CHAPTER 27: THE THEORY OF ACTIVE PORTFOLIO MANAGEMENT
1. a.
Define R = r – rf. Note that we compute the estimates of standard deviation using 4
degrees of freedom (i.e., we divide the sum of the squared deviations from the mean by 4
despite the fact that we have 5 observations), since deviations are taken from the sample
mean, not the theoretical population mean.
E(RB) = 11.16%
E(RU) = 8.42%
B = 21.24%
U = 14.85%
 = .75
Risk neutral investors would prefer the Bull Fund because its performance suggests a
higher mean. Given the estimates of the variance of the series and the small number of
observations, the difference in the averages is too small to determine the superiority of
Bull with any confidence.
b.
Using the reward to volatility (Sharpe) measure,
SB =
E(rB) – rf
B
SU
=
=
E(RB)
B
E(RU)
U
11.16
= 21.24 = .5254
8.42
= 14.85 = .5670
The data suggest that the Unicorn Fund dominates for a risk averse investor.
c.
The decision rule for the proportion to be invested in the risky asset, given by the formula
E(r) – rf
E(R)
=

.01A2
maximizes a mean variance utility of the form, U = E(r) – .005A2 for which Sharpe's
measure is the appropriate criterion for the selection of optimal risky portfolios. In any
event, an investor with A = 3 would invest the following fraction in Unicorn:
y=
yU =
8.42
.01  3  14.852
= 1.2727
Note that the investor wants to borrow in order to invest in Unicorn. In that case, his
portfolio risk premium and standard deviation would be:
E(rP) – rf = 1.2727  8.42% = 10.72% ,
27-1
P = 1.2727  14.85% = 18.90%
and his utility level would be:
U(P) = rf + 10.72 – .005  3  18.902 = rf + 5.36.
If borrowing is not allowed, investing 100% in Unicorn would lead to:
E(rP) – rf = 8.42%
P = 14.85%
U(P) = rf + 8.42 – .005  3  14.852 = rf + 5.11
Note that if Bull must be chosen, then:
yB =
11.16
= .8246
.01  3  21.242
E(rP) – rf = .8246  11.16% = 9.20%
P = .8246  21.24% = 17.51%
U(P) = rf + 9.20 – .005  3  17.512 = rf + 4.60.
Thus, even with a borrowing restriction, Unicorn (with the lower mean) is still superior.
2.
 = 5.5%
and
rf = 1%
Write the Black-Scholes formula from Chapter 21 as:
C = SN(d1) – PV(X) N(d2)
In this application, where we express the value of timing per dollar of assets, we use S =
1.0 for the value of the stock; the present value of the exercise price also is 1. Note that:
d1 =
ln(S/X) + (r + 2/2)T
and d2 = d1 –  T
 T
When S = PV(X), and T = 1, the formula for d1 reduces to d1 = and d2 = –/2.
Therefore, C = N() – N(–). Finally recall that N(–x) = 1 – N(x). Therefore, we
can write the value of the call as:
27-2
C = N() – [1 – N()] = 2N(/2) – 1
Since = .055, the value is:
C = 2N(.0275) – 1.
Interpolating from the standard normal table (Table 21.2):
.0075
C = 2[.5080 + .0200(.5160 – .5080) ] – 1
C = 1.0220 – 1 = .0220
Hence the added value of a perfect timing strategy is 2.2% per month.
3. a.
Using the relative frequencies to estimate the conditional probabilities P1 and P2 for
timers A and B, we find:
Timer A
Timer B
P1
78/135 = .58
86/135 = .64
P2
57/92 = .62
50/92 = .54
P* = P1 + P2 – 1
.20
>
.18
The data suggest that timer A is the better forecaster
b.
Using the following equation to value the imperfect timing services of A and B,
C(P*) = C(P1 + P2 – 1)
CA(P*) = 2.2%  .20 = .44% per month
CB(P*) = 2.2%  .18 = .40% per month
Timer A's added value is greater by 4 basis points per month.
4
a.
Alphas, 
i = ri – [rf + i(rM – rf) ]
Expected excess return
E(ri) – rf
A = 20 – [8 + 1.3(16 – 8)] = 1.6%
B = 18 – [8 + 1.8(16 – 8)] = –4.4%
20 – 8 = 12%
C = 17 – [8 + 0.7(16 – 8)] = 3.4%
D = 12 – [8 + 1.0(16 – 8)] = –4.0%
17 – 8 = 9%
27-3
18 – 8 = 10%
12 – 8 = 4%
Stocks A and C have positive alphas, whereas stocks B and D have negative alphas. The
residual variances are:
2(eA)= 3364
2(eB)= 5041
b.
