In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio. The model takes into account the asset's sensitivity to non-diversifiable risk (also known as systematic risk or market risk), often represented by the quantity beta (β) in the financial industry, as well as the expected return of the market and the expected return of a theoretical risk-free asset.
CAPM assumes a particular form of utility functions (in which only first and second moments matter, that is risk is measured by variance, for example a quadratic utility) or alternatively asset returns whose probability distributions are completely described by the first two moments (for example, the normal distribution) and zero transaction costs (necessary for diversification to get rid of all idiosyncratic risk). Under these conditions, CAPM shows that the cost of equity capital is determined only by beta. Despite its failing numerous empirical tests, and the existence of more modern approaches to asset pricing and portfolio selection (such as arbitrage pricing theory and Merton's portfolio problem), the CAPM still remains popular due to its simplicity and utility in a variety of situations.
Inventors The CAPM was introduced by economists Jack Treynor (1961, 1962), William F. Sharpe (1964), John Lintner (1965) and Jan Mossin (1966) independently, building on the earlier work of Harry Markowitz on diversification and modern portfolio theory. Sharpe, Markowitz and Merton Miller jointly received the 1990 Nobel Memorial Prize in Economic Sciences for this contribution to the field of financial economics. Fischer Black (1972) developed another version of CAPM, called Black CAPM or zero-beta CAPM, that does not assume the existence of a riskless asset. This version was more robust against empirical testing and was influential in the widespread adoption of the CAPM.
Black CAPM 'Fischer Black (1972) developed another version of CAPM, called Black CAPM or zero-beta CAPM, that does not assume the existence of a riskless asset. This version replaces the risk-free rate with the return of a "zero-beta portfolio"—a portfolio that has no correlation with the market. This version was found to be more robust against empirical testing, particularly in explaining why the security market line is often flatter than the standard model predicts.
Formula
The CAPM is a model for pricing an individual security or portfolio. For individual securities, it makes use of the security market line (SML) and its relation to expected return and systematic risk (beta) to show how the market must price individual securities in relation to their security risk class.
The SML enables us to calculate the reward-to-risk ratio for any security in relation to that of the overall market. Therefore, when the expected rate of return for any security is deflated by its beta coefficient, the reward-to-risk ratio for any individual security in the market is equal to the market reward-to-risk ratio, thus:
In equilibrium, all assets should plot directly on the SML. An asset located above the line is considered undervalued, as it offers a higher return for its level of systematic risk, while an asset below is overvalued. The slope of the SML represents the market risk premium, illustrating the trade-off between risk and return. By determining the position of a security relative to this line, investors can identify whether the expected return justifies the asset's market-related volatility. +CAPM Mathematical FrameworkComponentEquation / DerivationDescription & LogicReward-to-Risk Ratio:Shows that the risk premium per unit of beta is constant for all assets in the market.Standard CAPM:Obtained by rearranging the reward-to-risk equation to solve for . This is the primary model for determining the required return of an assetJensen's Alpha:Measures the abnormal return of an asset. A positive indicates outperformance relative to the SML.Black CAPM:A version for restricted borrowing; replaces with the return of a zero-beta portfolio ().Beta Definition:Measures the sensitivity of the expected excess asset returns to the expected excess market returns. Risk Premium Form : Restates that the individual risk premium equals the market premium multiplied by beta; this form is often used in empirical testing to isolate the excess return. Variable Glossary SymbolNameDefinition / RoleExpected ReturnThe expected return on the capital asset.Risk-Free RateInterest arising from government bonds or other riskless assets.AlphaThe active return measuring the distance from the SML, where indicates an undervalued asset (SCL y-intercept above zero),
fair valuation (SCL y-intercept at origin), and
an overvalued asset (SCL y-intercept below zero).Market ReturnThe expected return of the market portfolio.Market PremiumThe extra return required for market risk exposure.Indiv. Risk PremiumThe specific reward for an asset's systematic risk.CorrelationCorrelation coefficient between investment i and market m.Standard DeviationVolatility of the investment and the market, respectively.
Modified betas There has also been research into a mean-reverting beta often referred to as the adjusted beta, as well as the consumption beta. However, in empirical tests, the traditional CAPM has been found to do as well as or outperform these modified beta models. Specifically, research by Mankiw and Shapiro (1986) found that the market beta of the traditional CAPM outperformed the consumption beta in explaining the cross-section of stock returns. Furthermore, while the adjusted beta (Blume's method) improves forecast accuracy, it does not consistently resolve the underlying empirical failures of the model's predictive power over long horizons.
