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PORTFOLIO BETAS

Diversification decreases variability from unique risk but not from market risk. The beta of a portfolio is just an average of the betas of the securities in the portfolio, weighted by the investment in each security. For example, a portfolio comprised of only two stocks would have a beta as follows:

Beta of portfolio = (fraction of portfolio in first stock ГЧ beta of first stock) + (fraction of portfolio in second stock ГЧ beta of second stock)

Thus a portfolio invested 50-50 in MCI and Exxon would have a beta of (.5 ГЧ 1.3) + (.5 ГЧ .61) = .95.

A well-diversified portfolio of stocks all with betas of 1.3, like MCI, would still have a portfolio beta of 1.3. However, most of the individual stocks` unique risk would be diversified away. The market risk would remain, and such a portfolio would end up 1.3

times as variable as the market. For example, if the market has an annual standard deviation of 20 percent (about the historical average reported earlier), a fully diversified portfolio with beta of 1.3 has a standard deviation of 1.3 ГЧ 20 = 26 percent.

Portfolios with betas between 0 and 1.0 tend to move in the same direction as the market but not as far. A well-diversified portfolio of low-beta stocks like Exxon, all with betas of .61, has almost no unique risk and is relatively unaffected by market movements. Such a portfolio is .61 times as variable as the market. Of course, on average stocks have a beta of 1.0. A well-diversified portfolio including all kinds of stocks, with an average beta of 1, has the same variability as the market index.

How Risky Are Mutual Funds?

You don`t have to be wealthy to own a diversified portfolio. You can buy shares in one of the more than 6,000 mutual funds in the United States. Investors buy shares of the funds, and the funds use the money to buy portfolios of securities. The returns on the portfolios are passed back to the funds` owners in proportion to their shareholdings. Therefore, the funds act like investment cooperatives, offering even the smallest investors diversification and professional management at low cost.

Let`s look at the betas of two mutual funds that invest in stocks. Figure 4.9a plots the monthly returns of Vanguard`s Windsor II mutual fund and of the S&P index from July 1994 to June 1999. You can see that the stocks in the Windsor II fund had nearly average sensitivity to market changes: they had on average a beta of .87. If the Windsor II fund had no unique risk, its portfolio would have been .87 times as variable as the market portfolio. But the fund had not diversified away quite all the unique risk; there is still some scatter about the line in Figure 4.9a. As a result, the variability of the fund was somewhat more than .87 times that of the market.

Figure 4.9b shows the same sort of plot for Vanguard`s Index Trust 500 Portfolio mutual fund. Notice that this fund has a beta of 1.0 and only a tiny residual of unique risk ¤ the fitted line fits almost exactly because an index fund is designed to track the market as closely as possible. The managers of the fund do not attempt to pick good stocks but just work to achieve full diversification at very low cost. (The Vanguard index fund

takes investments of as little as $3,000 and manages the fund for an annual fee of less than .20 percent of the fund`s assets.) The index fund is fully diversified. Investors in this fund buy the market as a whole and don`t have to worry at all about unique risk.

FIGURE 4.9

(a) The slope of the fitted line shows that investors in the Windsor II mutual fund bore market risk slightly below that of the S&P 500 portfolio. Windsor II`s beta was .87. This was the average beta of the individual common stocks held by the fund. They also bore some unique risk, however; note the scatter of Windsor II`s returns above and below the fitted line.

(b) The Vanguard 500 Portfolio is a fully diversified index fund designed to track the performance of the market. Note the fund`s beta (1.0) and the absence of unique risk. The fund`s returns lie almost precisely on the fitted line relating its returns to those of the S&P 500 portfolio.

FIGURE 4.10

(a) Here we begin the plot of expected rate of return against beta. The first benchmarks are Treasury bills (beta = 0) and the market portfolio (beta = 1.0). We assume a Treasury bill rate of 5 percent and a market return of 14 percent. The market risk premium is 14 Ј 5 = 9 percent. (b) A portfolio split evenly between Treasury bills and the market will have beta = .5 and an expected return of 9.5 percent (point X). A portfolio invested 80 percent in the market and 20 percent in Treasury bills has beta = .8 and an expected rate of return of 12.2 percent (point

Y). Note that the expected rate of return on any portfolio mixing Treasury bills and the market lies on a straight line. The risk premium is proportional to the portfolio beta.