Question

You have $1 million to invest, and you must invest all your money. The following are assets you can invest: Expected Return Beta Risk-free asset 6% Not given Stock X 30% 1.8 Stock Y 20% 1.3

(a) Suppose you want to create a portfolio that has an expected return of 12% by investing in the risk-free asset and Stock X. Calculate how much money you will invest in Stock X.

(b) Suppose you want to create a portfolio with a systematic risk that is 70% of the risk of the overall market, by investing in the risk-free asset and Stock X. Calculate how much money you will invest in Stock X.

(c) Suppose you are told that the beta of Stock X is correct, but there is uncertainty whether the beta of Stock Y is correct. Using the CAPM framework, determine whether the beta of Stock Y is correct.

(d) You have learned in this course that if you invest only in X and Y, this 2-asset portfolio will have a higher return than the portfolio return when a risk-free asset (such as Treasury bills) is also included. This being the case, appraise why anyone would want to hold Treasury bills in their portfolio.

Answer 1

Solution :-

(i)

Expected Return of Portfolio = 12%

Stocks in Portfolio = Stock X and Risk Free Asset

Return of Stock X = 30%

Return of Risk Free Asset = 6%

Let the Amount invest in Risk free asset be X

therefore invest in Stock X = ( 1 - X )

Now ( X * 6% ) + 0.30 * ( 1-X ) = 12%

0.06 X + 0.30 - 0.30 X = 0.12

0.18 = 0.24 X

X = 0.18 / 0.24 = 0.75

75% in Risk Free Asset = $1,000,000 * 75% = $750,000

25% in Stock X = $1,000,000 * 25% = $250,000

(ii)

Overall risk of market = 1

Therefore 70% of overall market = 0.70

Beta of Risk Free Asset = 0

Beta of Stock X = 1.8

Let the Amount invest in Risk free asset be X

therefore invest in Stock X = ( 1 - X )

( X * 0 ) + 1.8 * ( 1 - X ) = 0.70

1.8 - 1.8 X = 0.70

X = 1.10 / 1.8

X = 0.611

(1 - X ) = 1 - 0.611 = 0.389

Means 61.1% in Risk Free Asset = $1,000,000 * 61.1% = 611,000

and 38.90 in Stock X = $1,000,000 * 38.9% = 389,000

(iii)

As per CAPM

in case of Stock X

Ke = Rf + Beta* ( Rm - Rf )

30% = 6% + 1.80 * ( Rm - 6% )

( Rm - 6% ) = 24% / 1.80

Rm = 13.33% + 6% = 19.33%

Now in case of Stock Y , Beta =

Ke = Rf + Beta* ( Rm - Rf )

20% = 6% + Beta* ( 19.33% - 6% )

Beta of Stock Y = 14% / 13.33% = 1.05

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