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 · (1 ’ 0.4) · (360 ’ 200)

1

k= +

12 360

= 0.0833 + 0.04

= 12.33 %

3.2 PORTFOLIO DIVERSIFICATION AND MANAGEMENT

3.2.1 Principles of diversi¬cation

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,

• Which of these equities should I choose?

• 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 ¬‚uctuations in the various portfolio components. This, in

sharp contrast to the approach described above, can help reduce the portfolio risk without

reducing its expected return too much.

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

σP = X1 σ1 + X2 σ2 + 2X1 X2 σ1 σ2 ρ

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 diversi¬cation (the impact of correlation on risk),

let us ¬rst consider the case in which the two securities have the same expected return

(E1 = E2 = E) and the same risk (σ1 = σ2 = σ ). Since X1 + X2 = 1, the equations will

become:

EP = E

σP = (X1 + X2 + 2X1 X2 ρ)σ 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 ρ = 1 for which σP = σ and

when ρ < 1, σP < σ . Note that in the case of a perfect negative correlation (ρ = ’1),

the risk can be written as σP = (X1 ’ X2 )2 σ 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 , σi ) for i = 1 or 2 and

can therefore be represented as a point in space (E, σ ); of course the same applied for the

portfolio, which corresponds to the point (EP , σP ). Depending on the values given to X1

(and therefore to X2 ), the representative point for the portfolio will describe a curve in

(E, σ ) plane. Let us now study in brief the shape of the curve with respect to the values

for the correlation coef¬cient ρ.

When ρ = 1, the portfolio variance15 becomes σP = (X1 σ1 + X2 σ2 )2 .

2

By eliminating X1 and X2 from the three equations

±

EP = X1 E1 + X2 E2

σ = X1 σ1 + X2 σ2

P

X1 + X2 = 1

we arrive at the relation

EP ’ E2 E1 ’ EP

σP = σ1 + σ2

E1 ’ E2 E1 ’ E2

This expresses σP as a function of EP , a ¬rst-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

•

(1) •

(2)

• •

(1) (2)

σ σ

Figure 3.2 Two-equity portfolio (ρ = 1 case)

15

Strictly speaking, one should say ˜the portfolio return variance™.

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 ρ = ’1, the variance in the portfolio will be: σP = (X1 σ1 ’ X2 σ2 )2 .

2

In other words, σP = |X1 σ1 ’ X2 σ2 |. 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 (σ = 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, ’1 < ρ < 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 ρ = ±1, as shown in Figure 3.4.

If one expresses σP as a function of EP , as was done in the ρ = 1 case, a second-

2

degree function is obtained. The curve obtained in the (E, σ ) plane will therefore be a

hyperbolic branch.

The term ef¬cient 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

• (2)

• (1)

σ

Figure 3.3 Two-equity portfolio (ρ = ’1 case)

E

• (2)

A

• (1)

σ

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 σP 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 ¬xed covariances σ12 , σ13 and σ23 , 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, σ ).

The concept of ˜ef¬cient portfolio™ is de¬ned in the same way as for two secu-

rities. The full range of ef¬cient 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 ef¬-

cient frontier.

The last part of this Section 3.2 is given over to the various techniques used to determine

the ef¬cient frontier, according to various restrictions and hypotheses.

An investor™s choice of a portfolio on the ef¬cient 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 ef¬cient frontier (the least risky portfolio, very

diversi¬ed), while a taste for risk will move him towards the portfolios located on the

right part of the ef¬cient frontier (acceptance of increased risk with hope of higher return,

generally obtained in portfolios made up of a very few pro¬table but highly volatile

securities).18

E

(1)

(2)

(3)

σ

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 ˜Ch 3™ directory.

18

This question is examined further in Section 3.2.6.

