Question 9.RQ.23: The expected return (r) and standard deviation (σ) of shares......
The expected return (r) and standard deviation (σ) of shares of X Ltd and Y Ltd are:
Required:
\, If the expected correlation between the two shares (\rho_\mathrm{XY}) is (a) 0.1, (b)–1, compute the return and risk for each of the following portfolios:
\, (i) X, 100 per cent, (ii) Y, 100 per cent, (iii) X, 50 per cent–Y, 50 per cent.
| σ | r | |
| 0.20 | 0.014 | X Ltd |
| 0.30 | 0.09 | Y Ltd |
(a) \rho_{xy} = 0.1
\, (i) X, 100 per cent: return = 0.14 = 14 per cent; risk = 0.20 = 20 per cent.
\, (ii) Y, 100 per cent: return = 0.09 = 9 per cent; risk = 0.30 = 30 per cent.
\, (iii) X, 50 per cent; Y, 50 per cent:
\qquad \qquad \sigma_p= \sqrt{w_x^2 \sigma^2_x + w_y^2 \sigma^2_y + 2w_xw_yPxy \sigma_ x \sigma_y}
\qquad \qquad=\sqrt{(0.5)^2\, (0.2)^2 +(0.5)^2 \,(0.3)^2+ 2(0.5)\, ((0.5) \,\rho_{xy} \,(0.2 (0.3))}
\qquad \qquad=\sqrt{ 0.01+0.0225+0.03 \,\rho_{xy} }
\qquad \qquad=\sqrt{0.0325 + 0.03(0.1)}+\sqrt{0.0355}= 0.1884 = 18.84 per cent
(b) P_{xy} = –1
\, (i) and (ii) same as in (a) (i) and (ii).
\qquad \qquad \sigma_p= \sqrt{0.0325 +0.03(-1)}=\sqrt{0.0025}= 0.05 = 5 per cent.