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j =1 j =1
пЈ« пЈ¶ пЈ« пЈ¶
N N
+ m1 В· пЈ­ Xj bj в€’ Y пЈё + m2 В· пЈ­ Xj в€’ 1пЈё
j =1 j =1

Calculation of the partial derivatives of this lagrangian function leads to the equality
MXв€— = О»E в€— + F , where we have:
пЈ«2 пЈ¶ пЈ« пЈ¶
. . b1 1 X1
2ПѓОµ1
пЈ¬ .пЈ· пЈ¬.пЈ·
. .
.. . . .пЈ· пЈ¬.пЈ·
пЈ¬ . . . .пЈ· пЈ¬.пЈ·
пЈ¬
пЈ¬. пЈ· пЈ¬ пЈ·
. bN 1 пЈ· Xв€— = пЈ¬ XN пЈ·
2
M=пЈ¬ 2ПѓОµN
пЈ¬. 2ПѓI2 в€’1 . пЈ· пЈ¬YпЈ·
В·В·В· .
пЈ¬ пЈ· пЈ¬ пЈ·
пЈ­b пЈё пЈ­ m1 пЈё
В· В· В· bN в€’1 . .
1
m2
В·В·В· . . .
1 1
пЈ«пЈ¶ пЈ«пЈ¶
a1 .
пЈ¬.пЈ· пЈ¬.пЈ·
пЈ¬.пЈ· пЈ¬.пЈ·
пЈ¬.пЈ· пЈ¬.пЈ·
пЈ¬ aN пЈ· пЈ¬пЈ·
F =пЈ¬.пЈ·
Eв€— = пЈ¬ пЈ·
пЈ¬ EI пЈ· пЈ¬.пЈ·
пЈ¬пЈ· пЈ¬пЈ·
пЈ­.пЈё пЈ­.пЈё
. 1
72 Asset and Risk Management

The solution for this system is written as: Xв€— = О»(M в€’1 E в€— ) + (M в€’1 F ).
Example
Let us take the same data as those used in the п¬Ѓrst formulation, namely:32
E1 = 0.05 E2 = 0.08 E3 = 0.10
Пѓ1 = 0.10 Пѓ2 = 0.12 Пѓ3 = 0.15
ПЃ12 = 0.3 ПЃ13 = 0.1 ПЃ23 = 0.4
Let us then suppose that the regression relations and the estimated residual variances are
given by:
R1 = 0.014 + 0.60RI (ПѓОµ21 = 0.0060)
R2 = в€’0.020 + 1.08RI (ПѓОµ22 = 0.0040)
R3 = 0.200 + 1.32RI (ПѓОµ23 = 0.0012)
Let us also suppose that the expected return and index variance represent respectively
EI = 0.04 and ПѓI2 = 0.0045.
These data allow us to write:
пЈ« пЈ¶
. . .
0.0120 0.60 1
пЈ¬. . . 1.08 1 пЈ·
0.0080
пЈ¬ пЈ·
пЈ¬. . . 1.32 1 пЈ·
0.0024
M=пЈ¬ пЈ·
пЈ¬. . . 0.0090 в€’1 . пЈ·
пЈ­ 0.60 .пЈё
в€’1 .
1.08 1.32
. . .
1 1 1
пЈ¶ пЈ«пЈ¶
пЈ«
.
0.014
пЈ¬ в€’0.020 пЈ· пЈ¬.пЈ·
пЈ· пЈ¬пЈ·
пЈ¬
пЈ¬.пЈ·
пЈ¬ 0.200 пЈ·
в€—
E =пЈ¬ F = пЈ¬ пЈ·.
пЈ·
пЈ¬.пЈ·
пЈ¬ 0.040 пЈ·
пЈё пЈ­.пЈё
пЈ­ .
. 1
We can therefore calculate:
пЈ«
пЈ¶ пЈ«пЈ¶
в€’7.46 0.513
пЈ¬ в€’18.32 пЈ· пЈ¬ 0.295 пЈ·
пЈ¬ пЈ· пЈ¬ пЈ·
пЈ¬ 25.79 пЈ· пЈ¬ 0.192 пЈ·
в€’1 в€— в€’1
M E =пЈ¬ M F =пЈ¬
пЈ· пЈ·
пЈ¬ 9.77 пЈ· пЈ¬ 0.880 пЈ·
пЈ­ 0.05 пЈё пЈ­ 0.008 пЈё
в€’0.011
0.07
The portfolios for the different values of О» are shown in Table 3.6.
The efп¬Ѓcient frontier is represented in Figure 3.14.
We should point out that although the efп¬Ѓcient frontier has the same appearance as in
MarkowitzвЂ™s model, there is no need to compare the proportions here as the regression
equations that have been relied upon are arbitrary and do not arise from an effective
analysis of the relation between the returns on securities and the returns on the index.
32
These values are clearly not necessary to determine the proportions using SharpeвЂ™s model (in addition, one reason
for this was to avoid the need to calculate the varianceвЂ“covariance matrix). We will use them here only to calculate the
efп¬Ѓcient frontier.
Equities 73
Table 3.6 Solution for SharpeвЂ™s simple index model

О» X1 X2 X3 EP ПѓP

в€’0.2332 в€’1.5375
0.100 2.7706 0.1424 0.3829
в€’0.1958 в€’1.4458
0.095 2.6417 0.1387 0.3647
в€’0.1585 в€’1.3542
0.090 2.5127 0.1350 0.3465
в€’0.1212 в€’1.2626
0.085 2.3838 0.1313 0.3284
в€’0.0839 в€’1.1710
0.080 2.2548 0.1276 0.3104
в€’0.0465 в€’1.0793
0.075 2.1259 0.1239 0.2924
в€’0.0092 в€’0.9877
0.070 1.9969 0.1202 0.2746
в€’0.8961
0.065 0.0281 1.8680 0.1165 0.2568
в€’0.8045
0.060 0.0654 1.7390 0.1128 0.2392
в€’0.7129
0.055 0.1028 1.6101 0.1091 0.2218
в€’0.6212
0.050 0.1401 1.4812 0.1054 0.2046
в€’0.5296
0.045 0.1774 1.3522 0.1017 0.1877
в€’0.4380
0.040 0.2147 1.2233 0.0980 0.1711
в€’0.3464
0.035 0.2521 1.0943 0.0943 0.1551
в€’0.2547
0.030 0.2894 0.9654 0.0906 0.1397
в€’0.1631
0.025 0.3267 0.8364 0.0869 0.1252
в€’0.0715
0.020 0.3640 0.7075 0.0832 0.1119
0.015 0.4014 0.0201 0.5785 0.0795 0.1003
0.010 0.4387 0.1118 0.4496 0.0758 0.0912
0.005 0.4760 0.2034 0.3206 0.0721 0.0853
0.000 0.5133 0.2950 0.1917 0.0684 0.0832

