Question

1. Consider the following information about three stocks:

State of Economy	Probability of State of Economy	Rate of Return If State Occurs
		Stock A	Stock B	Stock C
Boom	.25	.21	.36	.55
Normal	.60	.17	.13	.09
Bust	.15	.00	−.28	−.45

1. If your portfolio is invested 40 percent each in A and B and 20 percent in C, what is the portfolio expected return? The variance? The standard deviation?
2. If the expected T-bill rate is 3.80 percent, what is the expected risk premium on the portfolio?
3. If the expected inflation rate is 3.50 percent, what are the approximate and exact expected real returns on the portfolio? What are the approximate and exact expected real risk premiums on the portfolio?

Answer 1

Answer a.

Weight of Stock A = 0.40
Weight of Stock B = 0.40
Weight of Stock C = 0.20

Boom:

Expected Return = 0.40 * 0.21 + 0.40 * 0.36 + 0.20 * 0.55
Expected Return = 0.3380

Normal:

Expected Return = 0.40 * 0.17 + 0.40 * 0.13 + 0.20 * 0.09
Expected Return = 0.1380

Bust:

Expected Return = 0.40 * 0.00 + 0.40 * (-0.28) + 0.20 * (-0.45)
Expected Return = -0.2020

Expected Return of Portfolio = 0.25 * 0.3380 + 0.60 * 0.1380 + 0.15 * (-0.2020)
Expected Return of Portfolio = 0.1370 or 13.70%

Variance of Portfolio = 0.25 * (0.3380 - 0.1370)^2 + 0.60 * (0.1380 - 0.1370)^2 + 0.15 * (-0.2020 - 0.1370)^2
Variance of Portfolio = 0.027339

Standard Deviation of Portfolio = (0.027339)^(1/2)
Standard Deviation of Portfolio = 0.1653 or 16.53%

Answer b.

Expected Risk Premium = Expected Return - Risk-free Rate
Expected Risk Premium = 13.70% - 3.80%
Expected Risk Premium = 9.90%

Answer c.

Approximate Expected Real Return = Expected Return - Expected Inflation Rate
Approximate Expected Real Return = 13.70% - 3.50%
Approximate Expected Real Return = 10.20%

Exact Expected Real Return = (Expected Return - Expected Inflation Rate) / (1 + Expected Inflation Rate)
Exact Expected Real Return = (0.1370 - 0.0350) / (1 + 0.0350)
Exact Expected Real Return = 0.1020 / 1.0350
Exact Expected Real Return = 0.0986 or 9.86%

Approximate Expected Real Risk Premium = Expected Risk Premium - Expected Inflation Rate
Approximate Expected Real Risk Premium = 9.90% - 3.50%
Approximate Expected Real Risk Premium = 6.40%

Exact Expected Real Risk Premium = (Expected Real Risk Premium - Expected Inflation Rate) / (1 + Expected Inflation Rate)
Exact Expected Real Risk Premium = (0.0990 - 0.0350) / (1 + 0.0350)
Exact Expected Real Risk Premium = 0.0640 / 1.0350
Exact Expected Real Risk Premium = 0.0618 or 6.18%