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50 000

360
and the effectively observed price is 360, we arrive at: PER 0 = = 12.
30
This allows the rate of output13 to be determined as follows:

0.15 В· (1 в€’ 0.4) В· (360 в€’ 200)
1
k= +
12 360
= 0.0833 + 0.04
= 12.33 %

3.2 PORTFOLIO DIVERSIFICATION AND MANAGEMENT
3.2.1 Principles of diversiп¬Ѓcation
Putting together an optimum equity portfolio involves an answer to the following two
questions, given that a list of N equities is available on the market,

вЂў Which of these equities should I choose?
вЂў In what quantity (number or proportion)?

The aim is to look for the portfolio that provides the greatest return. This approach would
logically lead to holding a portfolio consisting of just one security, that with the greatest
expected return. Unfortunately, it misses out the risk aspect completely and can lead to a
catastrophe scenario if the price for the adopted security falls.
The correlations between the returns on the various available securities can, on the other
hand, help compensate for the п¬‚uctuations in the various portfolio components. This, in
sharp contrast to the approach described above, can help reduce the portfolio risk without
It is this phenomenon that we will be analysing here and use at a later stage to put
together an optimum portfolio.

3.2.1.1 The two-equity portfolio
According to what was stated above, the expected return and variance for a two-equity
portfolio represented in proportions14 X1 and X2 are given as follows:

EP = X1 E1 + X2 E2
ПѓP = X1 Пѓ1 + X2 Пѓ2 + 2X1 X2 Пѓ1 Пѓ2 ПЃ
2 22 22

Of course, if the rate had been equal to the intrinsic value V0 = 400, we arrive at k = 12 %.
13
14
It is implicitly supposed in this paragraph that the proportions are between 0 and 1, that is to say, there are no short sales.
52 Asset and Risk Management

In order to show clearly the effect of diversiп¬Ѓcation (the impact of correlation on risk),
let us п¬Ѓrst consider the case in which the two securities have the same expected return
(E1 = E2 = E) and the same risk (Пѓ1 = Пѓ2 = Пѓ ). Since X1 + X2 = 1, the equations will
become:
EP = E
ПѓP = (X1 + X2 + 2X1 X2 ПЃ)Пѓ 2
2 2 2

The expected return on the portfolio is equal to that on the securities, but the risk is lower
because the maximum value that it can take corresponds to ПЃ = 1 for which ПѓP = Пѓ and
when ПЃ < 1, ПѓP < Пѓ . Note that in the case of a perfect negative correlation (ПЃ = в€’1),
the risk can be written as ПѓP = (X1 в€’ X2 )2 Пѓ 2 .
2

This cancels itself out if one chooses X1 = X2 = 1/2; in this case, the expected return
is retained but the risk is completely cancelled.
Let us now envisage the more general case in which the expected return and the risk is
of whatever quantity. An equity is characterised by a couple (Ei , Пѓi ) for i = 1 or 2 and
can therefore be represented as a point in space (E, Пѓ ); of course the same applied for the
portfolio, which corresponds to the point (EP , ПѓP ). Depending on the values given to X1
(and therefore to X2 ), the representative point for the portfolio will describe a curve in
(E, Пѓ ) plane. Let us now study in brief the shape of the curve with respect to the values
for the correlation coefп¬Ѓcient ПЃ.
When ПЃ = 1, the portfolio variance15 becomes ПѓP = (X1 Пѓ1 + X2 Пѓ2 )2 .
2

By eliminating X1 and X2 from the three equations
пЈ±
пЈґ EP = X1 E1 + X2 E2
пЈІ
Пѓ = X1 Пѓ1 + X2 Пѓ2
пЈґP
пЈі
X1 + X2 = 1

we arrive at the relation
EP в€’ E2 E1 в€’ EP
ПѓP = Пѓ1 + Пѓ2
E1 в€’ E2 E1 в€’ E2
This expresses ПѓP as a function of EP , a п¬Ѓrst-degree function, and the full range of
portfolios is therefore the sector of the straight line that links the representative points for
the two securities (see Figure 3.2).

E E

вЂў
(1) вЂў
(2)

вЂў вЂў
(1) (2)

Пѓ Пѓ

Figure 3.2 Two-equity portfolio (ПЃ = 1 case)
15
Strictly speaking, one should say вЂ˜the portfolio return varianceвЂ™.
Equities 53

Faced with the situation shown on the left, the investor will choose a portfolio located
on the sector according to his attitude to the matter of risk: portfolio (1) will give a low
expected return but present little risk, while portfolio (2) is the precise opposite. Faced
with a situation shown on the right-hand graph, there is no room for doubting that portfolio
(2) is better than portfolio (1) in terms of both expected return and risk incurred.
When ПЃ = в€’1, the variance in the portfolio will be: ПѓP = (X1 Пѓ1 в€’ X2 Пѓ2 )2 .
2

In other words, ПѓP = |X1 Пѓ1 в€’ X2 Пѓ2 |. Applying the same reasoning as above leads to
the following conclusion: the portfolios that can be constructed make up two sectors of a
straight line from points (1) and (2), meet together at a point on the vertical axis (Пѓ = 0),
and have equal slopes, excepted the sign (see Figure 3.3).
Of these portfolios, of course, only those located in the upper sector will be of interest;
those in the lower sector will be less attractive from the point of view of both risk and
expected return.
In the general case, в€’1 < ПЃ < 1, and it can be shown that all the portfolios that can be
put together form a curved arc that links points (1) and (2) located between the extreme
case graphs for ПЃ = В±1, as shown in Figure 3.4.
If one expresses ПѓP as a function of EP , as was done in the ПЃ = 1 case, a second-
2

degree function is obtained. The curve obtained in the (E, Пѓ ) plane will therefore be a
hyperbolic branch.
The term efп¬Ѓcient portfolio is applied to a portfolio that is included among those that
can be put together with two equities and cannot be improved from the double viewpoint
of risk and expected return.
Graphically, we are looking at portfolios located above contact point A16 of the vertical
tangent to the portfolio curve. In fact, between A and (2), it is not possible to improve

E
вЂў (2)

вЂў (1)

Пѓ

Figure 3.3 Two-equity portfolio (ПЃ = в€’1 case)

E

вЂў (2)

A

вЂў (1)

Пѓ

Figure 3.4 Two-equity portfolio (general case)
16
This contact point corresponds to the minimum risk portfolio.
54 Asset and Risk Management

EP without increasing the risk or to decrease ПѓP without reducing the expected return.
In addition, any portfolio located on the arc that links A and (1) will be less good than
the portfolios located to its left.