0.16
0.14

0.12
Expected return

0.1

0.08
0.06
0.04

0.02
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Standard deviation

Figure 3.14 Efп¬Ѓcient frontier for SharpeвЂ™s simple index model
Note 1
The saving in data required, compared to MarkowitzвЂ™s model, is considerable: in the
last model the expected returns, variances and covariances (two by two) corresponds
N (N в€’ 1) N (N + 3)
to N + N + = , while in the simple index model we need only
2 2
the regression coefп¬Ѓcients and residual variances as well as the expected return and the
variance in the index, namely: 2N + N + 2 = 3N + 2.
For example, on a market on which there is a choice between 100 securities, the number
of items of information required is 5150 in the п¬Ѓrst case and just 302 in the second.

Note 2
в€’ +
If, in addition to the restrictions envisaged above, the inequality restrictions Bj в‰¤ Xj в‰¤ Bj
j = 1, . . . , N are imposed, the simple model index can still be used by applying the same
74 Asset and Risk Management

principles as for MarkowitzвЂ™s model (alteration of matrix elements according to the вЂ˜downвЂ™,
вЂ˜inвЂ™ and вЂ˜upвЂ™ status of the various securities, calculation of critical О» and corner portfolios).

3.2.4.3 Multi-index model
One criticism that can be made of the simple index model is that the behaviour of every
security is made according to just one index. Probably a more consistent way of proceeding
is to divide all the market securities into sectors and express the return on each security
in the same sector as a п¬Ѓrst-degree function of the return on a sectorial index.
The general method for writing this model is heavy and complex. We will be showing
it in relation to two sectors, the п¬Ѓrst corresponding to securities j = 1, . . . , N1 and the
second to j = N1 + 1, . . . , N1 + N2 = N . The sectorial indices will be noted as I1 and
I2 respectively.
The regression equations take the form:

Rj t = aj + bj RI1 t + Оµj t j = 1, . . . , N1
Rj t = aj + bj RI2 t + Оµj t j = N1 + 1, . . . , N1 + N2 = N

The return on the portfolio, and its expected return and variance, are shown as
N
RP = Xj Rj
j =1

N N
= Xj aj + Y1 RI1 + Y2 RI2 + Xj Оµj
j =1 j =1

N
EP = Xj aj + Y1 EI1 + Y2 EI2
j =1

N
ПѓP = Xj ПѓОµ2j + Y1 ПѓI21 + Y2 ПѓI22 + 2Y1 Y2 ПѓI1 I2
2 2 2 2

j =1
пЈ« пЈ¶пЈ« пЈ¶
ПѓОµ21 . X1
пЈ¬ пЈ·пЈ¬ . пЈ·
..
пЈ¬ пЈ·пЈ¬ . пЈ·
.
пЈ¬ пЈ·пЈ¬ . пЈ·
Y2 ) пЈ¬ пЈ·пЈ¬
= ( X1 В·В·В· XN Y1 пЈ· пЈ¬ XN пЈ·
ПѓОµ2N
пЈ¬ пЈ·
пЈ¬ пЈ·
ПѓI1 I2 пЈё пЈ­ Y1 пЈё
ПѓI21
пЈ­
Y2
. ПѓI2 I1 ПѓI2 2

Here, we have introduced the parameters
пЈ±
N1
пЈґ
пЈґ
пЈґ Y1 =
пЈґ Xj bj
пЈґ
пЈІ
j =1
пЈґ N
пЈґ
пЈґY =
пЈґ2 Xj bj
пЈґ
пЈі
j =N1 +1
Equities 75

to the relation: MXв€— = О»E в€— + F . This resolves
The usual reasoning leads once again
into Xв€— = О»(M в€’1 E в€— ) + (M в€’1 F ), with the notations:
пЈ«2 пЈ¶
. . . b1 .
2ПѓОµ1 1
пЈ¬ .пЈ·
. . . .
..
пЈ¬ .пЈ·
. . . .
. . . . . .пЈ·
пЈ¬
пЈ¬ 1пЈ·
. . bN1 .
2ПѓОµ2N
пЈ¬ пЈ·
пЈ¬ пЈ·
1
пЈ¬ bN1 +1 1 пЈ·
. . .
2ПѓОµ2N +1
пЈ¬ пЈ·
пЈ¬ .пЈ·
1
. . . .
..
пЈ¬ .пЈ·
. . . .
. . . . . .пЈ·
M=пЈ¬ пЈ¬ пЈ·
пЈ¬. . . . bN 1пЈ·
2ПѓОµ2N
пЈ¬ пЈ·
пЈ¬. .пЈ·
В·В·В· . . В·В·В· . 2ПѓI21 2ПѓI1 I2 в€’1 .
пЈ¬ пЈ·
пЈ¬. .пЈ·
В·В·В· . . В·В·В· . . в€’1
2ПѓI2 I1 2ПѓI22
пЈ¬ пЈ·
пЈ¬ b1 В· В· В· bN .пЈ·
. В·В·В· . в€’1 . . .
пЈ¬ пЈ·
1
пЈ­. .пЈё
В·В·В· . bN1 +1 В· В· В· bN . в€’1 . .
В·В·В· В·В·В· . . . . .
1 1 1 1
пЈ¶ пЈ«пЈ¶
пЈ« пЈ¶ пЈ«
X1 a1 .
пЈ¬.пЈ·
пЈ¬.пЈ· пЈ¬.пЈ·
пЈ¬.пЈ· пЈ¬.пЈ· пЈ¬.пЈ·
пЈ¬.пЈ·
пЈ¬.пЈ· пЈ¬.пЈ·
пЈ¬.пЈ·
пЈ¬X пЈ· пЈ¬a пЈ·
пЈ¬пЈ·
пЈ¬ NпЈ· пЈ¬ NпЈ·
пЈ¬ Y1 пЈ· пЈ¬ пЈ· пЈ¬пЈ·
F =пЈ¬.пЈ·
E = пЈ¬ EI1 пЈ·
в€— в€—
X =пЈ¬ пЈ·
пЈ¬ Y2 пЈ· пЈ¬ EI пЈ· пЈ¬.пЈ·
пЈ¬пЈ·
пЈ¬ пЈ· пЈ¬ 2пЈ·
пЈ¬.пЈ·
пЈ¬ m1 пЈ· пЈ¬.пЈ·
пЈ· пЈ¬пЈ·
пЈ¬ пЈ· пЈ¬
пЈ­.пЈё
пЈ­ m2 пЈё пЈ­.пЈё
m3 . 1

