San José State University
Department of Economics
& Tornado Alley
One major step advancing from Portfolio Analysis to the Capital Asset Pricing Model comes from a realization about what determines the risk of a well-diversified portfolio. Our intuition suggests that the individual riskiness of the stocks in the portfolio would the most important factor. Here our intuition is wrong.
If we work out the algebra of the standard deviation of a portfolio of N different stocks with an average variance of Var and average covariance of Cov the result is:
This function is plotted below.
We see from the graph that the risk of the portfolio, as measured by the standard deviation of its rate of return, approaches a lower limit as N increases. What is this lower limit? Well, the term Var/N goes to zero as N increase without bound, so the average variance drops out of the picture. The other term, (N-1)/N, goes to one as N increases without limit so the lower limit is:
Thus the only thing that affects the riskiness of a well-diversified portfolio is the average covariance of the stocks in the portfolio. This suggests that the relationship between the rate of return on a stock and the rate of return on other stocks in the market is an important characteristic of a stock.
The lower limit on the risk of a portfolio is then that risk which cannot be diversified away. It is called the market risk.
Another major line of development for the CAPM is through the notion of an optimal portfolio of common stocks, as emerged in Portfolio Analysis. William Sharpe and others asked what the market equilibrium would imply about the composition of the optimal portfolio. The Separation Theorem implies that all investors, to the extent that they buy any common stocks, would always buy in the proportions of the optimal portfolio. This means that if there was any stock that was not in the optimal portfolio there would be no buyers for it. Consequently the price of any stock not in the optimal portfolio would fall. As the price of a stock falls the expected rate of return, dividends divided by price, will rise. (Also the variance of the rate of return and the covariance of its rate of return with other stocks will rise.) This changes the picture and results in a new optimal portfolio. The price of a stock left out of the optimal portfolio will keep falling until it is in the optimal portfolio. Therefore for equilibrium the optimal portfolio must include all stocks in the market.
But the conditions for equilibrium imply even stronger restrictions on the optimal portfolio. Suppose that GM stock represents 2 percent of the value of all stock in the stock market but the optimal portfolio is composed of 1 percent GM stock. This would mean that there would be buyers for only half of the GM stock on the market. This would mean the price of GM stock would fall resulting in a change in the composition of the optimal portfolio. The market could not be in equilibrium until the proportion of GM or any other stock in the optimal portfolio was exactly the same as the proportion in the overall market. A portfolio that has the same proportions of stocks as in the overall market is called a market portfolio.
The market line is the relationship between expected rate of return of a portfolio of stocks and its risk. Risk in the CAPM is measured by beta β. This β is the regression coefficient for the rate of return on a portfolio and the rate of return on the market portfolio.
The β for a portfolio is the weighted sum of the β's of the stocks in the portfolio with the weights being the portions of the values of the stocks in the portfolio.
When the risk is zero the expected rate of return is just the risk-free interest rate. The beta for the market portfolio is the regression coefficient for the regression of rm on rm , which of course is equal to 1.0. This means the equation for the market line is:
The quantity (rm-rf) is the risk-premium on the market portfolio, which has been about 8.4 percent for the American financial market.
(To be continued.)
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