3.2.1.2 Portfolio with more than two equities
A portfolio consisting of three equities17 can be considered as a mixture of one of the
securities and a portfolio consisting of the two others. For example, a portfolio with
the composition X1 = 0.5, X2 = 0.2 and X3 = 0.3 can also be considered to consist of
security (1) and a portfolio that itself consists of securities (2) and (3) at rates of 40 %
and 60 % respectively. Therefore, for the п¬Ѓxed covariances Пѓ12 , Пѓ13 and Пѓ23 , the full range
of portfolios that can be constructed using this process corresponds to a continuous range
of curves as shown in Figure 3.5.
All the portfolios that can be put together using three or more securities therefore form
an area within the plane (E, Пѓ ).
The concept of вЂ˜efп¬Ѓcient portfolioвЂ™ is deп¬Ѓned in the same way as for two secu-
rities. The full range of efп¬Ѓcient portfolios is therefore the part of the boundary of
this area limited by security (1) and the contact point of the vertical tangent to the
area, corresponding to the minimum risk portfolio. This arc curve is known as the efп¬Ѓ-
cient frontier.
The last part of this Section 3.2 is given over to the various techniques used to determine
the efп¬Ѓcient frontier, according to various restrictions and hypotheses.
An investorвЂ™s choice of a portfolio on the efп¬Ѓcient frontier will be made according to
his attitude to risk. If he adopts the most cautious approach, he will choose the portfolio
located at the extreme left point of the efп¬Ѓcient frontier (the least risky portfolio, very
diversiп¬Ѓed), while a taste for risk will move him towards the portfolios located on the
right part of the efп¬Ѓcient frontier (acceptance of increased risk with hope of higher return,
generally obtained in portfolios made up of a very few proп¬Ѓtable but highly volatile
securities).18

E
(1)

(2)

(3)

Пѓ

Figure 3.5 Three-equity portfolio

The passage from two to three shares is a general one: the results obtained are valid for N securities. The attached
17

CD-ROM shows some more realistic examples of the various models in the Excel sheets contained in the вЂ˜Ch 3вЂ™ directory.
18
This question is examined further in Section 3.2.6.
Equities 55

3.2.2 Diversiп¬Ѓcation and portfolio size
We have just seen that diversiп¬Ѓcation has the effect of reducing the risk posed by a
portfolio through the presence of various securities that are not perfectly correlated. Let
us now examine the limits of this diversiп¬Ѓcation; up to what point, for a given correlation
structure, can diversiп¬Ѓcation reduce the risk?

3.2.2.1 Mathematical formulation
To simplify the analysis, let us consider a portfolio of N securities in equal proportions:

1
Xj = j = 1, . . . , N
N
The portfolio risk can therefore be developed as:
N N
ПѓP = Xi Xj Пѓij
2

i=1 j =1
пЈ± пЈј
пЈґ пЈґ
1пЈІ пЈЅ
N N N
=2 Пѓi2 + Пѓij
NпЈґ пЈґ
пЈі пЈѕ
i=1 i=1 j =1
j =i

This double sum contains N (N в€’ 1) terms, and it is therefore natural to deп¬Ѓne the average
variance and the average covariance as:
N
1
var = Пѓi2
N i=1
N N
1
cov = Пѓij
N (N в€’ 1) i=1 j =1
j =i

As soon as N reaches a sufп¬Ѓcient magnitude, these two quantities will almost cease to
depend on N . They will then allow the portfolio variance to be written as follows:

N в€’1
1
ПѓP = var +
2
cov
N N

3.2.2.2 Asymptotic behaviour
When N becomes very large, the п¬Ѓrst term will decrease back towards 0 while the second,
now quite stable, converges towards cov. The portfolio risk, despite being very diversiп¬Ѓed,
never falls below this last value, which corresponds to:

N в€’1
1
cov = lim var + cov = lim ПѓP = ПѓM
2 2
N N
Nв€’в†’в€ћ Nв€’в†’в€ћ

In other words, it corresponds to the market risk.
56 Asset and Risk Management

sP
2

cov

N

Figure 3.6 Diversiп¬Ѓcation and portfolio size

The behaviour of the portfolio variance can be represented according to the number of
securities by the graph shown in Figure 3.6.
The effects of diversiп¬Ѓcation are initially very rapid (the п¬Ѓrst term loses 80 % of its
value if the number of securities increases from 1 to 5) but stabilise quickly somewhere
near the cov value.

3.2.3 Markowitz model and critical line algorithm
3.2.3.1 First formulation
The efп¬Ѓcient frontier is the вЂ˜North-WestвЂ™ part of the curve, consisting of portfolios deп¬Ѓned
by this principle: for each п¬Ѓxed value r of EP , the proportions for which ПѓP is minimal
2

Xj (j = 1, . . . , N ) are determined. The efп¬Ѓcient frontier is this deп¬Ѓned by giving r all
the possible values.
Mathematically, the problem is therefore presented as a search for the minimum with
respect to X1 , . . . , XN of the function:
N N
ПѓP = Xi Xj Пѓij
2

i=1 j =1

пЈ±
under the double restriction:
N
пЈґ
пЈґ
пЈґ Xj Ej = r
пЈґ
пЈґ
пЈІ
j =1
пЈґ N
пЈґ
пЈґ
пЈґ Xj = 1
пЈґ
пЈі
j =1

The Lagrangian function19 for the problem can thus be written as:
пЈ« пЈ¶ пЈ« пЈ¶
N N N N
Xi Xj Пѓij + m1 В· пЈ­ X j E j в€’ r пЈё + m2 В· пЈ­ Xj в€’ 1пЈё
L(X1 , . . . , XN ; m1 , m2 ) =
i=1 j =1 j =1 j =1