It should be noted that compared to the simple index model, the two-index model requires
only three additional items of information: expected return, variance and covariance for
the second index.

3.2.5 Model with risk-free security
3.2.5.1 Modelling and resolution
Let us now examine the case in which the portfolio consists of a certain number N of
equities (of returns R1 , . . . , RN ) in proportions X1 , . . . , XN and a risk-free security with
a return of RF that is in proportion XN+1 with X1 + . . . + XN + XN+1 = 1.
This risk-free security is seen as a hypothesis formulated as follows. The investor has
the possibility of investing or loaning (XN+1 > 0) or of borrowing (XN+1 < 0) funds at
the same rate RF .
Alongside the returns on equities, which are the random variables that we looked at in
previous paragraphs (with their expected returns Ej and their varianceвЂ“covariance matrix
V ), the return on the risk-free security is a degenerated random variable:
пЈ±
пЈґ EN+1 = RF
пЈІ
ПѓN+1 = 0
2
пЈґ
пЈі
Пѓj,N+1 = 0 (j = 1, . . . , N )
76 Asset and Risk Management

Note
We will now study the effect of the presence of a risk-free security in the portfolio on
the basis of MarkowitzвЂ™s model without inequality restriction. We can easily adapt the
presentation to cover SharpeвЂ™s model, or take account of the inequality restrictions. The
result in relation to the shape of the efп¬Ѓciency curve (see below) is valid in all cases and
only one presentation is necessary.
The return on the portfolio is written as RP = X1 R1 + . . . + XN RN + XN+1 RF .
This allows the expected return and variance to be calculated:
пЈ±
N
пЈґ
пЈґ
пЈґ EP = Xj Ej + XN+1 RF
пЈґ
пЈґ
пЈІ
j =1
пЈґ N N
пЈґ2
пЈґПѓ =
пЈґP Xi Xj Пѓij
пЈґ
пЈі
i=1 j =1

We must therefore solve the problem, for the different values of О» between 0 and +в€ћ,
of minimisation with respect to the proportions X1 , . . . , XN and XN+1 of the expression
ПѓP в€’ О»EP , under the restriction:
2

N
Xj + XN+1 = 1
j =1

The Lagrangian function for this problem can be written as
пЈ« пЈ¶
N N N
Xi Xj Пѓij в€’ О» В· пЈ­ Xj Ej + XN+1 RF пЈё
L(X1 , . . . , XN , XN+1 ; m) =
i=1 j =1 j =1
пЈ« пЈ¶
N
+mВ·пЈ­ Xj + XN+1 в€’ 1пЈё
j =1

Calculation of its partial derivatives leads to the system of equations MXв€— = О»E в€— + F ,
where we have:
пЈ« пЈ¶ пЈ« пЈ¶
2Пѓ12 В· В· В· 2Пѓ1N . 1
2
X1
2Пѓ1
пЈ¬ 2Пѓ21 2Пѓ 2 В· В· В· 2Пѓ2N . 1 пЈ· пЈ¬ X2 пЈ·
пЈ¬ пЈ· пЈ¬ пЈ·
2
пЈ¬. . .пЈ· пЈ¬.пЈ·
. .
.. пЈ¬.пЈ·
пЈ¬. . . . .пЈ·
. Xв€— = пЈ¬ . пЈ·
M=пЈ¬ . . . . .пЈ·
пЈ¬ 2Пѓ пЈ· пЈ¬ XN пЈ·
пЈ¬ N1 2ПѓN2 В· В· В· 2ПѓN . 1 пЈ·
2
пЈ¬ пЈ·
пЈ­. пЈё пЈ­ XN+1 пЈё
. В·В·В· . .1
m
В·В·В· 1.
1 1 1
пЈ« пЈ¶
пЈ«пЈ¶
E1 .
пЈ¬ E2 пЈ·
пЈ¬.пЈ·
пЈ¬ пЈ·
пЈ¬.пЈ· пЈ¬пЈ·
пЈ¬.пЈ· пЈ¬.пЈ·
E =пЈ¬ . пЈ·
в€—
F =пЈ¬ пЈ·
пЈ¬.пЈ·
пЈ¬ EN пЈ·
пЈ¬ пЈ· пЈ­.пЈё
пЈ­ RF пЈё
1
.
Equities 77

The solution for this system is of course written as: Xв€— = О»(M в€’1 E в€— ) + (M в€’1 F ).

Example
Let us take the same data as those used in the п¬Ѓrst formulation, namely:

E1 = 0.05 E2 = 0.08 E3 = 0.10
Пѓ1 = 0.10 Пѓ2 = 0.12 Пѓ3 = 0.15
ПЃ12 = 0.3 ПЃ13 = 0.1 ПЃ23 = 0.4
Let us suppose that the risk-free interest rate is RF = 0.03.
We therefore have:
пЈ« пЈ¶ пЈ« пЈ¶ пЈ«пЈ¶
0.0200 0.0072 0.0030 . 1 .
0.05
пЈ¬ 0.0072 0.0288 0.0144 . 1 пЈ· пЈ¬.пЈ·
пЈ¬ 0.08 пЈ·
пЈ¬ пЈ· пЈ¬ пЈ· пЈ¬пЈ·
в€—
M = пЈ¬ 0.0030 0.0144 0.0450 . 1 пЈ· F =пЈ¬.пЈ·
E = пЈ¬ 0.10 пЈ·
пЈ­. . 1пЈё пЈ­ 0.03 пЈё пЈ­.пЈё
. .
1. .
1 1 1 1
and therefore: пЈ« пЈ¶ пЈ«пЈ¶
.
0.452
пЈ¬.пЈ·
пЈ¬ 1.024 пЈ·
пЈ¬ пЈ· пЈ¬пЈ·
в€’1 в€— в€’1
M F =пЈ¬.пЈ·
M E = пЈ¬ 1.198 пЈ·
пЈ­ в€’2.674 пЈё пЈ­1пЈё
.
0.030
This leads to the portfolios shown in Table 3.7.

