Question

Consider the following information on three stocks:

  Rate of Return If State OccursState of EconomyProbability of State
of EconomyStock AStock BStock CBoom .20  .20  .32  .54 Normal .45  .18  .16  .14 Bust .35  .02  −.34  −.42

a-1 If your portfolio is invested 40 percent each in A and B and 20 percent in C, what is the portfolio expected return? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)

Portfolio expected return             %

a-2 What is the variance? (Do not round intermediate calculations and round your answer to 5 decimal places, e.g., 32.16161.)

Variance         %   

a-3 What is the standard deviation? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)

Standard deviation             %

b. If the expected T-bill rate is 3.80 percent, what is the expected risk premium on the portfolio? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)

Expected risk premium             %

Answer 1

The information provided on the three stocks are as follows:

a.1)

The portfolio mix given is 40% in Stock A, 40% in Stock B and 20% in Stock C

The expected return in Boom State = Weight of Stock A X Return of Stock A in Boom State + Weight of Stock B X Return of Stock B in Boom State + Weight of Stock C X Return of Stock C in Boom State

= 0.4 X 0.2 + 0.4 X 0.32 + 0.2 X 0.54

= 0.08 + 0.128 + 0.108 = 0.316 = 31.6%

Similarly,

Expected Return for Normal State =  Weight of Stock A X Return of Stock A in Normal State + Weight of Stock B X Return of Stock B in Normal State + Weight of Stock C X Return of Stock C in Normal State

= 0.4 X 0.18 + 0.4 X 0.16 + 0.2 X 0.14

= 0.072 + 0.064 + 0.028 = 0.164 = 16.4%

Expected Return for Bust State =  Weight of Stock A X Return of Stock A in Bust State + Weight of Stock B X Return of Stock B in Bust State + Weight of Stock C X Return of Stock C in Bust State

= 0.4 X 0.02 + 0.4 X -0.34 + 0.2 X -0.42

= 0.008 - 0.136 - 0.084 = -0.212 = -21.2%

Expected Return for Portfolio = Probability of Boom State X Expected Return for Boom State + Probability of Normal State X Expected Return for Normal State + Probability of Bust State X Expected Return for Bust State

= 0.2 X 0.316 + 0.45 X 0.164 + 0.35 X -0.212

= 0.0632 + 0.0738 - 0.0742 = 0.0628 = 6.28%

a. 2)

Variance = Probability of Boom State X (Expected Return for Portfolio - Expected Return for Boom State)² + Probability of Normal State X (Expected Return for Portfolio - Expected Return for Normal State)^{2 +} Probability of Bust State X (Expected Return for Portfolio - Expected Return for Bust State)²

  = 0.2 X (0.0628 - 0.316)² + 0.45 X (0.0628 - 0.164)² + 0.35 X (0.0628 - (-0.212))² ​​​​​​​

  = 0.2 X 0.06411024+ 0.45 X 0.01024144​​​​​​​+ 0.35 X 0.07551504

  = 0.012822048+ 0.004608648​​​​​​​+ 0.026430264

  = 0.04386096 = 4.386096%

a.3)

Standard Deviation = Square Root of Variance

= √ 0.04386096

= 0.20943008 = 20.943008%

b)

T Bill Rate = 3.8%

Expected Risk Premium = Expected Return for Portfolio - T Bill Rate

= 6.28% - 3.8% = 2.48%