EXAMPLE 7.4

Stock portfolio diversification? An investor with a stock portfolio worth several hundred thousand dollars sued his broker and brokerage firm because lack of diversification in his portfolio led to poor performance. Table 7.1 gives the rates of return for the 39 months that the account was managed by the broker.2

Figure 7.5 gives a histogram for these data, and Figure 7.6 gives the Normal quantile plot. There are no outliers and the distribution shows no strong skewness. We are reasonably confident that the distribution of is approximately Normal, and we proceed with our inference based on Normal theory.

Table : TABLE 7.1 Monthly Rates of Return on a Portfolio (%)
−8.36 1.63 −2.27 −2.93 −2.70 −2.93 −9.14 −2.64
6.82 −2.35 −3.58 6.13 7.00 −15.25 −8.66 −1.03
−9.16 −1.25 −1.22 −10.27 −5.11 −0.80 −1.44 1.28
−0.65 4.34 12.22 −7.21 −0.09 7.34 5.04 −7.24
−2.14 −1.01 −1.41 12.03 −2.56 4.33 2.35

416

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FIGURE 7.5 Histogram of monthly rates of return for a stock portfolio, Example 7.4.
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FIGURE 7.6 Normal quantile plot, Example 7.4.

The arbitration panel compared these returns with the average of the Standard & Poor’s 500 stock index for the same period. Consider the 39 monthly returns as a random sample from the population of monthly returns the brokerage firm would generate if it managed the account forever. Are these returns compatible with a population mean of %, the S&P 500 average? Our hypotheses are

H0: μ = 0.95

Ha: μ ≠ 0.95

Minitab and SPSS outputs appear in Figure 7.7. Output from other software will be similar.

Here is one way to report the conclusion: the mean monthly return on investment for this client’s account was %. This is significantly worse than the performance of the S&P 500 stock index for the same period (, df = 38, ).