- Portfolio Management 111
From Proposition 5.9 we can compute the weights in the minimum variance
portfolio. Since
uC−^1 ∼=[
12 .102 16.818 9. 382
]
,
uC−^1 uT∼= 38. 302 ,we obtain
w=uC−^1
uC−^1 uT∼=[ 0 .316 0.439 0. 245 ].
The expected return and standard deviation of this portfolio are
μV=mwT∼= 0. 146 ,σV=√
wCwT∼= 0. 162.The minimum variance line can be computed using Proposition 5.10. To this
end we compute
uC−^1 ∼=[
12 .102 16.818 9. 382
]
,
mC−^1 ∼=[
0 .898 2.182 2. 530
]
,
uC−^1 uT∼= 38. 302 , mC−^1 mT∼= 0. 923 ,
uC−^1 mT=mC−^1 uT∼= 5. 609.Substituting these into the formula forwin Proposition 5.10, we obtain the
weights in the portfolio with minimum variance among all portfolios with ex-
pected returnμV:
w∼=[
1. 578 − 8. 614 μV 0. 845 − 2. 769 μV − 1 .422 + 11. 384 μV]
.
The standard deviation of this portfolio is
σV=√
wCwT∼=√
0. 237 − 2. 885 μV+9. 850 μ^2 V.Exercise 5.12
Among all attainable portfolios constructed using three securities with
expected returnsμ 1 =0.20,μ 2 =0.13,μ 3 =0.17, standard deviations of
returnsσ 1 =0.25,σ 2 =0.28,σ 3 =0.20, and correlations between returns
ρ 12 =0.30,ρ 23 =0.00,ρ 31 =0.15, find the minimum variance portfolio.
What are the weights in this portfolio? Also compute the expected return
and standard deviation of this portfolio.Exercise 5.13
Among all attainable portfolios with expected returnμV = 20% con-
structed using the three securities in Exercise 5.12 find the portfolio with
the smallest variance. Compute the weights and the standard deviation
of this portfolio.