1.
7.
9.
C H A P T E R
2 7
The Theory of Active Portfolio Management
953
uncovered by the security analysts (see Table C). Notice that the position in the active port-
folio amounts to 17%, financed in part by a combined short position in Dell and Walmart
of about 10%. Because the figures in Spreadsheet 27.1 are annualized, this performance is
equivalent to a 1-year holding-period return (HPR).
The alpha values we used in Spreadsheet 27.1 are actually small by the standard of
typical analysts’ forecasts. On June 1, we downloaded the current prices of the six stocks
in the example, as well as analysts’ 1-year target prices for each firm. These data and the
implied annual alpha values are shown in Table 27.2 . Notice that all alphas are positive,
indicating an optimistic view for this group of stocks. Figure 27.1 shows the graphs of the
stock prices, as well as the S&P 500 index (ticker 5 GSPC), for the previous year. The
graph shows that the optimistic views in Table 27.2 are not a result of extrapolating rates
from the past.
Spreadsheet 27.1
Active portfolio management with a universe of six stocks
σ
2
(e)
α/σ
2
(e)
w
0
(i)
[w
0
(i)]
2
α
A
σ
2
(e
A
)
w
0
w*
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
A
C
D
E
F
G
H
I
J
Table A: Risk Parameters of the Investable Universe (annualized)
SD of
Excess
Return
Beta
SD of
Systematic
Component
SD of
Residual
Correlation
with the S&P
500
S&P 500
0.1358
1.00
0.1358
0
1
HP
0.3817
2.03
0.2762
0.2656
0.72
DELL
0.2901
1.23
0.1672
0.2392
0.58
WMT
0.1935
0.62
0.0841
0.1757
0.43
TARGET
0.2611
1.27
0.1720
0.1981
0.66
BP
0.1822
0.47
0.0634
0.1722
0.35
SHELL
0.1988
0.67
0.0914
0.1780
0.46
Table B: The Index Model Covariance Matrix
SP 500
HP
DELL
WMT
TARGET
BP
SHELL
Beta
1.00
2.03
1.23
0.62
1.27
0.47
0.67
S&P 500
1.00
0.0184
0.0375
0.0227
0.0114
0.0234
0.0086
0.0124
HP
2.03
0.0375
0.1457
0.0462
0.0232
0.0475
0.0175
0.0253
DELL
1.23
0.0227
0.0462
0.0842
0.0141
0.0288
0.0106
0.0153
WMT
0.62
0.0114
0.0232
0.0141
0.0374
0.0145
0.0053
0.0077
TARGET
1.27
0.0234
0.0475
0.0288
0.0145
0.0682
0.0109
0.0157
BP
0.47
0.0086
0.0175
0.0106
0.0053
0.0109
0.0332
0.0058
SHELL
0.67
0.0124
0.0253
0.0153
0.0077
0.0157
0.0058
0.0395
Table C: Macro Forecast (S&P 500) and Forecasts of Alpha Values
SP 500
HP
DELL
WMT
TARGET
BP
SHELL
Alpha
0
0.0150
−0.0100
−0.0050
0.0075
0.012
0.0025
Risk premium
0.0600
0.1371
0.0639
0.0322
0.0835
0.0400
0.0429
Table D: Computation of the Optimal Risky Portfolio
S&P 500
Active Pf A
HP
DELL
WMT
TARGET
BP
SHELL
0.0705
0.0572
0.0309
0.0392
0.0297
0.0317
0.5505
0.2126
−0.1748
−0.1619
0.1911
0.4045
0.0789
1.0000
0.3863
−0.3176
−0.2941
0.3472
0.7349
0.1433
0.1492
0.1009
0.0865
0.1205
0.5400
0.0205
0.0222
0.0404
0.1691
Overall
0.8282
0.1718
Portfolio
0.0663
−0.0546
−0.0505
0.0596
0.1262
0.0246
Beta
1
1.0922
1.0158
0.0663
−0.0546
−0.0505
0.0596
0.1262
0.0246
Risk premium
0.06
0.0878
0.0648
0.0750
0.1121
0.0689
0.0447
0.0880
0.0305
SD
0.1358
0.2497
0.1422
0.3817
0.2901
0.1935
0.2611
0.1822
0.1988
Sharpe ratio
0.44
0.35
0.4556
M-square
0
−0.0123
0.0019
Benchmark risk
0.0346
B
e
X
c e l
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954
P A R T V I I
Applied Portfolio Management
Table 27.3 shows the optimal portfolio using the analysts’ forecasts rather than the
original alpha values in Table D in Spreadsheet 27.1 . The difference in performance is
striking. The Sharpe ratio of the new optimal portfolio has increased from the bench-
mark’s .44 to 2.32, amounting to a huge risk-adjusted return advantage. This shows up
in an M -square of 25.53%! However, these results also expose a potential major problem
with the Treynor-Black model. The optimal portfolio calls for extreme long/short positions
that may be infeasible for a real-world portfolio manager. For example, the model calls for
a position of 5.79 (579%) in the active portfolio, largely financed by a short position of
2 4.79 in the S&P 500 index. Moreover, the standard deviation of this optimal portfolio
is 52.24%, a level of risk that only extremely aggressive hedge funds would be willing to
bear. It is important to notice that this risk is largely nonsystematic because the beta of the
active portfolio, at .95, is less than 1.0, and the beta of the overall risky portfolio is even
lower, only .73, because of the short position in the passive portfolio. Only hedge funds
may still be interested in this portfolio.
One approach to this problem is to restrict extreme portfolio positions, beginning with
short sales. When the short position in the S&P 500 index is eliminated, forcing us to con-
strain the position in the active portfolio to be no more than 1.0, the position in the passive
portfolio (the S&P 500) is zero, and the active portfolio comprises the entire risky posi-
tion. Table 27.4 shows that the optimal portfolio now has a standard deviation of 15.68%,
not overwhelmingly greater than the SD of the passive portfolio (13.58%). The beta of
the overall risky portfolio is now that of the active portfolio (.95), still a slightly defensive
portfolio in terms of systematic risk. Despite this severe restriction, the optimization pro-
cedure is still powerful, and the M -square of the optimal risky portfolio (now the active
portfolio) is a very large 16.42%.
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