Question

Problem 11-25 Portfolio Returns and Deviations [LO 1, 2]

Consider the following information on a portfolio of three stocks:

State of	Probability of			Stock A			Stock B			Stock C
Economy	State of Economy			Rate of Return			Rate of Return			Rate of Return
Boom		.12			.11			.36			.41
Normal		.51			.19			.31			.29
Bust		.37			.20		?	.30		?	.39

a. If your portfolio is invested 44 percent each in A and B and 12 percent in C, what is the portfolio’s expected return, the variance, and the standard deviation? (Do not round intermediate calculations. Round your variance answer to 5 decimal places, e.g., 32.16161. Enter your other answers as a percent rounded to 2 decimal places, e.g., 32.16.)

Expected return		%
Variance
Standard deviation		%

b. If the expected T-bill rate is 4.7 percent, what is the expected risk premium on the portfolio? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)

Expected risk premium             %

Answer 1

We need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get:

Boom:

E(Rp) = 0.44(0.11) + 0.44(0.36) + .0.12(0.41)

E(Rp) = 0.2560 or 25.60%

Normal:

E(Rp) = 0.44(0.19) + 0.44(0.31) + 0.12(0.29)

E(Rp) = 0.2548 or 25.48%

Bust:

E(Rp) = 0.44(0.20) + 0.44(–0.30) + 0.12(–0.39)

E(Rp) = –0.0908 or –9.08%

And the expected return of the portfolio is:

E(Rp) = 0.12(0.2560) + 0.51(0.2548) + 0.37(–0.0908)

E(Rp) = 0.12707 or 12.71%

To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then sum. The result is the variance. So, the variance and standard deviation of the portfolio are:

?²p= 0.12(0.2560 – 0.12707)2+ 0.51(0.2548 – 0.12707)²+ 0.37(–0.0908 – 0.12707)²

?2p= 0.0804

?p= (0.0804)^1/2

?p= 0.2835 or 28.35%

B)

The risk premium is the return of a risky asset, minus the risk-free rate. T-bills are often used as the risk-free rate, so:

RP_i= E(R_p) – R_f

RP_i= 0.12707 – 0.047

= 0.08 or 8%