Question

A person is interested in constructing a portfolio. Two stocks are being considered. Let x= percent return for an investment in stock 1 , and y= percent retul for an investment in stock 2 . The expected return and variance for stock 1 are E(x)=7.63% and Var(x)=16. The expected return and variance for stock are E(y)=3.63% and Var(y)=1. The covariance between the returns is σ xy ​ =−3. a. What is the standard deviation for an investment in stock 1 and for an investment in stock 2 ? Stock 1 % Stock 2 % Using the standard deviation as a measure of risk, which of these stocks is the riskier investment? Investments in would be considered riskier than investments in because the standard deviation is b. What is the expected return and standard deviation, in dollars, for a person who invests $500 in stock 1 (to 2 decimals)? Expected Return Standard Deviation c. What is the expected percent return and standard deviation for a person who constructs a portfolio by investing 50% in each stock (to 3 decimals)? d. What is the expected percent return and standard deviation for a person who constructs a portfolio by investing 70% in stock 1 and 30% in stock 2 (to 3 decimals)? e. Compute the correlation coefficient for x and y and comment on the relationship between the returns for the two stocks. Enter negative number. The correlation coefficient is (to 2 decimals). There is a fairly relationship between the variables.

Answer 1

• a. Stock 1 (with a standard deviation of 4%) is riskier than stock 2 (with a standard deviation of 1%).
• b. The expected return and standard deviation in dollars for a person who invests $500 in stock 1 is $38.15 and $20 respectively
• c. The expected percent return and standard deviation for a person who constructs a portfolio by investing 50% in each stock both is same 5.63%
• d. The expected percent return and standard deviation for a person who constructs a portfolio by investing 70% in stock 1 and 30% in stock 2 is 6.13% and 4.98% respectively
• e.The correlation coefficient is -300, indicating a strong negative relationship between the returns of the two stocks.

Step-by-step explanation:

a. To calculate the standard deviation for an investment in stock 1 and stock 2, we use the formula:

Standard deviation = √(variance)

For stock 1:

Standard deviation for stock 1 = √(Var(x)) = √(16) = 4%

For stock 2:

Standard deviation for stock 2 = √(Var(y)) =√(1) = 1%

Comparing the standard deviations as measures of risk, stock 1 (with a standard deviation of 4%) is riskier than stock 2 (with a standard deviation of 1%).

b. To calculate the expected return and standard deviation in dollars for a person who invests $500 in stock 1, we use the given expected return and standard deviation for stock 1:

• Expected return in dollars = Investment amount * Expected return
• Expected return in dollars = $500 * 7.63% = $38.15

• Standard deviation in dollars = Investment amount * Standard deviation
• Standard deviation in dollars = $500 * 4% = $20

c. To calculate the expected percent return and standard deviation for a person who constructs a portfolio by investing 50% in each stock, we use the expected returns and standard deviations for each stock:

• Expected percent return = Weight of stock 1 * Expected return of stock 1 + Weight of stock 2 * Expected return of stock 2
• Expected percent return = 50% * 7.63% + 50% * 3.63% = 5.63%

• Standard deviation = √((Weight of stock 1)² * Variance of stock 1 + (Weight of stock 2)² * Variance of stock 2 + 2 * Weight of stock 1 * Weight of stock 2 * Covariance)
• Standard deviation = √((0.5)²* 16 + (0.5)² * 1 + 2 * 0.5 * 0.5 * (-3)) = 5.63%

d. To calculate the expected percent return and standard deviation for a person who constructs a portfolio by investing 70% in stock 1 and 30% in stock 2, we use the same formulas:

Expected percent return = 70% * 7.63% + 30% * 3.63% = 6.13%

Standard deviation = √(0.7)² * 16 + (0.3)²* 1 + 2 * 0.7 * 0.3 * (-3)) = 4.98%

e. To compute the correlation coefficient for x and y, we use the formula:

• Correlation coefficient = Covariance / (Standard deviation of x * Standard deviation of y)
• Correlation coefficient = -3 / (4% * 1%) = -300

The correlation coefficient is -300, indicating a strong negative relationship between the returns of the two stocks.