2(eC)= 3600
2(eD)= 3025
To construct the optimal risky portfolio, we first need to determine the active portfolio.
Using the Treynor-Black technique, we construct the active portfolio

2(e)
A
B
C
D
Total
.000476
–.000873
.000944
–.001322
–.000775
 / 2(e)
  2(e)
–.6142
1.1265
–1.2181
1.7058
1.0000
Do not be disturbed by the fact that the positive alpha stocks get negative weights and
vice versa. The entire position in the active portfolio will turn out to be negative,
returning everything to good order. With these weights, the forecast for the active
portfolio is:
 = –.6142  1.6 + 1.1265  (–4.4) – 1.2181  3.4 + 1.7058  (–4.0) = –16.90%
= –.6142  1.3 + 1.1265  1.8 – 1.2181  .70 + 1.7058  1 = 2.08
The high beta (higher than any individual beta) results from the short position in
relatively low beta stocks and long position in relatively high beta stocks.
2(e) = (–.6142)2  3364 + 1.12652  5041 + (–1.2181)2  3600 + 1.70582  3025
= 21809.6
e = 147.68%
Here, again, the levered position in stock B [with the high 2(e)] overcomes the
diversification effect, and results in a high residual standard deviation. The optimal risky
portfolio has a proportion w* in the active portfolio as follows:
w0 =
 / 2(e)
[E(rM) – rf] /
2
M
=
–16.90 / 21809.6
= –.05124
8 / 232
The negative position is justified for the reason given earlier.
27-4
The adjustment for beta is
w* =
w0
–.05124
= 1 + (1 – 2.08)(–.05124) = –.0486
1 + (1 – )w0
Because w* is negative, we end up with a positive position in stocks with positive alphas
and vice versa. The position in the index portfolio is:
1 – (–.0486) = 1.0486
c.
To calculate Sharpe's measure for the optimal risky portfolio we need the appraisal ratio
for the active portfolio and Sharpe's measure for the market portfolio. The appraisal ratio
of the active portfolio is:
A =  / e = –16.90/147.68 = –.1144
and
A2 =.0131
Hence, the square of Sharpe's measure, S, of the optimized risky portfolio is :
8
2
S2 = SM + A2 = ( 23 )2 + .0131 = .1341
and
S = .3662
Compare this to the market's Sharpe measure,
SM = 8/23 = .3478
The difference is .0184.
Note that the only-moderate improvement in performance results from the fact that only a
small position is taken in the active portfolio A because of its large residual variance.
We calculate the "Modigliani-squared", or M2 measure, as follows:
E(rP*) = rf + SP M = 8% + .3662  23% = 16.423%
M2 = E(rP*) – E(rM) = 16.423% – 16% = 0.423%
27-5
d.
To calculate the exact makeup of the complete portfolio, we need the mean excess return
of the optimal risky portfolio and its variance. The risky portfolio beta is given by:
P = wM + wA  A = 1.0486 + (–.0486)2.08 = .95
E(RP) = P+ PE(RM) = –.0486(–16.90%) + .95  8% = 8.42%
2
2
2 2
P = PM + e = (.95  23)2 + (–.0486)2  21809.6 = 528.94
P
P = 23.00%
Since A = 2.8, the optimal position in this portfolio is:
8.42
y = .01  2.8  528.94 = .5685
In contrast, with a passive strategy:
y=
8
= .5401
.01  2.8  232
which is a difference of .0284.
The final positions of the complete portfolio are:
Bills
M:
A:
B:
C:
D:
1 – .5685 =
.5685  l.0486 =
.5685 (–.0486)(–.6142) =
.5685 (–.0486)(1.1265) =
.5685 (–.0486)(–1.2181) =
.5685 (–.0486)(1.7058) =
43.15%
59.61
1.70
– 3.11
3.37
– 4.71
––––––
100.00
[sum is subject to rounding error]
Note that M may include positive proportions of stocks A through D.
27-6
5.
If a manager is not allowed to sell short he will not include stocks with negative alphas in
his portfolio, so that A and C are the only ones he will consider.
