108156.pdf

Solutions 285

5.13The weights in the portfolio with the minimum variance among all attainable

portfolios with expected returnμV= 20% arew∼=

[

0. 672 − 0 .246 0. 574

]

.

The standard deviation of this portfolio isσV∼= 0 .192.

5.14The weights and standard deviations of portfolios along the minimum variance

line, parametrised by the expected returnμV,are

w∼=

[

− 2 .027 + 13. 492 μV 2. 728 − 14. 870 μV 0 .298 + 1. 376 μV

]

,

σV=

√

0. 625 − 6. 946 μV+20. 018 μ^2 V.

This minimum variance line is presented in Figure S.8, along with the set of

attainable portfolios with short selling (light shading) and without (darker

shading).

Figure S.8 Minimum variance line and attainable portfilios on thew 2 ,w 3 and
σ, μplanes

5.15Letmbe the one-row matrix formed by the expected returns of the three

securities. By multiplying theγwC=m−μuequality byC−^1 uTand, re-

spectively,C−^1 mT,weget

μV(m−μu)C−^1 uT=(m−μu)C−^1 mT,

sincewuT=1andwmT=μV. This can be solved forμto get

μ=mC

− (^1) (mT−μVuT)
uC−^1 (mT−μVuT)
∼= 0 .142.
Then,γcan be computed as follows:
γ=(m−μu)C−^1 uT∼= 1. 367.
5.16The market portfolio weights arew∼=
[
0 .438 0.012 0. 550
]

. The expected
return on this portfolio isμM∼= 0 .183 and the standard deviation isσM∼=
0 .156.
5.17βV∼= 0 .857,αV∼=− 0 .102.