22

92 Asset and Risk Management

3.2.8.2 Portfolio beta

By using the regression expression for Rj , RP can be developed easily:

N

RP = Xj Rj

j =1

N

= Xj (±j + βj RM + µj )

j =1

«

N N N

Xj ±j + Xj βj RM +

= Xj µj

j =1 j =1 j =1

This shows that as for the portfolio return, the portfolio beta is the average of the betas

of all the constituent securities, weighted for the proportions expressed in terms of equity

market capitalisation:

N

βP = Xj βj

j =1

3.2.8.3 Link between market model and portfolio diversi¬cation

As for the simple index model, it is supposed here that the regression residuals relative

to the various securities are not correlated: cov (µi , µj ) = 0 for i = j . The portfolio risk

is written as:

«

N N

= var Xj µj

σP Xj ±j + βP RM +

2

j =1 j =1

N

= βP σ M + Xj σµ2j

22 2

j =1

If, to simplify matters, one considers a portfolio consisting of N securities in equal

proportions:

1

Xj = j = 1, . . . , N

N

the portfolio risk can develop as follows:

«

N N

σP = var Xj µj

Xj ±j + βP RM +

2

j =1 j =1

N

1

= βP σ M +2 σµ2j

22

N j =1

12

= βP σ M + σ

22

Nµ

Equities 93

Here, the average residual variance has been introduced:

N

1

σµ2 = σµ2j

N j =1

The ¬rst term of the decomposition is independent of N , while the second tends towards

0 when N becomes very large. This analysis therefore shows that the portfolio risk σP 2

can be broken down into two terms:

• The systematic component βP σM (non-diversi¬able risk).

22

• The speci¬c component Xj σµ2j (diversi¬able risk).

2

3.3 MODEL OF FINANCIAL ASSET EQUILIBRIUM

AND APPLICATIONS

3.3.1 Capital asset pricing model

Unlike the previous models, this model, developed independently by W. Sharpe39 and

J. Lintner40 and known as CAPM (MEDAF in French) is interested not in choosing a

portfolio for an individual investor but in the behaviour of a whole market when the

investors act rationally41 and show an aversion to risk. The aim, in this situation, is to

determine the exact value of an equity.

3.3.1.1 Hypotheses

The model being examined is based on a certain number of hypotheses. The hypotheses

relating to investor behaviour are:

• They put together their portfolio using Markowitz™s portfolio theory, that is, relying on

the mean“variance pairing.

• They all have the same expectations, that is, none of them has any privileged informa-

tion and they agree on the value of the parameters Ei , σi and σij to be used.

Hypotheses can also be laid down with regard to the transactions:

• They are made without cost.

• The purchase, sale and holding times are the same for all investors.

Finally, it is assumed that the following conditions have been veri¬ed in relation to

the market:

• There is no taxation either on increases in value, dividends or interest income.

• There are very many purchasers and sellers on the market and they do not have any

in¬‚uence on the market other than that exerted by the law of supply and demand.

39

Sharpe W., Capital assets prices, Journal of Finance, Vol. 19, 1964, pp. 435“42.

40

Lintner J., The valuation of risky assets and the selection of risky investments, Review of Economics and Statistics, Vol.

47, 1965, pp. 13“37.

41

That is, according to the portfolio theory based on the mean“variance analysis.

94 Asset and Risk Management

• There is a risk-free interest rate, RF , which is used for both borrowings and investments.

• The possibilities of borrowing and investing at this rate are not limited in terms

of volume.

These hypotheses are of course not realistic. However, there are extensions of the

model presented here, which make some of the hypotheses formulated more ¬‚exible. In

addition, even the basic model gives good results, as do the applications that arise from

it (see Sections 3.3.3, 3.3.4 and 3.3.5).

3.3.1.2 Separation theorem

This theorem states that under the conditions speci¬ed above, all the portfolios held

by the investors are, in terms of equilibrium, combinations of a risk-free asset and a

market portfolio.

According to the hypotheses, all the investors have the same ef¬cient frontier for the

equities and the same risk-free rate RF . Therefore, according to the study of Markowitz™s

model with the risk-free security (Section 3.2.5), each investor™s portfolio is located on

the straight line issuing from point (0, RF ) and tangential to the ef¬cient frontier. This

portfolio consists (see Figure 3.25) of:

• The risk-free equity, in proportion X.

• The portfolio A, corresponding to the tangent contact point, in proportion 1 ’ X.