Table 3.7 Solution for model with risk-free security

О» X1 X2 X3 X(RF ) EP ПѓP

в€’4.3472
2.0 0.9031 2.0488 2.3953 0.3182 0.5368
в€’4.0799
1.9 0.8580 1.9464 2.2755 0.3038 0.5100
в€’3.8125
1.8 0.8128 1.8439 2.1558 0.2894 0.4831
в€’3.5451
1.7 0.7677 1.7415 2.0360 0.2749 0.4563
в€’3.2778
1.6 0.7225 1.6390 1.9162 0.2605 0.4295
в€’3.0104
1.5 0.6774 1.5366 1.7965 0.2461 0.4026
в€’2.7431
1.4 0.6322 1.4342 1.6767 0.2317 0.3758
в€’2.4757
1.3 0.5870 1.3317 1.5569 0.2173 0.3489
в€’2.2083
1.2 0.5419 1.2293 1.4372 0.2029 0.3221
в€’1.9410
1.1 0.4967 1.1268 1.3174 0.1885 0.2952
в€’1.6736
1.0 0.4516 1.0244 1.1976 0.1741 0.2684
в€’1.4063
0.9 0.4064 0.9220 1.0779 0.1597 0.2416
в€’1.1389
0.8 0.3613 0.8195 0.9581 0.1453 0.2147
в€’0.8715
0.7 0.3161 0.7171 0.8384 0.1309 0.1879
в€’0.6042
0.6 0.2709 0.6146 0.7186 0.1165 0.1610
в€’0.3368
0.5 0.2258 0.5122 0.5988 0.1020 0.1342
в€’0.0694
0.4 0.1806 0.4098 0.4791 0.0876 0.1074
0.3 0.1355 0.3073 0.3593 0.1979 0.0732 0.0805
0.2 0.0903 0.2049 0.2395 0.4653 0.0588 0.0537
0.1 0.0452 0.1024 0.1198 0.7326 0.0444 0.0268
0.0 0.0000 0.0000 0.0000 1.0000 0.0300 0.0000
78 Asset and Risk Management
0.35

0.3

0.25

Expected return
0.2

0.15

0.1

0.05

0
0 0.1 0.2 0.3 0.4 0.5 0.6
Standard deviation

Figure 3.15 Efп¬Ѓcient frontier for model with risk-free security

0.35

0.3

0.25
Expected return

0.2

0.15

0.1

0.05

0
0 0.1 0.2 0.3 0.4 0.5 0.6
Standard deviation

Figure 3.16 Comparison of efп¬Ѓcient frontiers with and without risk-free security

The efп¬Ѓcient frontier is shown in Figure 3.15.
If the efп¬Ѓcient frontier obtained above and the frontier obtained using MarkowitzвЂ™s
model (without risk-free security) are superimposed, Figure 3.16 is obtained.

3.2.5.2 Efп¬Ѓcient frontier
The graphic phenomenon that appears in the previous example is general. In fact, a
portfolio consisting of N securities and the risk-free security can be considered to consist
of the risk-free security in the proportion X = XN+1 and a portfolio of equities with a
proportion of 1 в€’ X, and the return R (of parameters E and Пѓ ). The return of the risk-free
security has a zero variance and is not correlated with the equity portfolio. The parameters
for the portfolio are given by

EP = XRF + (1 в€’ X)E
ПѓP = (1 в€’ X)2 Пѓ 2
2

which gives, after X has been eliminated:

E в€’ RF
EP = RF В± ПѓP
Пѓ
Equities 79
EP
(X в‰¤ 1)

RF

(X в‰Ґ 1)

ПѓP

Figure 3.17 Portfolios with risk-free security

EP

A

RF

ПѓP

Figure 3.18 Efп¬Ѓcient frontier with risk-free security present

following that X в‰¤ 1 or X в‰Ґ 1. The equations for these straight lines show that the
portfolios in question are located on two semi-straight lines with the same slope, with the
opposite sign (see Figure 3.17).
The lower semi-straight line (X в‰Ґ 1) corresponds to a situation in which the portfolio
of equities is sold at a short price in order to invest more in the risk-free security. From
now on, we will be interested in the upper part.
If the efп¬Ѓcient frontier consisting only of equities is known, the optimum semi-straight
line, which maximises EP for a given ПѓP , is the line located the highest, that is, the
tangent on the efп¬Ѓcient frontier of the equities (see Figure 3.18).
The portfolios located between the vertical axis and the contact point A are characterised
by 0 в‰¤ X в‰¤ 1, and those beyond A are such that X в‰¤ 0 (borrowing at rate RF to invest
further in contact portfolio A).

3.2.6 The Elton, Gruber and Padberg method of portfolio management
The Elton, Gruber and Padberg or EGP method33 was developed34 to supply a quick
and coherent solution to the problem of optimising portfolios. Instead of determining
33
Or more precisely, methods; in fact, various models have been developed around a general idea according to the
hypotheses laid down.
34
Elton E., Gruber M. and Padberg M., Simple criteria for optimal portfolio selection, Journal of Finance, Vol. XI, No.
5, 1976, pp, 1341вЂ“57.
80 Asset and Risk Management

the efп¬Ѓcient frontier as in MarkowitzвЂ™s or SharpeвЂ™s models, this new technique simply
determines the portfolio that corresponds to the contact point of the tangent with the
efп¬Ѓcient frontier, produced by the point (0, RF ).

3.2.6.1 Hypotheses
The method now being examined assumes that:

вЂў The meanвЂ“variance approach is relevant, which will allow a certain number of results
from MarkowitzвЂ™s theory to be used.
вЂў There is a risk-free asset with a return indicated as RF .