19
Please refer to Appendix 1 for the theory of extrema.
Equities 57

Taking partial derivatives with respect to the variables X1 , . . . , Xn and to the Lagrange
multipliers m1 and m2 leads to the system of N + 2 equations with N + 2 unknowns:
пЈ±
N
пЈґ
пЈґ
пЈґ L (X1 , . . . , XN ; m1 , m2 ) = 2 Xi Пѓij + m1 Ej + m2 = 0 (j = 1, . . . , N )
пЈґ Xj
пЈґ
пЈґ
пЈґ
пЈґ i=1
пЈґ
пЈґ
пЈІ N
L (X , . . . , XN ; m1 , m2 ) = Xi Ei в€’ r = 0
пЈґ m1 1
пЈґ
пЈґ i=1
пЈґ
пЈґ
пЈґ N
пЈґ
пЈґ
пЈґ Lm (X1 , . . . , XN ; m1 , m2 ) = Xi в€’ 1 = 0
пЈґ2
пЈі
i=1

This can be written in a matrix form:
пЈ« пЈ¶пЈ« пЈ¶ пЈ«пЈ¶
В· В· В· 2Пѓ1N E1
2
X1 .
2Пѓ1 2Пѓ12 1
пЈ¬ 2Пѓ21 1 пЈ· пЈ¬ X2 пЈ· пЈ¬ . пЈ·
В· В· В· 2Пѓ2N E2
2
2Пѓ2
пЈ¬ пЈ·пЈ¬ пЈ· пЈ¬пЈ·
пЈ¬. . пЈ·пЈ¬ . пЈ· пЈ¬ . пЈ·
. . .
.. . пЈ·пЈ¬ . пЈ· пЈ¬ . пЈ·
пЈ¬. . . .
. . пЈ·пЈ¬ . пЈ· = пЈ¬ . пЈ·
пЈ¬. . . .
пЈ¬ 2Пѓ 1 пЈ· пЈ¬ XN пЈ· пЈ¬ . пЈ·
В· В· В· 2ПѓN EN
2
пЈ¬ N1 пЈ·пЈ¬ пЈ· пЈ¬пЈ·
2ПѓN2
пЈ­ E1 пЈё пЈ­ m1 пЈё пЈ­ r пЈё
E2 В· В· В· EN . .
m2
В·В·В· . . 1
1 1 1

By referring to the matrix of coefп¬Ѓcients,20 the vector of unknowns21 and the vector of
second members as M, Xв€— and G respectively, we give the system the form MXв€— = G.
The resolution of this system passes through the inverse matrix of M:Xв€— = M в€’1 G.
Note 1
In reality, this vector only supplies one stationary point of the Lagrangian function; it
can be shown (although we will not do this here) that it constitutes the solution to the
problem of minimisation that is concerning us.
Note 2
This relation must be applied to the different possible values for r to п¬Ѓnd the frontier,
of which only the efп¬Ѓcient (вЂ˜North-WestвЂ™) part will be retained. The interesting aspect of
this result is that if r is actually inside the vector G, it does not appear in the matrix M,
which then has to be inverted only once.22

Example
We now determine the efп¬Ѓcient frontier that can be constructed with three securities with
the following characteristics:
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

In its order N zone of the upper left corner, this contains the 2V matrix in which V is the varianceвЂ“covariance matrix.
20
21
The vector of unknowns does not contain the proportions only; it also involves the Lagrange multipliers (which will
not be of use to us later). For this reason we will use the notation Xв€— instead of X (which is reserved for the vector of
proportions). This remark applies to all the various models developed subsequently.
22
The attached CD-ROM contains a series of more realistic examples of the various models in an Excelп›› п¬Ѓle known as
Ch 3.
58 Asset and Risk Management

The varianceвЂ“covariance matrix is given by:
пЈ« пЈ¶
0.0100 0.0036 0.0015
V = пЈ­ 0.0036 0.0144 0.0072 пЈё
0.0015 0.0072 0.0225
The matrix M is therefore equal to:
пЈ« пЈ¶
0.0200 0.0072 0.0030 0.05 1
пЈ¬ 0.0072 0.0288 0.0144 0.08 1 пЈ·
пЈ¬ пЈ·
M = пЈ¬ 0.0030 0.0144 0.0450 0.10 1 пЈ·
пЈ­ 0.05 .пЈё
.
0.08 0.010
. .
1 1 1
This matrix inverts to:
пЈ« пЈ¶
31.16 в€’24.10 в€’ 7.06 0.57
пЈ¬ в€’24.10 0.24 пЈ·
40.86 в€’16.76
M в€’1 = пЈ¬ пЈ·
пЈ­ в€’ 7.06 в€’16.76 0.19 пЈё
23.82
0.19 в€’0.01
0.57 0.24
пЈ«пЈ¶
.
.
пЈ¬.пЈ·
By applying this matrix to the vector G = пЈ­ r пЈё, for different values of r, we п¬Ѓnd a
1
range of vectors Xв€— , the п¬Ѓrst three components of which supply the composition of the
portfolios (see Table 3.2).
These proportions allow ПѓP to be calculated23 for the various portfolios (Table 3.3).
It is therefore possible, from this information, to construct the representative curve for
these portfolios (Figure 3.7).

Table 3.2 Composition of portfolios

r X1 X2 X3

в€’0.3233 в€’0.8060
0.00 2.1293
в€’0.2391 в€’0.6565
0.01 1.8956
в€’0.1549 в€’0.5071
0.02 1.6620
в€’0.0707 в€’0.3576
0.03 1.4283
в€’0.2801
0.04 1.1946 0.0135
в€’0.0586
0.05 0.9609 0.0933
0.06 0.7272 0.1820 0.0908
0.07 0.4935 0.2662 0.2403
0.08 0.2598 0.3504 0.3898
0.09 0.0262 0.4346 0.5392
в€’0.2075
0.10 0.5188 0.6887
в€’0.4412
0.11 0.6030 0.8382
в€’0.6749
0.12 0.6872 0.9877
в€’0.9086
0.13 0.7714 1.1371
в€’1.1423
0.14 0.8556 1.2866
в€’1.3759
0.15 0.9398 1.4361

23
The expected return is of course known.
Equities 59
Calculation of ПѓP
Table 3.3

EP ПѓP

0.00 0.2348
0.01 0.2043
0.02 0.1746
0.03 0.1465
0.04 0.1207
0.05 0.0994
0.06 0.0857
0.07 0.0835
0.08 0.0937
0.09 0.1130
0.10 0.1376
0.11 0.1651
0.12 0.1943
0.13 0.2245
0.14 0.2554
0.15 0.2868

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
Standard deviation

Figure 3.7 Efп¬Ѓcient frontier

The efп¬Ѓcient part of this frontier is therefore the вЂ˜North-WestвЂ™ part, the lower limit of
which corresponds to the minimum risk portfolio. For this portfolio, we have values of
EP = 0.0667 and ПѓP = 0.0828.

