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14 Capital Asset Pricing Model (CAPM)

Because investors are risk averse, they will choose to hold a portfolio of securities to take advantage of the benefits of Diversification. Therefore, when they are deciding whether or not to invest in a particular stock, they want to know how the stock will contribute to the risk and expected return of their portfolios.

The standard deviation of an individual stock does not indicate how that stock will contribute to the risk and return of a diversified portfolio. Thus, another measure of risk is needed; a measure of a security's systematic risk. This measure is provided by the Capital Asset Pricing Model (CAPM).

Systematic and Unsystematic Risk

An asset's total risk consists of both systematic and unsystematic risk. Systematic risk, which is also called market risk or undiversifiable risk, is the portion of an asset's risk that cannot be eliminated via diversification. The systematic risk indicates how including a particular asset in a diversified portfolio will contribute to the riskiness of the portfolio Unsystematic risk, which is also called firm-specific or diversifiable risk, is the portion of an asset's total risk that can be eliminated by including the security as part of a diversifiable portfolio.

The Capital Asset Pricing Model (CAPM) provides an expression which relates the expected return on an asset to its systematic risk. The relationship is known as the Security Market Line (SML) equation and the measure of systematic risk in the CAPM is called Beta.

14.1.1 The Security Market Line (SML)

The SML equation is expressed as follows:

where

• E[R_i] = the expected return on asset i,
• R_f = the risk-free rate,
• E[R_m] = the expected return on the market portfolio,
• b_i = the Beta on asset i, and
• E[R_m] - R_f = the market risk premium.

The graph below depicts the SML. Note that the slope of the SML is equal to (E[R_m] - R_f) which is the market risk premium and that the SML intercepts the y-axis at the risk-free rate.

In capital market equilibrium, the required return on an asset must equal its expected return. Thus, the SML equation can also be used to determine an asset's required return given its Beta.

14.1.2 The Beta (B_i)

The beta for a stock is defined as follows:

where

• s_im = the Covariance between the returns on asset i and the market portfolio and
• s²_m = the Variance of the market portfolio.

Note that, by definition, the beta of the market portfolio equals 1 and the beta of the risk-free asset equals 0.

An asset's systematic risk, therefore, depends upon its covariance with the market portfolio. The market portfolio is the most diversified portfolio possible as it consists of every asset in the economy held according to its market portfolio weight.

Example Problems

1. Find the expected return on a stock given that the risk-free rate is 6%, the expected return on the market portfolio is 12%, and the beta of the stock is 2. 2. Find the beta on a stock given that its expected return is 16%, the risk-free rate is 4%, and the expected return on the market portfolio is 12%.

14.2 Critical assumptions of CAPM

The CAPM is simple and elegant. Consider the many assumptions that underlie the model. Are they valid?

• Zero transaction costs. The CAPM assumes trading is costless so investments are priced to all fall on the capital market line. If not, some investments would hover below and above the line -- with transaction costs discouraging obvious swaps. But we know that many investments (such as acquiring a small business) involve significant transaction costs. Perhaps the capital market line is really a band whose width reflects trading costs.
• Zero taxes. The CAPM assumes investment trading is tax-free and returns are unaffected by taxes. Yet we know this to be false: (1) many investment transactions are subject to capital gains taxes, thus adding transaction costs; (2) taxes reduce expected returns for many investors, thus affecting their pricing of investments; (3) different returns (dividends versus capital gains, taxable versus tax-deferred) are taxed differently, thus inducing investors to choose portfolios with tax-favored assets; (4) different investors (individuals versus pension plans) are taxed differently, thus leading to different pricing of the same assets.
• Homogeneous investor expectations. The CAPM assumes invests have the same beliefs about expected returns and risks of available investments. But we know that there is massive trading of stocks and bonds by investors with different expectations. We also know that investors have different risk preferences. Again, it may be that the capital market line is a fuzzy amalgamation of many different investors' capital market lines.
• Available risk-free assets. The CAPM assumes the existence of zero-risk securities, of various maturities and sufficient quantities to allow for portfolio risk adjustments. But we know even Treasury bills have various risks: reinvestment risk -- investors may have investment horizons beyond the T-bill maturity date; inflation risk -- fixed returns may be devalued by future inflation; currency risk -- the purchasing power of fixed returns may diminish compared to that of other currencies. (Even if investors could sell assets short -- by selling an asset she does not own, and buying it back later, thus profiting from price declines -- this method of reducing portfolio risk has costs and assumes unlimited short-selling ability.)
• Borrowing at risk-free rates. The CAPM assumes investors can borrow money at risk-free rates to increase the proportion of risky assets in their portfolio. We know this is not true for smaller, non-institutional investors. In fact, we would predict that the capital market line should become kinked downward for riskier portfolios (ß > 1) to reflect the higher cost of risk-free borrowing compared to risk-free lending.
• Beta as full measure of risk. The CAPM assumes that risk is measured by the volatility (standard deviation) of an asset's systematic risk, relative to the volatility (standard deviation) of the market as a whole. But we know that investors face other risks: inflation risk -- returns may be devalued by future inflation; and liquidity risk -- investors in need of funds or wishing to change their portfolio's risk profile may be unable to readily sell at current market prices. Moreover, standard deviation does not measures risk when returns are not evenly distributed around the mean (non-bell curve). This uneven distribution describes our stock markets where winning companies, like Dell and Walmart, have positive returns (35,000% over ten years) that greatly exceed losing companies' negative returns (which are capped at a 100% loss).