Alongside these general hypotheses, Elton, Gruber and Padberg have developed resolution
algorithms in two speciп¬Ѓc cases:

вЂў Constant correlations. In this п¬Ѓrst model, it is assumed that the correlation coefп¬Ѓcients
for the returns on the various securities are all equal: ПЃij = ПЃ в€Ђi, j .
вЂў SharpeвЂ™s simple index model can be used.

The п¬Ѓrst of these two simpliп¬Ѓcations is quite harsh and as such not greatly realistic,
and we will instead concentrate on the second case. Remember that it is based on the
following two conditions.

1. The returns on the various securities are expressed as п¬Ѓrst-degree functions of the
return on a market-representative index: Rj t = aj + bj RI t = Оµj t . j = 1, . . ., N .
It is also assumed that the residuals verify the classic hypotheses of linear regression,
including the hypothesis that the residuals have zero-expected return and are not cor-
related with the explanatory variable Rit .
2. The residuals of the regressions relative to the various securities are not correlated:
cov (Оµit , Оµj t ) = 0 for all the different i and j values.

3.2.6.2 Resolution of case in which short sales are authorised
First of all, we will carry out a detailed analysis of a case in which the proportions
are not subject to inequality restrictions. Here, the reasoning is more straightforward35
than in cases where short sales are prohibited. Nevertheless, as will be seen (but without
demonstration), applying the algorithm is scarcely any more complex in the second case.
If one considers a portfolio P consisting solely of equities in proportions X1 , X2 , . . .,
XN , the full range of portfolios consisting partly of P and partly of risk-free securities

Elton E., Gruber M. and Padberg M., Optimal portfolios from simple ranking devices, Journal of Portfolio Management,
Vol. 4, No. 3, 1978, pp. 15вЂ“19.
Elton E., Gruber M. and Padberg M., Simple criteria for optimal portfolio selection; tracing out the efп¬Ѓcient frontier,
Journal of Finance, Vol. XIII No. 1, 1978, pp. 296вЂ“302.
Elton E., Gruber M. and Padberg M., Simple criteria for optimal portfolio selection with upper bounds, Operation Research,
1978.
Readers are also advised to read Elton E. and Gruber M., Modern Portfolio Theory and Investment Analysis, John Wiley
& Sons, Inc, 1991.
35
In addition, it starts in the same way as the demonstration of the CAPM equation (see В§3.3.1).
Equities 81

EP

A
P

RF

ПѓP

Figure 3.19 EGP method
RF shall make up the straight line linking the points (0, RF ) and (ПѓP , EP ) as illustrated
in Figure 3.19.
EP в€’ RF
The slope of the straight line in question is given by P = , which may be
ПѓP
interpreted as a risk premium, as will be seen in Section 3.3.1.
According to the reasoning set out in the previous paragraph, the ideal portfolio P
corresponds to the contact point A of the tangent to the efп¬Ѓcient frontier coming from
the point (0, RF ) for which the slope is the maximum. We are therefore looking for
proportions that maximise the slope P or, which amounts to the same thing, maximise
2
P . Such as:
пЈ« пЈ¶
пЈ±
N N N
пЈґ
пЈґ
пЈґE в€’ R = Xj Ej в€’ пЈ­ Xj пЈё RF =
пЈґP Xj (Ej в€’ RF )
пЈґ F
пЈІ
j =1 j =1 j =1
пЈґ N N
пЈґ2
пЈґ
пЈґПѓ = Xi Xj Пѓij
пЈґP
пЈі
i=1 j =1

the derivative of:
пЈ« пЈ¶2
N
пЈ­ Xj (Ej в€’ RF )пЈё
(EP в€’ RF )2 j =1
= =
2
P
ПѓP N N
2
Xi Xj Пѓij
i=1 j =1

with respect to Xk is given by:
пЈ« пЈ¶ пЈ« пЈ¶2
N N N
2пЈ­ Xj (Ej в€’ RF )пЈё (Ek в€’ RF ) В· в€’пЈ­ Xj (Ej в€’ RF )пЈё В· 2
ПѓP Xj Пѓkj
2

j =1 j =1 j =1
( P )Xk =
2
ПѓP
4

N
2(EP в€’ RF )(Ek в€’ RF )ПѓP в€’ 2(EP в€’ RF ) Xj Пѓkj
2 2

j =1
=
ПѓP
4
82 Asset and Risk Management
пЈ« пЈ¶
N
2(EP в€’ RF ) пЈ­
Xj Пѓkj пЈё
= В· (Ek в€’ RF ) в€’ Оі В·
ПѓP
2
j =1

In which we have provisionally Оі = (Ep в€’ RF )/ПѓP .
2

This derivative will be zero if:
N
Ek в€’ RF = Оі В· Xj Пѓkj
j =1

By introducing Zj = Оі В· Xj (j = 1, . . ., N ), the system to be resolved with respect to
Z1 , . . ., ZN is therefore
N
Ek в€’ RF = Zj Пѓkj k = 1, . . . , N
j =1

Before proceeding with the resolution, note that п¬Ѓnding the Zk quantities allows the Xk
quantities to be found, as

Zk Zk Zk
Xk = = =
Оі N N
ОіВ· Xj Zj
j =1 j =1

The hypotheses from SharpeвЂ™s model allow the following to be written:

Пѓkj = cov(ak + bk RI + Оµk , aj + bj RI + Оµj )
ПѓОµ2k si j = k
= bk bj ПѓI2 +
si j = k
0

