« «

N N

+ m1 · Xj bj ’ Y + m2 · Xj ’ 1

j =1 j =1

Calculation of the partial derivatives of this lagrangian function leads to the equality

MX— = »E — + F , where we have:

«2 «

. . b1 1 X1

2σµ1

¬ .· ¬.·

. .

.. . . .· ¬.·

¬ . . . .· ¬.·

¬

¬. · ¬ ·

. bN 1 · X— = ¬ XN ·

2

M=¬ 2σµN

¬. 2σI2 ’1 . · ¬Y·

··· .

¬ · ¬ ·

b m1

· · · bN ’1 . .

1

m2

··· . . .

1 1

« «

a1 .

¬.· ¬.·

¬.· ¬.·

¬.· ¬.·

¬ aN · ¬·

F =¬.·

E— = ¬ ·

¬ EI · ¬.·

¬· ¬·

. .

. 1

72 Asset and Risk Management

The solution for this system is written as: X— = »(M ’1 E — ) + (M ’1 F ).

Example

Let us take the same data as those used in the ¬rst formulation, namely:32

E1 = 0.05 E2 = 0.08 E3 = 0.10

σ1 = 0.10 σ2 = 0.12 σ3 = 0.15

ρ12 = 0.3 ρ13 = 0.1 ρ23 = 0.4

Let us then suppose that the regression relations and the estimated residual variances are

given by:

R1 = 0.014 + 0.60RI (σµ21 = 0.0060)

R2 = ’0.020 + 1.08RI (σµ22 = 0.0040)

R3 = 0.200 + 1.32RI (σµ23 = 0.0012)

Let us also suppose that the expected return and index variance represent respectively

EI = 0.04 and σI2 = 0.0045.

These data allow us to write:

«

. . .

0.0120 0.60 1

¬. . . 1.08 1 ·

0.0080

¬ ·

¬. . . 1.32 1 ·

0.0024

M=¬ ·

¬. . . 0.0090 ’1 . ·

0.60 .

’1 .

1.08 1.32

. . .

1 1 1

«

«

.

0.014

¬ ’0.020 · ¬.·

· ¬·

¬

¬.·

¬ 0.200 ·

—

E =¬ F = ¬ ·.

·

¬.·

¬ 0.040 ·

.

.

. 1

We can therefore calculate:

«

«

’7.46 0.513

¬ ’18.32 · ¬ 0.295 ·

¬ · ¬ ·

¬ 25.79 · ¬ 0.192 ·

’1 — ’1

M E =¬ M F =¬

· ·

¬ 9.77 · ¬ 0.880 ·

0.05 0.008

’0.011

0.07

The portfolios for the different values of » are shown in Table 3.6.

The ef¬cient frontier is represented in Figure 3.14.

We should point out that although the ef¬cient frontier has the same appearance as in

Markowitz™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 Sharpe™s model (in addition, one reason

for this was to avoid the need to calculate the variance“covariance matrix). We will use them here only to calculate the

ef¬cient frontier.

Equities 73

Table 3.6 Solution for Sharpe™s simple index model

» X1 X2 X3 EP σP

’0.2332 ’1.5375

0.100 2.7706 0.1424 0.3829

’0.1958 ’1.4458

0.095 2.6417 0.1387 0.3647

’0.1585 ’1.3542

0.090 2.5127 0.1350 0.3465

’0.1212 ’1.2626

0.085 2.3838 0.1313 0.3284

’0.0839 ’1.1710

0.080 2.2548 0.1276 0.3104

’0.0465 ’1.0793

0.075 2.1259 0.1239 0.2924

’0.0092 ’0.9877

0.070 1.9969 0.1202 0.2746

’0.8961

0.065 0.0281 1.8680 0.1165 0.2568

’0.8045

0.060 0.0654 1.7390 0.1128 0.2392

’0.7129

0.055 0.1028 1.6101 0.1091 0.2218

’0.6212

0.050 0.1401 1.4812 0.1054 0.2046

’0.5296

0.045 0.1774 1.3522 0.1017 0.1877

’0.4380

0.040 0.2147 1.2233 0.0980 0.1711

’0.3464

0.035 0.2521 1.0943 0.0943 0.1551

’0.2547

0.030 0.2894 0.9654 0.0906 0.1397

’0.1631

0.025 0.3267 0.8364 0.0869 0.1252

’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 Ef¬cient frontier for Sharpe™s simple index model

Note 1

The saving in data required, compared to Markowitz™s model, is considerable: in the

last model the expected returns, variances and covariances (two by two) corresponds

N (N ’ 1) N (N + 3)

to N + N + = , while in the simple index model we need only

2 2

the regression coef¬cients 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 ¬rst case and just 302 in the second.

