Question

Here are some historical data on the risk characteristics of Bank of America and Starbucks.

                                                 Bank of America                                  Starbucks

β (beta)                                                 1.23                                                .76

Yearly standard deviation of return (%) 32.6                                              15.9

Assume the standard deviation of the return on the market was 17%. (Use decimals, not percents, in your calculations.)

a. The correlation coefficient of Bank of America's return versus Starbucks is .26. What is the standard deviation of a portfolio invested half in each stock? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

Standard deviation %

b. What is the standard deviation of a portfolio invested one-third in Bank of America, one-third in Starbucks, and one-third in risk-free Treasury bills? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

Standard deviation %

c. What is the standard deviation if the portfolio is split evenly between Bank of America and Starbucks and is financed at 50% margin, that is, the investor puts up only 50% of the total amount and borrows the balance from the broker? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

Standard deviation %

d-1. What is the approximate standard deviation of a portfolio comprised of 100 stocks with betas of 1.23 like Bank of America? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

Standard deviation %

d-2. What is the approximate standard deviation of a portfolio comprised of 100 stocks with betas of .76 like Starbucks? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

Standard deviation %

That is everything from the question.

Answer 1

Let's first summarize the information we have in a tabular form:

	Bank of America	Starbucks	Market
Beta, β	βB = 1.23	βS = 0.76	βM = 1
Standard deviation, σ	σB = 0.326	σS = 0.159	σM = 0.17
correlation coefficient of Bank of America's return versus Starbucks	ρBS = 0.26

If w_B and w_S are the proportion invested in Bank of America and Starbucks respectively then variance of the portfolio is given by: σ_P² = (w_Bσ_B)² + (w_Sσ_S)² + 2ρ_BS.(w_Bσ_B).(w_Sσ_S)

Standard deviation is square root of variance.

Let's now proceed to solve your questions one by one.

a. The correlation coefficient of Bank of America's return versus Starbucks is .26. What is the standard deviation of a portfolio invested half in each stock? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

w_B = 0.5; w_S = 0.5

Hence, σ_P² = (w_Bσ_B)² + (w_Sσ_S)² + 2ρ_BS.(w_Bσ_B).(w_Sσ_S) = (0.5 x 0.326)² + (0.5 x 0.159)² + 2 x 0.26 x (0.5 x 0.326) x ( 0.5 x 0.159) =  0.039628

Hence, σ_P =  0.199067 = 19.91%
Standard deviation: 19.91%

-----------------------------------------------

b. What is the standard deviation of a portfolio invested one-third in Bank of America, one-third in Starbucks, and one-third in risk-free Treasury bills? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

Treasury bills are risk free. They have zero risk and, hence, zero standard deviation.

w_B = 1/3; w_S = 1/3

Hence, σ_P² = (w_Bσ_B)² + (w_Sσ_S)² + 2ρ_BS.(w_Bσ_B).(w_Sσ_S) = (1/3 x 0.326)² + (1/3 x 0.159)² + 2 x 0.26 x (1/3 x 0.326) x ( 1/3 x 0.159) =   0.017612

Hence, σ_P = 0.132711 = 13.27%

Standard deviation: 13.27%

-----------------------------------------

c. What is the standard deviation if the portfolio is split evenly between Bank of America and Starbucks and is financed at 50% margin, that is, the investor puts up only 50% of the total amount and borrows the balance from the broker? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

Since, the investor puts up only 50% of the total amount and borrows the balance from the broker, that means investor has invested a total amount equal to twice the investment in part (a) of the question.

Hence, the risk now will be = 2 x risk calculated in part (a)

Standard deviation = 2 x standard deviation in part (a) = 2 x  0.199067 = 0.398133998 = 39.81%

Standard deviation: 39.81%

--------------------------------------

d-1. What is the approximate standard deviation of a portfolio comprised of 100 stocks with betas of 1.23 like Bank of America? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

The portfolio comprised of 100 stocks. This means portfolio is well diversified. In case a portfolio is well diversified then the approximate standard deviation of a portfolio comprised of 100 stocks with betas of 1.23 like Bank of America = β_B x σ_M = 1.23 x 0.17 = 0.2091 = 20.91%

Standard deviation: 20.91%

------------------------------

d-2. What is the approximate standard deviation of a portfolio comprised of 100 stocks with betas of .76 like Starbucks? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

The portfolio comprised of 100 stocks. This means portfolio is well diversified. In case a portfolio is well diversified then the approximate standard deviation of a portfolio comprised of 100 stocks with betas of .76 like Starbucks = β_S x σ_M = 0.76 x 0.17 = 0.1292 = 12.92%

Standard deviation: 12.92%