Investment Analysis: HW week 9 Solutions
You are given the following information: Stock Expected return (in %) (in %) A 10 10 B 5 5 The covariance between these returns is 16%2. The risk-free rate is 6%
1. Find the expected return and standard deviation of the following portfolios:
(a) 50% in A,50% in B The expected return of a portfolio E(rp) is given by the formula:
E(rp)=w1E(r1)+w2E(r2)
With these numbers, I get E(rp) = 0.5 x 10 + 0.5 x 5 = 7.5% For the variance, apply the formula: the variance of a portfolio o? is given by
02= w?o?+w3o2+2w1W2C0v1,2
where Cov1.2 is the covariance between the returns of assets 1 and 2. Plugging in the values, I find the variance is 39.25%2, and the standard deviation is 6.26%.
(b) 50% in A, 50% in the risk-free asset The expected return is the usual formula, so the expected return is 0.5 x 10 + 0.5 x 6 = 8%. In this case, where the portfolio consists of one risky and one riskfree asset, the standard deviation can be calculated directly, as [w1|1, where wi is the weight of the risky asset, and [. | means absolute value. Therefore, the standard deviation of the portfolio is .5 x 10=5%
(c) 150% in A, financed by borrowing at the risk-free rate Applying the formulas above, E(rp) = 12%, op = 1.5 10 = 15%. (d) --100% in A, rest invested at the risk-free rate Applying the formulas above, E(rp) = 2%, p = 1 x 10 = 10%. Notice that the number is 1, not --1, since it is the absolute value.
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(e) 20% in A, 20% in B, rest invested at the risk-free rate We've only learned to calculate the expected return and standard deviation of portfolios of two assets. The portfolio in this question can be thought of as a portfolio of two assets, with one asset being the portfolio in part (a) and the second being the risk-free asset. Call the portfolio from part (a) PortA. Then the portfolio in this question can be thought of as a portfolio where we invest 0.4 in PortA and 0.6 in the risk-free asset. Check this. Imagine you invest $1 in the portfolio in this question. This means. you will invest 20 cents in stock A, 20 cents in stock B, and 60 cents in the risk- free asset. Alternatively, you can think of this as investing 40 cents in PortA and 60 cents in the riskfree asset. But investing 40 cents in PortA is the same as investing 20 cents in stock A and 20 cents in stock B (since PortA is half in stock A and half in stock B). So the two are the same thing. Once we have said that the required portfolio is the same as investing 0.4 in PortA and 0.6 in the riskfree asset, we can easily calculate its expected return and standard deviation of return. The expected return is 0.4 7.5+0.