Equities 55

3.2.2 Diversi¬cation and portfolio size

We have just seen that diversi¬cation 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 diversi¬cation; up to what point, for a given correlation

structure, can diversi¬cation 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

σP = Xi Xj σij

2

i=1 j =1

±

1

N N N

=2 σi2 + σij

N

i=1 i=1 j =1

j =i

This double sum contains N (N ’ 1) terms, and it is therefore natural to de¬ne the average

variance and the average covariance as:

N

1

var = σi2

N i=1

N N

1

cov = σij

N (N ’ 1) i=1 j =1

j =i

As soon as N reaches a suf¬cient magnitude, these two quantities will almost cease to

depend on N . They will then allow the portfolio variance to be written as follows:

N ’1

1

σP = var +

2

cov

N N

3.2.2.2 Asymptotic behaviour

When N becomes very large, the ¬rst term will decrease back towards 0 while the second,

now quite stable, converges towards cov. The portfolio risk, despite being very diversi¬ed,

never falls below this last value, which corresponds to:

N ’1

1

cov = lim var + cov = lim σP = σM

2 2

N N

N’’∞ N’’∞

In other words, it corresponds to the market risk.

56 Asset and Risk Management

sP

2

cov

N

Figure 3.6 Diversi¬cation 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 diversi¬cation are initially very rapid (the ¬rst 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 ef¬cient frontier is the ˜North-West™ part of the curve, consisting of portfolios de¬ned

by this principle: for each ¬xed value r of EP , the proportions for which σP is minimal

2

Xj (j = 1, . . . , N ) are determined. The ef¬cient frontier is this de¬ned 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

σP = Xi Xj σij

2

i=1 j =1

±

under the double restriction:

N

Xj Ej = r

j =1

N

Xj = 1

j =1

The Lagrangian function19 for the problem can thus be written as:

« «

N N N N

Xi Xj σij + m1 · X j E j ’ r + m2 · Xj ’ 1

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:

±

N

L (X1 , . . . , XN ; m1 , m2 ) = 2 Xi σij + m1 Ej + m2 = 0 (j = 1, . . . , N )

Xj

i=1

N

L (X , . . . , XN ; m1 , m2 ) = Xi Ei ’ r = 0

m1 1

i=1

N

Lm (X1 , . . . , XN ; m1 , m2 ) = Xi ’ 1 = 0

2

i=1

This can be written in a matrix form:

« « «

· · · 2σ1N E1

2

X1 .

2σ1 2σ12 1

¬ 2σ21 1 · ¬ X2 · ¬ . ·

· · · 2σ2N E2

2

2σ2

¬ ·¬ · ¬·

¬. . ·¬ . · ¬ . ·

. . .

.. . ·¬ . · ¬ . ·

¬. . . .

. . ·¬ . · = ¬ . ·

¬. . . .

¬ 2σ 1 · ¬ XN · ¬ . ·

· · · 2σN EN

2

¬ N1 ·¬ · ¬·

2σN2

E1 m1 r

E2 · · · EN . .

m2

··· . . 1

1 1 1

By referring to the matrix of coef¬cients,20 the vector of unknowns21 and the vector of

second members as M, X— and G respectively, we give the system the form MX— = G.

The resolution of this system passes through the inverse matrix of M:X— = M ’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 ¬nd the frontier,

of which only the ef¬cient (˜North-West™) 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 ef¬cient frontier that can be constructed with three securities with

the following characteristics:

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

In its order N zone of the upper left corner, this contains the 2V matrix in which V is the variance“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 X— 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 Excel ¬le known as

Ch 3.

58 Asset and Risk Management

The variance“covariance matrix is given by:

«

0.0100 0.0036 0.0015

V = 0.0036 0.0144 0.0072

0.0015 0.0072 0.0225

The matrix M is therefore equal to:

«

0.0200 0.0072 0.0030 0.05 1

¬ 0.0072 0.0288 0.0144 0.08 1 ·

¬ ·

M = ¬ 0.0030 0.0144 0.0450 0.10 1 ·

0.05 .

.

0.08 0.010

. .

1 1 1

This matrix inverts to:

«

31.16 ’24.10 ’ 7.06 0.57

¬ ’24.10 0.24 ·

40.86 ’16.76

M ’1 = ¬ ·

’ 7.06 ’16.76 0.19

23.82

0.19 ’0.01

0.57 0.24

«

.