A:
C:


2(e)


2(e)
1.6
3.4
3364
3600
.000476
.000944
.3352
.6648


.001420
1.0000
 / 2(e)
  2(e)
The forecast for the active portfolio is:
 = .3352  1.6 + .6648  3.4 = 2.80%
 = .3352  1.3 + .6648  0.7 = 0.90
2(e) = .33522  3364 + .66482  3600 = 1969.03
e = 44.37%
The weight in the active portfolio is:
 / 2(e)
2.80 / 1969.03
w0 =
= .0940
2 =
8 / 232
E(RM) / M
and adjusting for beta:
w* =
w0
.094
= 1 + (1 – .90)(.094) = .0931
1 + (1 – )w0
The appraisal ratio of the active portfolio is:
A =  /e = 2.80 / 44.37 = .0631
and hence, the square of Sharpe's measure is:
S2 = (8/23)2 + .06312 = .1250
and S = .3535, compared to the market's Sharpe measure SM = .3478. When short sales
were allowed (problem 4) , the manager's Sharpe measure was higher, .3662. The
reduction in the Sharpe measure is the cost of the short sale restriction.
27-7
We calculate the "Modigliani-squared", or M2 measure, as follows:
E(rP*) = rf + SP M = 8% + .3535  23% = 16.1305%
M2 = E(rP*) – E(rM) = 16.1305% – 16% = 0.1305%
versus .423% when short sales were allowed.
The characteristics of the optimal risky portfolio are:
P = wM + wA  A = 1 – .0931 + (.0931  .9) = .99
E(RP) = P+ PE(RM) = .0931  2.8% + .99  8% = 8.18%
2
2
2
P = PM + 2(eP)= (.99  23)2 + .09312  1969.03 = 535.54
P = 23.14%
With A = 2.8, the optimal position in this portfolio is:
8.18
y = .01  2.8  535.54 = .5455
The final positions in each asset are:
Bills
M:
A:
C:
b.
1 – .5455 =
.5455  (1 – .0931) =
.5455  .0931  .3352 =
.5455  .0931  .6648 =
45.45%
49.47%
1.70%
3.38%
–––––––
100.00%
The mean and variance of the optimized complete portfolios in the unconstrained and
short-sales constrained cases, and for the passive strategy are:
Unconstrained
Constrained
Passive
E(RC)
–––––––
.5685  8.42 = 4.79
.5455  8.18 = 4.46
.5401  8.00 = 4.32
27-8
2
C
–––––––––
2
.5685  528.94 = 170.95
.54552  535.54 = 159.36
.54012  529.00 = 154.31
2
The utility level, E(rC) – .005AC is:
Unconstrained
Constrained
Passive
8 + 4.79 – .005  2.8  170.95 = 10.40
8 + 4.46 – .005  2.8  159.36 = 10.23
8 + 4.32 – .005  2.8  154.31 = 10.16
6. a. The optimal passive portfolio is obtained from equation (8.7) in Chapter 8 on Optimal
Risky Portfolios.
wM =
2
E(RM)H – E(RH)Cov(rH,rM)
2
2
E(RM)H + E(RH)M – [E(RH) + E(RM)]Cov(rH,rM)
where E(RM) = 8%, E(RH) = 2% and Cov(rH,rM) = MH =.6  23  18 = 248.4
8  182 – 2  248.4
wM =
= 1.797
8  182 + 2  232 – (8 + 2)248.4
and
wH = – .797
If short sales are not allowed, portfolio H would have to be left out of the passive
portfolio because the weight on H is negative.
b.
With short sales allowed,
E(Rpassive) = 1.797  8% + (–.797)  2% = 12.78%
2
passive = (1.797  23)2 + [(–.797)  18]2 + 2  1.797  (–.797)  248.4 = 1202.54
passive = 34.68%
Sharpe's measure in this case is:
Spassive = 12.78 / 34.68 = .3685
compared with the market's Sharpe measure of
SM = 8 / 23 = .3478
27-9
c.
The improvement in utility for the expanded model of H and M versus a portfolio of M
alone is calculated below for A = 2.8.
12.78
y = .01  2.8  1202.54 = .3796
Therefore,
Upassive = 8 + (12.78  .3796) – (.005  2.8  .37962  1202.54) = 10.43
which is greater than Upassive = 10.16 from problem 5.
7.
The first step is to find the beta of the stocks relative to the optimized passive portfolio.