Note 2

’ +

If, in addition to the restrictions envisaged above, the inequality restrictions Bj ¤ Xj ¤ 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 Markowitz™s model (alteration of matrix elements according to the ˜down™,

˜in™ and ˜up™ status of the various securities, calculation of critical » 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 ¬rst-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 ¬rst 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 + µj t j = 1, . . . , N1

Rj t = aj + bj RI2 t + µj 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 µj

j =1 j =1

N

EP = Xj aj + Y1 EI1 + Y2 EI2

j =1

N

σP = Xj σµ2j + Y1 σI21 + Y2 σI22 + 2Y1 Y2 σI1 I2

2 2 2 2

j =1

« «

σµ21 . X1

¬ ·¬ . ·

..

¬ ·¬ . ·

.

¬ ·¬ . ·

Y2 ) ¬ ·¬

= ( X1 ··· XN Y1 · ¬ XN ·

σµ2N

¬ ·

¬ ·

σI1 I2 Y1

σI21

Y2

. σI2 I1 σI2 2

Here, we have introduced the parameters

±

N1

Y1 =

Xj bj

j =1

N

Y =

2 Xj bj

j =N1 +1

Equities 75

to the relation: MX— = »E — + F . This resolves

The usual reasoning leads once again

into X— = »(M ’1 E — ) + (M ’1 F ), with the notations:

«2

. . . b1 .

2σµ1 1

¬ .·

. . . .

..

¬ .·

. . . .

. . . . . .·

¬

¬ 1·

. . bN1 .

2σµ2N

¬ ·

¬ ·

1

¬ bN1 +1 1 ·

. . .

2σµ2N +1

¬ ·

¬ .·

1

. . . .

..

¬ .·

. . . .

. . . . . .·

M=¬ ¬ ·

¬. . . . bN 1·

2σµ2N

¬ ·

¬. .·

··· . . ··· . 2σI21 2σI1 I2 ’1 .

¬ ·

¬. .·

··· . . ··· . . ’1

2σI2 I1 2σI22

¬ ·

¬ b1 · · · bN .·

. ··· . ’1 . . .

¬ ·

1

. .

··· . bN1 +1 · · · bN . ’1 . .

··· ··· . . . . .

1 1 1 1

«

« «

X1 a1 .

¬.·

¬.· ¬.·

¬.· ¬.· ¬.·

¬.·

¬.· ¬.·

¬.·

¬X · ¬a ·

¬·

¬ N· ¬ N·

¬ Y1 · ¬ · ¬·

F =¬.·

E = ¬ EI1 ·

— —

X =¬ ·

¬ Y2 · ¬ EI · ¬.·

¬·

¬ · ¬ 2·

¬.·

¬ m1 · ¬.·

· ¬·

¬ · ¬

.

m2 .

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 variance“covariance matrix

V ), the return on the risk-free security is a degenerated random variable:

±

EN+1 = RF

σN+1 = 0

2

σj,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 Markowitz™s model without inequality restriction. We can easily adapt the

presentation to cover Sharpe™s model, or take account of the inequality restrictions. The

result in relation to the shape of the ef¬ciency 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:

±

N

EP = Xj Ej + XN+1 RF

j =1

N N

2

σ =

P Xi Xj σij

i=1 j =1

We must therefore solve the problem, for the different values of » between 0 and +∞,

of minimisation with respect to the proportions X1 , . . . , XN and XN+1 of the expression

σP ’ »EP , under the restriction:

2

N

Xj + XN+1 = 1

j =1

The Lagrangian function for this problem can be written as

«

N N N

Xi Xj σij ’ » · Xj Ej + XN+1 RF

L(X1 , . . . , XN , XN+1 ; m) =

i=1 j =1 j =1

«

N

+m· Xj + XN+1 ’ 1

j =1

Calculation of its partial derivatives leads to the system of equations MX— = »E — + F ,

where we have:

« «

2σ12 · · · 2σ1N . 1

2

X1

2σ1

¬ 2σ21 2σ 2 · · · 2σ2N . 1 · ¬ X2 ·

¬ · ¬ ·

2

¬. . .· ¬.·

. .

