第三次作业答案

第三次作业答案

Question 1

a. Expected returns

State of the Economy Probability Return on Security I Return on

Security II

Low Growth 0.4 -2% -10% Medium Growth 0.5 28% 40 High Growth 0.1 48% 60

()()%18%481.0%285.0%24.0=⨯+⨯+-⨯=I R E

()()%22%601.0%405.0%104.0=⨯+⨯+-⨯=II R E

b. Variances and standard deviations

()()()()03.0%18%481.0%18%285.0%18%24.02

222=-⨯+-⨯+--⨯=I R σ ()()()()0716.0%22%601.0%22%405.0%22%104.02222=-⨯+-⨯+--⨯=II R σ

()%32.1703.0==I R σ

()%76.260716.0==II R σ

c. Portfolio expected returns and standard deviations Portfolio I = 90% invested in security I and 10% in security II

Portfolio II = 10% invested in security I and 90% in security II

State of the Economy

Probability

Return on portfolio I Return on Portfolio II Low Growth 0.4 -2.8% -9.2% Medium Growth 0.5 29.2% 38.8 High Growth

0.1

49.2%

58.8

()()%4.18%2.491.0%2.295.0%8.24.0=⨯+⨯+-⨯=PI R E ()()%6.21%8.581.0%8.385.0%2.94.0=⨯+⨯+-⨯=PI R E

()()()()%25.18%4.18%2.491.0%4.18%2.295.0%4.18%8.24.022=-⨯+-⨯+--⨯=PI R σ

()()()()%

80.25%6.21%8.581.0%6.21%8.385.0%6.21%2.94.022=-⨯+-⨯+--⨯=PI R σ

The general expression for the expected return on a portfolio of two securities is

()()()()II I P R E R E R E θθ-+=1 with 10≤≤θ.

Assume E[R I ] > E[R II ]. The highest value E[R P ] can occur when you invest 100% of the portfolio in security I (θ = 1), giving a portfolio expected return equal to the expected return for security I. It cannot be any higher. Similarly, the lowest value occurs when you invest 100% of the portfolio in security II (θ = 0), giving a portfolio expected return equal to the expected return for security II. It cannot be any lower. For values of θ between zero and one, the portfolio return will lie between the expected returns on securities I and II. Mathematically speaking, the expected return on the portfolio is a weighted average of the security expected returns. In the example above, the expected return on portfolio I is 90 percent of the expected return on security I plus 10 percent of the expected return on security II. This sort of relation always holds.

As long as the two securities are not perfectly (positively) correlated, the standard

deviation works differently. The standard deviation of a portfolio will always be less than a weighted average of the security standard deviations. In the example above, the standard deviation of portfolio I is 18.25%. A weighted average of the standard deviations is [(0.9 ×17.32) +(0.1 ×26.76)] = 18.26%. Here the reduction is not very great because although not perfectly correlated, the returns on securities I and II are very highly correlated. If the correlation is low enough, it is possible for the standard deviation of a portfolio of two securities to be lower than the standard deviations of each individual security. However, the portfolio standard deviation cannot be higher than the standard deviation on either security. It will equal the higher of the two security standard deviations only when the portfolio is 100 percent invested in the corresponding security.

Question 2

a. Expected returns

()%5.9=A R E , ()%5.18=B R E , ()%1.34=C R E

b. Variances are

()02=A R σ, ()030525.02=B R σ, ()074589.02=C R σ

giving standard deviations

()0=A R σ, ()%47.17=B R σ, ()%31.27=C R σ

c. Security A is a risk-free security, since it has zero standard deviation (no risk).

d. The total investment is (3 ×2025) + (2.25 ×900) = £ 8,100

The portfolio weights are therefore 75% in B and 25% in C.

The portfolio's expected return is 22.4%, and the standard deviation is 17.27%. This

standard deviation is less than three-quarters of the standard deviation of B plus one-quarter of the standard deviation of C, which would be 19.93%. It is less even than the standard deviation of B alone. This is the effect of risk diversification working again.