States. Table 4.9 shows the percentage of risk that can be eliminated by holding a widely diversified portfolio in each of several countries as well as an internationally diversified portfolio. As can be seen, the effectiveness of diversification in reducing the risk of a portfolio varies from country to country. From the previous equation we know why. The average covariance relative to the variance varies from country to country. Thus, in Switzerland and Italy securities have relatively high covariance, indicating that stocks tend to move together. On the other hand, the security markets in Belgium and the Netherlands tend to have stocks with relatively low covariances. For these latter security markets, much more of the risk of holding individual securities can be diversified away. Diversification is especially useful in reducing the risk on a portfolio in these markets.

Number of Stocks U.S.A

Figure 4.2 The effect of number of securities on risk of the portfolio in the United States [13].

Number of stocks U.K.

Figure 4.3 The effect of securities on risk in the U.K. [13]. TWO CONCLUDING EXAMPLES

We will close this chapter and several chapters that follow with realistic applications of the principles discussed in the chapter. These applications serve both to review the concepts presented and to demonstrate their usefulness. The two examples that follow are applications to the asset allocation decision. The first application analyzes the allocation between stocks and bonds; the second analyzes the allocation between domestic and foreign stocks.

Bond Stock Allocation

One of the major decisions facing an investor is the allocation of funds between stocks and bonds. In order to make this allocation one needs to have estimates of mean returns, standard deviations of return, and either correlation coefficients or covariances. In order to estimate these variables it is useful to begin by looking at historical data. Even in allocating among managed portfolios it is useful to start by assuming that the stock and bond portfolio managers you are allocating between have performance similar to that of broad representative indexes.

The principal index used to represent common stock portfolios is the Standard and Poor's index. As described in Chapter 2, the Standard and Poor's index is a value weighted index of 500 large stocks. Value weighting means that the weight each stock represents of the portfolio is the market value of that stock (price times number of shares) divided by the aggregate market value of all shares in the index. Thus large stocks are weighted more heavily.

The version of the Standard and Poor's index reported in the newspapers is a capital appreciation index and as such doesn't include the return from dividends. In order to get total return one has to add dividend income. We will use the S&P index plus dividends for examining the characteristics of stock returns.

The standard index used to represent bond performance is the Lehman Brothers aggregate bond index. It is a value weighted index of almost all bonds in the market, and includes both capital appreciation and interest income. Thus it is a total return index.

In Table 4.10 we report the standard deviation and correlation coefficients calculated using monthly data but expressed in annual terms. The data is for a 15-year period and three 5-year periods. The month of the major market crash, October 1987, was omitted in the belief that it was atypical. Examining Table 4.10 shows that the standard deviation over each of the five-year periods is fairly constant for the S&P index; thus, using the overall average is a reasonable estimate and we will use 14.9%. Because the standard deviation for bonds has declined as markets have become less volatile, an estimate closer to the latest five-year results is probably appropriate and we will use 4.8%. The correlation coefficient has risen over time. Placing more emphasis on recent data, .45 is a reasonable estimate. At the time of the revision of this book the average forecast by security analysts surveyed was a return of 12.5% for the S&P index and 6% for the Lehman Brothers aggregate index. Thus our inputs are

RS&P = 12.5% <js&p= 14.9% ps&PB = 0.45 Rb = 6% ctb=4.8%

The means and standard deviation of return for combinations of stocks and bonds varying from 100% in the S&P, which is Xs&p = 1 and XSL = 0 to 0% in the S&P are presented in Table 4.11. Note that the expected return varies linearly from 12.5% to 6% as we decrease the amount in the S&P and increase it in bonds. Also the risk decreases as we put more in the bonds, but not linearly. Figure 4.4 shows the various choices diagrammatically.

Table 4.10 Historical Data on Bonds and Stocks
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