You are managing a portfolio of 10 stocks that are held in equal dollar amounts. The current beta of the portfolio is 1.8, and the beta of Stock A is 2.0. If Stock A is sold and the proceeds are used to purchase a replacement stock, what does the beta of the replacement stock have to be to lower the portfolio beta to 1.7

The Beta of Replacement Stock is 1

Explanation:

First we need to find the Beta of Portfolio after selling the Stock A.

using following formula

it is assume that all stock has equally weighted hence stock A has weight of 10% since and remaining portfolio weight (100%-10%) = 90%

Portfolio beta = (R Pb x Weight) + (Stock A Beta x Weight)

1.8 = (R Pb x 0.9) + (2.0 x 0.1)

1.8 = (R Pb x 0.9) + 0.2

1.8 - 0.2 = (R Pb x 0.9)

1.6 = 0.9 R Pb

R Pb = 1.6 / 0.9 = 1.778

Now Using the same formula we will find beta of replacement stock.

Portfolio beta = (R Pb x Weight) + (Replacement stock Beta x Weight)

1.7 = (1.778 x 0.9) + (Replacement stock Beta x 0.1)

1.7 = 1.6 + (Replacement stock Beta x 0.1)

1.7 - 1.6 = (Replacement stock Beta x 0.1)

0.1 = (Replacement stock Beta x 0.1)

Replacement Stock beta = 0.1 / 0.1

Replacement stock beta = 1