Question

Consider the following information about three stocks:

  

				Rate of Return If State Occurs
State of	Probability of
Economy	State of Economy			Stock A			Stock B			Stock C
Boom		.20			.28			.40			.56
Normal		.45			.22			.20			.18
Bust		.35			.00			−.20			−.48

  

a-1.	If your portfolio is invested 30 percent each in A and B and 40 percent in C, what is the portfolio expected return? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
a-2.	What is the variance? (Do not round intermediate calculations and round your answer to 5 decimal places, e.g., .16161.)
a-3.	What is the standard deviation? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
b.	If the expected T-bill rate is 4.20 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.)
c-1.	If the expected inflation rate is 3.80 percent, what are the approximate and exact expected real returns on the portfolio? (Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.)
c-2.	What are the approximate and exact expected real risk premiums on the portfolio?(Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.)

Answer 1

Part a - 1

Please see the table below for portfolio return Rp_i across three states. Figures in parenthesis mean negative value.

		Rate of Return If State Occurs
State of	Probability of	Stock A	Stock B	Stock C	Portfolio
Economy	State of Economy
	Pi	Rai	Rbi	Rci	Rpi = 0.3Rai + 0.3Rbi + 0.4Rci
Boom	0.20	0.28	0.40	0.56	0.43
Normal	0.45	0.22	0.20	0.18	0.20
Bust	0.35	-	(0.20)	(0.48)	(0.25)

Portfolio expected return = E(Rp)

=0.20 x 0.43 + 0.45 x 0.20 + 0.35 x (-0.25) = 0.0865 = 8.65%

Part a - 2

Variance

= 0.20 x (0.43 - 0.0865)² + 0.45 x (0.20 - 0.0865)² + 0.35 x (-0.25 - 0.0865)² = 0.06902

Part a - 3

Standard deviation = (Variance)^1/2 = (0.06902)^1/2 = 0.2627 = 26.27%

Part (b)

Expected risk premium on the portfolio = E(Rp) - Risk free rate = 8.65% - 4.20% = 4.45%

Part (c) - 1

inflation = i = 3.80%

The approximate expected real returns on the portfolio = E(Rp) - i = 8.65% - 3.80% = 4.85%

The exact expected real returns on the portfolio = [1 + E(Rp)] / (1 + i) - 1 = (1 + 8.65%) / (1 + 3.80%) - 1 = 4.67%

Part (c) - 2

Approximate real risk free rate = Risk free rate - inflation = 4.20% - 3.80% = 0.40%

The approximate expected real risk premiums on the portfolio = 4.85% (calculated in part (c) - 1 above) - 0.40% = 4.45%

Exact real risk free rate = (1 + Risk free rate) / (1 + inflation) - 1 = (1 + 4.20%) / (1 + 3.80%) - 1 = 0.39%

The exact expected real risk premiums on the portfolio = 4.85% (calculated in part (c) - 1 above) - 0.39% = 4.46%