The k th equation in the system can then be written:
пЈ« пЈ¶
N
Ek в€’ RF = bk пЈ­ Zj bj пЈё ПѓI2 + Zk ПѓОµ2k
j =1

or also, by resolving with respect to Zk :
пЈ± пЈ¶пЈј
пЈ«
пЈІ пЈЅ
N
1
Zk = 2 (Ek в€’ RF ) в€’ bk пЈ­ Zj bj пЈё ПѓI
2
ПѓОµ k пЈі пЈѕ
j =1
пЈ± пЈ¶пЈј
пЈ«
bk пЈІ пЈЅ
N
пЈ­ пЈё ПѓI2
= 2 Оёk в€’ Zj bj
ПѓОµ k пЈі пЈѕ
j =1

where we have:
Ek в€’ RF
Оёk =
bk
Equities 83

All that now remains is to determine the sum between the brackets. On the basis of the
last result, we п¬Ѓnd:
пЈ± пЈ¶пЈј
пЈ«
пЈІ пЈЅ
N N N
bk2
Оёk в€’ пЈ­ Zj bj пЈё ПѓI
Zk bk = 2
ПѓОµ2k пЈі пЈѕ
k=1 k=1 j =1
пЈ« пЈ¶
N N N
bk bk пЈ­
2 2
Zj bj пЈё ПѓI2
= Оёk в€’
Пѓ Пѓ
2 2
k=1 Оµk k=1 Оµk j =1

the resolution of which gives
N
bk2
Оёk
ПѓОµ2k
N
k=1
Zj bj = N
bk2
j =1
1+ ПѓI2
ПѓОµ2k
k=1

By introducing the new notation
N
bk2
пЈ« пЈ¶ Оёk
ПѓОµ2k
N
k=1
П†=пЈ­ Zj bj пЈё В· ПѓI2 = В· ПѓI2
N
bk2
j =1
1+ ПѓI2
ПѓОµ2k
k=1

and by substituting the sum just calculated within the expression of Zk , we п¬Ѓnd
bk
Zk = (Оёk в€’ П†) k = 1, . . . , N
ПѓОµ2k

Example
Let us take the same data as those used in the simple index model (only essential data
mentioned here).

E1 = 0.05 E2 = 0.08 E3 = 0.10

with the regression relations and the estimated residual variances:

R1 = 0.014 + 0.60RI (ПѓОµ1 = 0.0060)
2

R2 = в€’0.020 + 1.08RI (ПѓОµ2 = 0.0040)
2

R3 = 0.200 + 1.32RI (ПѓОµ3 = 0.0012)
2

Assume that the variance of the index is equal to ПѓI2 = 0.0045. Finally, assume also
that as for the model with the risk-free security, this last value is RF = 0.03. These data
allow calculation of:

Оё1 = 0.0333 Оё2 = 0.0463 Оё3 = 0.0530.
84 Asset and Risk Management

Therefore, П† = 0.0457. The Zk values are deduced:

Z1 = в€’1.2327 Z2 = 0.1717 Z3 = 8.1068

The proportions of the optimum portfolio are therefore deduced:

X1 = в€’0.1750 X2 = 0.0244 X3 = 1.1506

3.2.6.3 Resolution of case in which short sales are prohibited
Let us now examine cases in which restrictions are introduced. These are less general
than those envisaged in MarkowitzвЂ™s model, and are written simply as 0 в‰¤ Xj в‰¤ 1(j =
1, . . . , N ).
The method, which we are showing here without supporting calculations, is very similar
to that used for cases in which short sales are authorised. As above, the following are
calculated:
Ek в€’ RF
Оёk = k = 1, . . . , N
bk
The securities are then sorted in decreasing order of Оёk and this order is preserved until the
end of the algorithm. Instead of having just one parameter П†, one parameter is calculated
for each security:
k
bj
2
Оёj
ПѓОµ2j
j =1
пЈ« пЈ¶
П†k = В· ПѓI2 k = 1, . . . , N
k
bj
2
пЈ­ пЈё ПѓI2
1+
Пѓ2
j =1 Оµj

It can be shown that the sequence of П†k numbers п¬Ѓrst increases, then passes through a max-
imum and п¬Ѓnally ends with a decreasing phase. The value K of the k index corresponding
to the maximum П†k , is noted. The П†K number is named the вЂ˜cut-off rateвЂ™ and it can be
shown that the calculation of the Zk values for the same relation as before (replacing П† by
П†K ) produces positive values for k = 1, . . . , K and negative values for k = K + 1, . . . , N .
Only the п¬Ѓrst K securities are included in the portfolio. The calculations to be made are
therefore:
bk
Zk = 2 (Оёk в€’ П†K ) k = 1, . . . , K
ПѓОµ k
This, for the proportions of integrated K securities, gives:
Zk
Xk = k = 1, . . . , K
K
Zj
j =1

Example
Let us take the same data as above. Of course, we still have:

Оё1 = 0.0333 Оё2 = 0.0463 Оё3 = 0.0530
Equities 85

This allows the securities to be classiп¬Ѓed in the order (3), (2), (1). We will provisionally
renumber the securities in this new order, thus producing:

П†1 = 0.04599 П†2 = 0.04604 П†3 = 0.04566

This shows that K = 2 and the cut-off rate is П†2 = 0.04604. The Zk values will there-
fore be deduced:
Z1 = 7.6929 Z2 = 0.0701

The proportions of the optimum portfolio are therefore deduced:

X1 = 0.9910 X2 = 0.0090

If one then reverts to the initial order, the securities to be included in the portfolio shall
therefore be securities (2) and (3) with the following relative proportions:

X2 = 0.0090 X3 = 0.9910

3.2.7 Utility theory and optimal portfolio selection
Once the efп¬Ѓcient frontier has been determined, the question that faces the investor is
that of choosing from all the efп¬Ѓcient portfolios the one that best suits him. The portfolio
chosen will differ from one investor to another, and the choice made will depend on his
attitude and behaviour towards the risk. The efп¬Ѓcient frontier, in fact, contains as many
prudent portfolios (low expected return and risk, located at the left end of the curve) as
more risky portfolios (higher expected return and risk, located towards the right end).

