91Ó°ÊÓ

Do bonds reduce the overall risk of an investment portfolio? Let \(x\) be a random variable representing annual percent return for Vanguard Total Stock Index (all stocks). Let \(y\) be a random variable representing annual return for Vanguard Balanced Index \((60 \% \text { stock and } 40 \%\) bond). For the past several years, we have the following data (Reference: Morning star Research Group, Chicago). $$\begin{array}{ccccccccccc} x: & 11 & 0 & 36 & 21 & 31 & 23 & 24 & -11 & -11 & -21 \\ y: & 10 & -2 & 29 & 14 & 22 & 18 & 14 & -2 & -3 & -10 \end{array}$$ (a) Compute \(\Sigma x, \Sigma x^{2}, \Sigma y,\) and \(\Sigma y^{2}\). (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for \(x\) and for \(y\). (c) Compute a \(75 \%\) Chebyshev interval around the mean for \(x\) values and also for \(y\) values. Use the intervals to compare the two funds. (d) Compute the coefficient of variation for each fund. Use the coefficients of variation to compare the two funds. If \(s\) represents risks and \(\bar{x}\) represents expected return, then \(s / \bar{x}\) can be thought of as a measure of risk per unit of expected return. In this case, why is a smaller CV better? Explain.

Step by step solution

01

Calculate Sums and Squares

First, let's compute the sums and squares for each dataset. For the x-values:- \( \Sigma x = 11 + 0 + 36 + 21 + 31 + 23 + 24 - 11 - 11 - 21 = 103 \)- \( \Sigma x^2 = 11^2 + 0^2 + 36^2 + 21^2 + 31^2 + 23^2 + 24^2 + (-11)^2 + (-11)^2 + (-21)^2 = 3559 \)For the y-values:- \( \Sigma y = 10 - 2 + 29 + 14 + 22 + 18 + 14 - 2 - 3 - 10 = 90 \)- \( \Sigma y^2 = 10^2 + (-2)^2 + 29^2 + 14^2 + 22^2 + 18^2 + 14^2 + (-2)^2 + (-3)^2 + (-10)^2 = 2331 \)

02

Compute Means

Next, let's calculate the sample mean for each dataset. For the x-values:- There are 10 data points, so the mean is \( \bar{x} = \frac{\Sigma x}{10} = \frac{103}{10} = 10.3 \)For the y-values:- Also, there are 10 data points, so the mean is \( \bar{y} = \frac{\Sigma y}{10} = \frac{90}{10} = 9.0 \)

03

Calculate Variance and Standard Deviation

For variance, we use the formula \[ s^2 = \frac{\Sigma(x_i^2) - \frac{(\Sigma x)^2}{n}}{n - 1} \].For x-values:- \( s_x^2 = \frac{3559 - \frac{103^2}{10}}{9} = \frac{3559 - 1059.7}{9} = \frac{2499.3}{9} \approx 277.7 \)- Standard deviation, \( s_x = \sqrt{277.7} \approx 16.66 \)For y-values:- \( s_y^2 = \frac{2331 - \frac{90^2}{10}}{9} = \frac{2331 - 810}{9} = \frac{1521}{9} \approx 169 \)- Standard deviation, \( s_y = \sqrt{169} = 13 \)

04

Compute Chebyshev Intervals

To find the Chebyshev interval for each dataset that captures at least 75% of the data:For x-values:- The interval \( [\bar{x} - 2s_x, \bar{x} + 2s_x] \) is because Chebyshev's inequality states that \( 1 - \frac{1}{k^2} \) of data lies within \( k \) standard deviations.- \( [10.3 - 2(16.66), 10.3 + 2(16.66)] = [-23.02, 43.62] \)For y-values:- \( [\bar{y} - 2s_y, \bar{y} + 2s_y] = [9 - 2(13), 9 + 2(13)] = [-17, 35] \)These intervals show that both funds cover similar ranges, but the x-values (all stocks) have a wider spread.

05

Calculate Coefficient of Variation

The coefficient of variation (CV) is \( \frac{s}{\bar{x}} \) and measures risk per unit of return.For x-values:- \( CV_x = \frac{16.66}{10.3} \approx 1.62 \)For y-values:- \( CV_y = \frac{13}{9} \approx 1.44 \)These CVs indicate that the stock-alone index has a higher risk per unit of return than the balanced portfolio. A smaller CV is better as it implies less risk per unit of return.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chebyshev's Inequality

Chebyshev's Inequality is a fundamental concept in probability and statistics, providing a useful bound on how a distribution of data is spread around its mean. Unlike the normal distribution, which assumes data symmetry around the mean, Chebyshev’s theorem applies to any dataset, regardless of its distribution.
Chebyshev’s Inequality states: For any number of standard deviations \( k \) greater than 1, the proportion of data within \( k \) standard deviations from the mean is at least \( 1 - \frac{1}{k^2} \).
This is crucial for analyzing portfolios since it allows us to establish an expected range of returns without assuming normal distribution. For instance, with a \( 75\% \) probability, data will fall within approximately 2 standard deviations from the mean, indicating wider variability for the stock-only fund compared to the balanced fund.

Coefficient of Variation

The Coefficient of Variation (CV) is a vital statistical measure used to assess the risk relative to expected return, particularly valuable in finance for comparing different investments.
The formula for CV is \( \frac{s}{\bar{x}} \), where \( s \) is the standard deviation and \( \bar{x} \) is the mean.
A smaller CV is preferable as it implies lower risk per unit of expected return, which is crucial when assessing portfolios. In our exercise, the stock-only index fund has a CV of about 1.62, higher than the balanced index fund's CV of 1.44, indicating a higher risk. Such analyses help investors make informed decisions based on the volatility of returns in relation to expected performance.

Sample Mean and Variance

Understanding sample mean and variance is pivotal in data analysis, providing insights into the central tendency and dispersion of a dataset.
The sample mean, \( \bar{x} \), is calculated as the sum of all data points divided by the number of observations \( n \). It represents the average return when analyzing financial data.
Variance, \( s^2 \), measures the spread of data points from the mean using the formula \( s^2 = \frac{\Sigma(x_i^2) - \frac{(\Sigma x)^2}{n}}{n - 1} \). A higher variance indicates greater dispersion, implying more uncertainty around the expected return.
Together, these metrics help quantify the potential volatility within an investment, such as the stock-only and balanced index funds in our dataset.

Data Analysis in Statistics

Data analysis in statistics encompasses various methods for examining and interpreting datasets to extract meaningful insights. This process is crucial for decision-making in fields such as finance.
Key steps in data analysis include:

• Collecting data
• Calculating descriptive statistics like mean and variance
• Assessing variability through measures like standard deviation
• Employing probabilistic theories like Chebyshev's Inequality

In investment, these steps assist in evaluating the risk and return of portfolios. By leveraging statistical tools, investors can better understand potential outcomes and optimize their strategies to meet financial goals.