.

¬.·

By applying this matrix to the vector G = r , for different values of r, we ¬nd a

1

range of vectors X— , the ¬rst three components of which supply the composition of the

portfolios (see Table 3.2).

These proportions allow σP 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

’0.3233 ’0.8060

0.00 2.1293

’0.2391 ’0.6565

0.01 1.8956

’0.1549 ’0.5071

0.02 1.6620

’0.0707 ’0.3576

0.03 1.4283

’0.2801

0.04 1.1946 0.0135

’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

’0.2075

0.10 0.5188 0.6887

’0.4412

0.11 0.6030 0.8382

’0.6749

0.12 0.6872 0.9877

’0.9086

0.13 0.7714 1.1371

’1.1423

0.14 0.8556 1.2866

’1.3759

0.15 0.9398 1.4361

23

The expected return is of course known.

Equities 59

Calculation of σP

Table 3.3

EP σP

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 Ef¬cient frontier

The ef¬cient part of this frontier is therefore the ˜North-West™ part, the lower limit of

which corresponds to the minimum risk portfolio. For this portfolio, we have values of

EP = 0.0667 and σP = 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 ˜classic™ portfolios (0 ¤ Xj ¤ 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 ¬xed percentage. In this case,

the problem must be resolved by putting together portfolios in which proportions of the

60 Asset and Risk Management

’ +

type Bj ¤ Xj ¤ 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 σP as the x-axis and

Ep as the y-axis (as in Figure 3.7), EP is now shown as the x-axis and σP as the y-axis,

2

the ef¬cient frontier graph now appears as shown in Figure 3.8.

A straight line in this graph has the equation σ 2 = a + »E in which a represents the

intercept and » the slope of the straight line. We are looking speci¬cally at a straight

line at a tangent to the ef¬cient frontier. If the slope of this straight line is zero (» = 0),

the contact point of the tangent shows the least risky portfolio in the ef¬cient frontier.

Conversely, the more » increases, the further the contact point moves away from the

ef¬cient frontier towards the risky portfolios. The » parameter may vary from 0 to +∞

and is therefore representative of the portfolio risk corresponding to the contact point of

the tangent with this » value for a slope.

For a ¬xed » value, the tangent to the ef¬cient frontier with slope » is, of all the straight

lines with that slope and with at least one point in common with the ef¬cient frontier,

that which is located farthest to the right, that is, the one with the smallest coordinate at

the origin a = σ 2 ’ »E.

The problem is therefore reformulated as follows: for the various values of » between

0 and ∞, minimise with respect to the proportions X1 , . . . , XN the expression:

N N N

σP ’ »EP = Xi Xj σij ’ » Xj Ej

2

i=1 j =1 j =1

under the restriction N=1 Xj = 1.

j

Once the solution, which will depend on », has been found, it will be suf¬cient to make

this last parameter vary between 0 and +∞ to arrive at the ef¬cient frontier.

sP

2

EP

a

Figure 3.8 Reformulation of problem

Equities 61

The Lagrangian function for the problem can be written as:

«

N N N N

Xj Ej + m · Xj ’ 1

L(X1 , . . . , XN ; m) = Xi Xj σij ’ »

i=1 j =1 j =1 j =1

A reasoning similar to that used in the ¬rst formulation allows the following matrix

expression to be deduced from the partial derivatives:

MX— = »E — + F

Here, it has been noted that24

« «

«

«

2σ12 · · · 2σ1N

2

X1 E1 .

2σ1 1

¬ 2σ21 2σ 2 · · · 2σ2N 1· ¬ X2 · ¬ E2 · ¬.·

¬ · ¬ · ¬ · ¬·

2

¬. .· ¬.· ¬.· ¬·

F =¬.·

. .

.. — —

X =¬ . · E =¬ . · .

M=¬ . . . .·

. ¬.· ¬.· ¬.·

. . . .·

¬

2σ 1 XN EN .

