91Ó°ÊÓ

In Section \(5.1,\) we studied linear combinations of independent random variables. What happens if the variables are not independent? A lot of mathematics can be used to prove the following: Let \(x\) and \(y\) be random variables with means \(\mu_{x}\) and \(\mu_{y},\) variances \(\sigma_{x}^{2}\) and \(\sigma_{y}^{2}\) and population correlation coefficient \(\rho \text { (the Greek letter } r h o) .\) Let \(a\) and \(b\) be any constants and let \(w=a x+b y .\) Then, $$\begin{array}{l}\mu_{w}=a \mu_{x}+b \mu_{y} \\\\\sigma_{w}^{2}=a^{2} \sigma_{x}^{2}+b^{2} \sigma_{y}^{2}+2 a b \sigma_{x} \sigma_{y} \rho\end{array}$$ In this formula, \(\rho\) is the population correlation coefficient, theoretically computed using the population of all \((x, y)\) data pairs. The expression \(\sigma_{x} \sigma_{y}, \rho\) is called the covariance of \(x\) and \(y .\) If \(x\) and \(y\) are independent, then \(\rho=0\) and the formula for \(\sigma_{w}^{2}\) reduces to the appropriate formula for independent variables (see Section 5.1 ). In most real-world applications, the population parameters are not known, so we use sample estimates with the understanding that our conclusions are also estimates. Do you have to be rich to invest in bonds and real estate? No, mutual fund shares are available to you even if you aren't rich. Let \(x\) represent annual percentage return (after expenses) on the Vanguard Total Bond Index Fund, and let \(y\) represent annual percentage return on the Fidelity Real Estate Investment Fund. Over a long period of time, we have the following population estimates (based on Morningstar Mutual Fund Report). $$ \mu_{x} \approx 7.32 \sigma_{x} \approx 6.59 \mu_{y} \approx 13.19 \sigma_{y} \approx 18.56, \rho \approx 0.424 $$ (a) Do you think the variables \(x\) and \(y\) are independent? Explain. (b) Suppose you decide to put \(60 \%\) of your investment in bonds and \(40 \%\) in real estate. This means you will use a weighted average \(w=0.6 x+0.4 y\) Estimate your expected percentage return \(\mu_{w}\) and risk \(\sigma_{w}\) (c) Repeat part (b) if \(w=0.4 x+0.6 y\) (d) Compare your results in parts (b) and (c). Which investment has the higher expected return? Which has the greater risk as measured by \(\sigma_{w} ?\)

Step by step solution

01

Evaluate Independence (Part a)

The correlation coefficient \( \rho = 0.424 \) suggests that the random variables \( x \) and \( y \) are not independent because independence corresponds to \( \rho = 0 \). A correlation of 0.424 indicates some degree of linear relationship between \( x \) and \( y \).

02

Calculate Expected Return for w = 0.6x + 0.4y (Part b)

Given \( w = 0.6x + 0.4y \), the expected return \( \mu_w \) is given by \( \mu_w = a \mu_x + b \mu_y \) where \( a = 0.6 \) and \( b = 0.4 \). Plugging in the values, we get: \( \mu_w = 0.6 \times 7.32 + 0.4 \times 13.19 = 9.428 \).

03

Calculate Risk for w = 0.6x + 0.4y (Part b)

The variance of \( w \), \( \sigma_w^2 \), is given by \( a^2 \sigma_x^2 + b^2 \sigma_y^2 + 2ab\sigma_x\sigma_y \rho \). Calculating this: - \( a^2 \sigma_x^2 = 0.6^2 \times 6.59^2 = 15.5628 \)- \( b^2 \sigma_y^2 = 0.4^2 \times 18.56^2 = 55.0352 \)- \( 2ab\sigma_x\sigma_y \rho = 2 \times 0.6 \times 0.4 \times 6.59 \times 18.56 \times 0.424 = 25.8944 \)Thus, \( \sigma_w^2 = 15.5628 + 55.0352 + 25.8944 = 96.4924 \) and \( \sigma_w = \sqrt{96.4924} \approx 9.82 \).

04

Calculate Expected Return for w = 0.4x + 0.6y (Part c)

With \( w = 0.4x + 0.6y \), expected return \( \mu_w = a \mu_x + b \mu_y \). Hence, \( \mu_w = 0.4 \times 7.32 + 0.6 \times 13.19 = 11.082 \).

