Question

You want to compile a $1,000 portfolio which will be invested in Stocks A and B plus a risk-free asset. Stock A has a beta of 1.2 and Stock B has a beta of .7. If you invest $300 in Stock A and want a portfolio beta of .9, how much should you invest in Stock B?

Answer 1

This question is related to the Calculation of Beta of Portfolio:

Beta of a portfolio is the sum of the weighted beta of all the stocks included in the portfolio.

Formula to calculate Portfolio Beta: ( W1 * 1 + W2 * 2 + W3 * 3 + . . . . . . . )

where, W1 = weight of stock 1, W2 = weight of stock 2 . . . .

and, 1 = beta of stock 1, 2 = beta of stock 2, . . . . . . . .

Now, from the information provided in the question we have,

Total amount invested in the portfolio = $1,000

Beta of stock A = 1.2

beta of stock B = 0.7

beta of risk - free asset = 0( risk- free assets have zero beta)

Portfolio beta = 0.9 (given)

Amount invested in stock A = $300

Amount invested in stock B = $X(suppose)

therefore, amount invested in stock C(risk - free asset) = $1000 - $300 - X = ($700 - $X)

Now to findout the value of X(amount invested in stock B) we will have to calculate the weight of each stock in the portfolio.

(i). Weight of stock A = amount invested in stock A / Total value of Investment

therefore, stock A's weight = $300 / $1000 = 0.3 or 30%

(ii). Weight of stock B = $X / $1000

(iii). Weight of stock C(risk-free asset) = ($700 - $X) / $1000

From the formula of Portfolio beta:

Portfolio beta = 0.3 * 1.2 + (X / 1000) * 0.7 + ($700 - $X) * 0

portfolio beta is given as 0.9 in the question itself, so we will put the value against it.

therefore, 0.9 = 0.36 + ( X / 1000 ) * 0.7 + 0

or, 0.9 - 0.36 = (X / 1000)

or, 0.54 = X / 1,000

or, X = 0.54 * 1000

or, X = $540

Hence, the amount invested in the stock "B" is $540 so as to maintain portfolio beta of 0.9 .