# Question: The year 1987 featured extreme volatility on the stock market

The year 1987 featured extreme volatility on the stock market, including a loss of over 20 percent of the market’s value on a single day. Figure 7.6 (a) shows the percent frequency histogram of the percentage returns for the entire year 1987 for the population of all 1,815 stocks listed on the New York Stock Exchange. The mean and the standard deviation of the population of percentage returns are 3.5 percent and 26 percent, respectively. Consider drawing a random sample of n 5 stocks from the population of 1,815 stocks and calculating the mean return of the sampled stocks. If we use a computer, we can generate all the different samples of five stocks that can be obtained (there are trillions of such samples) and calculate the corresponding sample mean returns. A per-cent frequency histogram describing the population of all possible sample mean returns is given in Figure 7.6

(b). Comparing Figures 7.6( a) and ( b), we see that, although the histogram of individual stock returns and the histogram of sample mean returns are both bell- shaped and centered over the same mean of 3.5 percent, the histogram of sample mean returns looks less spread out than the histogram of individual returns. A sample of 5 stocks is a portfolio of stocks, where the average return of the 5 stocks is the portfolio’s return if we invest equal amounts of money in each of the 5 stocks. Because the sample mean returns are less spread out than the individual stock returns, we have illustrated that diversification reduces risk. Find the standard deviation of the population of all sample mean returns, and assuming that this population is normally distributed, find an interval that contains 95.44 percent of all sample mean returns.

(b). Comparing Figures 7.6( a) and ( b), we see that, although the histogram of individual stock returns and the histogram of sample mean returns are both bell- shaped and centered over the same mean of 3.5 percent, the histogram of sample mean returns looks less spread out than the histogram of individual returns. A sample of 5 stocks is a portfolio of stocks, where the average return of the 5 stocks is the portfolio’s return if we invest equal amounts of money in each of the 5 stocks. Because the sample mean returns are less spread out than the individual stock returns, we have illustrated that diversification reduces risk. Find the standard deviation of the population of all sample mean returns, and assuming that this population is normally distributed, find an interval that contains 95.44 percent of all sample mean returns.

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