05

Calculate Risk for w = 0.4x + 0.6y (Part c)

The variance \( \sigma_w^2 \) is \( a^2 \sigma_x^2 + b^2 \sigma_y^2 + 2ab\sigma_x\sigma_y \rho \). Calculating:- \( a^2 \sigma_x^2 = 0.4^2 \times 6.59^2 = 6.9168 \)- \( b^2 \sigma_y^2 = 0.6^2 \times 18.56^2 = 123.8292 \)- \( 2ab\sigma_x\sigma_y \rho = 2 \times 0.4 \times 0.6 \times 6.59 \times 18.56 \times 0.424 = 25.8944 \)Therefore, \( \sigma_w^2 = 6.9168 + 123.8292 + 25.8944 = 156.64 \) and so \( \sigma_w = \sqrt{156.64} \approx 12.52 \).

06

Compare Returns and Risks (Part d)

Comparing the results:- Part (b): \( \mu_w = 9.428 \), \( \sigma_w \approx 9.82 \).- Part (c): \( \mu_w = 11.082 \), \( \sigma_w \approx 12.52 \).The investment in Part (c) has a higher expected return but also a higher risk (\( \sigma_w \)).

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Key Concepts

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

Random Variables

In statistics, random variables are a key concept and can be seen as numerical outcomes that are dependent on the outcome of a random event. They are essentially a way to assign numerical values to the results of a random process. For example, if you were to roll a six-sided die, the result of that roll would be a random variable. It could be a 1, 2, 3, 4, 5, or 6, each with an equal probability.

• Discreteness and Continuity: Random variables can be discrete, like the die roll, where the outcomes are countable and distinct.
• Probability Distributions: Each random variable has a probability distribution that tells you the probabilities of the outcomes. For discrete variables, this is a probability mass function, while continuous variables use a probability density function.

Understanding these distributions helps us to analyze events quantitatively. They allow us to compute measures like mean (expected value) and variance, giving us tools to understand variability and central tendencies in datasets.

Population Correlation Coefficient

The population correlation coefficient, often symbolized by the Greek letter \( \rho \), shows the strength and direction of a linear relationship between two random variables. It ranges from -1 to 1.

• Value Interpretation: A \( \rho \) value of 1 indicates a perfect positive linear relationship, meaning as one variable increases, the other one does too. A \( \rho \) of -1 indicates a perfect negative linear relationship, meaning as one variable increases, the other decreases.
• Zero Correlation: If \( \rho \) is 0, there's no linear relationship between the variables, suggesting independence.

In our context, \( \rho = 0.424 \) reveals some positive correlation between the bond and real estate funds, showing that as one increases, the other tends to also increase, but not perfectly. This statistic allows investors to understand and predict how two different investments might behave relative to one another.

Expected Return and Risk

Expected return and risk are crucial in understanding investments. Expected return, represented by the mean \( \mu \), indicates the average outcome you hope to achieve, while risk is quantified by variance or standard deviation, which shows how much the returns can vary.

• Expected Return (\( \mu \)): Using constants to create a weighted average, you can predict the future returns of an investment portfolio. In the exercise, the calculation \( \mu_w = a \mu_x + b \mu_y \) helped determine the expected return of mixed investments in bonds and real estate.
• Standard Deviation (\( \sigma \)): This measures the risk or variability of investment returns. A higher standard deviation means more risk, as the actual returns can vary widely from the expected return.

These calculations are central to strategic investment decisions, allowing investors to align risks with potential gains meaningfully.

Weighted Averages

Weighted averages are used when different components contribute unequally to the total. In finance, they help determine the average return of an investment portfolio composed of multiple assets, each having different importance.

• Calculation Basics: A weighted average is calculated by multiplying each component by a factor (its weight), summing these products, and dividing by the total weight. In the case of investment, if the weights are percentages, they should add up to 1 or 100%.
• Application in Portfolios: In the investment context, weighted averages help you understand combined returns and risks of different assets. If 60% of your money is in bonds and 40% in real estate, the weighted average return \( w = 0.6x + 0.4y \) gives the expected return \( \mu_w \).

Weighted averages simplify complex aggregate calculations, providing a more personalized insight into overall portfolio performance based on personal investment distributions.