.. ¬.·

¬. . . . .·

. X— = ¬ . ·

M=¬ . . . . .·

¬ 2σ · ¬ XN ·

¬ N1 2σN2 · · · 2σN . 1 ·

2

¬ ·

. XN+1

. ··· . .1

m

··· 1.

1 1 1

«

«

E1 .

¬ E2 ·

¬.·

¬ ·

¬.· ¬·

¬.· ¬.·

E =¬ . ·

—

F =¬ ·

¬.·

¬ EN ·

¬ · .

RF

1

.

Equities 77

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

Example

Let us take the same data as those used in the ¬rst formulation, namely:

E1 = 0.05 E2 = 0.08 E3 = 0.10

σ1 = 0.10 σ2 = 0.12 σ3 = 0.15

ρ12 = 0.3 ρ13 = 0.1 ρ23 = 0.4

Let us suppose that the risk-free interest rate is RF = 0.03.

We therefore have:

« « «

0.0200 0.0072 0.0030 . 1 .

0.05

¬ 0.0072 0.0288 0.0144 . 1 · ¬.·

¬ 0.08 ·

¬ · ¬ · ¬·

—

M = ¬ 0.0030 0.0144 0.0450 . 1 · F =¬.·

E = ¬ 0.10 ·

. . 1 0.03 .

. .

1. .

1 1 1 1

and therefore: « «

.

0.452

¬.·

¬ 1.024 ·

¬ · ¬·

’1 — ’1

M F =¬.·

M E = ¬ 1.198 ·

’2.674 1

.

0.030

This leads to the portfolios shown in Table 3.7.

Table 3.7 Solution for model with risk-free security

» X1 X2 X3 X(RF ) EP σP

’4.3472

2.0 0.9031 2.0488 2.3953 0.3182 0.5368

’4.0799

1.9 0.8580 1.9464 2.2755 0.3038 0.5100

’3.8125

1.8 0.8128 1.8439 2.1558 0.2894 0.4831

’3.5451

1.7 0.7677 1.7415 2.0360 0.2749 0.4563

’3.2778

1.6 0.7225 1.6390 1.9162 0.2605 0.4295

’3.0104

1.5 0.6774 1.5366 1.7965 0.2461 0.4026

’2.7431

1.4 0.6322 1.4342 1.6767 0.2317 0.3758

’2.4757

1.3 0.5870 1.3317 1.5569 0.2173 0.3489

’2.2083

1.2 0.5419 1.2293 1.4372 0.2029 0.3221

’1.9410

1.1 0.4967 1.1268 1.3174 0.1885 0.2952

’1.6736

1.0 0.4516 1.0244 1.1976 0.1741 0.2684

’1.4063

0.9 0.4064 0.9220 1.0779 0.1597 0.2416

’1.1389

0.8 0.3613 0.8195 0.9581 0.1453 0.2147

’0.8715

0.7 0.3161 0.7171 0.8384 0.1309 0.1879

’0.6042

0.6 0.2709 0.6146 0.7186 0.1165 0.1610

’0.3368

0.5 0.2258 0.5122 0.5988 0.1020 0.1342

’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 Ef¬cient 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 ef¬cient frontiers with and without risk-free security

The ef¬cient frontier is shown in Figure 3.15.

If the ef¬cient frontier obtained above and the frontier obtained using Markowitz™s

model (without risk-free security) are superimposed, Figure 3.16 is obtained.

3.2.5.2 Ef¬cient 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 ’ X, and the return R (of parameters E and σ ). 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 ’ X)E

σP = (1 ’ X)2 σ 2

2

which gives, after X has been eliminated:

E ’ RF

EP = RF ± σP

σ

Equities 79

EP

(X ¤ 1)

RF

(X ≥ 1)

σP

Figure 3.17 Portfolios with risk-free security

EP

A

RF

σP

Figure 3.18 Ef¬cient frontier with risk-free security present

following that X ¤ 1 or X ≥ 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 ≥ 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 ef¬cient frontier consisting only of equities is known, the optimum semi-straight

line, which maximises EP for a given σP , is the line located the highest, that is, the

tangent on the ef¬cient frontier of the equities (see Figure 3.18).

