Question

You have $3 million to form a portfolio containing two stocks A & B, and the risk-free asset.  You will not leave any cash and this portfolio should give an expected return of 13%, with a risk equivalent to 70% of the overall market.  Stock A gives an expected return of 31% and a beta of 1.8 while stock B gives an expected return of 20% and a beta of 1.3, and the risk-free asset gives 7% return.  How should you form this portfolio ?  Interpret your answers.

Answer 1

In the given question we need to compute the proportion in which we need to invest $ 3million in two Stocks A & B besides a risk free asset. Let the weights (proporions be Wa ,Wb & Wc) such that Wa +Wb +Wc = 1

We are also given that the expected return of the portfolio should be 13% It means weighted average returns of the portofio is 13% mathematically it can be written as

0.31Wa + 0.2Wb + 0..07Wc = 0.13

It is also given that the overall beta should be .70% or 0.70 this can also be written as

1.8Wa + 1.3 Wb + 0Wc = 0.70 ( Beta of risk free asset is always 0)

From above we get 3 equations

Wa +Wb +Wc = 1 or Wc = 1 - (Wa+Wb) (1)

0.31 Wa + 0.2Wb + .07Wc = 0.13 (2)

1.8Wa +1.3Wb + 0Wc = 0.70 (3)  

Substituting the value of Wc in equation (2) to get

0.31Wa + 0.2Wb + 0.07[1-(Wa+Wb)]

Wa(0.31-.0.07) + Wb (0.2- 0.07) = 0.13-.0.07

Wa(0.24) +Wb (0.13) = 0.06 ...........................(4)

Wa (1.8) + Wb (1.3) = 0.70 (bringning the equation (3) under this) .....(5)

Multiplying Equation (4) by 10 & subtract equation (5) from it to get

Wa( 2.4-1.8) + Wb (1.3 - 1.3) = 0.6 - 0.7

Wa(0.6) = - 0.10 or Wa = -1/6 substituting the value of Wa in equation (5) to get

-(1.8 / 6) + Wb(1.3) = 0.7

Wb( 1.3) = 0.7 + 0.3 =1.0 or

Wb = 1/1.3 = 0.769

substituting the values of Wa &,Wb in equation (1) to get

-1/6 + 0.769 + Wc =1 or

Wc = 0.4

From the above we get the weights as Wa = -1 /6 of the portfolio i.e Stock A

While Wb would be 1/1.3 or 76.9% of portfolio i,e Stock

Finally Wc would be 40% of portfolio i.e Risk free asset

Negative weight of Stock A indicates that we need to take a short (Sell) position as far as Stock A is convcerned. & long (Buy) position for StockB & Risk free asset