The method just presented does not require the proportions to be positive. Moreover,
a look at the preceding diagram will show that negative values (and values over 1) are
sometimes obtained, as the вЂ˜classicвЂ™ portfolios (0 в‰¤ Xj в‰¤ 1 for any j ) correspond only
to expected return values between 0.06 and 0.09.
A negative value for a proportion corresponds to a short sale. This type of transaction,
which is very hazardous, is not always authorised, especially in the management of invest-
ment funds. Symmetrically, a proportion of over 1 indicates the purchase of a security
for an amount greater than the total invested.
In addition, many portfolios contain regulatory or internal restrictions stating that certain
types of security cannot be represented for a total over a п¬Ѓxed percentage. In this case,
the problem must be resolved by putting together portfolios in which proportions of the
60 Asset and Risk Management
в€’ +
type Bj в‰¤ Xj в‰¤ Bj for j = 1, . . . , N are subject to regulations. We will examine this
problem at a later stage.

3.2.3.2 Reformulating the problem
We now continue to examine the problem without any regulations on inequality of pro-
portions. We have simply altered the approach slightly; it will supply the same solution
but can be generalised more easily into the various models subsequently envisaged.
If instead of representing the portfolios graphically by showing ПѓP as the x-axis and
Ep as the y-axis (as in Figure 3.7), EP is now shown as the x-axis and ПѓP as the y-axis,
2

the efп¬Ѓcient frontier graph now appears as shown in Figure 3.8.
A straight line in this graph has the equation Пѓ 2 = a + О»E in which a represents the
intercept and О» the slope of the straight line. We are looking speciп¬Ѓcally at a straight
line at a tangent to the efп¬Ѓcient frontier. If the slope of this straight line is zero (О» = 0),
the contact point of the tangent shows the least risky portfolio in the efп¬Ѓcient frontier.
Conversely, the more О» increases, the further the contact point moves away from the
efп¬Ѓcient frontier towards the risky portfolios. The О» parameter may vary from 0 to +в€ћ
and is therefore representative of the portfolio risk corresponding to the contact point of
the tangent with this О» value for a slope.
For a п¬Ѓxed О» value, the tangent to the efп¬Ѓcient frontier with slope О» is, of all the straight
lines with that slope and with at least one point in common with the efп¬Ѓcient frontier,
that which is located farthest to the right, that is, the one with the smallest coordinate at
the origin a = Пѓ 2 в€’ О»E.
The problem is therefore reformulated as follows: for the various values of О» between
0 and в€ћ, minimise with respect to the proportions X1 , . . . , XN the expression:

N N N
ПѓP в€’ О»EP = Xi Xj Пѓij в€’ О» Xj Ej
2

i=1 j =1 j =1

under the restriction N=1 Xj = 1.
j
Once the solution, which will depend on О», has been found, it will be sufп¬Ѓcient to make
this last parameter vary between 0 and +в€ћ to arrive at the efп¬Ѓcient frontier.

sP
2

EP

a

Figure 3.8 Reformulation of problem
Equities 61

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

A reasoning similar to that used in the п¬Ѓrst formulation allows the following matrix
expression to be deduced from the partial derivatives:

MXв€— = О»E в€— + F

Here, it has been noted that24
пЈ« пЈ¶ пЈ«
пЈ¶ пЈ«
пЈ¶ пЈ«пЈ¶
2Пѓ12 В· В· В· 2Пѓ1N
2
X1 E1 .
2Пѓ1 1
пЈ¬ 2Пѓ21 2Пѓ 2 В· В· В· 2Пѓ2N 1пЈ· пЈ¬ X2 пЈ· пЈ¬ E2 пЈ· пЈ¬.пЈ·
пЈ¬ пЈ· пЈ¬ пЈ· пЈ¬ пЈ· пЈ¬пЈ·
2
пЈ¬. .пЈ· пЈ¬.пЈ· пЈ¬.пЈ· пЈ¬пЈ·
F =пЈ¬.пЈ·
. .
.. в€— в€—
X =пЈ¬ . пЈ· E =пЈ¬ . пЈ· .
M=пЈ¬ . . . .пЈ·
. пЈ¬.пЈ· пЈ¬.пЈ· пЈ¬.пЈ·
. . . .пЈ·
пЈ¬
пЈ­ 2Пѓ 1пЈё пЈ­ XN пЈё пЈ­ EN пЈё пЈ­.пЈё
В· В· В· 2ПѓN 2
N1 2ПѓN2
m .
В·В·В· . 1
1 1 1

The solution to this system of equations is therefore supplied by: Xв€— = О»(M в€’1 E в€— ) +
(M в€’1 F ).
As for the п¬Ѓrst formulation, the matrix M is independent of the parameter О», which
must be variable; it only needs to be inverted once.

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

The same varianceвЂ“covariance matrix V as above will be used, and the matrix M can be
expressed as: пЈ« пЈ¶
0.0200 0.0072 0.0030 1
пЈ¬ 0.0072 0.0144 1 пЈ·
0.0288
M=пЈ¬ пЈ·
пЈ­ 0.0030 0.0450 1 пЈё
0.0144
.
1 1 1

This matrix inverts to:
пЈ« пЈ¶
31.16 в€’24.10 в€’ 7.06 0.57
пЈ¬ в€’24.10 0.24 пЈ·
40.86 в€’16.76
=пЈ¬ пЈ·
M в€’1 пЈ­ в€’ 7.06 в€’16.76 0.19 пЈё
23.82
0.19 в€’0.01
0.57 0.24

In the same way as the function carried out for Xв€— , we are using the E в€— notation here as E is reserved for the
24

N-dimensional vector for the expected returns.
62 Asset and Risk Management
Solutions for different values of О»
Table 3.4

