Question 9.PP.18: Rashid Jitani, a renowned leading investment manager, has id......
Rashid Jitani, a renowned leading investment manager, has identified two shares East and West and the perfect negative correlation between their returns. From the characteristics of these shares determine the risk-free rate. What inference do you draw from your results?
| Standard \>deviation | Expected \>return | Share |
| 6% | 12% | East |
| 10% | 15% | West |
The risk-free rate is one that corresponds to zero risk (standard deviation). Since East and West have perfect negative correlation, it is possible to determine their portfolio proportions, which will produce minimum (zero) portfolio variance and standard deviation. The expected portfolio return for the minimum-variance weights constitutes the risk-free rate.
\qquad \quad \mathrm{W}^*_{\>\,x} = \sigma^2_{\>\,y} – \mathrm{COV}_{xy}/\sigma^2_{\>\,x} + \sigma^2_{\>\,y} – 2\> \mathrm{COV}_{xy}
\qquad \mathrm{COV}_{xy}= \rho\> \sigma_x \sigma_y
\qquad \>\, \mathrm{W}^*_\mathrm{\>\,East} = (10)^2 − (–1)\, (6)\, (10)/(6)^2 + (10)^2 − 2\, (–1)\, (6)\, (10)
\qquad \qquad \quad= 100 + 60/36 + 100 + 120 = 160/256 = 0.625
\qquad \mathrm{W}^*_\mathrm{\>\,Wast}= 1 − 0.625 = 0.375
\, Verification for zero–variance:
\qquad \sigma^ 2 _{\>p} = (\sigma x\> Wx)^2 + (\sigma y\> Wy)^2 + 2 \>Wx\> \sigma x \>Wy \>\sigma y\>\rho
\qquad \quad = (0.625 × 6)^2 + (0.375 × 10)^2 + 2(0.625 × 6)\, (0.375 × 10) \>(–1)
\qquad \quad= (3.75)^2 + (3.75)^2 –2\> (3.75) \,(3.75) = 2\> (3.75)^2 – 2\> (3.75)^2 = 0
\qquad r_f = E\> (r_p) = w_xr_x + w_yr_y
\qquad \quad = 0.625 × 12 + 0.375 × 15 = 7.5 + 5.625 = 13.125 per cent
\, Share East lies below the efficient frontier as the risk-free rate is higher
\, than the expected return on East.