Question

In: Finance

Suppose that the index model for stocks A and B is estimated from excess returns with...

Suppose that the index model for stocks A and B is estimated from excess returns with the following results:

RA = 2.5% + 0.95RM + eA

RB = –1.8% + 1.10RM + eB

σM = 27%; R-squareA = 0.23; R-squareB = 0.11

Assume you create a portfolio Q, with investment proportions of 0.50 in a risky portfolio P, 0.30 in the market index, and 0.20 in T-bill. Portfolio P is composed of 60% Stock A and 40% Stock B.

a. What is the standard deviation of portfolio Q? (Calculate using numbers in decimal form, not percentages. Do not round intermediate calculations. Round your answer to 2 decimal places.)

b.What is the beta of portfolio Q? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

c, What is the "firm-specific" risk of portfolio Q? (Calculate using numbers in decimal form, not percentages. Do not round intermediate calculations. Round your answer to 4 decimal places.)

d. What is the covariance between the portfolio and the market index? (Calculate using numbers in decimal form, not percentages. Do not round intermediate calculations. Round your answer to 2 decimal places.)

Solutions

Expert Solution

(a) Standard Deviation of Portfolio Q(σQ):

Let proportion of stock A be represented by Px and Proportion of stock B be Py

Standard deviation of stock A be represented by σx

and standard deviation of stock B be σy

and correlation coefficient of stock A and B be represented by r (xy)

Covariance of stock A and Market index be r(xm)

Covariance of stock B and Market index be r(ym)

Covariance of stock A and Market index be Cov(xm)

Covariance of stock B and Market index be Cov(ym)

σQ = √( Px^2)(σx^2 )+ (Py^2)(σy^2) + 2(Px)(Py)(σx)(σy)(r(xy))

β of Stock A = Cov(xm) ÷ σΜ^2

0.95 = Cov(xm) ÷ 0.0729

Cov(xm) = 0.95×0.0729 = 0.069

β of stock B = Cov(ym) ÷ σM^2

1.10 = Cov(ym) ÷ 0.0729

Cov(ym) = 1.10×0.0729 = 0.080

r(xm) = Cov(xm) ÷ σx × σΜ

Given r^2 (xm) = 0.23, r(xm) = √0.23 = 0.479

r^2 (ym) = 0.11, r (ym) = √0.11 = 0.332

σΜ = 27%, σΜ^2 = 0.0729

r(xm) = Cov(xm) ÷ (σx)(σM)

0.479 = 0.069 ÷( σx)(0.27)

(σx)(0.27) = 0.144

σx = 0.144÷0.27 = 53.333%

r(ym) = Cov(ym) ÷ (σy)(σM)

0.332 = 0.080 ÷ (σy)(0.27)

(σy)(0.27) = 0.241

σy = 0.241÷0.27 = 89.259%

Cov(xy) = β1×β2×(σM)^2

Cov(xy) = 0.95×1.10×0.0729

Cov(xy) = 0.076

r(xy) = Cov(xy) ÷ σx × σy

r(xy) = 0.076 ÷ (0.53333 × 0.89259)

r(xy) = 0.076 ÷ 0.476 = 0.160

Standard deviation of Portfolio Q

σQ = √(0.60)^2(0.53333)^2 + (0.40)^2(0.89259)^2 + 2(0.6)(0.4)(0.53333)(0.89259)(0.160)

σQ = √(0.36)(0.284) + (0.16)(0.797) + 0.036

σQ = √0.102 + 0.127 + 0.036

σQ = √0.265

σQ = 51.478%

(b) Beta of Portfolio Q:

β of stock A = 0.95, Proportion of stock A = 0.60

β of stock B = 1.10, Proportion of stock B = 0.40

β of Portfolio Q = Beta × Weights

β =( 0.95)(0.60) + (1.10)(0.40)

β = 1.01

(c) Firm specific risk of portfolio:

i.e., Unsystematic Risk(ei)

Variance of Portfolio (σQ)^2

=( 0.51478)^2 = 0.265

Variance of Portfolio = (β)^2 ×(σM)^2 + (ei)

0.265 = (1.01)^2 ×( 0.27)^2 + (ei)

0.265 = 1.0201 × 0.0729 + ei

0.265 = 0.074 + ei

ei = 0.191

(d) Covariance between Portfolio and Market index:

β = Cov(xy,m) ÷ (σM)^2

1.01 = Cov(xy,m) ÷ 0.0729

Cov(xy,m) = 1.01×0.0729

Cov(xy,m) = 0.074


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