For any stock i, the covariance with a portfolio is the sum of the covariances with the
portfolio components, accounting for the weights of the components. Thus,
i =
Cov(ri, rpassive)
2
passive
2
=
2
iMwMM + iHwHH
2
passive
Therefore,
A =
1.2  1.797  232 + 1.8  (–.797)182
= .5621
1202.54
B =
1.4  1.797  232 + 1.1  (–.797)182
= .8705
1202.54
C =
0.5  1.797  232 + 1.5  (–.797)182
= .0731
1202.54
D =
1.0  1.797  232 + 0.2  (–.797)182
= .7476
1202.54
27-10
Now the alphas relative to the optimized portfolio are:
i = E(ri) – rf – i,passive  E(Rpassive)
A = 20 –8 –.5621  12.78 = 4.82%
B = 18 –8 –.8705  12.78 = –1.12%
C = 17 – 8 – .0731  12.78 = 8.07%
D = 12 –8 –.7476  12.78 = –5.55%
The residual variances are now obtained from:
2
2(ei; passive)= i –
2
2
2
(i:passive
2
 passive )
2
where i = MM + 2(ei) from Problem 4.
2(eA) = (1.3  23)2 + 582 – (.5621  34.68)2 = 3878.01
2(eB) = (1.8  23)2 + 712 – (.8705  34.68)2 = 5843.59
2(eC) = (0.7  23)2 + 602 – (.0731  34.68)2 = 3852.78
2(eD) = (1.0  23)2 + 552 – (.7476  34.68)2 = 2881.80
From this point, the procedure is identical to that of problem 6:
A
B
C
D
Total

2(e)
 / 2(eA)
  2(eA)
.001243
–.000192
.002095
–.001926
.001220
1.0189
–0.1574
1.7172
–1.5787
1.0000
The active portfolio parameters are:
 1.0189  4.82 + (–0.1574)(–1.12) + 1.7172  8.07 + (–1.5787)(–5.55) = 27.71%
 1.0189  .5621 + (–0.1574)(.8705) + 1.7172  .0731 + (–1.5787)(.7476) = –0.6190
2(e) = 1.01892  3878.01 +(–0.1574)2  5843.59
+ 1.71722  3852.78 + (–1.5787)2  2881.80 = 22,714.03
27-11
The proportions in the overall risky portfolio can now be determined.
w0 =
/2(e)
2
E(Rpassive)/passive
27.71 / 22,714.03
= 12.78 / 1202.54 = .1148
.1148
w* = 1 + (1 + 0.6190).1148 = .0968
a.
Sharpe's measure for the optimal risky portfolio is
27.712
2
S2 = Spassive + [2(e)]2 = .36852 + 22,714.03 = .1696
S = .4118 compared to Spassive = .3685
The difference in the Sharpe measure is therefore .0433.
b.
The beta of the optimal risky portfolio is:
P = w*A + (1 – w*) = .0968(–0.6190) + .9032 = .8433
The mean excess return of this portfolio is:
E(R) = .0968  27.71% + .8433  12.78% = 13.46%
and its variance and standard deviation are:
2 = .84332  1202.54 + .09682  22714.03 = 1068.03
 = 32.68%
Therefore, the position in it would be:
13.46
y = .01  2.8  1068.03 = .4501
The utility value for this portfolio is:
U = 8 + (.4501  13.46) – (.005  2.8  .45012  1068.03) = 11.03
which is superior to all previous alternatives.
27-12
8.
If short sales are not allowed, then the passive portfolio reverts to M, and the solution
mimics the solution to problem 5.
9.
All alphas are reduced to .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 .3 times its previous value, .3  16.90% = 5.07%, and 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:
w0 =
 / 2(e)
[E(rM)–rf] /
2
M
=
–5.07 / 21,809.6
= –.01537
8 / 232
The negative position is justified for the reason given earlier.
The adjustment for beta is
w* =
w0
–.01537
= 1 + (1 – 2.08)(–.01537) = –.0151
1 + (1 – )w0
Because w* is negative, we end up with a positive position in stocks with positive alphas
and vice versa. The position in the index portfolio is:
1 – (–.0151) = 1.0151
To calculate Sharpe's measure for the optimal risky portfolio we need the appraisal ratio
for the active portfolio and Sharpe's measure for the market portfolio. The appraisal ratio
of the active portfolio is .3 times its previous value:
A =  / (e)= –5.07/147.68 = –.0343
and
A2 =.00118
Hence, the square of Sharpe's measure of the optimized risky portfolio is :
8
2
S2 = SM + A2 = ( 23 )2 + .00118 = .1222
and
S = .3495
27-13
Compare this to the market's Sharpe measure,
SM = 8/23 = .3478
The difference is .0017.
Note that the reduction of the forecast alphas by a factor of .3 reduced the squared
appraisal ratio and improvement in the squared Sharpe ratio by a factor of (.3)2 = .09.
27-14