О» X1 X2 X3 EP ПѓP

в€’1.5810
2.0 1.0137 1.5672 0.0588 0.3146
в€’1.4734
1.9 0.9750 1.4984 0.1542 0.3000
в€’1.3657
1.8 0.9362 1.4296 0.1496 0.2854
в€’1.2581
1.7 0.8974 1.3607 0.1450 0.2709
в€’1.1505
1.6 0.8586 1.2919 0.1404 0.2565
в€’1.0429
1.5 0.8198 1.2231 0.1357 0.2422
в€’0.9353
1.4 0.7810 1.1542 0.1311 0.2280
в€’0.8276
1.3 0.7423 1.0854 0.1265 0.2139
в€’0.7200
1.2 0.7035 1.0165 0.1219 0.2000
в€’0.6124
1.1 0.6647 0.9477 0.1173 0.1863
в€’0.5048
1.0 0.6259 0.8789 0.1127 0.1729
в€’0.3972
0.9 0.5871 0.8100 0.1081 0.1597
в€’0.2895
0.8 0.5484 0.7412 0.1035 0.1470
в€’0.1819
0.7 0.5096 0.6723 0.0989 0.1347
в€’0.0743
0.6 0.4708 0.6035 0.0943 0.1231
0.5 0.0333 0.4320 0.5437 0.0897 0.1123
0.4 0.1409 0.3932 0.4658 0.0851 0.1027
0.3 0.2486 0.3544 0.3970 0.0805 0.0945
0.2 0.3562 0.3157 0.3282 0.0759 0.0882
0.1 0.4638 0.2769 0.2593 0.0713 0.0842
0.0 0.5714 0.2381 0.1905 0.0667 0.0828

пЈ«
пЈ¶ пЈ«пЈ¶
В·
0.05
пЈ¬ 0.08 пЈ· пЈ¬В·пЈ·
As the vectors E в€— and F are given by E в€— = пЈ¬ пЈ· F = пЈ¬ пЈ·, the solutions
пЈ­ 0.10 пЈё пЈ­В·пЈё
. 1
to the problem for the different values of О» are shown in Table 3.4.
The efп¬Ѓcient frontier graph then takes the form shown in Figure 3.9.
The advantage of this new formulation is twofold. On one hand, it only shows the
truly efп¬Ѓcient portfolios instead of the boundary for the range of portfolios that can be
put together, from which the upper part has to be selected. On the other hand, it readily

0.18
0.16
0.14
Expected return

0.12
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
Standard deviation

Figure 3.9 Efп¬Ѓcient frontier for the reformulated problem
Equities 63

lends itself to generalisation in the event of problems with inequality restrictions, as well
as to the simple index models with non-risk titles.

3.2.3.3 Constrained Markowitz model

The problem to be solved is formulated here as follows: for the different values of О»
between 0 and +в€ћ, minimise with respect to the proportions X1 , . . ., XN the expression

N N N
ПѓP в€’ О»EP = Xi Xj Пѓij в€’ О» Xj Ej
2

i=1 j =1 j =1

with the restrictions:
N
j =1 Xj =1
в€’ +
Bj в‰¤ Xj в‰¤ Bj j = 1, . . . , N

We will п¬Ѓrst of all introduce the concept of a securityвЂ™s вЂ˜statusвЂ™. The security (j ) is
deп¬Ѓned as вЂ˜downвЂ™ (resp. вЂ˜upвЂ™) if its proportion is equal to the вЂ˜lowerвЂ™ (resp. вЂ˜upperвЂ™)
в€’ +
bound imposed on it: Xj = Bj (resp. Xj = Bj ). For an efп¬Ѓcient portfolio (that is, one
that minimises the Lagrangian function), the partial derivative of the Lagrangian function
with respect to Xj is not zero in an optimum situation; it is strictly positive (resp. strictly
negative) as can be seen in Figure 3.10.
In the system of equations produced by the partial derivatives of the Lagrangian func-
tion, the equations relating to the вЂ˜downвЂ™ (resp. вЂ˜upвЂ™) securities should therefore be
в€’ +
replaced here by Xj = Bj (resp. Xj = Bj ).
в€’ +
The other securities are deп¬Ѓned as вЂ˜inвЂ™, and are such that Bj < Xj < Bj , and in an
optimum situation, the partial derivative of the Lagrangian function with respect to Xj is
zero. The equations relating to these securities should not be altered.
The adaptation to the system of equations produced by the partial derivatives of
the Lagrangian function MXв€— = О»E в€— + F , will therefore consist of not altering the
components that correspond to the вЂ˜inвЂ™ securities, and if the security (j ) is вЂ˜downвЂ™ or

L
L

вЂ“ + вЂ“ +
Xj Xj
Bj Bj Bj Bj

Figure 3.10 вЂ˜UpвЂ™ security and вЂ˜downвЂ™ security
64 Asset and Risk Management

j th component of E в€— and F , as follows:
вЂ˜upвЂ™, of altering the j th line of M and the
пЈ« пЈ¶ пЈ« пЈ¶ пЈ« пЈ¶
2Пѓ1 В· В· В· 2Пѓ1j В· В· В· 2Пѓ1N E1
2 0
1
пЈ¬. .пЈ· пЈ¬.пЈ· пЈ¬.пЈ·
. .
.. пЈ¬.пЈ·
пЈ¬. . . .пЈ· пЈ¬.пЈ·
. пЈ¬ .В± пЈ·
. . . .пЈ· .пЈ·
пЈ¬ пЈ¬
пЈ¬0 0пЈ· пЈ¬0пЈ· пЈ¬B пЈ·
В·В·В· В·В·В·
1 0
M=пЈ¬ . пЈ· Eв€— = пЈ¬ пЈ· F =пЈ¬ j пЈ·
пЈ¬ .пЈ· пЈ¬.пЈ· пЈ¬.пЈ·
. .
пЈ¬. . . .пЈ· пЈ¬.пЈ· пЈ¬.пЈ·
. . В·В·В· . .пЈ· .пЈ· пЈ¬.пЈ·
пЈ¬ пЈ¬
пЈ­ 2Пѓ пЈё пЈ­ EN пЈё пЈ­0пЈё
В· В· В· 2ПѓNj В· В· В· 2ПѓN 2
1
N1
В·В·В· В·В·В· 0
1 1 1 0 1
В±
With this alteration, in fact, the j th equation becomes Xj = Bj . In addition, when con-
sidering the j th line of the equality M в€’1 M = I , it is evident that M в€’1 has the same jth
line as M and the j th component of the solution Xв€— = О»(M в€’1 E в€— ) + (M в€’1 F ). This is also
В±
written as Xj = Bj .
If (j ) has an вЂ˜inвЂ™ status, this j th component can of course be written as Xj = О»uj + vj ,
в€’ +
a quantity that is strictly included between Bj and Bj .
в€—
The method proceeds through a series of stages and we will note M0 , E0 and F0 , the
matrix elements as deп¬Ѓned in the вЂ˜unconstrainedвЂ™ case. The index develops from one stage
to the next.
The method begins with the major values for О» (+в€ћ ideally). As we are looking to
minimise ПѓP в€’ О»EP , EP needs to be as high as possible, and this is consistent with
2