The portfolios located between the vertical axis and the contact point A are characterised

by 0 ¤ X ¤ 1, and those beyond A are such that X ¤ 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, 1341“57.

80 Asset and Risk Management

the ef¬cient frontier as in Markowitz™s or Sharpe™s models, this new technique simply

determines the portfolio that corresponds to the contact point of the tangent with the

ef¬cient frontier, produced by the point (0, RF ).

3.2.6.1 Hypotheses

The method now being examined assumes that:

• The mean“variance approach is relevant, which will allow a certain number of results

from Markowitz™s theory to be used.

• 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 speci¬c cases:

• Constant correlations. In this ¬rst model, it is assumed that the correlation coef¬cients

for the returns on the various securities are all equal: ρij = ρ ∀i, j .

• Sharpe™s simple index model can be used.

The ¬rst of these two simpli¬cations 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 ¬rst-degree functions of the

return on a market-representative index: Rj t = aj + bj RI t = µj 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 (µit , µj 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. 15“19.

Elton E., Gruber M. and Padberg M., Simple criteria for optimal portfolio selection; tracing out the ef¬cient frontier,

Journal of Finance, Vol. XIII No. 1, 1978, pp. 296“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 §3.3.1).

Equities 81

EP

A

P

RF

σP

Figure 3.19 EGP method

RF shall make up the straight line linking the points (0, RF ) and (σP , EP ) as illustrated

in Figure 3.19.

EP ’ RF

The slope of the straight line in question is given by P = , which may be

σP

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 ef¬cient 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:

«

±

N N N

E ’ R = Xj Ej ’ Xj RF =

P Xj (Ej ’ RF )

F

j =1 j =1 j =1

N N

2

σ = Xi Xj σij

P

i=1 j =1

the derivative of:

« 2

N

Xj (Ej ’ RF )

(EP ’ RF )2 j =1

= =

2

P

σP N N

2

Xi Xj σij

i=1 j =1

with respect to Xk is given by:

« « 2

N N N

2 Xj (Ej ’ RF ) (Ek ’ RF ) · ’ Xj (Ej ’ RF ) · 2

σP Xj σkj

2

j =1 j =1 j =1

( P )Xk =

2

σP

4

N

2(EP ’ RF )(Ek ’ RF )σP ’ 2(EP ’ RF ) Xj σkj

2 2

j =1

=

σP

4

82 Asset and Risk Management

«

N

2(EP ’ RF )

Xj σkj

= · (Ek ’ RF ) ’ γ ·

σP

2

j =1

In which we have provisionally γ = (Ep ’ RF )/σP .

2

This derivative will be zero if:

N

Ek ’ RF = γ · Xj σkj

j =1

By introducing Zj = γ · Xj (j = 1, . . ., N ), the system to be resolved with respect to

Z1 , . . ., ZN is therefore

N

Ek ’ RF = Zj σkj k = 1, . . . , N

j =1

Before proceeding with the resolution, note that ¬nding the Zk quantities allows the Xk

quantities to be found, as

Zk Zk Zk

Xk = = =

γ N N

γ· Xj Zj

j =1 j =1

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

σkj = cov(ak + bk RI + µk , aj + bj RI + µj )

σµ2k si j = k

= bk bj σI2 +

si j = k

0

The k th equation in the system can then be written:

«

N

Ek ’ RF = bk Zj bj σI2 + Zk σµ2k

j =1

or also, by resolving with respect to Zk :

±

«

N

1

Zk = 2 (Ek ’ RF ) ’ bk Zj bj σI

2

σµ k

j =1

±

«

bk

N

σI2

= 2 θk ’ Zj bj

σµ k

j =1

where we have:

Ek ’ RF

θk =

bk

Equities 83

All that now remains is to determine the sum between the brackets. On the basis of the

last result, we ¬nd:

±

«

N N N

bk2

θk ’ Zj bj σI

Zk bk = 2

σµ2k

k=1 k=1 j =1

«

N N N

bk bk

2 2

Zj bj σI2

= θk ’

