You are considering investing $1,000 in a T-bill that pays 0.05 and a risky port

Question

You are considering investing $1,000 in a T-bill that pays 0.05 and a risky portfolio, P, constructed with 2 risky securities, X and Y. The weights of X and Y in P are 0.60 and 0.40, respectively. X has an expected rate of return of 0.14 and variance of 0.01, and Y has an expected rate of return of 0.10 and a variance of 0.0081. If you want to form a portfolio with an expected rate of return of 0.10, what percentages of your money must you invest in the T-bill, X, and Y, respectively if you keep X and Y in the same proportions to each other as in portfolio P?

Explanation / Answer

Let the % of total investment in T-bill be t

So,

0.05t+(0.6*0.14 + 0.4*0.1)(1 - t) = 0.10

or, 0.05t + 0.124(1-t) = 0.10

or, 0.05t + 0.124 - 0.124t = 0.10

or, 0.074t = 0.024

or, t = 0.024/0.074

or, t = 0.324 i.e 32.4 %

So Investment in T-bill is 32.4%

In securities X = (1 - .324) * 0.6 = 40.5%

In securities Y = (1 - .324) * 0.4 = 27.1%

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