The risked portfolio A is therefore the same for all investors. The market will therefore,

in accordance with the principle of supply and demand, adapt the prices so that the

proportions in the portfolio are those of the whole market (A = M) and the portfolios

held by the investors are perfectly diversi¬ed.

The investor™s choice will therefore be made only on the proportion X of the market

portfolio (and therefore the 1 ’ X proportion of the risk-free equity). If the portfolio

chosen is located to the left of the point M (0 < X < 1), we are in fact looking at

a combination of the two investments. If it is to the right of M (X > 1), the investor

borrows at the rate RF in order to acquire more than 100 % of the market portfolio. The

line in question is known as the market straight line.

EP

A=M

RF

σP

Figure 3.25 Separation theorem and market straight line

Equities 95

Interpretation of the separation theorem is simple. The market straight line passes

through the points (0, RF ) and (σM , EM ). Its equation is therefore given by:

EM ’ RF

EP = RF + · σP

σM

The expected return EP on a portfolio is equal to the risk-free rate RF plus the risk

premium collected by the investor when he agrees to take a risk σP . The coef¬cient of σP

(the slope of the market straight line) is therefore the increase in expected return obtained

to support one unit of risk: this is the unit price of the risk on the market.

3.3.1.3 CAPM equation

We will now determine a relation very similar to the previous one “ that is, a relation

between expected return and risk “ but in connection with a security instead of a portfolio.

For any portfolio of equities B, the straight line that connects the points (0, RF ) and

(σB , EB ) has the slope

EB ’ RF

B=

σB

This slope is clearly at its maximum when B = M (see Figure 3.26) and, in the same

2 2

way, the maximum value of B is M . Therefore, if one terms the proportions of the

various equities in the market portfolio X1 , X2 , . . ., XN , ( Xi = 1) we will have:

( M )Xk =0 k = 1, . . . , N

2

«

Like ±

N N N

E ’ R = Xj Ej ’ Xj RF =

M Xj (Ej ’ RF )

F

j =1 j =1 j =1

N N

2

σ = Xi Xj σij

M

i=1 j =1

EP

M

B

RF

σP

Figure 3.26 CAPM

96 Asset and Risk Management

the derivative of

« 2

N

Xj (Ej ’ RF )

(EM ’ RF ) 2 j =1

= =

2

M

σM 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

σM Xj σkj

2

j =1 j =1 j =1

( M )Xk =

2

σM

4

N

2(EM ’ RF )(Ek ’ RF )σM ’ 2(EM ’ RF ) Xj σkj

2 2

j =1

=

σM4

2(EM ’ RF ) · ((Ek ’ RF )σM ’ (EM ’ RF )σkM )

2

=

σM4

This will be zero if

σkM

Ek ’ RF = (EM ’ RF )

σM2

or

Ek = RF + βk .(EM ’ RF )

This is termed the CAPM equation, which is interpreted in a similar way to the relation

in the previous paragraph. The expected return Ek on the security (k) is equal to the

risk-free rate RF , plus a risk premium collected by the investor who agrees to take the

risk. This risk premium is the increase in the expected return, to which more importance

is given as the risk of the security within the market in question increases (βk ).

Note

As we have said, the hypotheses used as a basis for the model just developed are not

realistic. Empirical studies have been carried out in order to determine whether the results

obtained from the application of the CAPM model are valid. One of the most detailed

analyses is that carried out by Fama and Macbeth,42 which, considering the relation

Ek = RF + βk (EM ’ RF ) as an expression of Ek according to βk , tested the following

hypotheses on the New York Stock Exchange (Figure 3.27):

• The relation Ek = f (βk ) is linear and increasing.

• βk is a complete measurement of the risk of the equity (k) on the market; in other

words, the speci¬c risk σµ2k is not a signi¬cant explanation of Ek .

42

Fama E. and Macbeth J., Risk, return and equilibrium: empirical tests, Journal of Political Economy, Vol. 71, No. 1,

1974, pp. 606“36.

Equities 97

Ek

EM

RF

bk

1

Figure 3.27 CAPM test

To do this, they used generalisations of the equation Ek = f (βk ), including powers of βk

of a degree greater than 1 and a term that takes the speci¬c risk into consideration. Their

conclusion is that the CAPM model is in most cases acceptable.

3.3.2 Arbitrage pricing theory

In the CAPM model, the risk premium Ek ’ RF for an equity is expressed as a multiple

of the risk premium EM ’ RF for the market:

Ek ’ RF = βk (EM ’ RF )

The proportionality coef¬cient is the β of the security. It can therefore be considered that

this approach allows the risk premium for an equity to be expressed on the basis of the

risk premium for a single explanatory macroeconomic factor, or, which amounts to the

same thing, on the basis of an aggregate that includes all the macroeconomic factors that

interact with the market.

