# Question

The Fall 1995 issue of Investment Digest, a publication of The Variable Annuity Life Insurance Company of Houston, Texas, discusses the importance of portfolio diversification for long- term investors. The article states:

While it is true that investment experts generally advise long- term investors to invest in variable investments, they also agree that the key to any sound investment portfolio is diversification. That is, investing in a variety of investments with differing levels of historical return and risk.

Investment risk is often measured in terms of the volatility of an investment over time. When volatility, sometimes referred to as standard deviation, increases, so too does the level of return. Conversely, as risk ( standard deviation) declines, so too do returns.

In order to explain the relationship between the return on an investment and its risk, Investment Digest presents a graph of mean return versus standard deviation ( risk) for nine investment classes over the period from 1970 to 1994. This graph, which Investment Digest calls the “ risk/ return trade- off,” is shown in Figure 3.26. The article says that this graph . . . illustrates the historical risk/ return trade- off for a variety of investment classes over the 24- year period between 1970 and 1994. In the chart, cash equivalents and fixed annuities, for instance, had a standard deviation of 0.81% and 0.54% respectively, while posting returns of just over 7.73% and 8.31%. At the other end of the spectrum, domestic small- cap stocks were quite volatile— with a standard deviation of 21.82%— but compensated for that increased volatility with a return of 14.93%. The answer seems to lie in asset allocation. Investment experts know the importance of asset allocation. In a nutshell, asset allocation is a method of creating a diversified portfolio of investments that minimize historical risk and maximize potential returns to help you meet your retirement goals and needs. Suppose that, by reading off the graph of Figure 3.26, we obtain the mean return and standard deviation combinations for the various investment classes as shown in Table 3.9. Further suppose that future returns in each investment class will behave as they have from 1970 to 1994. That is, for each investment class, regard the mean return and standard deviation in Table 3.9 as the population mean and the population standard deviation of all possible future returns. Then do the following:

a. Assuming that future returns for the various investment classes are mound- shaped, for each investment class compute intervals that will contain approximately 68.26 percent and 99.73 percent of all future returns.

b. Making no assumptions about the population shapes of future returns, for each investment class compute intervals that will contain at least 75 percent and 88.89 percent of all future returns.

c. Assuming that future returns are mound- shaped, find (1) An estimate of the maximum return that might be realized for each investment class. (2) An estimate of the minimum return (or maximum loss) that might be realized for each investment class.

d. Assuming that future returns are mound- shaped, which two investment classes have the highest estimated maximum returns? What are the estimated minimum returns (maximum losses) for these investment classes?

e. Assuming that future returns are mound- shaped, which two investment classes have the smallest estimated maximum returns? What are the estimated minimum returns for these investment classes?

While it is true that investment experts generally advise long- term investors to invest in variable investments, they also agree that the key to any sound investment portfolio is diversification. That is, investing in a variety of investments with differing levels of historical return and risk.

Investment risk is often measured in terms of the volatility of an investment over time. When volatility, sometimes referred to as standard deviation, increases, so too does the level of return. Conversely, as risk ( standard deviation) declines, so too do returns.

In order to explain the relationship between the return on an investment and its risk, Investment Digest presents a graph of mean return versus standard deviation ( risk) for nine investment classes over the period from 1970 to 1994. This graph, which Investment Digest calls the “ risk/ return trade- off,” is shown in Figure 3.26. The article says that this graph . . . illustrates the historical risk/ return trade- off for a variety of investment classes over the 24- year period between 1970 and 1994. In the chart, cash equivalents and fixed annuities, for instance, had a standard deviation of 0.81% and 0.54% respectively, while posting returns of just over 7.73% and 8.31%. At the other end of the spectrum, domestic small- cap stocks were quite volatile— with a standard deviation of 21.82%— but compensated for that increased volatility with a return of 14.93%. The answer seems to lie in asset allocation. Investment experts know the importance of asset allocation. In a nutshell, asset allocation is a method of creating a diversified portfolio of investments that minimize historical risk and maximize potential returns to help you meet your retirement goals and needs. Suppose that, by reading off the graph of Figure 3.26, we obtain the mean return and standard deviation combinations for the various investment classes as shown in Table 3.9. Further suppose that future returns in each investment class will behave as they have from 1970 to 1994. That is, for each investment class, regard the mean return and standard deviation in Table 3.9 as the population mean and the population standard deviation of all possible future returns. Then do the following:

a. Assuming that future returns for the various investment classes are mound- shaped, for each investment class compute intervals that will contain approximately 68.26 percent and 99.73 percent of all future returns.

b. Making no assumptions about the population shapes of future returns, for each investment class compute intervals that will contain at least 75 percent and 88.89 percent of all future returns.

c. Assuming that future returns are mound- shaped, find (1) An estimate of the maximum return that might be realized for each investment class. (2) An estimate of the minimum return (or maximum loss) that might be realized for each investment class.

d. Assuming that future returns are mound- shaped, which two investment classes have the highest estimated maximum returns? What are the estimated minimum returns (maximum losses) for these investment classes?

e. Assuming that future returns are mound- shaped, which two investment classes have the smallest estimated maximum returns? What are the estimated minimum returns for these investment classes?

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