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 \(\rho\) (the Greek letter rho). Let \(a\) and \(b\) be any constants and let \(w=a x+b y .\) Then, 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). $$ \mu_{x} \approx 7.32 \quad \sigma_{x} \approx 6.59 \quad \mu_{y} \approx 13.19 \quad \sigma_{y} \approx 18.56 \quad \rho \approx 0.424 $$ (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

Determine Independence

The independence of two random variables, such as \(x\) and \(y\), is determined by the correlation coefficient \(\rho\). If \(\rho = 0\), the variables are independent. However, given \(\rho \approx 0.424\), \(x\) and \(y\) are not independent because \(\rho\) is not zero. So, they have some degree of linear relationship.

02

Calculate Expected Return for Part (b)

To find the expected percentage return \(\mu_{w}\) when \(w = 0.6x + 0.4y\), use linear combinations of expected values: \[\mu_{w} = 0.6\mu_{x} + 0.4\mu_{y} = 0.6(7.32) + 0.4(13.19)\]. Compute this to find \(\mu_{w}\).

03

Calculation for Part (b):

Plug in the values: \[\mu_{w} = 0.6 \times 7.32 + 0.4 \times 13.19 = 4.392 + 5.276 = 9.668\]. So, \(\mu_{w} \approx 9.668\%\).

04

Calculate Risk for Part (b)

To find the risk \(\sigma_{w}\) for \(w = 0.6x + 0.4y\), use the formula: \[\sigma_{w}^{2} = (0.6^{2})\sigma_{x}^{2} + (0.4^{2})\sigma_{y}^{2} + 2(0.6)(0.4)\sigma_{x}\sigma_{y}\rho\]. Substitute known values.

05

Risk Calculation for Part (b):

Given \(\sigma_{x} \approx 6.59\), \(\sigma_{y} \approx 18.56\), \(\rho \approx 0.424\): \(\sigma_{w}^{2} = 0.36(6.59^{2}) + 0.16(18.56^{2}) + 0.48(6.59)(18.56)(0.424)\). Calculate \(\sigma_{w}\) by finding the square root of \(\sigma_{w}^{2}\).

06

Calculate Expected Return for Part (c)

For \(w = 0.4x + 0.6y\), \(\mu_{w} = 0.4\mu_{x} + 0.6\mu_{y}\). Use the same formula to compute: \[\mu_{w} = 0.4(7.32) + 0.6(13.19)\]. Compute this to find \(\mu_{w}\).

07

Calculation for Part (c):

Plug in the values: \[\mu_{w} = 0.4 \times 7.32 + 0.6 \times 13.19 = 2.928 + 7.914 = 10.842\]. So, \(\mu_{w} \approx 10.842\%\).

08

Calculate Risk for Part (c)

Using \(w = 0.4x + 0.6y\), apply the variance formula again: \[\sigma_{w}^{2} = (0.4^{2})\sigma_{x}^{2} + (0.6^{2})\sigma_{y}^{2} + 2(0.4)(0.6)\sigma_{x}\sigma_{y}\rho\]. Calculate \(\sigma_{w}\).

09

Compare Results from Parts (b) and (c)

Compare the expected returns \(\mu_{w}\) and risks \(\sigma_{w}\) calculated in parts (b) and (c). Identify which investment strategy provides a higher expected return and which has more risk based on the calculated \(\sigma_{w}\).

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Key Concepts

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

Linear Combinations

Linear combinations are an essential concept in statistics, particularly when dealing with random variables. It involves creating a new variable from two or more existing variables by multiplying each by a constant and summing the results. This process allows us to combine variables to study their collective behavior.
For example, if you have two random variables, \(x\) and \(y\), you can form a linear combination by defining a new variable \(w = ax + by\), where \(a\) and \(b\) are constants. This new variable \(w\) represents a weighted sum of \(x\) and \(y\), allowing us to analyze things like risks and returns in combined investment portfolios.
Linear combinations are useful in many fields, such as finance, where they help in creating diversified investment strategies. In this context, you might adjust \(a\) and \(b\) to reflect your desired investment distribution across different asset types.
Understanding linear combinations helps you determine how changing the proportions of your investments could alter overall performance metrics like expected return and risk.

Random Variables

In statistics, a random variable is a numerical representation of a random phenomenon. It assigns a real number to each outcome of a random event. Random variables can be discrete, taking on a finite or countable number of values, or continuous, taking on any value within a certain range.
In our exercise, we have \(x\) and \(y\) as random variables representing the annual percentage returns of different investment funds. They provide a framework to model the uncertainty and variability in investment returns. This allows us to apply statistical methods to predict future performance and make informed decisions.
The values of a random variable describe the possible outcomes and each outcome's likelihood in the context of the random event we're studying. These values can be used to compute other statistical properties such as mean (expected value), variance, and standard deviation, which are necessary for understanding risk and return in investments.

Covariance

Covariance is a crucial measure that quantifies the degree to which two random variables change together. It indicates whether an increase in one variable tends to be accompanied by an increase or decrease in another variable.
If the covariance is positive, it signifies that the variables generally move in the same direction. Conversely, a negative covariance implies that as one variable increases, the other tends to decrease. When evaluating investments, understanding covariance helps us assess how different assets might interact with one another in a portfolio.
Mathematically, the covariance of variables \(x\) and \(y\) is expressed as \(\text{Cov}(x, y) = \sigma_x \sigma_y \rho\), where \(\rho\) is the correlation coefficient. This formula shows how variance and correlation interplay—the stronger the correlation, the bigger the covariance, influencing the total risk of an investment portfolio when considering linear combinations.

Correlation Coefficient

The correlation coefficient, represented by \(\rho\), measures the strength and direction of a linear relationship between two random variables. Its values range from -1 to 1. A value of 1 indicates a perfect positive linear relationship, 0 implies no linear correlation, and -1 indicates a perfect negative linear relationship.
In our example, \(\rho \approx 0.424\), indicating a moderate positive relationship between the returns on the Vanguard Total Bond Index Fund and the Fidelity Real Estate Investment Fund. This means that, to some extent, when the return on one fund increases, the return on the other tends to increase as well.
Understanding the correlation coefficient is vital for portfolio management. It helps investors minimize risk and optimize returns by selecting a mix of investments with correlations that lower the overall portfolio variance. This concept, when combined with covariance and linear combinations, forms the basis of Modern Portfolio Theory, aiming to achieve the best possible investment returns for a given level of risk.