a major value for the risk parameter О». The п¬Ѓrst portfolio will therefore consist of the
securities that offer the highest expected returns, in equal proportions to the upper bounds
+ в€’
Bj , until (with securities in proportions equal to Bj ) the sum of the proportions equals
1.25 This portfolio is known as the п¬Ѓrst corner portfolio.
At least one security will therefore be вЂ˜upвЂ™; one will be вЂ˜inвЂ™, and the others will be
вЂ˜downвЂ™. The matrix M and the vectors E в€— and F are altered as shown above. This brings
us to M1 , E1 and F1 , and we calculate: Xв€— = О» (M1 в€’1 E1 ) + (M1 в€’1 F1 ).
в€— в€—

The parameter О» is thus decreased until one of the securities changes its status.26 This
(1)
п¬Ѓrst change will occur for a value of О» equal to О»c , known as the п¬Ѓrst critical О». To
determine this critical value, and the security that will change its status, each of the various
securities for which a potentially critical О»j is deп¬Ѓned will be examined.
A вЂ˜downвЂ™ or вЂ˜upвЂ™ security (j ) will change its status if the equation corresponding to it
becomes LXj = 0, that is:

N
Xk Пѓj k в€’ О»j Ej + m = 0
2
k=1

в€—
This is none other than the j th component of the equation M0 Xв€— = О»E0 + F0 , in which the
different Xk and m are given by the values obtained by Xв€— = О» (M1 в€’1 E1 ) + (M1 в€’1 F1 ).
в€—

If the inequality restrictions are simply 0 в‰¤ Xj в‰¤ 1 в€Ђj (absence of short sales), the п¬Ѓrst portfolio will consist only of
25

the security with the highest expected return.
For the restrictions 0 в‰¤ Xj в‰¤ 1 в€Ђj , the п¬Ѓrst corner portfolio consists of a single вЂ˜upвЂ™ security, all the others being
26

вЂ˜downвЂ™. The п¬Ѓrst change of status will be a transition to вЂ˜inвЂ™ of the security that was вЂ˜upвЂ™ and of one of the securities that
в€—
were вЂ˜downвЂ™. In this case, on one hand the matrix elements M1 , E1 and F1 are obtained by making the alteration required for
the вЂ˜downвЂ™ securities but for the one that it is known will pass to вЂ˜inвЂ™ status, and on the other hand there is no equation for
determining the potential critical О» for this security.
Equities 65

For an вЂ˜inвЂ™ security (j ), it is known that Xj = О»j uj + vj and it will change its status
if it becomes a вЂ˜downвЂ™ (uj > 0 as О» decreases) or вЂ˜upвЂ™ security (uj < 0), in which case
В±
we have Bj = О»j uj + vj . This is none other than the j th component of the relation
в€’1 в€— в€’1
Xв€— = О»(M1 E1 ) + (M1 F1 ), in which the left member is replaced by the lower or upper
bound depending on the case.
We therefore obtain N equations for N values of potentially critical О»j . The highest
(1)
of these is the п¬Ѓrst critical О»j or О»c . The proportions of the various securities have not
(1)
changed between О» = +в€ћ and О» = О»c . The corresponding portfolio is therefore always
the п¬Ѓrst corner portfolio.
The security corresponding to this critical О» therefore changes its status, thus allowing
в€— (2)
M2 , E2 and F2 to be constructed and the second critical О», О»c , to be determined together
(1) (2)
with all the portfolios that correspond to the values of О» between О»c and О»c . The
(2)
portfolio corresponding to О»c is of course the second corner portfolio.
The process is then repeated until all the potentially critical О» values are negative, in
which case the last critical О» is equal to 0. The last and least risky corner portfolio, located
at the extreme left point of the efп¬Ѓcient frontier, corresponds to this value.
The corner portfolios are of course situated on the efп¬Ѓcient frontier. Between two
consecutive corner portfolios, the status of the securities does not change; only the pro-
(kв€’1)
and О»(k) , using the
portions change. These proportions are calculated, between О»c c
в€’1 в€— в€’1
relation Xв€— = О»(Mk Ek ) + (Mk Fk ).
The various sections of curve thus constructed are connected continuously and with
same derivative27 and make up the efп¬Ѓcient frontier.

Example
Let us take the same data as were processed before:

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 impose the requirement of absence of short sales: 0 в‰¤ Xj в‰¤ 1 (j = 1, 2, 3).
We have the following basic matrix elements:
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¬ 0.08 пЈ· пЈ¬.пЈ·
M0 = пЈ¬ пЈ· пЈ¬ пЈ· пЈ¬пЈ·
в€—
пЈ­ 0.0030 0.0144 0.0450 1 пЈё E0 = пЈ­ 0.10 пЈё F0 = пЈ­ . пЈё
. .
1 1 1 1

The п¬Ѓrst corner portfolio consists only of security (3), the one with the highest expected
return. As securities (1) and (2) are вЂ˜downвЂ™, we construct:
пЈ« пЈ¶ пЈ« пЈ¶ пЈ«пЈ¶
. . . . .
1
пЈ¬. пЈ· пЈ¬.пЈ· пЈ¬.пЈ·
. .пЈ·
1
M1 = пЈ¬ пЈ¬ пЈ· пЈ¬пЈ·
в€—
пЈ­ 0.0030 0.0144 0.0450 1 пЈё E1 = пЈ­ 0.10 пЈё F1 = пЈ­ . пЈё
. .
1 1 1 1