3.2.7.1 Utility function
The concept of utility function can be introduced generally36 to represent from an indi-
vidual personвЂ™s viewpoint the utility and interest that he п¬Ѓnds in a project, investment,
strategy etc., the elements in question presenting a certain level of risk. The numerical
values of this risk function are of little importance, as it is essentially used to compare
projects, investments, strategies etc. Here, we will present the theory of utility in the
context of its application to a return (which, remember, is random) of, for example, a
portfolio of equities.
Because of the presence of the risk, it is evident that we cannot be content with taking
E(R) as utility of return U (R). This was clearly shown by D. Bernoulli in 1732 through
the вЂ˜St Petersburg paradoxвЂ™. The question is: How much would you be prepared to stake
to participate in the next game? I toss a coin a number of times and I give you two \$
if tails comes up on the п¬Ѓrst throw, four \$ if tails comes up for the п¬Ѓrst time on the
second throw, eight \$ if tails appears for the п¬Ѓrst time on the third throw, and so on. I
will therefore give you 2n \$ if tails comes up for the п¬Ѓrst time on the nth throw. Most
people would lay down a small sum (at least two \$), but would be reluctant to invest more
because of the increased risk in the game. A player who put down 20 \$ would have a

36
An excellent presentation on the general concepts of behaviour in the face of risk (not necessarily п¬Ѓnancial) and the
concept of вЂ˜utilityвЂ™ is found in Eeckhoudt L. and Gollier C., Risk, Harvester Wheatsheaf, 1995.
86 Asset and Risk Management

probability of losing of 1/2 + 1/4 + 1/8 + 1/16 = 15/16 = 0.9375, and would therefore
only win on 6.25 stakes out of every 100. The average gain in the game, however, is
в€ћ n
1
n
= 1 + 1 + 1 + ... = в€ћ
2
2
n=1

It is the aversion to the risk that justiп¬Ѓes the decision of the player. The aim of the utility
function is to represent this attitude.
In utility theory, one compares projects, investments, strategies etc. (in our case, returns)
through a relation of preference (R1 is preferable to R2 : R1 > R2 ) and a relation of
indifference (indifference between R1 and R2 : R1 в€ј R2 ). The behaviour of the investor
can be expressed if these two relations obey the following axioms:

вЂў (Comparability): The investor can always compare two returns. в€ЂR1 , R2 . We always
have R1 > R2 , R2 < R1 , or R1 в€ј R2 .
вЂў (Reп¬‚exivity): в€ЂR R в€ј R.
вЂў (Transitivity): в€ЂR1 , R2 , R3 , if R1 > R2 and R2 > R3 , then R1 > R3 .
вЂў (Continuity): в€ЂR1 , R2 , R3 , if R1 > R2 > R3 , there is a single X в€€ [0; 1] such as [X.R1 +
(1 в€’ X).R3 ] в€ј R2 .
вЂў (Independence): в€ЂR1 , R2 , R3 and в€ЂX в€€ [0; 1], if R1 > R2 , then [X.R1 + (1 в€’ X).R3 ] >
[X.R2 + (1 в€’ X).R3 ].

Von Neumann and Morgenstern37 have demonstrated a theorem of expected utility,
which states that if the preferences of an investor obey the axioms set out above, there is
a function U so that в€ЂR1 , R2 , R1 > R2 в‡” E[U (R1 )] > E[U (R2 )].
This utility function is clearly a growing function. We have noted that its numerical
values are not essential as it is only used to make comparisons of returns. The theorem
of expected utility allows this concept to be deп¬Ѓned more accurately: if an investorвЂ™s
preferences are modelled by the utility function U , there will be the same system of
preferences based on the function aU + b with a > 0. In fact, if R1 > R2 is expressed as
E[U (R1 )] > E[U (R2 )], we have:

E[U в€— (R1 )] = E[aU (R1 ) + b]
= aE[U (R1 )] + b
> aE[U (R2 )] + b
= E[aU (R2 ) + b]
= E[U в€— (R2 )]

The utility function is an element that is intrinsically associated with each investor (and is
also likely to evolve with time and depending on circumstances). It is not easy or indeed
even very useful to know this function. If one wishes to estimate it approximately, one
has to deп¬Ѓne a list of possible values R1 < R2 < . . . < Rn for the return, and then for
i = 2, . . . , n в€’ 1, ask the investor what is the probability of it being indifferent to obtain

37
Von Neumann J. and Morgenstern O., Theory of Games and Economic Behaviour, Princeton University Press, 1947.
Equities 87

a deп¬Ѓnite return Ri or play in a lottery that gives returns of R1 and Rn with the respective
probabilities (1 в€’ pi ) and pi . If one chooses arbitrarily U (R1 ) = 0 and U (Rn ) = 100,
then U (Ri ) = 100 pi (i = 2, . . . , n в€’ 1).

3.2.7.2 Attitude towards risk
For most investors, an increase in return of 0.5 % would be of greater interest if the
current return is 2 % than if it is 5 %. This type of attitude is called risk aversion. The
opposite attitude is known as taste for risk, and the middle line is termed risk neutrality.
How do these behaviour patterns show in relation to utility function?
Let us examine the case of aversion. Generally, if one wishes to state that the utility of
return U (R) must increase with R and give less weight to the same variations in return
when the level of return is high, we will have: R1 < R2 в‡’ U (R1 + R) в€’ U (R1 ) >
U (R2 + R) в€’ U (R2 ).
This shows the decreasing nature of the marginal utility. In this case, the derivative of
the utility function is a decreasing function and the second derivative is therefore negative;
the utility function is concave.
The results obtained from these considerations are summarised in Table 3.8, and a
representation of the utility function in the various cases is shown in Figure 3.20.
Let us now deп¬Ѓne this concept more precisely. We consider an investor who has a
choice between a certain return totalling R on one hand and a lottery that gives him a
random return that may have two values (R в€’ r) and (R + r), each with a probability
of 1/2. If he shows an aversion to risk, the utility of the certain return will exceed the
expected utility of the return on the lottery:

U (R) > 1 [U (R в€’ r) + U (R + r)]
2

This is shown in graphic form in Figure 3.21.