· · · 2σN 2

N1 2σN2

m .

··· . 1

1 1 1

The solution to this system of equations is therefore supplied by: X— = »(M ’1 E — ) +

(M ’1 F ).

As for the ¬rst formulation, the matrix M is independent of the parameter », which

must be variable; it only needs to be inverted once.

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

The same variance“covariance matrix V as above will be used, and the matrix M can be

expressed as: «

0.0200 0.0072 0.0030 1

¬ 0.0072 0.0144 1 ·

0.0288

M=¬ ·

0.0030 0.0450 1

0.0144

.

1 1 1

This matrix inverts to:

«

31.16 ’24.10 ’ 7.06 0.57

¬ ’24.10 0.24 ·

40.86 ’16.76

=¬ ·

M ’1 ’ 7.06 ’16.76 0.19

23.82

0.19 ’0.01

0.57 0.24

In the same way as the function carried out for X— , we are using the E — 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 »

Table 3.4

» X1 X2 X3 EP σP

’1.5810

2.0 1.0137 1.5672 0.0588 0.3146

’1.4734

1.9 0.9750 1.4984 0.1542 0.3000

’1.3657

1.8 0.9362 1.4296 0.1496 0.2854

’1.2581

1.7 0.8974 1.3607 0.1450 0.2709

’1.1505

1.6 0.8586 1.2919 0.1404 0.2565

’1.0429

1.5 0.8198 1.2231 0.1357 0.2422

’0.9353

1.4 0.7810 1.1542 0.1311 0.2280

’0.8276

1.3 0.7423 1.0854 0.1265 0.2139

’0.7200

1.2 0.7035 1.0165 0.1219 0.2000

’0.6124

1.1 0.6647 0.9477 0.1173 0.1863

’0.5048

1.0 0.6259 0.8789 0.1127 0.1729

’0.3972

0.9 0.5871 0.8100 0.1081 0.1597

’0.2895

0.8 0.5484 0.7412 0.1035 0.1470

’0.1819

0.7 0.5096 0.6723 0.0989 0.1347

’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

«

«

·

0.05

¬ 0.08 · ¬··

As the vectors E — and F are given by E — = ¬ · F = ¬ ·, the solutions

0.10 ·

. 1

to the problem for the different values of » are shown in Table 3.4.

The ef¬cient 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 ef¬cient 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 Ef¬cient 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 »

between 0 and +∞, minimise with respect to the proportions X1 , . . ., XN the expression

N N N

σP ’ »EP = Xi Xj σij ’ » Xj Ej

2

i=1 j =1 j =1

with the restrictions:

N

j =1 Xj =1

’ +

Bj ¤ Xj ¤ Bj j = 1, . . . , N

We will ¬rst of all introduce the concept of a security™s ˜status™. The security (j ) is

de¬ned as ˜down™ (resp. ˜up™) if its proportion is equal to the ˜lower™ (resp. ˜upper™)

’ +

bound imposed on it: Xj = Bj (resp. Xj = Bj ). For an ef¬cient 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 ˜down™ (resp. ˜up™) securities should therefore be

’ +

replaced here by Xj = Bj (resp. Xj = Bj ).

’ +

The other securities are de¬ned as ˜in™, 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 MX— = »E — + F , will therefore consist of not altering the

components that correspond to the ˜in™ securities, and if the security (j ) is ˜down™ or

L

L

“ + “ +

Xj Xj

Bj Bj Bj Bj

Figure 3.10 ˜Up™ security and ˜down™ security

64 Asset and Risk Management

j th component of E — and F , as follows:

˜up™, of altering the j th line of M and the

« « «

2σ1 · · · 2σ1j · · · 2σ1N E1

2 0

1

¬. .· ¬.· ¬.·

. .

.. ¬.·

¬. . . .· ¬.·

. ¬ .± ·

. . . .· .·

¬ ¬

¬0 0· ¬0· ¬B ·

··· ···

1 0

M=¬ . · E— = ¬ · F =¬ j ·

¬ .· ¬.· ¬.·

. .

