Financial Risk Manager Part 1
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An executive on the in-house trading floor of a financial institution is evaluating the performance of two bond traders, trader A and trader B, by analyzing their annual outcomes over a ten-year period. Throughout these ten years, trader A has generated an average profit of 7%, with a standard deviation of 15%. In contrast, trader B has attained an average profit of 12% with a standard deviation of 20%. The executive intends to perform a hypothesis test to compare the traders' performance, where the null hypothesis (H0) claims that both traders have identical performance levels, and the alternative hypothesis (H1) asserts that trader B's mean return is higher than trader A's mean return. Assuming the results from both traders are independent, which of the following options correctly establishes the test statistic and the critical value at the 5% significance level for this alternative hypothesis?
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A
The test statistic is -0.63 and the critical value is 2.31.
B
The test statistic is -0.27 and the critical value is 2.31.
C
The test statistic is 0.27 and the critical value is 1.86.
D
The test statistic is 0.63 and the critical value is 1.86.
Explanation:
D is correct. If we let X and Y denote traders B and A respectively (noting that it is safest to work with proportions rather than percentages) the test statistic for testing that the means are equal (i.e., Ho: μx = μy) is: [ T = \frac{\bar{x} - \bar{y}}{s_{\bar{x},\bar{y}}} = \frac{0.12 - 0.07}{\sqrt{\frac{0.20^2}{10} + \frac{0.15^2}{10}}} = 0.63 ] Note that the denominator no longer has the (2\bar{oxy}/n) component in the calculation; this is because the two performance results are independent - as such, the correlation between X and Y is 0. The statistic 0.63 follows a t-distribution with 8 degrees of freedom and a 5% (one-sided) size of test uses the critical value of 1.86._
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