σ σ

2 2

k=1 µk k=1 µk j =1

the resolution of which gives

N

bk2

θk

σµ2k

N

k=1

Zj bj = N

bk2

j =1

1+ σI2

σµ2k

k=1

By introducing the new notation

N

bk2

« θk

σµ2k

N

k=1

φ= Zj bj · σI2 = · σI2

N

bk2

j =1

1+ σI2

σµ2k

k=1

and by substituting the sum just calculated within the expression of Zk , we ¬nd

bk

Zk = (θk ’ φ) k = 1, . . . , N

σµ2k

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 (σµ1 = 0.0060)

2

R2 = ’0.020 + 1.08RI (σµ2 = 0.0040)

2

R3 = 0.200 + 1.32RI (σµ3 = 0.0012)

2

Assume that the variance of the index is equal to σI2 = 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:

θ1 = 0.0333 θ2 = 0.0463 θ3 = 0.0530.

84 Asset and Risk Management

Therefore, φ = 0.0457. The Zk values are deduced:

Z1 = ’1.2327 Z2 = 0.1717 Z3 = 8.1068

The proportions of the optimum portfolio are therefore deduced:

X1 = ’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 Markowitz™s model, and are written simply as 0 ¤ Xj ¤ 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 ’ RF

θk = k = 1, . . . , N

bk

The securities are then sorted in decreasing order of θk and this order is preserved until the

end of the algorithm. Instead of having just one parameter φ, one parameter is calculated

for each security:

k

bj

2

θj

σµ2j

j =1

«

φk = · σI2 k = 1, . . . , N

k

bj

2

σI2

1+

σ2

j =1 µj

It can be shown that the sequence of φk numbers ¬rst increases, then passes through a max-

imum and ¬nally ends with a decreasing phase. The value K of the k index corresponding

to the maximum φk , is noted. The φK number is named the ˜cut-off rate™ and it can be

shown that the calculation of the Zk values for the same relation as before (replacing φ by

φK ) produces positive values for k = 1, . . . , K and negative values for k = K + 1, . . . , N .

Only the ¬rst K securities are included in the portfolio. The calculations to be made are

therefore:

bk

Zk = 2 (θk ’ φK ) k = 1, . . . , K

σµ 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:

θ1 = 0.0333 θ2 = 0.0463 θ3 = 0.0530

Equities 85

This allows the securities to be classi¬ed in the order (3), (2), (1). We will provisionally

renumber the securities in this new order, thus producing:

φ1 = 0.04599 φ2 = 0.04604 φ3 = 0.04566

This shows that K = 2 and the cut-off rate is φ2 = 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 ef¬cient frontier has been determined, the question that faces the investor is

that of choosing from all the ef¬cient 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 ef¬cient 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 person™s viewpoint the utility and interest that he ¬nds 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 ˜St Petersburg paradox™. 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 ¬rst throw, four $ if tails comes up for the ¬rst time on the

second throw, eight $ if tails appears for the ¬rst time on the third throw, and so on. I

will therefore give you 2n $ if tails comes up for the ¬rst 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 ¬nancial) and the

concept of ˜utility™ 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

∞ n

1

n

= 1 + 1 + 1 + ... = ∞

2

2

n=1

It is the aversion to the risk that justi¬es 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 ∼ R2 ). The behaviour of the investor

can be expressed if these two relations obey the following axioms:

• (Comparability): The investor can always compare two returns. ∀R1 , R2 . We always

have R1 > R2 , R2 < R1 , or R1 ∼ R2 .

• (Re¬‚exivity): ∀R R ∼ R.

• (Transitivity): ∀R1 , R2 , R3 , if R1 > R2 and R2 > R3 , then R1 > R3 .

• (Continuity): ∀R1 , R2 , R3 , if R1 > R2 > R3 , there is a single X ∈ [0; 1] such as [X.R1 +

(1 ’ X).R3 ] ∼ R2 .