Table 3.8 Attitude to risk

U U
Marginal utility

<0
Risk aversion Decreasing Concave
=0
Risk neutrality Increasing Linear
>0
Taste for risk Increasing Convex

U(R)
Aversion
Neutrality
Taste

R

Figure 3.20 Utility function
88 Asset and Risk Management
U(x)

U(R + r)

U(R)

1
[U(R вЂ“ r) + U(R + r)]
2

U(R вЂ“ r)

p

RвЂІ
RвЂ“r R+r
R x

Figure 3.21 Aversion to risk

This п¬Ѓgure shows R , the certain return, for which the utility is equal to the expected
return on the lottery. The difference p = R в€’ R represents the price that the investor
is willing to pay to avoid having to participate in the lottery; this is known as the
Taylor expansions for U (R + r), U (R в€’ r) and U(R ) = U(R в€’ p) readily lead to the
relation:
U (R) r 2
p=в€’ В·
U (R) 2

The п¬Ѓrst factor in this expression is the absolute risk aversion coefп¬Ѓcient:

U (R)
О±(R) = в€’
U (R)

The two most frequently used examples of the utility function corresponding to the risk
aversion are the exponential function and the quadratic function.
If U (R) = a.ebR , with a and b < 0, we will have О±(R) = в€’b.
If U (R) = aR 2 + bR + c, with a < 0 and b < 0, we of course have to limit ourselves
to values for R that do not exceed в€’b/2a in order for the utility function to remain an
increasing function. The absolute risk aversion coefп¬Ѓcient is then given by:

1
О±(R) =
b
в€’ в€’R
2a

When this last form can be accepted for the utility function, we have another justiп¬Ѓcation
for deп¬Ѓning the distribution of returns by the two parameters of mean and variance
alone, without adding a normality hypothesis (see Section 3.1.1). In this case, in fact,
the expected utility of a return on a portfolio (the quantity that the investor wishes to
Equities 89

optimise) is given as:

E[U (RP )] = E[aRP + bRP + c]
2

= aE(RP ) + bE(RP ) + c
2

= a(ПѓP + EP ) + bEP + c
2 2

This quantity then depends on the п¬Ѓrst two moments only.

3.2.7.3 Selection of optimal portfolio
Let us now consider an investor who shows an aversion for risk and has to choose a
portfolio from those on the efп¬Ѓcient frontier.
We begin by constructing indifference curves in relation to its utility function, that is,
the curves that correspond to the couples (expectation, standard deviation) for which the
expected utility of return equals a given value (see Figure 3.22). These indifference curves
are of close-п¬Ѓtting convex form, the utility increasing as the curve moves upwards and
to the left.
By superimposing the indifference curves and the efп¬Ѓcient frontier, it is easy to deter-
mine the portfolio P that corresponds to the maximum expected utility, as shown in
Figure 3.23.

E
U=4 U=3 U=2 U=1

Пѓ

Figure 3.22 Indifference curves

E

EP

ПѓP Пѓ

Figure 3.23 Selection of optimal portfolio
90 Asset and Risk Management

3.2.7.4 Other viewpoints
Alongside the efп¬Ѓcient portfolio based on the investor preference system, shown through
the utility function, other objectives or restrictions can be taken into consideration.
Let us examine, for example, the case of deп¬Ѓcit constraint. As well as optimising the
couple (E, Пѓ ), that is, determining the efп¬Ѓcient frontier, and before selecting the portfolio
(through the utility function), the return on the portfolio here must not be less than a п¬Ѓxed
threshold38 u except with a very low probability p, say: Pr[RP в‰¤ u] в‰¤ p.
If the hypothesis of normality of return is accepted, we have:

RP в€’ EP u в€’ EP
в‰¤ в‰¤p
Pr
ПѓP ПѓP

that is:
u в€’ EP
в‰¤ zp
ПѓP

Here, zp is the p-quantile of the standard normal distribution (zp < 0 as p is less than
1/2). The condition can thus be written as EP в‰Ґ u в€’ zp .ПѓP .
The portfolios that obey the deп¬Ѓcit constraint are located above the straight line for the
equation EP = u = zp .ПѓP (see Figure 3.24).
The portion of the efп¬Ѓcient frontier delimited by this straight line of constraint is the
range of portfolios from which the investor will make his selection.
If p is п¬Ѓxed, an increase of u (higher required return) will cause the straight line
of constraint to move upwards. In the same way, if u is п¬Ѓxed, a reduction in p (more
security with respect for restriction) will cause the straight line of constraint to move
upwards while pivoting about the point (0, u). In both cases, the section of the efп¬Ѓcient
frontier that obeys the restriction is limited.
One can also, by making use of these properties, determine the optimal portfolio on
the basis of one of the two criteria by using the straight line tangential to the efп¬Ѓcient
frontier.

E

u

Пѓ

Figure 3.24 Deп¬Ѓcit constraint

If u = 0, this restriction means that except in a low probability event, the capital invested must be at least maintained.
38
Equities 91

3.2.8 The market model
Some developments in the market model now include the reasoning contained in the
construction of SharpeвЂ™s model, where the index is replaced by the market in its totality.
This model, however, contains more of a macroeconomic thought pattern than a search
for efп¬Ѓcient portfolios.

3.2.8.1 Systematic risk and speciп¬Ѓc risk
We have already encountered the concept of a systematic security risk in Section 3.1.1:

Пѓj M
ОІj =
ПѓM2

This measures the magnitude of the risk of the security (j ) in comparison to the risk
of the average security on the market. It appears as a regression coefп¬Ѓcient when the
return on this security is expressed as a linear function of the market return: Rj t =
О±j + ОІj RMt + Оµj t .
It is, of course, supposed that the residuals verify the classical hypotheses of the linear
regression, establishing among other things that the residuals are of zero expectation and
constant variance and are not correlated with the explanatory variable RMt .
Alongside the systematic risk ОІj , which is the same for every period, another source
of п¬‚uctuation in Rj is the residual Оµj t , which is speciп¬Ѓc to the period t. The term speciп¬Ѓc
risk is given to the variance in the residuals: ПѓОµ2j = var(Оµj t ).

Note
In practice, the coefп¬Ѓcients О±j and ОІj for the regression are estimated using the least square
Л†
method. For example: ОІj = sj M /sM . The residuals are then estimated by Оµj t = Rj t в€’
Л†
2
1T2
Л†
(О±j + ОІj RMt ) and the speciп¬Ѓc risk is estimated using its ergodic estimator
Л† t=1 Оµj t .
Л†
T
In the rest of this paragraph, we will omit the index t relating to time.

We will see how the risk Пѓj2 for a security consists of a systematic component and a
speciп¬Ѓc component. We have:

Пѓj2 = var(Rj )
= E[(О±j + ОІj RM + Оµj в€’ E(О±j + ОІj RM + Оµj ))2 ]
= E[(ОІj (RM в€’ EM ) + Оµj )2 ]
= ОІj E[(RM в€’ EM )2 ] + E(Оµj ) + 2ОІj E[(RM в€’ EM )Оµj ]
2 2

= ОІj var(RM ) + var(Оµj )
2

Hence the announced decomposition relation of:

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