You have $136,000 to invest in a portfolio containing Stock X, Stock Y, and a ri

Question

You have $136,000 to invest in a portfolio containing Stock X, Stock Y, and a risk-free asset. You must invest all of your money. Your goal is to create a portfolio that has an expected return of 12 percent and that has only 74 percent of the risk of the overall market. If X has an expected return of 33 percent and a beta of 1.7, Y has an expected return of 18 percent and a beta of 1.1, and the risk-free rate is 6 percent, how much money will you invest in Stock Y? (Do not round intermediate calculations. Round your answer to the nearest whole dollar.)

Explanation / Answer

Amount to be invested = $ 136,000

Stock X

Expected return = 33% or 0.33

Beta = 1.7

Stock Y

Expected Return = 18% or 0.18

Beta = 1.1

Risk Free asset

Rate of return = 6% or 0.06

Let X1 be the weight of stock X in the portfolio and X2 be the weight of risk-free asset in the portfolio. Then the weight of stock Y will be (1-X1-X2)

Target return on portfolio = 12% or 0.12

Return on portfolio = w1* return on stock X + w2 * return on stock Y + w3 * return on risk-free asset

0.12 = X1 * 0.33 + (1-X1-X2) * 0.18 + X2 * 0.06

0.12 = 0.33 * X1 + 0.18 – 0.18*X1 – 0.18 * X2 + 0.06 * X2

0.12 = 0.15* X1 – 0.12 * X2 + 0.18

0.15*X1 – 0.12 * X2 = 0.12 – 0.18

0.15 *X1 – 0.12*X2 = - 0.06

- 0.12 * X2 = - 0.06 – 0.15*X1

X2 = (0.06+0.15*X1)/0.12

Target risk = 74% of the market = 0.74

Since beta of risk-free asset is zero

Portfolio risk = w1 * Beta of Stock X + w2 * Beta of stock Y

0.74 = X1 * 1.7 + (1-X1-X2) * 1.1

0.74 = 1.7 * X1 + 1.1 – 1.1 * X1 – 1.1*X2

0.74 = 0.6 * X1 + 1.1 – 1.1*X2

0.6 * X1 – 1.1*X2 = 0.74 – 1.1

0.6 * X1 – 1.1*X2 = - 0.36

Substituting value of X2 in above equation

0.6 * X1 – 1.1 * (0.06+0.15*X1)/0.12 = - 0.36

0.6 * X1 – (1.1 * 0.06)/0.12 – (1.1 * 0.15 * X1)/0.12 = -0.36

0.6 * X1 – 0.55 – 1.375 * X1 = -0.36

-0.775 * X1 - 0.55 = -0.36

-0.775 * X1 = 0.55-0.36

X1 = -0.24516 or – 24.52%

Substituting Value of X1 in expected return calculation

0.12 = -0.2452 * 0.33 + (1-(0.2452)-X2) * 0.18 + 0.06 * X2

0.12 = -0.080916 + (1.2452 – X2) * 0.18 + 0.06 * X2

0.12 = -0.080916 + 0.224136 – 0.18 * X2 + 0.06 * X2

0.12 = 0.14322 – 0.12 * X2

0.12 – 0.14322 = - 0.12 * X2

-0.02322 = - 0.12 * X2

X2 = 0.02322/0.12 = 0.1935 or 19.35%

Weight of stock X = -24.52%

Weight of Risk-free asset = 19.35%

Weight of Stock Y = 100 – (24.52%) – 19.35% = 105.17%

Expected return = -0.2452 * 0.33 + 1.0517 * 18 + 0.1935 * 0.06

                               = -0.080916 + 0.189306 + 0.01161

                               = 0.12

Beta of Portfolio = -0.2452 * 1.7 + 1.0517 * 1.1 = -0.41684 + 1.15687 = 0.74

Amount to be invested

Risk-free Asset = $ 136000 * 19.35%   = $ 26,316

Stock X = $ 136000 * -24.52% = -$ 33, 347 (sell)

Stock Y = $ 136,000 * 105.17% = $ 143,031

Related Questions

Navigate

• Browse (All)
• Browse Y
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.