Security market line The SML graphs the results from the capital asset pricing model (CAPM) formula. The x-axis represents the risk (beta), and the y-axis represents the expected return. The market risk premium is determined from the slope of the SML.
The relationship between and required return is plotted on the security market line (SML), which shows expected return as a function of . The intercept is the nominal risk-free rate available for the market, while the slope is the market premium, . The security market line can be regarded as representing a single-factor model of the asset price, where is the exposure to changes in the value of the Market.
It is a useful tool for determining if an asset being considered for a portfolio offers a reasonable expected return for its risk. Individual securities are plotted on the SML graph. If the security's expected return versus risk is plotted above the SML, it is undervalued since the investor can expect a greater return for the inherent risk. Conversely, a security plotted below the SML is overvalued since the investor would be accepting less return for the amount of risk assumed.
Asset pricing Once the expected/required rate of return is calculated using CAPM, we can compare this required rate of return to the asset's estimated rate of return over a specific investment horizon to determine whether it would be an appropriate investment. To make this comparison, you need an independent estimate of the return outlook for the security based on either fundamental or technical analysis techniques, including P/E, M/B etc.
Assuming that the CAPM is correct, an asset is correctly priced when its estimated price is the same as the present value of future cash flows of the asset, discounted at the rate suggested by CAPM. If the estimated price is higher than the CAPM valuation, then the asset is overvalued (and undervalued when the estimated price is below the CAPM valuation). When the asset does not lie on the SML, this could also suggest mis-pricing. Since the expected return of the asset at time is , a higher expected return than what CAPM suggests indicates that is too low (the asset is currently undervalued), assuming that at time the asset returns to the CAPM suggested price.
The asset price using CAPM, sometimes called the certainty equivalent pricing formula, is a linear relationship given by
where is the future price of the asset or portfolio.
Asset-specific required return The CAPM returns the asset-appropriate required return or discount rate—the rate at which future cash flows produced by the asset should be discounted given that asset's relative riskiness. Betas exceeding one signify more than average "riskiness", while betas below one indicate lower than average risk. Thus, a more risky stock will have a higher beta and will be discounted at a higher rate; less sensitive stocks will have lower betas and be discounted at a lower rate. Given the accepted concave utility function, the CAPM is consistent with intuition that investors require a higher return for holding a more risky asset.
Since beta reflects asset-specific sensitivity to non-diversifiable market risk, the market as a whole, by definition, has a beta of one. Stock market indices are frequently used as local proxies for the market—and in that case, by definition, have a beta of one. An investor in a large, diversified portfolio, such as a mutual fund designed to track the total market, therefore expects performance in line with the market.
Fundamental Valuation (Present Value)
Once the required expected return is established via CAPM, it is used as the discount rate to determine an asset's intrinsic value based on future cash flows (CF).Damodaran, A. (2012). Investment Valuation, Wiley.
The intrinsic value is calculated using the following present value formula:
An asset is considered undervalued if its calculated is higher than the current market price, and overvalued''' if the price exceeds this intrinsic value.Graham, B. and Dodd, D. (1934). Security Analysis. +Fundamental Valuation VariablesVariableDescriptionPVThe Present Value (intrinsic fair price) today.Bodie, Z., et al. (2021). Investments, McGraw-Hill.E(CFt)The expected cash flow in period t.E(Ri)The CAPM-derived required return (or WACC) used as the discount rate.Sharpe, W. F. (1964). "Capital Asset Prices," Journal of Finance.nThe total number of periods (time horizon).
Risk and diversification The risk of a portfolio comprises systematic risk, also known as undiversifiable risk, and unsystematic risk, which is also known as idiosyncratic risk or diversifiable risk. Systematic risk refers to the risk common to all securities—i.e., market risk. Unsystematic risk is the risk associated with individual assets.
Unsystematic risk can be diversified away to smaller levels by including a greater number of assets in the portfolio as specific risks "average out". The same is not possible for systematic risk within one market. Depending on the market, a portfolio of approximately 30–40 securities in developed markets such as the UK or US will render the portfolio sufficiently diversified such that risk exposure is limited to systematic risk only. This number may vary depending on the way securities are weighted in a portfolio which alters the overall risk contribution of each security. For example, market cap weighting means that securities of companies with larger market capitalization will take up a larger portion of the portfolio, making it effectively less diversified. In developing markets, a larger number of securities is required for diversification due to higher asset volatilities.