Question

You invest $100 in a risky asset with an expected rate of return of 0.14 and a standard deviation of 0.20 and a T-bill with a rate of return of 0.06.

What percentages of your money must be invested in the risk-free asset and the risky asset, respectively, to form a portfolio with a standard deviation of 0.08?

	A.	30% and 70%
	B.	50% and 50%
	C.	60% and 40%
	D.	cannot be determined
	E.	40% and 60%

Answer 1

T-bills are risk-free asset and have standard devaition of 0.

Weight of the risk-free asset = w1, Return on the risk-free asset = R1 = 0.06, Standard deviation of the risk-free asset = σ1 = 0

Weight of the risky asset = w2, Return on the risky asset = R2 = 0.14, Standard deviation of the risky asset = σ2 = 0.20

Standard deviation of the portfolio = σP = 0.08

The portfolio consists of risk-free asset with weight w1 and risky asset of weight w2. Sum of the weights of risk free asset and risky asset is 100% or 1, i.e., w1 + w2 = 1. Variance of the portfolio consisting of risk-free asset and risky asset is given by the formula:

Vraiance of portfolio = σP2 = w12*σ12 + w22*σ22 + 2*ρ*w1*w2*σ1*σ2

We know that σ1 = 0

Therefore, σP2 = 0 + w22*σ22 + 0

Standard devaition of the portfolio is square-root of Variance

Standard deviation of the portfolio = σP = (w22*σ22)1/2 = w2*σ2

Standard deviation of the portfolio consisting of a risk-free asset and risky asset is given by the formula:

Standard deviation of the portfolio = σP = w2*σ2 = 0.08

w2*0.2 = 0.08

w2 = 0.08/0.2 = 0.4 = 40%

w1 = 1 - w2 = 1 - 0.4 = 0.6 = 60%

Weight of the risk-free asset = 60%

Weight of the risky-asset = 40%

Answer -> 60% and 40% (Option C)