigma_i}{\sigma_M} = 0.6 \times \frac{0.30}{0.20} = 0.9

Corollary 2.1: Risk Decomposition

Total variance can be decomposed:

\sigma_i^2 = \beta_i^2 \sigma_M^2 + \sigma_{\epsilon_i}^2

Systematic risk: $\beta_i^2 \sigma_M^2$ - cannot be diversified
Idiosyncratic risk: $\sigma_{\epsilon_i}^2$ - diversifiable, not priced

R-squared from regression: $R^2 = \beta_i^2 \sigma_M^2 / \sigma_i^2$ = fraction of variance explained by market.

3. Applications of CAPM

Definition 3.1: Cost of Equity

CAPM provides a method to estimate the cost of equity capital:

r_E = r_f + \beta_E (\mathbb{E}[R_M] - r_f)

This is the required return shareholders demand, used in:

NPV calculations and capital budgeting
Valuation (as discount rate in DCF models)
Weighted Average Cost of Capital (WACC)

Example 3.1: Cost of Equity Calculation

A company has beta 1.3, risk-free rate is 4%, expected market return is 10%. What is its cost of equity?

r_E = 4\% + 1.3(10\% - 4\%) = 4\% + 1.3(6\%) = 4\% + 7.8\% = 11.8\%

The company must earn at least 11.8% on equity-financed projects to satisfy shareholders.

Definition 3.2: Jensen's Alpha

Jensen's alpha measures abnormal performance:

\alpha_i = \bar{R}_i - [r_f + \beta_i(\bar{R}_M - r_f)]

where $\bar{R}_i$ is the realized average return. Positive alpha suggests outperformance (after adjusting for risk).

Definition 3.3: Treynor Ratio

The Treynor ratio measures excess return per unit of systematic risk:

T_i = \frac{\bar{R}_i - r_f}{\beta_i}

Compare to Sharpe ratio $(\bar{R}_i - r_f)/\sigma_i$ which uses total risk. Treynor is appropriate when evaluating a component of a diversified portfolio.

Example 3.2: Performance Evaluation

Fund A: return 15%, beta 1.2. Fund B: return 13%, beta 0.8. Risk-free rate 3%, market return 11%.

CAPM predictions:

E[R_A] = 3\% + 1.2(8\%) = 12.6\%

E[R_B] = 3\% + 0.8(8\%) = 9.4\%

Jensen's alphas:

\alpha_A = 15\% - 12.6\% = 2.4\%

\alpha_B = 13\% - 9.4\% = 3.6\%

Fund B has higher alpha despite lower raw return - better risk-adjusted performance!

Remark 3.1: Empirical Evidence

CAPM has mixed empirical support:

Supporting evidence:

Positive relationship between beta and average returns (broadly)
Market risk premium is positive historically
Simple and widely used in practice

Anomalies (violations):

Size effect: Small-cap stocks outperform beyond what beta predicts
Value premium: High book-to-market stocks earn higher returns
Momentum: Past winners continue to outperform
Low-beta anomaly: Low-beta stocks earn higher risk-adjusted returns than predicted

These findings led to multi-factor models like Fama-French three-factor and five-factor models.

4. Extensions and Limitations

Proposition 4.1: Zero-Beta CAPM

Without risk-free borrowing/lending, CAPM becomes:

\mathbb{E}[R_i] = \mathbb{E}[R_Z] + \beta_i(\mathbb{E}[R_M] - \mathbb{E}[R_Z])

where $R_Z$ is the zero-beta portfolio (minimum variance portfolio orthogonal to market). When risk-free asset exists, $R_Z = r_f$ .

Remark 4.1: Key Limitations of CAPM

Single-period model: Real investors have multi-period horizons
Market portfolio unobservable: Roll's critique - true market includes all assets (stocks, bonds, real estate, human capital), not just stock indices
Homogeneous expectations unrealistic: Investors have different information and beliefs
Mean-variance may not capture all relevant risk: Ignores higher moments (skewness, kurtosis), downside risk
Empirical failures: Anomalies suggest missing risk factors

Proposition 4.2: Conditional CAPM

Allows for time-varying betas and risk premiums:

\mathbb{E}_t[R_{i,t+1}] = r_{f,t} + \beta_{i,t}(\mathbb{E}_t[R_{M,t+1}] - r_{f,t})

where $\beta_{i,t}$ varies with economic conditions. More realistic but harder to implement.

Key Takeaways

1.CAPM predicts linear relationship between expected return and beta (systematic risk)
2.Only systematic risk is priced - idiosyncratic risk earns no premium because it's diversifiable
3.In equilibrium, all investors hold market portfolio combined with risk-free asset
4.Beta measures sensitivity to market - calculated as Cov(Ri, RM)/Var(RM)
5.CAPM widely used for cost of equity estimation despite empirical limitations
6.Anomalies (size, value, momentum) motivated multi-factor asset pricing models

Frequently Asked Questions

What is the main prediction of CAPM?

CAPM predicts that expected return is linearly related to systematic risk (beta). The Security Market Line states: E[Ri] = Rf + βi(E[RM] - Rf). Only systematic risk is priced - investors are not compensated for idiosyncratic risk, which can be diversified away.

What does beta measure?

Beta measures an asset's sensitivity to market movements. β = Cov(Ri, RM)/Var(RM). A stock with β = 1.5 moves 1.5% on average when the market moves 1%. High-beta stocks are more volatile with respect to the market and require higher expected returns.

Why does CAPM only price systematic risk?

Because investors hold diversified portfolios (the market portfolio in equilibrium), idiosyncratic risk is eliminated. Since investors don't bear idiosyncratic risk, they don't demand compensation for it. Only non-diversifiable systematic risk commands a risk premium.

What are the key assumptions of CAPM?

CAPM assumes: (1) investors are mean-variance optimizers, (2) homogeneous expectations, (3) single period, (4) risk-free borrowing/lending, (5) no taxes or transaction costs, (6) all assets are tradeable and divisible. These strong assumptions limit real-world applicability.

How do we test CAPM empirically?

Test whether high-beta stocks earn higher average returns. Run cross-sectional regression: Ri = α + βγ + ε. CAPM predicts α = Rf and γ = E[RM] - Rf. However, empirical evidence shows size effect, value premium, and momentum anomalies that CAPM can't explain.

Practice Quiz

Capital Asset Pricing Model

Questions

Correct

Accuracy

If risk-free rate is 3%, market return is 10%, and a stock has β = 1.2, what is the required return per CAPM?

Easy

Not attempted

A stock has expected return 14%, the market has expected return 11% and std 18%. If Rf = 4% and stock std is 25%, what is beta?

Medium

Not attempted

What is the beta of the market portfolio?

Easy

Not attempted

If a portfolio has 40% in a stock with β = 0.8 and 60% in a stock with β = 1.4, what is the portfolio beta?

Easy

Not attempted

A stock's actual return was 15%. CAPM predicted 12%. What is Jensen's alpha?

Easy

Not attempted

According to CAPM, an asset with zero beta should have expected return equal to:

Medium

Not attempted

The Treynor ratio is:

Medium

Not attempted

In CAPM equilibrium, all investors hold:

Medium

Not attempted

If Cov(Ri, RM) = 0.04 and Var(RM) = 0.03, what is the beta?

Easy

Not attempted

Which empirical finding violates CAPM?

Hard

Not attempted