layout: default title: Capital Asset Pricing Model (CAPM) —————————————–

Capital Asset Pricing Model (CAPM)

Sukrit Mittal Franklin Templeton Investments

Outline

  1. Why CAPM?
  2. Assumptions and market setting
  3. From portfolio choice to equilibrium
  4. Derivation of CAPM
  5. Security Market Line (SML)
  6. Systematic vs idiosyncratic risk
  7. Characteristic line
  8. Empirical interpretation and limits
  9. Exercises

1. Why CAPM?

Up to now we solved investor problems.

CAPM answers a different question:

How are assets priced in equilibrium?

This is not about optimal portfolios.

It is about consistency across the entire market.

What CAPM Is—and Is Not

CAPM is:

  • A logical consequence of mean–variance optimization
  • An equilibrium restriction

CAPM is not:

  • A law of nature
  • A trading strategy

It is a benchmark.

Benchmarks can be wrong—and still useful.

2. Market Setting and Assumptions

We assume:

  • All investors are mean–variance optimizers
  • Homogeneous expectations
  • A single risk-free rate $R_f$
  • Frictionless markets (no taxes, no transaction costs)

These assumptions are strong.

They are also transparent.

Why These Assumptions?

Not because they are realistic.

But because they allow us to isolate:

The pricing role of risk.

CAPM is a controlled experiment in theory.

3. From Portfolio Choice to Equilibrium

From Lecture 07:

  • All investors hold the same risky portfolio
  • Differences arise only via leverage

In equilibrium:

The risky portfolio held by everyone must be the market portfolio.

This is the critical step.

The Market Portfolio

The market portfolio contains:

  • All risky assets
  • In proportion to their market values

No asset can escape the market.

If it exists, it is priced.

4. Risk Decomposition

Consider an asset $i$ with return $R_i$.

Decompose its risk relative to the market $R_M$.

Only part of this risk matters.

The rest is diversifiable noise.

Beta: Measuring Systematic Risk

Define beta:

\[\beta_i = \frac{\text{Cov}(R_i, R_M)}{\text{Var}(R_M)}\]

Beta measures:

Sensitivity to market movements.

This is the only risk investors are paid for.

5. Derivation of CAPM

Consider the market portfolio $M$.

For any asset $i$:

  • Adding $i$ to $M$ must not improve the Sharpe ratio

Otherwise, $M$ would not be optimal.

This restriction pins down expected returns.

Mathematical Statement

The equilibrium condition yields:

\[\mathbb{E}[R_i] - R_f = \beta_i\big( \mathbb{E}[R_M] - R_f \big)\]

This is the CAPM equation.

Nothing mystical happened.

6. Security Market Line (SML)

Plot expected return against beta.

The CAPM predicts a straight line:

\[\mathbb{E}[R] = R_f + \beta (\mathbb{E}[R_M] - R_f)\]

This line is the Security Market Line.

Interpretation of the SML

  • Intercept: risk-free rate
  • Slope: market risk premium

Assets:

  • Above the line: underpriced
  • Below the line: overpriced

In theory.

7. Systematic vs Idiosyncratic Risk

Total risk decomposes into:

  • Systematic risk (market-related)
  • Idiosyncratic risk (asset-specific)

Diversification eliminates only the latter.

Markets pay only for what cannot be diversified.

8. Characteristic Line

The characteristic line relates an asset’s return to the market return:

\[R_i - R_f = \alpha_i + \beta_i (R_M - R_f) + \varepsilon_i\]

This is a regression equation.

Interpretation

  • $\beta_i$: systematic exposure
  • $\alpha_i$: abnormal return
  • $\varepsilon_i$: idiosyncratic noise

In CAPM:

\[\alpha_i = 0\]

Nonzero alpha is a claim.

Extraordinary claims require extraordinary evidence.

9. Empirical Reality

CAPM is elegant.

Reality is less cooperative.

Empirically:

  • Betas are unstable
  • Many anomalies exist

But CAPM survives as a benchmark.

Bad models die.

Useful models endure.

10. Exercises

Exercise 1

Given:

  • $R_f = 4%$
  • $\mathbb{E}[R_M] = 10%$
  • $\beta_i = 1.2$

Compute $\mathbb{E}[R_i]$ under CAPM.

Exercise 2

Estimate $\beta$ for a stock using historical returns.

Discuss limitations of this approach.

Exercise 3

An asset lies persistently above the SML.

List three possible explanations.

Final Takeaways

  • CAPM links risk to expected return
  • Only systematic risk is priced
  • The SML is an equilibrium restriction
  • The characteristic line connects theory to data

Next, we move beyond CAPM.

Because markets did.