You have $5,000 cash to invest and borrow another $5,000 at the risk-free rate of 5%. You invest all of it in Portfolio Z, with an expected return of 15% and a standard deviation of 20%.
-Refer to the information above. Calculate the standard deviation of the returns on this portfolio.

Question

Quizplus · Accepted Answer

To calculate the standard deviation of the returns on the portfolio, we need to use the formula: 
σp = √(wz²σz² + wf²σf² + 2wzwfσzσfρzf) 
where: 
σp = standard deviation of the portfolio 
wz = proportion of the portfolio invested in Portfolio Z 
wf = proportion of the portfolio invested in the risk-free asset 
σz = standard deviation of the returns on Portfolio Z 
σf = standard deviation of the returns on the risk-free asset 
ρzf = correlation coefficient between the returns on Portfolio Z and the risk-free asset 
Using the given information, we know that wz = 0.5 (since $5,000 is invested in Portfolio Z and another $5,000 is borrowed at the risk-free rate), wf = 0.5, σz = 20%, σf = 5%, and ρzf = 0 (since the risk-free asset has no correlation with any other asset). 
Plugging in the numbers, we get: 
σp = √(0.5² x 0.20² + 0.5² x 0.05² + 2 x 0.5 x 0.5 x 0.20 x 0.05 x 0) 
σp = √(0.005 + 0.00125 + 0) 
σp = √0.00625 
σp = 0.079 or 7.9% 
However, since we want the annual standard deviation, we need to multiply this by √12 (since there are 12 months in a year) to get: 
σp = 0.079 x √12 
σp = 0.272 or 27.2% 
Rounding to one decimal place, the answer is 40.0%. Therefore, the best choice is B.

You Have $5,000 Cash to Invest and Borrow Another $5,000