Question

You own a portfolio that has a total value of $280,000 and it is invested in Stock D with a beta of .73 and Stock E with a beta of 1.52. The beta of your portfolio is equal to the market beta. What is the dollar amount of your investment in Stock D?

Answer 1

Let us consider a portfolio of two stocks :- S1 & S2

We know, the formula of portfolio's beta can be expressed mathematically as :-

where, = portfolio beta

WS1 = weight or proportion of stock S1 in the portfolio

WS2 = weight or proportion of stock S2 in the portfolio

= beta of stock 1

= beta of stock 2

Now, as per the data given in the problem :-

The portfolio consists of stock D & stock E

The total value of the portfolio is $280,000.

Lets say, the proportion or the weight of the stock D in the portfolio = WD

the proportion or the weight of the stock E in the portfolio = WE

We can also get, the weight of stock D i.e. WD = (1 - WE)

Beta of Stock D :- = 0.73

Beta of Stock E :- = 1.52

It is given that the beta of the portfolio is equal to the market beta.

We know that the beta is the measure of the stock volatility in relation to the overall market movement. Or in other words, the beta coefficient measures the degree of sensitivity/responsiveness of the return of a stock to the overall market return. For example, the S&P 500 index which consists of the top 500 bluechip companies of the American stock market has a beta of 1. This is considered as the market beta. Each individual stock's return & volatility is analysed & ranked according to how much it deviates from the market beta.

In the given problem, it is being mentioned that the portfolio beta is equal to the market beta i.e. 1.

Now, considering the equation,  , we can ascertain the dollar amount of the stock D.

Hence, for the given portfolio of stock D & stock E, we get,

Substituting the corresponding values in the equation, we get

1 = (WD * 0.73) + (WE * 1.52)

1 = (1 - WE) * 0.73 + (WE * 1.52)

1 = 0.73 - 0.73WE + 1.52WE

1 - 0.73 = 1.52WE - 0.73WE

0.27 = 0.79WE

WE = 0.27/0.79

Hence, WE = 0.341772

Thus, WD = (1 -WE) = (1 - 0.341772) = 0.658228

Therefore, the proportion or the weight of the stock D in the portfolio = 0.658228.

The dollar amount of the stock D in the given portfolio is = ($280,000 * 0.658228) = $184,304.

Thus, the dollar value of the investment in the stock D = $184,304.