《金融理论与公司政策(第四版)》课后答案

2 2 var = a 2 σ2 x + 2ab cov(x, x) + b σ x
The long position means 100% in X and the short position means –100% in X. Therefore, a=1 b = –1 Therefore, the variance is
r = P1 − P0 , P0 P0 = 50
pi(ri – E(r)) .014 .003 .000 .002 .009 ∑ = .028
2
E(r) = Var(r) =
∑p r
i
i i
= .009
i
∑ p (r
- E(r))2 = .028
Range (r) = rmax – rmin = .20 – (–.30) = .50 Semi-interquartile range = 1/2(r.75 – r.25) = 1/2(.10 – (–.16)) = .13
2 var = σ2 x − 2 cov(x, x) + σ x 2 2 σ2 x − 2σ x + σ x = 0
and the position is perfectly risk free. 6. Using matrix notation, the variance is V = X' Σ X where V = variance X = vector of weights X’ = transpose of the vector of weights Σ = the variance-covariance matrix (a) Therefore, the variance of an equally weighted portfolio is 24 −10 25 1/ 3 V = [1/ 3 1/ 3 1/ 3] −10 75 32 1/ 3 25 32 12 1/ 3 − 10 / 3 + 25/ 3 8 V = [1/ 3 1/ 3 1/ 3] −10 / 3 + 25 + 32 / 3 25/ 3 + 32 / 3 + 4 13 V = [1/ 3 1/ 3 1/ 3] 321/ 3 23 V = 13 321/ 3 23 + + = 22.78 3 3 3
Thus, x and y are perfectly negatively correlated. 3. (a) Table S5.1 Prob. .15 .10 .30 .20 .25 ri –.30 –.16 0 .10 .20 piri –.045 –.016 0 .020 .050 ∑ = .009 where
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Copeland/Shastri/Weston. • Financial Theory and Corporate Policy, Fourth Edition
(b) Table S5.2 Prob. .01 .05 .07 .02 .10 .30 .20 .15 .05 .05 ri –1.00 –.30 –.23 –.20 –.16 0 .10 .14 .20 .38 piri –.010 –.015 –.016 –.004 –.016 0 .020 .021 .010 .019 ∑ = .009 pi (ri− − E(r))2 (semivariance) .010 .005 .004 .001 .003 .000 — — — — ∑ = .023
E(r) = ∑piri = .009 Range = rmax – rmin = .38 – (–1.0) = 1.38 Semi-interquartile range = 1/2(r.75 – r.25) = 1/2(.10 – (–.16)) = 0.13 The expected return and semi-interquartile range are the same, because future possible prices are drawn from the same probability distribution as in part a). Only n, the number of observations, has increased. The range increases with n. Semi-variance is given by E[(x i − − E(x)) 2 ] where xi − = xi if x i < E(x); x i + = 0 if x i ≥ E(x)
Chapter 5
Objects of Choice: Mean-Variance Portfolio Theory
47
5. Let σ2 x be the variance of the stock. The variance of a perfectly hedged portfolio is
2 2 Variance = a 2 σ2 x + 2ab cov(X, Y) + (1 − a) σ y
% in X 125 100 75 50 25 0 –25
% in Y –25 0 25 50 75 100 125
E(Rp) 10.0 9.0 8.0 7.0 6.0 5.0 4.0
var(Rp) 47.675 28.000 18.475 19.100 29.875 50.800 81.875
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(c) The minimum variance portfolio is given by equation 5.21. a* = σ2 y − rxy σ x σ y
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requires the calculation of var(x), var(y), and cov(x, y). Given y = a – bx, var(y) = var(a – bx) = E{[a – bx – E(a – bx)]2} = E{[a – bx – a – E(–bx)]2} = E{[–bx + bE(x)]2} = (–b)2E{[x – E(x)]2} = b2 var(x) Therefore, because the standard deviation must be positive σ y = bσ x and σ x σ y = bσ x 2
var(X) = 28.0 pi[Yi – E(Y)] 5.0 12.8 20.0 9.8 3.2 var(Y) = 50.8
2
pi[Xi – E(X)][Yi – E(Y)] –9.0 6.4 6.0 –7.0 2.4 cov(X, Y) = –1.2
(b) Mean = aE(X) + (1 – a)E(Y)
Chapter 5
Objects of Choice: Mean-Variance Portfolio Theory
1.
Figure S5.1 Skewed distribution of stock prices The reason stock prices are skewed right is because theoretically there is no upper bound to the price level a stock can attain, while, with limited liability, the probability distribution is bounded on the left by P = 0. 2. The equation for correlation between x and y, rx,y = cov(x, y) σx σy
σp 6.905 5.292 4.298 4.370 5.466 7.127 9.048
The opportunity set is shown in Figure S5.2 on the following page.
Chapter 5
Objects of Choice: Mean-Variance Portfolio Theory
Chapter 5
Objects of Choice: Mean-Variance Portfolio Theory
45
cov(x, y) = cov(x, a – bx) = E[(x – E(x)) (a – bx – E(a – bx))] = E[x(a – bx) – xE(a – bx) – E(x) (a – bx) + E(x)E(a– bx)] = E[ax – bx2 – ax – x(–b) E(x) – E(x)a – (–b)xE(x) + aE(x) – b(E(x))2] = E[(–b)(x2 – 2xE(x) + (E(x))2] = –b E(x – E(x))2 = –b var(x) Substitution into the correlation equation yields rxy = cov(x, y) − b var(x) = = −1 σx σy b var(x)
In the example above, semi-variance = ∑ pi (ri− − E(r))2 = 0.023 . Semi-variance is important to risk-averse investors who are more concerned with downside risk (losses) than gains. 4. First, recall the definition of covariance. cov(x, y) = E[(x – E(x))(y – E(y))] Multiplying the factors in brackets on the right-hand side, we have cov(x, y) = E[(xy – xE(y) – yE(x) + E(x)E(y)] and taking the expectation of the right-hand side we have cov(x, y) = E(xy) – E(x)E(y) – E(y)E(x) + E(x)E(y) cov(x, y) = E(xy) – E(x)E(y) Therefore, E(xy) = cov(x, y) + E(x)E(y)
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Copeland/Shastri/Weston. • Financial Theory and Corporate Policy, Fourth Edition
(b) The covariance is 24 −10 25 1.25 cov = [.1 .8 .1] −10 75 32 −.10 25 32 12 −.15 27.25 cov = [.1 .8 .1] −24.80 26.25 cov = [.1(27.25) – .8(24.80) + .1(26.25)] = –14.49 7. (a) pi .2 .2 .2 .2 .2 Xi 18 5 12 4 6 piXi 3.6 1.0 2.4 .8 1.2 E(X) = 9.0 Yi – E(Y) –5 –8 10 7 –4 [Yi – E(Y)] 25 64 100 49 16
2
Xi – E(X) 9.0 –4.0 3.0 –5.0 –3.0
[Xi – E(X)] 81.00 16.00 9.00 25.00 9.00
2
pi[Xi – E(X)] 16.2 3.2 1.8 5.0 1.8
2
Yi 0 –3 15 12 1
piYi 0 –.6 3.0 2.4 .2 E(Y) = 5.0