Question

Mr Watson has savings of £12,000.  Of this amount he has invested £6,000 in Treasury Bills which currently yield a return of 6%.  The remainder has been invested in a portfolio of four different companies’ shares. Details of this portfolio are as follows:

Company	Expected Return	b shares	Worth of share holding
W	7.6%	0.20	£1,200
X	12.4%	0.80	£1,200
Y	15.6%	1.20	£1,200
Z	18.8%	1.60	£2,400

1. Calculate the expected return and beta (b) value of Mr Watson’s savings portfolio.
2. Mr Watson has decided that he wants an expected return of 12% on his savings portfolio.  Show how he would achieve this by selling some of his Treasury Bills and investing the proceeds in the market portfolio.                                                                                      
3. If Mr Watson were only to invest in Treasury Bills and the market portfolio, what savings portfolio would be required to give him an expected return of 10.32%?

Answer 1

i).

Given the value of holdings in the portfolio as 1200,1200,1200,2400 for the 4 stocks in the portfolio, which gives a cumulative sum of 6000. So, we can arrive at individual weights as 1200/6000= 0.2 and 2400/6000= 0.4. So, Individual weights of the stocks are 0.2,0.2,0.2 and 0.4

Expected return of a portfolio is calculated as the weighted average of individual expected returns. So, Expected return of the portfolio= (0.2*7.60%)+(0.2*12.40%)+(0.2*15.60%)+(0.4*18.80%)= 14.64%

Beta of a portfolio is also calculated as weighted average of individual Betas. So, Beta of the portfolio= (0.2*0.2)+(0.2*0.8)+(0.2*1.2)+(0.4*1.6)= 1.08

ii).

Watson is getting 6% on Treasury bills in which he invested a total amount of 6000 and remaining 6000 in the portfolio, whose return we calculated as 14.64%.

So, total return of his savings portfolio is (0.5*6%)+(0.5*14.64%)= 10.32%

He wants an expected return of 12% on his portfolio, by investing some proceeds from Treasury bills to market portfolio. The return on market portfolio is 14%. This is calculated using CAPM model for a stock in the portfolio. For Company W, 7.6%= 6%+(0.2*Rm-6%). Which gives Rm= 14%.

Let the weights on Treasury bills and market portfolio be x and y. So, we have (x*6%)+(50%*14.64%)+(y*14%)= 12% and x+y should be equal to 1.

On solving these two equations, we get x= 29% and y= 21%

So, he can achieve the desired returm of 12% by selling 21% (50%-29%) of Treasury bills and investing the proceeds in market portfolio.

iii).

If Mr.Watson can invest in only Treasury bills and Market portfolio, his expected return would be (x*6%)+(y*14%), assuming x and y are weights of both respectively.

Given the expected return should be 10.32%.

So, (x*6%)+(y*14%)= 10.32% and x+y=1

So, 0.06x+(1-x)*0.14= 0.1032

= -0.08x= -0.0368

So, x= 46%.

So, To achieve 10.32%, His savings portfolio should be 46% of Treasury Bills and 54% of Market portfolio