¬. . . .· ¬.· ¬.·

. . ··· . .· .· ¬.·

¬ ¬

2σ EN 0

· · · 2σNj · · · 2σN 2

1

N1

··· ··· 0

1 1 1 0 1

±

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 ’1 M = I , it is evident that M ’1 has the same jth

line as M and the j th component of the solution X— = »(M ’1 E — ) + (M ’1 F ). This is also

±

written as Xj = Bj .

If (j ) has an ˜in™ status, this j th component can of course be written as Xj = »uj + vj ,

’ +

a quantity that is strictly included between Bj and Bj .

—

The method proceeds through a series of stages and we will note M0 , E0 and F0 , the

matrix elements as de¬ned in the ˜unconstrained™ case. The index develops from one stage

to the next.

The method begins with the major values for » (+∞ ideally). As we are looking to

minimise σP ’ »EP , EP needs to be as high as possible, and this is consistent with

2

a major value for the risk parameter ». The ¬rst portfolio will therefore consist of the

securities that offer the highest expected returns, in equal proportions to the upper bounds

+ ’

Bj , until (with securities in proportions equal to Bj ) the sum of the proportions equals

1.25 This portfolio is known as the ¬rst corner portfolio.

At least one security will therefore be ˜up™; one will be ˜in™, and the others will be

˜down™. The matrix M and the vectors E — and F are altered as shown above. This brings

us to M1 , E1 and F1 , and we calculate: X— = » (M1 ’1 E1 ) + (M1 ’1 F1 ).

— —

The parameter » is thus decreased until one of the securities changes its status.26 This

(1)

¬rst change will occur for a value of » equal to »c , known as the ¬rst critical ». To

determine this critical value, and the security that will change its status, each of the various

securities for which a potentially critical »j is de¬ned will be examined.

A ˜down™ or ˜up™ security (j ) will change its status if the equation corresponding to it

becomes LXj = 0, that is:

N

Xk σj k ’ »j Ej + m = 0

2

k=1

—

This is none other than the j th component of the equation M0 X— = »E0 + F0 , in which the

different Xk and m are given by the values obtained by X— = » (M1 ’1 E1 ) + (M1 ’1 F1 ).

—

If the inequality restrictions are simply 0 ¤ Xj ¤ 1 ∀j (absence of short sales), the ¬rst portfolio will consist only of

25

the security with the highest expected return.

For the restrictions 0 ¤ Xj ¤ 1 ∀j , the ¬rst corner portfolio consists of a single ˜up™ security, all the others being

26

˜down™. The ¬rst change of status will be a transition to ˜in™ of the security that was ˜up™ and of one of the securities that

—

were ˜down™. In this case, on one hand the matrix elements M1 , E1 and F1 are obtained by making the alteration required for

the ˜down™ securities but for the one that it is known will pass to ˜in™ status, and on the other hand there is no equation for

determining the potential critical » for this security.

Equities 65

For an ˜in™ security (j ), it is known that Xj = »j uj + vj and it will change its status

if it becomes a ˜down™ (uj > 0 as » decreases) or ˜up™ security (uj < 0), in which case

±

we have Bj = »j uj + vj . This is none other than the j th component of the relation

’1 — ’1

X— = »(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 »j . The highest

(1)

of these is the ¬rst critical »j or »c . The proportions of the various securities have not

(1)

changed between » = +∞ and » = »c . The corresponding portfolio is therefore always

the ¬rst corner portfolio.

The security corresponding to this critical » therefore changes its status, thus allowing

— (2)

M2 , E2 and F2 to be constructed and the second critical », »c , to be determined together

(1) (2)

with all the portfolios that correspond to the values of » between »c and »c . The

(2)

portfolio corresponding to »c is of course the second corner portfolio.

The process is then repeated until all the potentially critical » values are negative, in

which case the last critical » is equal to 0. The last and least risky corner portfolio, located

at the extreme left point of the ef¬cient frontier, corresponds to this value.

