In this section, we study a financial application, the Capital Asset Pricing Model, often referred to by the acronym CAPM. The name is something of a misnomer in that the model is really about returns based on capital assets, not the prices themselves. The types of assets that we examine are equity securities that are traded on an active market, such as the New York Stock Exchange (NYSE). For a stock on the exchange, we can relate returns to prices through the following expression:
begin{equation*}scriptsize mathrm{return=}frac{mathrm{ price~at~the~end~of~a~period+dividends-price~at~the~beginning~of~a~period}}{ mathrm{price~at~the~beginning~of~a~period}}. end{equation*}
If we can estimate the returns that a stock generates, then knowledge of the price at the beginning of a generic financial period allows us to estimate the value at the end of the period (ending price plus dividends). Thus, we follow standard practice and model returns of a security.
An intuitively appealing idea, and one of the basic characteristics of the CAPM, is that there should be a relationship between the performance of a security and the market. One rationale is simply that if economic forces are such that the market improves, then those same forces should act upon an individual stock, suggesting that it also improve. As noted above, we measure performance of a security through the return. To measure performance of the market, several market indices exist that summarize the performance of each exchange. We will use the “equally-weighted” index of the Standard & Poor’s 500. The Standard & Poor’s 500 is the collection of the 500 largest companies traded on the NYSE, where “large” is identified by Standard & Poor’s, a financial services rating organization. The equally-weighted index is defined by assuming a portfolio is created by investing one dollar in each of the 500 companies.
Another rationale for a relationship between security and market returns comes from financial economics theory. This is the CAPM theory, attributed to Sharpe (1964) and Lintner (1965) and based on the portfolio diversification ideas of Harry Markowitz (1959). Other things equal, investors would like to select a return with a high expected value and low standard deviation, the latter being a measure of risk. One of the desirable properties about using standard deviations as a measure of riskiness is that it is straight-forward to calculate the standard deviation of a portfolio. One only needs to know the standard deviation of each security and the correlations among securities. A notable security is a risk-free one, that is, a security that theoretically has a zero standard deviation. Investors often use a 30-day U.S. Treasury bill as an approximation of a risk-free security, arguing that the probability of default of the U.S. government within 30 days is negligible. Positing the existence of a risk-free asset and some other mild conditions, under the CAPM theory there exists an efficient frontier called the securities market line. This frontier specifies the minimum expected return that investors should demand for a specified level of risk. To estimate this line, we can use the equation begin{equation*} mathrm{E}~r = beta_0 + beta_1 r_m end{equation*} where (r) is the security return and (r_m) is the market return. We interpret (beta_1 r_m) as a measure of the amount of security return that is attributed to the behavior of the market.
Testing economic theory, or models arising from any discipline, involves collecting data. The CAPM theory is about ex-ante (before the fact) returns even though we can only test with ex-post (after the fact) returns. Before the fact, the returns are unknown and there is an entire distribution of returns. After the fact, there is only a single realization of the security and market return. Because at least two observations are required to determine a line, CAPM models are estimated using security and market data gathered over time. In this way, several observations can be made. For the purposes of our discussions, we follow standard practice in the securities industry and examine monthly prices.