Question

The market portfolio has an expected return of 9% and standard deviation of 25%. The risk-free rate is 3%. Bob has $300 to invest. What is the standard deviation of Bob's portfolio if he borrows $100 and invests $400 in the market portfolio?

a. 33.33%

b. 25.00%

c. 28.33%

d. 16.67%

e. 37.50%

Answer 1

It is given that $100 is borrowed at the risk-free rates. Note that borrowing is denoted by negative sign.

Amount invested in market portfolio = $400

Amount invested in the risk-free borrowing = -100

Total amount invested in the portfolio = $400 - $100 = $300

Weight of the market portfolio = wM = 400/300 = 4/3

Weight of the risk-free borrowing in the portfolio = wF = -100/300 = -1/3

The standard deviation of the market portfolio = σM = 25%

Standard deviation of the risk-free borrowing = σF = 0

The variance of the portfolio is calculated using the formula:

Portfolio variance = σP2 = wM2*σM2 + wF2*σF2 + 2*ρ*wM*wF*σM*σF

Putting, wM = 4/3, σM = 25% and σF = 0

Portfolio variance = σP2 = (4/3)2*(25%)2 + 0 + 0 = (4/3)2*(25%)2

We know that the standard deviation is the square root of the variance

Standard deviation = σP = (4/3)*25% = 33.3333333333333% ~ 33.33% (Rounded to two decimals)

Answer -> 33.33% (option a)