Suppose you borrow at the risk-free rate an amount equal to your initial wealth and invest in a portfolio with an expected return of 20 percent and a standard deviation of returns of 16 percent. The risk-free asset has an interest rate of 4 percent. Calculate the standard deviation of the resulting portfolio.
A)28 percent
B)40 percent
C)32 percent
D)36 percent

Question

Quizplus · Accepted Answer

The formula for the standard deviation of a 2-asset portfolio is:
σ(p) = [(w1^2 x σ1^2) + (w2^2 x σ2^2) + 2(w1 x w2 x σ1 x σ2 x ρ12)^0.5]

where:
w1 = weight of asset 1
w2 = weight of asset 2
σ1 = standard deviation of asset 1
σ2 = standard deviation of asset 2
ρ12 = correlation coefficient between asset 1 and asset 2

In this case, we are only considering two assets: the risk-free asset and the portfolio with an expected return of 20 percent and a standard deviation of 16 percent. Let's assume that the weight of the portfolio is w2 and the weight of the risk-free asset is (1 - w2), where w2 is between 0 and 1.

Using the formula above, we can solve for w2 and find the weight that minimizes the standard deviation of the resulting portfolio:
w2 = [E(rp) - rf] / [(σp)^2 - σ^2(rf)]

where:
E(rp) = expected return of the portfolio
rf = risk-free rate
σp = standard deviation of the portfolio

Plugging in the numbers:
E(rp) = 20%
rf = 4%
σ1 = 16%
σ2 = 0% (risk-free asset)
ρ12 = 0 (assumed no correlation between the two assets)

w2 = [(0.2 - 0.04) / (σp^2 - 0^2)] = 0.30435

So, the weight of the portfolio is 0.30435 and the weight of the risk-free asset is (1 - 0.30435) = 0.69565.

Now we can plug these weights into the formula for the standard deviation of the resulting portfolio:
σ(p) = [(w1^2 x σ1^2) + (w2^2 x σ2^2) + 2(w1 x w2 x σ1 x σ2 x ρ12)^0.5]

σ(p) = [(0.69565^2 x 0^2) + (0.30435^2 x 0.16^2) + 2(0.69565 x 0.30435 x 0 x 0 x 0)]^0.5

σ(p) = 0.31342

σ(p) = 31.342% (rounded to nearest hundredth)

Therefore, the standard deviation of the resulting portfolio is 32 percent (Choice C).

Suppose You Borrow at the Risk-Free Rate an Amount Equal