27
That is, with the same tangent.
66 Asset and Risk Management

пЈ« пЈ¶
We have:
. . .
1
пЈ¬. пЈ·
. .
1
=пЈ¬ пЈ·
в€’1
M1 пЈ­ в€’1 пЈё
в€’1 . 1
1 в€’0.0450
0.0420 0.0306

and therefore
пЈ« пЈ¶пЈ« пЈ¶
. .
пЈ¬.пЈ· пЈ¬ пЈ·
.
Xв€— = О»(M1 E1 ) + (M1 F1 ) = О» пЈ¬ пЈ·+пЈ¬ пЈ·
в€’1 в€— в€’1
пЈ­.пЈё пЈ­ 1 пЈё
в€’0.045
0.1
в€—
The п¬Ѓrst two components of M0 Xв€— = О»E0 and F0 , with the vector Xв€— obtained
above, give:

0.003 + (0.1 О»1 в€’ 0.045) = 0.05 О» О»1
0.0144 + (0.1 О»2 в€’ 0.045) = 0.08 О»2

This will give the two potential critical О» values: О»1 = 0.84 and О»2 = 1.53. The п¬Ѓrst
(1)
critical О» is therefore О»c = 1.53 and security (2) becomes вЂ˜inвЂ™ together with (3), while
(1) remains вЂ˜downвЂ™.
We can therefore construct:
пЈ« пЈ¶ пЈ« пЈ¶ пЈ«пЈ¶
. . . . .
1
пЈ¬ 0.0072 0.0288 0.0144 1 пЈ· пЈ¬ 0.08 пЈ· пЈ¬.пЈ·
M2 = пЈ¬ пЈ· E2 = пЈ¬ пЈ· F2 = пЈ¬ пЈ·
в€—
пЈ­ 0.0030 0.0144 0.0450 1 пЈё пЈ­ 0.10 пЈё пЈ­.пЈё
. .
1 1 1 1

This successively gives:
пЈ« пЈ¶
. . .
1
пЈ¬ в€’0.7733 0.68 пЈ·
22.22 в€’22.22
M2 = пЈ¬ пЈ·
в€’1
пЈ­ в€’0.2267 в€’22.22 0.32 пЈё
22.22
0.32 в€’0.0242
0.0183 0.68
пЈ¶
пЈ« пЈ¶пЈ«
. .
пЈ¬ в€’0.4444 пЈ· пЈ¬ 0.68 пЈ·
пЈ·пЈ¬ пЈ·
Xв€— = О»(M2 E2 ) + (M2 F2 ) = О» пЈ¬
в€’1 в€— в€’1
пЈ­ 0.4444 пЈё + пЈ­ 0.32 пЈё
в€’0.0242
0.0864

The п¬Ѓrst component of M0 Xв€— = О»E0 + F0 , with vector Xв€— obtained above, gives:
в€—

0.0072 В· (в€’0.4444О»1 + 0.68) + 0.0030 В· (0.4444О»1 + 0.32) + (0.0864О»1 в€’ 0.0242) =
0.05О»1 . This produces a potential critical О» of О»1 = 0.5312.
в€’1 в€— в€’1
The second and third components of the relation Xв€— = О»(M2 E2 ) + (M2 F2 ), in which
the left member is replaced by the suitable bound, produce

в€’0.4444 О»2 + 0.68 = 1
0.4444 О»3 + 0.32 = 0
Equities 67
(2)
In consequence, О»2 = О»3 = в€’0.7201. The second critical О» is therefore О»c = 0.5312 and
the three securities acquire an вЂ˜inвЂ™ status.
в€—
The matrix elements M3 , E3 and F3 are therefore the same as those in the base and
the problem can be approached without restriction. We therefore have:
пЈ« пЈ¶
31.16 в€’24.10 в€’7.06 0.57
пЈ¬ в€’24.10 0.24 пЈ·
40.86 в€’16.76
M3 = пЈ¬ пЈ·
в€’1
пЈ­ в€’ 7.06 в€’16.76 0.19 пЈё
23.82
0.19 в€’0.01
0.57 0.24

and therefore
пЈ«
пЈ¶пЈ« пЈ¶
в€’1.0762 0.5714
пЈ¬ 0.3878 пЈ· пЈ¬ 0.2381 пЈ·
Xв€— = О»(M3 E3 ) + (M3 F3 ) = О» пЈ¬ пЈ·пЈ¬ пЈ·
в€’1 в€— в€’1
пЈ­ 0.6884 пЈё + пЈ­ 0.1905 пЈё
в€’0.0137
0.0667

With suitable bounds, the п¬Ѓrst three components of this give: в€’1.0762 О»1 etc.
We therefore arrive at О»1 = в€’0.3983, О»2 = в€’0.6140 and О»3 = в€’0.2767. The last crit-
(3)
ical О» is therefore О»c = 0 and the three securities retain their вЂ˜inвЂ™ status until the end
of the process.28 The various portfolios on the efп¬Ѓcient frontier, as well as the expected
return and the risk, are shown in Table 3.5.
Of course, between О» = 0.5312 and О» = 0, the proportions obtained here are the same
as those obtained in the вЂ˜unrestrictedвЂ™ model as all the securities are вЂ˜inвЂ™. The efп¬Ѓcient
frontier graph therefore takes the form shown in Figure 3.11.

Table 3.5 Solution for constrained Markowitz model

О» X1 X2 X3 EP ПѓP

1.53 0 0 1 0.1000 0.1500
1.5 0 0.0133 0.9867 0.0997 0.1486
1.4 0 0.0578 0.9422 0.0988 0.1442
1.3 0 0.1022 0.8978 0.0980 0.1400
1.2 0 0.1467 0.8533 0.0971 0.1360
1.1 0 0.1911 0.8089 0.0962 0.1322
1.0 0 0.2356 0.7644 0.0953 0.1286
0.9 0 0.2800 0.7200 0.0944 0.1253
0.8 0 0.3244 0.6756 0.0935 0.1222
0.7 0 0.3689 0.6311 0.0926 0.1195
0.6 0 0.4133 0.5867 0.0917 0.1170
0.5312 0 0.4439 0.5561 0.0911 0.1155
0.5 0.0333 0.4320 0.5347 0.0897 0.1123
0.4 0.1409 0.3932 0.4658 0.0851 0.1027
0.3 0.2486 0.3544 0.3970 0.0805 0.0945
0.2 0.3562 0.3157 0.3282 0.0759 0.0882
0.1 0.4638 0.2769 0.2853 0.0713 0.0842
0.0 0.5714 0.2381 0.1905 0.0687 0.0828

28
It is quite logical to have signiп¬Ѓcant diversiп¬Ѓcation in the least risk-efп¬Ѓcient portfolio.
68 Asset and Risk Management

0.12

0.1

Expected return
0.08

0.06

0.04

0.02

0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Standard deviation

Figure 3.11 Efп¬Ѓcient frontier for the constrained Markowitz model

0.14

0.12

0.1
Expected return

0.08

0.06

0.04

0.02

0
0 0.05 0.1 0.15 0.2 0.25
Standard deviation

Figure 3.12 Comparison of unconstrained and constrained efп¬Ѓcient frontiers

Figure 3.12 superimposes the two efп¬Ѓcient frontiers (constrained and unconstrained).
The zones corresponding to the short sales, and those in which all the securities are вЂ˜inвЂ™,
can be clearly seen.