The corner portfolios are of course situated on the ef¬cient frontier. Between two

consecutive corner portfolios, the status of the securities does not change; only the pro-

(k’1)

and »(k) , using the

portions change. These proportions are calculated, between »c c

’1 — ’1

relation X— = »(Mk Ek ) + (Mk Fk ).

The various sections of curve thus constructed are connected continuously and with

same derivative27 and make up the ef¬cient frontier.

Example

Let us take the same data as were processed before:

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 impose the requirement of absence of short sales: 0 ¤ Xj ¤ 1 (j = 1, 2, 3).

We have the following basic matrix elements:

« « «

.

0.0200 0.0072 0.0030 1 0.05

¬ 0.0072 0.0288 0.0144 1 · ¬ 0.08 · ¬.·

M0 = ¬ · ¬ · ¬·

—

0.0030 0.0144 0.0450 1 E0 = 0.10 F0 = .

. .

1 1 1 1

The ¬rst corner portfolio consists only of security (3), the one with the highest expected

return. As securities (1) and (2) are ˜down™, we construct:

« « «

. . . . .

1

¬. · ¬.· ¬.·

. .·

1

M1 = ¬ ¬ · ¬·

—

0.0030 0.0144 0.0450 1 E1 = 0.10 F1 = .

. .

1 1 1 1

27

That is, with the same tangent.

66 Asset and Risk Management

«

We have:

. . .

1

¬. ·

. .

1

=¬ ·

’1

M1 ’1

’1 . 1

1 ’0.0450

0.0420 0.0306

and therefore

« «

. .

¬.· ¬ ·

.

X— = »(M1 E1 ) + (M1 F1 ) = » ¬ ·+¬ ·

’1 — ’1

. 1

’0.045

0.1

—

The ¬rst two components of M0 X— = »E0 and F0 , with the vector X— obtained

above, give:

0.003 + (0.1 »1 ’ 0.045) = 0.05 » »1

0.0144 + (0.1 »2 ’ 0.045) = 0.08 »2

This will give the two potential critical » values: »1 = 0.84 and »2 = 1.53. The ¬rst

(1)

critical » is therefore »c = 1.53 and security (2) becomes ˜in™ together with (3), while

(1) remains ˜down™.

We can therefore construct:

« « «

. . . . .

1

¬ 0.0072 0.0288 0.0144 1 · ¬ 0.08 · ¬.·

M2 = ¬ · E2 = ¬ · F2 = ¬ ·

—

0.0030 0.0144 0.0450 1 0.10 .

. .

1 1 1 1

This successively gives:

«

. . .

1

¬ ’0.7733 0.68 ·

22.22 ’22.22

M2 = ¬ ·

’1

’0.2267 ’22.22 0.32

22.22

0.32 ’0.0242

0.0183 0.68

« «

. .

¬ ’0.4444 · ¬ 0.68 ·

·¬ ·

X— = »(M2 E2 ) + (M2 F2 ) = » ¬

’1 — ’1

0.4444 + 0.32

’0.0242

0.0864

The ¬rst component of M0 X— = »E0 + F0 , with vector X— obtained above, gives:

—

0.0072 · (’0.4444»1 + 0.68) + 0.0030 · (0.4444»1 + 0.32) + (0.0864»1 ’ 0.0242) =

0.05»1 . This produces a potential critical » of »1 = 0.5312.

’1 — ’1

The second and third components of the relation X— = »(M2 E2 ) + (M2 F2 ), in which

the left member is replaced by the suitable bound, produce

’0.4444 »2 + 0.68 = 1

0.4444 »3 + 0.32 = 0

Equities 67

(2)

In consequence, »2 = »3 = ’0.7201. The second critical » is therefore »c = 0.5312 and

the three securities acquire an ˜in™ status.