• (Independence): ∀R1 , R2 , R3 and ∀X ∈ [0; 1], if R1 > R2 , then [X.R1 + (1 ’ X).R3 ] >

[X.R2 + (1 ’ 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 ∀R1 , R2 , R1 > R2 ” 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 de¬ned more accurately: if an investor™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 — (R1 )] = E[aU (R1 ) + b]

= aE[U (R1 )] + b

> aE[U (R2 )] + b

= E[aU (R2 ) + b]

= E[U — (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 de¬ne a list of possible values R1 < R2 < . . . < Rn for the return, and then for

i = 2, . . . , n ’ 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 de¬nite return Ri or play in a lottery that gives returns of R1 and Rn with the respective

probabilities (1 ’ pi ) and pi . If one chooses arbitrarily U (R1 ) = 0 and U (Rn ) = 100,

then U (Ri ) = 100 pi (i = 2, . . . , n ’ 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 ’ U (R1 + R) ’ U (R1 ) >

U (R2 + R) ’ 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 de¬ne 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 ’ 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 ’ 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 “ r) + U(R + r)]

2

U(R “ r)

p

R′

R“r R+r

R x

Figure 3.21 Aversion to risk

This ¬gure shows R , the certain return, for which the utility is equal to the expected

return on the lottery. The difference p = R ’ R represents the price that the investor

is willing to pay to avoid having to participate in the lottery; this is known as the

risk premium.

Taylor expansions for U (R + r), U (R ’ r) and U(R ) = U(R ’ p) readily lead to the

relation:

U (R) r 2

p=’ ·

U (R) 2

The ¬rst factor in this expression is the absolute risk aversion coef¬cient:

U (R)

±(R) = ’

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 ±(R) = ’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 ’b/2a in order for the utility function to remain an

increasing function. The absolute risk aversion coef¬cient is then given by:

1

±(R) =

b

’ ’R

2a

When this last form can be accepted for the utility function, we have another justi¬cation

for de¬ning 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(σP + EP ) + bEP + c

2 2

This quantity then depends on the ¬rst 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 ef¬cient 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-¬tting convex form, the utility increasing as the curve moves upwards and

to the left.

By superimposing the indifference curves and the ef¬cient 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

σ

Figure 3.22 Indifference curves

E

EP

σP σ

Figure 3.23 Selection of optimal portfolio

90 Asset and Risk Management

3.2.7.4 Other viewpoints

Alongside the ef¬cient 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 de¬cit constraint. As well as optimising the

couple (E, σ ), that is, determining the ef¬cient frontier, and before selecting the portfolio

(through the utility function), the return on the portfolio here must not be less than a ¬xed

threshold38 u except with a very low probability p, say: Pr[RP ¤ u] ¤ p.

If the hypothesis of normality of return is accepted, we have:

RP ’ EP u ’ EP

¤ ¤p

Pr

σP σP

that is:

u ’ EP

¤ zp

σP

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 ≥ u ’ zp .σP .

The portfolios that obey the de¬cit constraint are located above the straight line for the

equation EP = u = zp .σP (see Figure 3.24).

The portion of the ef¬cient frontier delimited by this straight line of constraint is the

range of portfolios from which the investor will make his selection.

If p is ¬xed, an increase of u (higher required return) will cause the straight line

of constraint to move upwards. In the same way, if u is ¬xed, 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 ef¬cient

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 ef¬cient

frontier.

E

u

σ

Figure 3.24 De¬cit 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 Sharpe™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 ef¬cient portfolios.

3.2.8.1 Systematic risk and speci¬c risk

We have already encountered the concept of a systematic security risk in Section 3.1.1:

σj M

βj =

σM2

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 coef¬cient when the

return on this security is expressed as a linear function of the market return: Rj t =

±j + βj RMt + µj 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 βj , which is the same for every period, another source

of ¬‚uctuation in Rj is the residual µj t , which is speci¬c to the period t. The term speci¬c

risk is given to the variance in the residuals: σµ2j = var(µj t ).

Note

In practice, the coef¬cients ±j and βj for the regression are estimated using the least square

ˆ

method. For example: βj = sj M /sM . The residuals are then estimated by µj t = Rj t ’

ˆ

2

1T2

ˆ

(±j + βj RMt ) and the speci¬c risk is estimated using its ergodic estimator

ˆ t=1 µj t .

ˆ

T

In the rest of this paragraph, we will omit the index t relating to time.

We will see how the risk σj2 for a security consists of a systematic component and a

speci¬c component. We have:

σj2 = var(Rj )

= E[(±j + βj RM + µj ’ E(±j + βj RM + µj ))2 ]

= E[(βj (RM ’ EM ) + µj )2 ]

= βj E[(RM ’ EM )2 ] + E(µj ) + 2βj E[(RM ’ EM )µj ]

2 2

= βj var(RM ) + var(µj )

2

Hence the announced decomposition relation of: