Question

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

	Bank of America	Starbucks
β (beta)	1.68	.73
Yearly standard deviation of return (%)	30.7	17.0

Assume the standard deviation of the return on the market was 19%. (Use decimals, not percents, in your calculations.)

     

a. The correlation coefficient of Bank of America's return versus Starbucks is .44. 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 1.68 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 .73 like Starbucks? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

Standard deviation             %

Answer 1

Standard deviation for portfolio of two stocks is calculated by = $\sqrt{(w1^{2} \sigma 1^{2})+(w2^{2} \sigma 2^{2})+ (2* w1* w2*\sigma 1 *\sigma 2*\rho )}$

where w1 and w2 are weights of stocks in the portfolio , $\sigma 1, \sigma 2$ are standard deviations of the stocks and $\rho$ is correlation between the two

a) $\sigma p$ = $\sqrt{(w1^{2}\sigma 1^{2})+(w2^{2}\sigma 2^{2})+ (2* w1* w2*\sigma 1 *\sigma 2*\rho )}$

= $\sqrt{(0.5^{2}* 30.7^{2})+(0.5^{2}*17^{2})+ (2* 0.5* 0.5*30.7 *17*0.44 )}$ =20.5594 %

b) As treasury bills are risk free , their standard deviation = 0 . Also correlation between treasury bill and stocks = 0

Standard deviation for portfolio of three assets is calculated by =

$\sqrt{(w1^{2} \sigma 1^{2})+(w2^{2} \sigma 2^{2})+(w3^{2} \sigma 3^{2})+ (2* w1* w2*\sigma 1 *\sigma 2*\rho(1,2)) +(2* w2* w3*\sigma 2*\sigma 3*\rho (2,3))+(2* w1* w3*\sigma 1 *\sigma 3*\rho (1,3))}$= $\sqrt{(0.33^{2}* 30.7^{2})+(0.33^{2}*17^{2})+ +(0.33^{2}*0^{2})+(2* 0.33* 0.33*30.7 *17*0.44 )+(2* 0.33* 0.33*30.7 *0*0 )+(2* 0.33* 0.33*0 *17*0 )}$ =13.569%

c) Where 50% of the amount is financed , weight of risk free treasury bills = -50%

so weight of each stock can be given by equation w1+ w2 + (-50%) = 100% here w1 = w2 as the portfolio is split evenly between Bank of America and Starbucks.

2w =150% , w = 75%

Standard deviation for portfolio of three assets is calculated by =

$\sqrt{(w1^{2} \sigma 1^{2})+(w2^{2} \sigma 2^{2})+(w3^{2} \sigma 3^{2})+ (2* w1* w2*\sigma 1 *\sigma 2*\rho(1,2)) +(2* w2* w3*\sigma 2*\sigma 3*\rho (2,3))+(2* w1* w3*\sigma 1 *\sigma 3*\rho (1,3))}$= $\sqrt{(0.75^{2}* 30.7^{2})+(0.75^{2}*17^{2})+ +(-0.5^{2}*0^{2})+(2* 0.75* 0.75*30.7 *17*0.44 )+(2* 0.75*-0.5*30.7 *0*0 )+(2*0.75* -0.5*0 *17*0 )}$ =30.839%

d) Standard deviation for portfolio of n similar assets = $n*\sqrt{((1/n)^2*\sigma ^2)}$ = n* 1/n * $\sigma$ = $\sigma$

d-1) standard deviation of a portfolio comprised of 100 stocks with betas of 1.68 like Bank of America = standard deviation of Bank of America = 30.7%

d-2) standard deviation of a portfolio comprised of 100 stocks with betas of .73 like Starbucks = standard deviation of Starbucks = 17%