—

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:

«

31.16 ’24.10 ’7.06 0.57

¬ ’24.10 0.24 ·

40.86 ’16.76

M3 = ¬ ·

’1

’ 7.06 ’16.76 0.19

23.82

0.19 ’0.01

0.57 0.24

and therefore

«

«

’1.0762 0.5714

¬ 0.3878 · ¬ 0.2381 ·

X— = »(M3 E3 ) + (M3 F3 ) = » ¬ ·¬ ·

’1 — ’1

0.6884 + 0.1905

’0.0137

0.0667

With suitable bounds, the ¬rst three components of this give: ’1.0762 »1 etc.

We therefore arrive at »1 = ’0.3983, »2 = ’0.6140 and »3 = ’0.2767. The last crit-

(3)

ical » is therefore »c = 0 and the three securities retain their ˜in™ status until the end

of the process.28 The various portfolios on the ef¬cient frontier, as well as the expected

return and the risk, are shown in Table 3.5.

Of course, between » = 0.5312 and » = 0, the proportions obtained here are the same

as those obtained in the ˜unrestricted™ model as all the securities are ˜in™. The ef¬cient

frontier graph therefore takes the form shown in Figure 3.11.

Table 3.5 Solution for constrained Markowitz model

» X1 X2 X3 EP σP

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 signi¬cant diversi¬cation in the least risk-ef¬cient 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 Ef¬cient 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 ef¬cient frontiers

Figure 3.12 superimposes the two ef¬cient frontiers (constrained and unconstrained).

The zones corresponding to the short sales, and those in which all the securities are ˜in™,

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 ≥ 0 (j = 1, . . . , N ). It is known as the critical line algorithm.

This algorithm starts with the ¬rst 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:

• A new security is introduced into the portfolio.

• A security is taken out of the portfolio.

• 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 ¤ Xj ¤ 1 j = 1, 2, 3

de¬ne, 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 » = »c , »c and »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 Sharpe™s simple index model

3.2.4.1 Principles

Determining the ef¬cient frontier within the Markowitz model is not an easy process. In

addition, the amount of data required is substantial as the variance“covariance matrix

is needed. For this reason, W. Sharpe30 has proposed a simpli¬ed version of Markowitz™s

model based on the following two hypotheses.

1. The returns of the various securities are expressed as ¬rst-degree functions of the return

of a market-representative index: Rj t = aj + bj RI t + µj 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 (µit , µj 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 simpli¬ed model for portfolio analysis, Management Science, Vol. 9, No. 1, 1963, pp. 277“93.

31

See Appendix 3 on this subject.

70 Asset and Risk Management

N

= Xj (aj + bj RI + µj )

j =1

«

N N N

Xj aj + Xj bj RI +

= Xj µj

j =1 j =1 j =1

N N

= Xj aj + Y RI + Xj µj

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

σP = Xj σµ2j + Y 2 σI2

2 2

j =1

Note 1

The variance of the portfolio can be written as a matrix using a quadratic form:

« «

σµ21 . X1

¬ ·¬ . ·

.. ·¬ . ·

¬ . ·¬ . ·

= ( X1 ··· XN Y )¬

σP

2

XN

σµ2N

Y

. σI2

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 simpli¬cation is not exces-

sive. It is not assumed that the returns from the various securities will not be correlated,

as

σij = cov(ai + bi RI + µi , aj + bj RI + µj )

= bi bj σI2

Note 2

In practice, the aj and bj coef¬cients for the various regressions are estimated using the

ˆ

least squares method: aj and bj . The residuals are estimated using the relation

ˆ

ˆ

µj t = Rj t ’ (aj + bj RI t )

ˆ ˆ

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 » between

0 and +∞, minimise the following expression with respect to the proportions X1 , . . . , XN

and the variable Y :

«

N N

’»· Xj aj + Y EI

σP ’ »EP = Xj σµ2j +Y σI2

2 2 2

j =1 j =1

±

with the restrictions

N

Xj bj = Y

j =1

N

Xj = 1

j =1

The Lagrangian function for the problem is written as:

L(X1 , . . . , XN , Y ; m1 , m2 )

«

N N

Xj σµ2j + Y 2 σI2 ’ » · Xj aj + Y EI

= 2