Problem 59 - An Introduction to Mathematical Statistics and Its Applications

Question

A stock analyst claims to have devised a mathematical technique for selecting high-quality mutual funds and promises that a client's portfolio will have higher average ten-year annualized returns and lower volatility; that is, a smaller standard deviation. After ten years, one of the analyst's twenty-four- stock portfolios showed an average ten-year annualized return of \(11.50 \%\) and a standard deviation of \(10.17 \%\). The benchmarks for the type of funds considered are a mean of \(10.10 \%\) and a standard deviation of \(15.67 \%\). (a) Let \(\mu\) be the mean for a twenty-four-stock portfolio selected by the analyst's method. Test at the \(0.05\) level that the portfolio beat the benchmark; that is, test \(H_{0}: \mu=10.1\) versus \(H_{1}: \mu>10.1\). (b) Let \(\sigma\) be the standard deviation for a twenty-fourstock portfolio selected by the analyst's method. Test at the \(0.05\) level that the portfolio beat the benchmark; that is, test \(H_{0}: \sigma=15.67\) versus \(H_{1}: \sigma<15.67\).

Step-by-Step Solution

Answer

To determine whether the analyst's stock selection technique is better than the benchmark, two hypotheses are tested using a one-sample t-test for the mean and a chi-square test for the standard deviation. Results will indicate whether the analyst was successful in outperforming the benchmark.

Step 1: Testing the Mean

A one-sample t-test can be used here. The null hypothesis (\(H_0\)) is \(\mu = 10.10\%\), and the alternative hypothesis (\(H_1\)): \(\mu > 10.10\%\). In this case, the sample mean (\(x̄ = 11.50\%\)) and the sample size (\(n = 24\)). The t-statistic can be calculated using the following formula: \[ t = \frac{x̄ - μ_{0}}{s/√n} \] where \(μ_{0}\) is the value in the null hypothesis and \(s\) is the standard deviation (\(10.17 \%\)). The calculated t-statistic is then compared to the critical t-value from the t-distribution table at the \(0.05\) level of significance.

Step 2: Testing the Standard Deviation

To test the standard deviation, the chi-square test can be used. The null hypothesis (\(H_0\)) is \(\sigma = 15.67\%\), and the alternative hypothesis (\(H_1\)): \(\sigma < 15.67\%\). The chi-square statistic can be calculated by: \[ χ^{2} = (n-1) * (s/σ_{0})^{2} \] where \(σ_{0} = 15.67\%\) is the null hypothesis standard deviation value, \(s = 10.17\%\) is the sample standard deviation, and \(n = 24\) is the sample size. Compare the calculated value with the critical chi-square value from the chi-square distribution table at the \(0.05\) significance level.

Key Concepts

t-test

The t-test is a popular statistical tool used to determine if there is a significant difference between the mean of a sample and a known value (or between the means of two samples). It helps us decide whether to accept or reject a hypothesis based on data collected from samples.
In the context of the original exercise, we are testing the claim that a portfolio's performance, with a sample mean of 11.50%, is actually better than a benchmark, which has an expected mean of 10.10%.
Here's a breakdown of the steps:

• Define the null hypothesis (\(H_{0}\)): The mean return of the portfolio is equal to the benchmark, i.e., \(\mu = 10.10\%\).
• Define the alternative hypothesis (\(H_{1}\)): The mean return of the portfolio is greater than the benchmark, i.e., \(\mu > 10.10\%\).
• Compute the t-statistic using the formula:\[ t = \frac{x̄ - μ_{0}}{s/\sqrt{n}}\]where \(x̄\) is the sample mean, \(μ_{0}\) is the benchmark mean, \(s\) is the sample standard deviation, and \(n\) is the sample size.
• Compare the t-statistic to the critical value from a t-distribution table at the 0.05 significance level to determine if the result is statistically significant.

This test helps decide if we can confidently say the portfolio outperforms the benchmark or if the observed difference might be due to random sampling variance.

chi-square test

The chi-square test is applied to study the variability of data, specifically the variance or standard deviation. It's typically used in cases where you are comparing the variance of a sample to a known value.
In our example, the chi-square test helps us understand whether the volatility of an investment portfolio is genuinely lower compared to a benchmark's standard deviation.
Here's how it works:

• State the null hypothesis (\(H_{0}\)): The portfolio's standard deviation is equal to the benchmark's standard deviation, i.e., \(\sigma = 15.67\%\).
• State the alternative hypothesis (\(H_{1}\)): The portfolio's standard deviation is less than the benchmark's standard deviation, i.e., \(\sigma < 15.67\%\).
• Calculate the chi-square statistic by:\[χ^{2} = (n-1) * (s/σ_{0})^{2}\]where \(σ_{0}\) is the benchmark standard deviation, \(s\) is the sample standard deviation, and \(n\) is the sample size.
• Look up the critical chi-square value in the chi-square distribution table for the 0.05 significance level to determine if the observed standard deviation is significantly lower.

This test allows analysts to decide if the portfolio really offers lower volatility than the benchmark, which can be a critical assessment for risk-averse investors.

statistical significance

Statistical significance is a measure of how likely it is that an observed difference or relationship in data occurred by chance. It relates to hypothesis testing in that it helps us determine whether to support or reject a null hypothesis.
In statistical testing, we compare test statistics (like t-statistic or chi-square) against a threshold value known as the critical value. If the calculated statistic exceeds this critical value, the result is deemed statistically significant.
Key considerations include:

• Significance Level: It is usually set as \(\alpha = 0.05\), meaning there is a 5% risk of rejecting the null hypothesis when it is actually true.
• P-Value: An output of many statistical tests, it tells us the probability of observing the results given that the null hypothesis is true. A p-value less than 0.05 typically indicates statistical significance.
• Decision Making: If the results are statistically significant, this means the hypothesis tests suggest a genuine effect or difference that isn't due solely to chance.

In practice, statistical significance helps guide decision-making based on data. In the context of the portfolio exercise, determining statistical significance helps investors decide if the analyst’s method truly provides a return above and beyond traditional benchmarks, or if it offers reduced volatility as claimed.