Question

an investor has a portfolio worth 12.3 million dollars, made up of ten different types of securities. the beta of the portfolio is 3.8. the investor sold security A whose beta is 1.25 for 4 million dollars. he then invested the full amount of the sale in security C whose beta is 1.6. interest rate on treasury bills is 8 percent and market rate is 11 percent.

1.compute the required rate or return of the new portfolio

2. two months later, the investor invested additional 2.7 million dollars in the portfolio by including security D. compute the beta of security D, assuming the required rate of return on the portfolio is now 18.143 percent.

Answer 1

Beta of portfolio is sum of products of individual bets and weight of each stock

1)

the new beta of portfolio = Old beta of portfolio + (weight of security C * beta of security C) - (weight of security A * beta of security A)

Weight of A/C security = value of stock / total value of portfolio = 4 / 12.3 = 0.33

(Weight of both security A and C is same as they were sold and purchased at same value)

beta of new security = 1.6

beta of old security = 1.25

old beta of portfolio = 3.8

the new beta of portfolio = 3.8 + (0.33 * 1.6) - (0.33 * 1.25) = 3.9

Expected rate of return of portfolio = Rf + beta*(Rm - Rf)

Rf = return on treasury bil = 8%

Rm = market rate = 11%

beta = 3.9

Expected rate of return of portfolio = 0.08 + 3.9*(0.11 - 0.08) = 19.7%

2)

Expected rate of return of portfolio = Rf + beta*(Rm - Rf)

18.143% = 0.08 + beta*(0.11 - 0.08)

beta of new portfolio = 3.38

worth of new portfolio = 12.3 + 2.7 = $15 mn

Weight of security D = 2.7 / 15 = 0.18

New beta = old beta + (weight of security D * beta of security D)

3.38 = 3.9 + (0.18 * beta of security D)

Beta of security D = -2.8