Capital Asset Pricing Model (CAPM)
The Capital Asset Pricing Model (CAPM) describes the relationship between systematic risk and expected return for assets, particularly equities. Developed independently by William Sharpe (1964), John Lintner (1965), and Jan Mossin (1966), it remains one of the most widely used models in finance for estimating the cost of equity and evaluating portfolio performance.
Overview
CAPM establishes that the expected return of an asset is determined by its sensitivity to systematic (market) risk, measured by beta. The model assumes investors are rational, markets are efficient, and all investors hold the same market portfolio. Any return above the risk-free rate is proportional to the asset's beta with the market.
The key insight is that only non-diversifiable (systematic) risk is rewarded in equilibrium. Idiosyncratic risk can be eliminated through diversification and therefore commands no premium.
Mathematical Formulation
The CAPM Equation
The expected return of asset is given by:
Beta Definition
Beta measures the sensitivity of an asset's return to the market return:
Alternative Beta Expression
Beta can also be expressed using the correlation coefficient and individual volatilities:
Regression Form (Market Model)
In practice, beta is estimated through time-series regression of excess returns:
Where is Jensen's alpha (abnormal return) and is the idiosyncratic error term.
Notation
- -- Expected return of asset
- -- Risk-free rate of return
- -- Beta of asset with respect to the market
- -- Expected return of the market portfolio
- -- Market risk premium (equity premium)
- -- Correlation between asset and the market
- -- Standard deviations of asset and the market
- -- Jensen's alpha (abnormal return)
- -- Idiosyncratic error term
Security Market Line (SML)
The Security Market Line plots expected return against beta. Assets above the SML are undervalued (positive alpha), those below are overvalued (negative alpha), and those on the line are correctly priced according to CAPM.
CAPM Assumptions
- Rational investors: All investors are risk-averse mean-variance optimizers.
- Homogeneous expectations: All investors share the same beliefs about expected returns, variances, and covariances.
- Single period: All investors plan for the same single holding period.
- Risk-free borrowing and lending: Investors can borrow and lend unlimited amounts at the risk-free rate.
- No taxes or transaction costs: Markets are frictionless.
- Perfectly divisible assets: All assets are infinitely divisible.
- Price takers: No individual investor can influence market prices.
- Publicly available information: All information is freely and simultaneously available to all investors.
Common Applications
| Application | Description |
|---|---|
| Cost of Equity | Estimate the required return on equity for corporate valuation and capital budgeting (WACC). |
| Performance Evaluation | Assess whether a portfolio manager generates alpha above the SML benchmark. |
| Security Valuation | Determine if a stock is over- or under-priced relative to its systematic risk. |
| Risk Budgeting | Decompose portfolio risk into systematic and idiosyncratic components. |
| Hedging | Use beta to determine hedge ratios for reducing market exposure. |
Limitations & Extensions
Limitations
- Single factor: Only captures market risk; ignores size, value, momentum, and other systematic factors.
- Static beta: Assumes beta is constant over time, which empirical evidence contradicts.
- Market portfolio:The true market portfolio is unobservable (Roll's critique, 1977).
- Normality assumption: Assumes returns are normally distributed; real returns exhibit skewness and fat tails.
- Empirical failures: The low-beta anomaly and flat SML in practice challenge the model.
Extensions
- Fama-French Three-Factor Model: Adds size (SMB) and value (HML) factors.
- Carhart Four-Factor Model: Adds a momentum factor (WML).
- Black CAPM (Zero-Beta): Replaces the risk-free asset with a zero-beta portfolio.
- Conditional CAPM: Allows beta and the market premium to vary over time.
- ICAPM:Merton's Intertemporal CAPM accounts for changing investment opportunities.
References
- Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19(3), 425-442.
- Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics, 47(1), 13-37.
- Mossin, J. (1966). Equilibrium in a capital asset market. Econometrica, 34(4), 768-783.
- Black, F. (1972). Capital market equilibrium with restricted borrowing. Journal of Business, 45(3), 444-455.
- Fama, E. F., & French, K. R. (2004). The Capital Asset Pricing Model: Theory and evidence. Journal of Economic Perspectives, 18(3), 25-46.
- Bodie, Z., Kane, A., & Marcus, A. J. (2021). Investments (12th ed.). McGraw-Hill.