When short sales are allowed (Problem 18), the manager’s Sharpe measure is
higher (0.3662).
The reduction in the Sharpe measure is the cost of the short
sale restriction.
The characteristics of the optimal risky portfolio are:
β
P
= w
M
+ w
A
×
β
A
= (1 – 0.0931) + (0.0931
×
0.9) = 0.99
E(R
P
) =
α
P
+
β
P
E(R
M
) = (0.0931
×
2.8%) + (0.99
×
8%) = 8.18%
54
.
535
)
03
.
969
,
1
0931
.
0
(
)
23
99
.
0
(
)
e
(
2
2
P
2
2
M
2
P
2
P
=
×
+
×
=
σ
+
σ
β
=
σ
%
14
.
23
P
=
σ
With A = 2.8, the optimal position in this portfolio is:
5455
.
0
54
.
535
8
.
2
01
.
0
18
.
8
y
=
×
×
=
The final positions in each asset are:
Bills
1 – 0.5455 =
45.45%
M
0.5455
×
(1
−
0.0931) =
49.47%
A
0.5455
×
0.0931
×
0.3352 =
1.70%
C
0.5455
×
0.0931
×
0.6648 =
3.38%
100.00%
811
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The mean and variance of the optimized complete portfolios in the
unconstrained and shortsales constrained cases, and for the passive strategy
are:
E(R
C
)
2
C
σ
Unconstrained
0.5685
×
8.42 = 4.79
0.5685
2
×
528.94 = 170.95
Constrained
0.5455
×
8.18 = 4.46
0.5455
2
×
535.54 = 159.36
Passive
0.5401
×
8.00 = 4.32
0.5401
2
×
529.00 = 154.31
The utility levels below are computed using the formula:
2
C
C
A
005
.
0
)
r
(
E
σ
−
Unconstrained
8 + 4.79 – (0.005
×
2.8
×
170.95) = 10.40
Constrained
8 + 4.46 – (0.005
×
2.8
×
159.36) = 10.23
Passive
8 + 4.32 – (0.005
×
2.8
×
154.31) = 10.16
20.
All alphas are reduced to 0.3 times their values in the original case.
Therefore, the
relative weights of each security in the active portfolio are unchanged, but the alpha
of the active portfolio is only 0.3 times its previous value: 0.3
×
−
16.90% =
−
5.07%
The investor will take a smaller position in the active portfolio.
The optimal risky
portfolio has a proportion w
*
in the active portfolio as follows:
01537
.
0
23
/
8
6
.
809
,
21
/
07
.
5
/
)
r
r
(
E
)
e
(
/
w
2
2
M
f
M
2
0
−
=
−
=
σ
−
σ
α
=
The negative position is justified for the reason given earlier.
The adjustment for beta is:
0151
.
0
)]
01537
.
0
(
)
08
.
2
1
[(
1
01537
.
0
w
)
1
(
1
w
*
w
0
0
−
=
−
×
−
+
−
=
β
−
+
=
Since w* is negative, the result is a positive position in stocks with positive alphas
and a negative position in stocks with negative alphas.
The position in the index
portfolio is:
1 – (–0.0151) = 1.0151
To calculate Sharpe’s measure for the optimal risky portfolio we compute the
information ratio for the active portfolio and Sharpe’s measure for the market
portfolio.
The information ratio of the
active portfolio
is 0.3 times its previous
value:
A =
α
/
σ
(e)= –5.07/147.68 = –0.0343 and A
2
=0.00118
812
813
Hence, the square of Sharpe’s measure of the
optimized risky portfolio
is:
S
2
= S
2
M
+ A
2
= (8/23)
2
+ 0.00118 = 0.1222
S = 0.3495
Compare this to the market’s Sharpe measure: S
M
= 8/23 = 0.3478
The difference is: 0.0017
Note that the reduction of the forecast alphas by a factor of 0.3 reduced the squared
information ratio and the improvement in the squared Sharpe ratio by a factor of:
0.3
2
= 0.09
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 Spring '13
 Ohk
 Regression Analysis, Modern portfolio theory, Sharpe, active portfolio

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