**William Sharpe** (1964) published the **capital asset pricing model** (CAPM). Parallel work was also performed by Treynor (1961) and Lintner (1965). The model extended Harry Markowitz’s portfolio theory to introduce the notions of systematic and specific risk. For his work on the capital asset pricing model, Sharpe shared the 1990 Nobel Prize in Economics with Harry Markowitz and Merton Miller.

The capital asset pricing model considers a simplified world where:

- There are no taxes or transaction costs.
- All investors have identical investment horizons.
- All investors have identical opinions about expectedreturns, volatilities and correlations of available investments.

In such a simple world, Tobin’s (1958) super-efficient portfolio must be the market portfolio. All investors will hold the market portfolio, leveraging or de-leveraging it with positions in the risk-free asset in order to achieve a desired level of risk.

The capital asset pricing model decomposes a portfolio’s risk into systematic and specific risk. **Systematic risk** is the risk of holding the market portfolio. As the market moves, each individual asset is more or less affected. To the extent that any asset participates in such general market moves, that asset entails systematic risk. **Specific risk** is the risk which is unique to an individual asset. It represents the component of an asset’s return which is uncorrelated with general market moves.

According to the capital asset pricing model, the marketplace compensates investors for taking systematic risk but not for taking specific risk. This is because specific risk can be diversified away. When an investor holds the market portfolio, each individual asset in that portfolio entails specific risk, but through diversification, the investor’s net exposure is just the systematic risk of the market portfolio.

Systematic risk can be measured using beta. According to the capital asset pricing model, the expected return of a stock equals the risk-free rate plus the portfolio’s beta multiplied by the expected excess return of the market portfolio. Specifically, let *Z _{s}* and

*Z*be random variables for the simple returns of the stock and the market over some specified period. Let

_{m}*z*be the known risk-free rate, also expressed as a simple return, and let β be the stock’s beta. Then

_{f}*E*(*Z _{s}*) =

*z*+ β[

_{f}*E*(

*Z*) –

_{m}*z*]

_{f}[1]

where *E* denotes an expectation.

Stated another way, the stock’s excess expected return over the risk-free rate equals its beta times the market’s expected excess return over the risk free rate.

For example, suppose a stock has a beta of 0.8. The market has an expected annual return of 0.12 (that is 12%) and the risk-free rate is .02 (2%). Then the stock has an expected one-year return of

*E*(*Z _{s}*) = .02 + .8[.12 – .02] = .10

[2]

Because [1] is linear, it generalizes to portfolios. Let *Z _{p}* be a portfolio’s simple return, and let β now denote the portfolio’s beta. We obtain

*E*(*Z _{p}*) =

*z*+ β[

_{f}*E*(

*Z*) –

_{m}*z*]

_{f}[3]

Formula [1] is the essential conclusion of the capital asset pricing model. It states that a stock’s (or portfolio’s) excess expected return depends on its beta and not its volatility. Stated another way, excess return depends upon systematic risk and not on total risk.

[]

We call CAPM a “capital asset pricing model” because, given a beta and an expected return for an asset, investors will bid its current price up or down, adjusting that expected return so that it satisfies formula [1]. Accordingly, the capital asset pricing model predicts the equilibrium price of an asset. This works because the model *assumes* that all investors agree on the beta and expected return of any asset. In practice, this assumption is unreasonable, so the capital asset pricing model is largely of theoretical value. It is the most famous example of an equilibrium pricing model.

## References

- Lintner, John (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets,
*Review of Economics and Statistics*, 47, 13-37. - Sharpe, William F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk,
*Journal of Finance*, 19 (3), 425-442. - Tobin, James (1958). Liquidity preference as behavior towards risk,
*The Review of Economic Studies*, 25, 65-86. - Treynor, Jack (1961). Towards a theory of market value of risky assets,
*unpublished manuscript*.

Offer a critical evaluation to the capital asset pricing model and arbitrage pricing model as models of equilibrium returns