Tuesday, 21 October 2014

The Evolution of the Capital Asset Pricing Model: How an Economic Model Evolves and Improves

by Calvin Price

In 1964, William Sharpe and John Lintner developed a formula called the Capital Asset Pricing Model (CAPM) for predicting the pricing of stocks. It was mathematically simplistic and intuitive; it asked the right questions; it was an immediate success in the economic community. It was taught in virtually all business schools as the stand-alone method for pricing stocks. There was just one problem with the model: it didn't work. 

Models attempt to simplify the real world, find a rule, and then apply that rule in reality. Through this stringent means of testing and retesting, hypothesizing and theorizing – applying the scientific method - researchers can find rules that govern economics. To reduce the complex system of variables that apply in any situation into a linear formula is no easy task – but it is especially difficult, and probably impossible to do so for the stock market. The stock market is subject to a complex scheme of crowd psychology, moods, global affairs, and seemingly unrelated topics all have the potential to shift prices. It is highly doubtful that any theory would be able to accurately measure these effects and create a theory from that.

In 1994, a famous investment company called Long-Term Capital Management (LTCM) was founded and was praised as being the largest gathering of academic and practical knowledge on the stock market in existence. It had several Nobel Prize winners, professors from prestigious academic institutions, and weathered pit traders and quantitative analysts from Wall Street. They enjoyed risk-free borrowing from banks and had access to the latest technology. Within three years they were hailed as possibly being the greatest investment company ever founded having made unheard of returns on investments. In 1998, the company flamed out of existence in a period of less than a month in a relatively non-volatile market. They are one of the most well known examples of trading errors, are frequently examined in case studies, have been the subject of a book and numerous academic articles, and relied heavily on the CAPM.

The CAPM’s mistake, as Fama and French contend, is in both its formula and in its underlying assumptions. One of the key assumptions underlying the Sharpe-Lintner model is that there is borrowing and lending at a risk-free rate. This is obviously an over-simplification of the market, there can never be sustained borrowing and lending at risk-free rates, but the motivation for including this is to contain the risk of the market to merely the movement of stocks and not interest rates. In 1972, another economist named Fischer Black, who would later become famous for the Black-Scholes model for pricing stock options, altered the CAPM that did not include the assumption of risk-free borrowing. Instead, he allowed for the possibility of unrestricted short sales of risky assets – essentially saying that if investors felt there was risk in a situation they could unload it onto another investor. Black’s model is not correct either, but it is more accurate than the original CAPM.
CAPM’s simplicity is derived from its brief algebraic formula (as defined in the original Sharpe-Lintner model):

E(Ri) is the expected return (how much money you expect to make) on any investment i. Rf is the assumption of risk-free borrowing and lending. E(RM) is the expected return of the market portfolio. The market portfolio is a theoretical construct, it is a selection of stocks that minimize the variance of the return (meaning profits will be relatively consistent) and maximize profit per risk. β is the correlation of a stock to the market average. If β = 1, the stock will be perfectly correlated with the market. If it is greater than 1, the stock will be more volatile than the market. If it is less than 1, it will move less than the market. Thus, if the beta of a stock is 1.50, it is considered 50% more volatile than the market. Betas that are equal to zero are uncorrelated to the market. For its apparent origins in a foreign language, the CAPM relies on relatively few variables to predict a potentially accurate price.

Unfortunately, the equation appears to fail immediately as soon empirical testing occurs. The CAPM predicts low profits for investments with low beta and high profits for investments with high betas. In reality, the returns for low beta portfolios are too high and the returns on high beta portfolios are too low. After extensive tests, Black’s theory that market betas can predict expected returns seems to hold. However, the precise predictions of Sharpe and Lintner beta calculation, expected market return minus the risk-free interest rate, is consistently rejected. Further debates over CAPM range from considering it to be completely useless and unreliable to be merely a slightly-flawed model. Fama and French attempt to prove the latter. 

In 1973, Robert Merton, who would later become a highly influential figure with LTCM, created the Intertemporal Capital Asset Pricing Model (ICAPM). The ICAPM assumes that investors are interested not only in the end-of period payoff, when they cash in on the stocks, (as had been assumed in the original CAPM) but also in the value of the asset at any specific time. Thus, the ICAPM introduces a generic element of time into the equation. The ICAPM also assumes that optimal portfolios are ‘multifactor efficient’, implying the assumption that risk-free borrowing and short-sales of risky assets are both possible.

Fama and French extend this theory with one final idea: the overall market movement is not necessarily correlated to the movement of specific portfolios. In other words, the betas that Sharpe, Lintner, Black, and Merton had been using were all potentially inaccurate. To deal with this Fama and French use the betas from the overall market, from specific portfolios with small and big stocks, and diversified portfolios with high and low book value/market value (B/M) stocks. Book value is the price at which that asset was bought and market value is the price the asset would sell for. This gives the Three-Factor Model:

In this equation, the first β is the original one that was preserved from the first CAPM adjusted for time. The second β is the expected return on a portfolio of small and big stocks (SMB means ‘small minus big’). The third β is the expected return on a portfolio with high and low B/M stocks (HML means ‘high minus low’). Fama and French’s model has been an empirical test and, since proposed, has withstood numerous tests.

Although the Three-Factor Model has been a success, it still does not predict everything in the market. The stock market is a complex system of pricing that, at times, relies as much upon human emotion and mistakes as it does empirical data and knowledge. Things as inane and intangible as stock momentum and popularity can have a huge and often nonsensical impact on the pricing of a stock. An equation can never accurately account and predict every movement that occurs on the market, but the Three-Factor Model is about as close as anyone has ever come to doing so.


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