Question

Your retirement fund consists of a $7,500 investment in each of 20 different common stocks. The portfolio's beta is 1.95. Now, suppose you sell one of the stocks with a beta of 1.0 for $7,500 and use the proceeds to buy another stock whose beta is 2.05. Calculate your portfolio's new beta. Do not round intermediate calculations. Round your answer to two decimal places.

Answer 1

Investment in Each Stock = $ 7500, Number of Stocks = 20, Portfolio Value = 20 x 7500 = $ 150000

Portfolio Beta = 1.95

A stock with beta 1 and investment value $ 7500 is removed from the portfolio such that the portfolio now has only 19 stocks.

Let the beta of the stocks be b1,b2,b3 and so on. Further, let the stock removed have a beta of b20 (=1)

Therefore, (7500/150000) x b1 + (7500/150000) x b2 +................+ (7500/150000) x b19 + (7500/150000) x b20 = 1.95

7500 x (b1+..........+b20) = 150000 x 1.95

Subtracting b20 x 7500 from both sides of the equation, we get:

7500 x (b1 + b2 +................+ b18 + b19) + 7500 x b20 - 7500 x b20 = 150000 x 1.95 - b20 x 7500 = $ 285000

7500 x (b1+b2+.......+b18+b19) = 285000

Dividing both sides of the equation with (7500 x 19 = $ 142500 or remaining portfolio value) we get:

Beta of Remaining Portfolio = 285000 / 142500 = 2

New Stock Beta = 2.05 and New Stock Value = $ 7500

Therefore, New Overall Beta = (142500 / 150000) x 2 + (7500/150000) x 2.05 = 2.0025 ~ 2