Question

In: Finance

                                          &nb

                                                                                          E(R)

                                                                                          Stock A                    Stock B

State                                           p(i)                             E(Ra)                         E(Rb)

Recession                                30%                          -30%                         -20%

Normal                                     40%                          10%                          10%                          

Expansion                               30%                          50%                          30%

Beta                                                                                1.4                             1.25

Rf                                                                                      3%

RM                                                                                    8%

  1. What is the expected return on Stock A across all market situations?

What is the expected return on Stock B across all market situations?

  1. What is the standard deviation of Stock A’s return?

What is the standard deviation of Stock B’s return?

  1. With a mix of 75% Stock A and 25% Stock B what is the portfolio return?

With a mix of 75% Stock A and 25% Stock B what is the portfolio standard deviation?

  1. Which of the two stocks has higher total risk?

Which of the two stocks has higher systematic risk?

  1. Which of the two stocks has a higher Reward-to-Risk Ratio?

Are the two stocks under or over-valued based on the Capital asset Pricing Model?

  1. What return would you expect on each stock based on the Capital Asset Pricing Model?

Solutions

Expert Solution

Expected Return on Stock A = PR * E (RR) + PN * E (RN) +                 PE * E (RE).

Where, PR is Probability of Recession State.

             PN is Probability of Normal State.

             PE is Probability of Expansion State.

             E (RR) is the Expected return of Recession State.

             E (RN) is the Expected return of Normal State.

             E (RE) is the Expected return of Expansion State.

Expected Return on Stock A = 0.3 * (-0.3) + 0.4 * 0.1 + 0.3 * 0.5

                                                = -0.09 + 0.04 + 0.15

                                                = 0.1 = 10%.

Expected Return on Stock A is 10%.

Expected Return on Stock B = PR * E (RR) + PN * E (RN) +                 PE * E (RE).

Where, PR is Probability of Recession State.

             PN is Probability of Normal State.

             PE is Probability of Expansion State.

             E (RR) is the Expected return of Recession State.

             E (RN) is the Expected return of Normal State.

             E (RE) is the Expected return of Expansion State.

Expected Return on Stock B = 0.3 * (-0.2) + 0.4 * 0.1 + 0.3 * 0.3

                                                = -0.06 + 0.04 + 0.09

                                                = 0.07 = 7%.

Expected Return on Stock B is 7%.

Variance of Stock A = PR * [E (RR) – E (RA)]2 + PN * [E (RN) – E (RA)]2 + PE * [E (RE) – E (RA)]2

Where, PR is Probability of Recession State.

             PN is Probability of Normal State.

             PE is Probability of Expansion State.

             E (RR) is the Expected return of Recession State.

             E (RN) is the Expected return of Normal State.

             E (RE) is the Expected return of Expansion State.

             E (RA) is the Expected return of Stock A.

Variance of Stock A = 0.3 * (-0.3 – 0.1)2 + 0.4 * (0.1 – 0.1)2 + 0.3 * (0.5 – 0.1)2

                                                = 0.048 + 0 + 0.048

                                                = 0.096

Variance of Stock A is 9.6%.

Standard Deviation of Stock A = (Variance of Stock A) 1/2

Standard Deviation of Stock A = (0.096)1/2

Standard Deviation of Stock A = 30.98%.

Variance of Stock B = PR * [E (RR) – E (RB)] 2 + PN * [E (RN) – E (RB)] 2 + PE * [E (RE) – E (RB)] 2

Where, PR is Probability of Recession State.

             PN is Probability of Normal State.

             PE is Probability of Expansion State.

             E (RR) is the Expected return of Recession State.

             E (RN) is the Expected return of Normal State.

             E (RE) is the Expected return of Expansion State.

             E (RB) is the Expected return of Stock B.

Variance of Stock B = 0.3 * (-0.2 – 0.07)2 + 0.4 * (0.1 – 0.07)2 + 0.3 * (0.3 – 0.07)2

                                                = 0.02187 + 0.00036 + 0.01587

                                                = 0.0381.

Variance of Stock B is 3.81%.

Standard Deviation of Stock B = (Variance of Stock B) 1/2

Standard Deviation of Stock B = (0.0381)1/2

Standard Deviation of Stock A = 19.51%.

Weight age of Stock A in portfolio = 75%.

Weight age of Stock B in portfolio = 25%.

Expected Return on Stock A is 10%.

Expected Return on Stock B is 7%.

Expected Return of Portfolio = WA * E (RA) + WB * E (RB)

                                                = 0.75 * 0.1 + 0.25 * 0.07

                                                = 9.25%.

  Expected Return of Portfolio is 9.25%.

Variance of Portfolio = WA * [E (RA) – E (RP)]2 + WB * [E (RB) – E (RP)]2

Where, WA is Weight of Stock A

             WB is Weight of Stock B

             E (RA) is the Expected return of Stock A.

             E (RB) is the Expected return of Stock B.

             E (RP) is the Expected return of Portfolio.

Variance of Portfolio = 0.75 * (0.1 – 0.0925)2 + 0.25 * (0.07 – 0.0925)2

                                   = 0.0548

Variance of Portfolio is 5.48%.

Standard Deviation of Portfolio = (Variance of Portfolio) 1/2

Standard Deviation of Portfolio = (0.0548)1/2

Standard Deviation of Portfolio = 23.41%.

Standard Deviation of Portfolio is 23.41%.

Stock A has more systematic risk as Stock A has more standard Deviation.

Sharpe Ratio of Stock A = (Expected Return of Stock A - Risk Free Rate) / Standard Deviation of A

                                      = (10% - 3%) / 30.98%

                                      = 0.225.

Sharpe Ratio of Stock A is 0.225.

Sharpe Ratio of Stock B = (Expected Return of Stock B - Risk Free Rate) / Standard Deviation of B

                                      = (7% - 3%) / 19.51%

                                      = 0.205.

Sharpe Ratio of Stock B is 0.205.

Stock A has higher reward to risk ratio.

CAPM for Stock A = Risk Free Rate + Beta * Market Return

                                 = 3% + 1.4 * 8%

                                = 14.2%.

CAPM for Stock B = Risk Free Rate + Beta * Market Return

                                 = 3% + 1.25 * 8%

                                = 13%.

The expected return on Stock A is 14.2% and for Stock B is 13%.

Both stocks are overpriced.

jeff jeffy answered 7 months ago
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