Question

Consider the following situation:

State of Economy	Probability of State of Economy	Returns if State Occurs
		Stock A	Stock B	Stock C
Boom	20%	25%	10%	5%
Recession	80%	-30%	5%	10%

The expected return on the market portfolio is 7% and the US Treasury bill yields 3%. The capital market is currently in equilibrium.

1. Which stock has the most systematic risk? Provide all the steps and equations.

2. Which stock has the most unsystematic risk? Explain why. Provide all the steps and equations.

3. What is the standard deviation of a portfolio which is comprised of $8,400 invested in stock A,$3,600 in stock B, and $8,000 in stock C?

4. If the expected inflation rate is 2.5%, what is the exactexpected real return on the portfolio of part (c)?

Answer 1

Part - a. Which has the most systematic risk

Stock A: E(Ra) = 0.2(0.25) + 0.8(-0.3) = -0.19 → -19%

-0.19 = 0.03 + 0.07

Beta = -3.142857143

SD = [0.2(0.25 + 0.19)^2] + [0.8(-0.3 + 0.19)^2]

SD = 0.03872 + 0.00968 = 0.0484

SD = (0.0484)^1/2 = 22%

Stock B:E(Rb) = 0.2(0.1) + 0.8(0.05) = 0.06 → 6%

0.06 = 0.03 + 0.07

beta = 0.4285714286

SD = [0.2(0.1 – 0.06)^2] + [0.8(0.05 – 0.06)^2]

SD = 0.00032 + 0.0000072 = 0.0003272(0.0003272)^1/2

SD= 0.0180886705 = 1.8%

Stock C:E(Rc) = 0.2(0.05) + 0.8(0.1) = 0.09 → 9%

0.09 = 0.03 + 0.07

beta = 0.8571428671

SD = [0.2(0.05 – 0.09)] + [0.8(0.1 – 0.09)]SD

SD= -0.008 + 0.008 = 0%

Answer for part A: STOCK C has the most systematic risk as it has the largest beta.

Part B: Which stock has the most unsystematic risk.

Answer for part B: STOCK A has the most systematic risk as it has the largest SD (standard deviation).Steps already shown in pervious part

Part C Standard deviation of portfolio:

Boom: 0.42(0.25) + 0.18(0.1) + 0.4(0.05)

0.105 + 0.018 + 0.02 = 0.143

Recesion: 0.42(-0.3) + 0.18(0.05) + 0.4(0.1)

-0.126 + 0.009 + 0.04 = -0.077

0.2(0.143) + 0.8(-0.077)

0.0286 + -0.616

-0.033 = -3.3%

State	Prob	Return	(R - ER)	(R - ER)^2	(R - ER)^2 * P
Boom	0.2	0.143	0.176	0.030976	0.0061952
Recession	0.8	-0.077	-0.044	0.001936	0.0015488
Expected Return		-3.30%
Standard deviation					8.80%

The standard deviation for the portfolio = 8.80%

Part - D: If inflation is 2.5% Real return = (-0.033-0.025)/(1.025) = -0.0566 = -5.66%