3.2.3.4 Critical line algorithm
H. Markowitz has proposed an algorithmic method for resolving the problem with the
restrictions Xj в‰Ґ 0 (j = 1, . . . , N ). It is known as the critical line algorithm.
This algorithm starts with the п¬Ѓrst corner portfolio, which of course consists of the
single security with the highest expected return. It then passes through the succes-
sive corner portfolios by testing, at each stage, the changes in the function to be min-
imised when:

вЂў A new security is introduced into the portfolio.
вЂў A security is taken out of the portfolio.
вЂў A security in the portfolio is replaced by one that was not previously present.

The development of the algorithm is outside the scope of this work and is instead covered in
specialist literature.29 Here, we will simply show the route taken by a three-security problem
29
For example Markowitz, H., Mean Variance Analysis in Portfolio Choice and Capital Markets, Basil Blackwell, 1987.
Equities 69
X3

A

B
C

X2
X1

Figure 3.13 Critical line

3
Xj = 1 j =1
such as the one illustrated in this section. The restrictions
0 в‰¤ Xj в‰¤ 1 j = 1, 2, 3
deп¬Ѓne, in a three-dimensional space, a triangle with points referenced (1, 0, 0) (0, 1, 0) and
(0, 0, 1) as shown in Figure 3.13. The critical line is represented in bold and points AB and
(1) (2) (3)
C correspond to the corner portfolios obtained for О» = О»c , О»c and О»c respectively.
In this algorithm, only the corner portfolios are determined. Those that are located
between two consecutive corner portfolios are estimated as linear combinations of the
corner portfolios.

3.2.4 SharpeвЂ™s simple index model
3.2.4.1 Principles
Determining the efп¬Ѓcient frontier within the Markowitz model is not an easy process. In
addition, the amount of data required is substantial as the varianceвЂ“covariance matrix
is needed. For this reason, W. Sharpe30 has proposed a simpliп¬Ѓed version of MarkowitzвЂ™s
model based on the following two hypotheses.

1. The returns of the various securities are expressed as п¬Ѓrst-degree functions of the return
of a market-representative index: Rj t = aj + bj RI t + Оµj t j = 1, . . . , N . It is also
assumed that the residuals verify the classical hypotheses of linear regression,31 which
are, among others, that the residuals have zero expectation and are not correlated to
the explanatory variable RI t .
2. The residuals for the regressions relative to the various securities are not correlated:
cov (Оµit , Оµj t ) = 0 for all different i and j .

By applying the convention of omitting the index t, the return on a portfolio will therefore
be written, in this case, as
N
RP = Xj Rj
j =1

30
Sharpe W., A simpliп¬Ѓed model for portfolio analysis, Management Science, Vol. 9, No. 1, 1963, pp. 277вЂ“93.
31
See Appendix 3 on this subject.
70 Asset and Risk Management
N
= Xj (aj + bj RI + Оµj )
j =1
пЈ« пЈ¶
N N N
Xj aj + пЈ­ Xj bj пЈё RI +
= Xj Оµj
j =1 j =1 j =1

N N
= Xj aj + Y RI + Xj Оµj
j =1 j =1

where we have inserted Y = Xj bj .
The expected return and portfolio variance can, on the basis of the hypotheses in the
model, be written

N
EP = Xj aj + Y EI
j =1

N
ПѓP = Xj ПѓОµ2j + Y 2 ПѓI2
2 2

j =1

Note 1
The variance of the portfolio can be written as a matrix using a quadratic form:
пЈ« пЈ¶пЈ« пЈ¶
ПѓОµ21 . X1
пЈ¬ пЈ·пЈ¬ . пЈ·
.. пЈ·пЈ¬ . пЈ·
пЈ¬ . пЈ·пЈ¬ . пЈ·
= ( X1 В·В·В· XN Y )пЈ¬
ПѓP
2
пЈ­ пЈё пЈ­ XN пЈё
ПѓОµ2N
Y
. ПѓI2

Because of the structure of this matrix, the simple index model is also known as a
diagonal model.
However, contrary to the impression the term may give, the simpliп¬Ѓcation is not exces-
sive. It is not assumed that the returns from the various securities will not be correlated,
as

Пѓij = cov(ai + bi RI + Оµi , aj + bj RI + Оµj )
= bi bj ПѓI2

Note 2
In practice, the aj and bj coefп¬Ѓcients for the various regressions are estimated using the
Л†
least squares method: aj and bj . The residuals are estimated using the relation
Л†

Л†
Оµj t = Rj t в€’ (aj + bj RI t )
Л† Л†
Equities 71

On the basis of these estimations, the residual variances will be determined using their
ergodic estimator.

3.2.4.2 Simple index model
We therefore have to resolve the following problem: for the different values of О» between
0 and +в€ћ, minimise the following expression with respect to the proportions X1 , . . . , XN
and the variable Y :
пЈ« пЈ¶
N N
в€’О»В·пЈ­ Xj aj + Y EI пЈё
ПѓP в€’ О»EP = Xj ПѓОµ2j +Y ПѓI2
2 2 2

j =1 j =1

пЈ±
with the restrictions
N
пЈґ
пЈґ
пЈґ Xj bj = Y
пЈґ
пЈґ
пЈІ
j =1
пЈґ N
пЈґ
пЈґ
пЈґ Xj = 1
пЈґ
пЈі
j =1

The Lagrangian function for the problem is written as:

L(X1 , . . . , XN , Y ; m1 , m2 )
пЈ« пЈ¶
N N
Xj ПѓОµ2j + Y 2 ПѓI2 в€’ О» В· пЈ­ Xj aj + Y EI пЈё
= 2

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