The arbitrage pricing theory43 or APT allows a more re¬ned analysis of the portfolio

than does the CAPM, as breaking down the risk according to the single market factor,

namely the beta, may prove insuf¬cient to describe all the risks in a portfolio of equities.

Hence the interest in resorting to risk breakdowns on the basis of several factors F1 , F2 ,

. . ., FP .

p

Ek ’ RF = ±kj (EFj ’ RF )

j =1

The APT theory shows that in an ef¬cient market the quoted equity prices will be

balanced by successive arbitrages, through the involvement of actors on the market. If

one makes a point of watching developments in relative prices, it is possible to extract

from the market a small number of arbitrage factors that allow the prices to balance out.

This is precisely what the APT model does.

43

Ross S. A., The arbitrage theory of capital asset pricing, Journal of Economic Theory, 1976, pp. 343“62.

98 Asset and Risk Management

The early versions44 of the APT model relied on a previously compiled list of basic

factors such as an industrial activity index, the spread between short-term and long-

term interest rates, the difference in returns on bonds with very different ratings (see

Section 4.2.1) etc. The coef¬cients ±k1 , . . . , ±kp are then determined by a regression

technique based on historical observations Rkt and RFj ,t (j = 1, . . . , p).

The more recent versions are based on more empirical methods that provide factors not

correlated by a statistical technique45 (factorial analysis), without the number of factors

being known beforehand and even without them having any economic interpretation at all.

The factors obtained46 from temporal series of returns on asset prices are purely sta-

tistical. Taken individually, they are not variables that are commonly used to describe

a portfolio construction process or management strategy. None of them represents an

interest, in¬‚ation or exchange rate. They are the equivalent of an orthogonal axis system

in geometry.

The sole aim is to obtain a referential that allows a description of the interrelations

between the assets studied on a stable basis over time. Once the referential is established,

the risk on any asset quoted (equities, bonds, investment funds etc.) is broken down into a

systematic part (common to all assets in the market) that can be represented in the factor

space, and a speci¬c part (particular to the asset). The systematic part is subsequently

explained by awareness coef¬cients (±kj ) for the different statistical factors.

The explanatory power of the model can be explained by the fact that the different

standard variables (economic, sectorial, fundamental etc.) used to understand the way

in which it behaves are also represented in the referential for the factors provided an

associated quoted support exists (price history).

The relation that links the return on a security to the various factors allows a breakdown

of its variance into a part linked to the systematic risk factors (the explicative statistical

factors) and a part that is speci¬c to the securities and therefore diversi¬able (regression

residues etc.), that is:

p

σk2 = ±kj var(RFj ) + σµ2k

2

j =1

Example

A technically developed version of this method, accompanied by software, has been

produced by Advanced Portfolio Technologies Inc. It extracts a series of statistical factors

(represented by temporal series of crossed returns on assets) from the market, using a form

search algorithm. In this way, if the left of Figure 3.28 represents the observed series of

returns on four securities, the straight line on the same ¬gure illustrates the three primary

factors that allow reconstruction of the previous four series by linear combination.

For example, the ¬rst series breaks down into: R1 ’ RF = 1 · (RF1 ’ RF ) + 1 · (RF2 ’

RF ) + 0.(RF3 ’ RF ) + µ1 .

44

Dhrymes P. J., Friend I. and Gultekin N. B., A critical re-examination of the empirical evidence on the arbitrage

pricing theory, Journal of Finance, No. 39, 1984, pp. 323“46. Chen N. F., Roll R. and Ross S. A., Economic forces of the

stock market, Journal of Business, No. 59, 1986, pp. 383“403. More generally, Grinold C. and Kahn N., Active Portfolio

Management, McGraw-Hill, 1998.

45

See for example Saporta G., Probabilities, Data Analysis and Statistics, Technip, 1990; or Morrison D., Multivariate

Statistical Methods, McGraw-Hill, 1976.

46

Readers interested in the mathematical developments produced by extracting statistical factors from historical series of

returns on assets should read Mehta M. L., Random Matrices, Academic Press, 1996. This work deals in depth with problems

of proper values and proper vectors for matrices with very large numbers of elements generated randomly.

Equities 99

Series 4 Factor 3

Series 3

Factor 2

Series 2

Series 1

Factor 1

Figure 3.28 Arbitrage pricing theory

3.3.3 Performance evaluation

3.3.3.1 Principle

The portfolio manager,47 of course, has an interest in the product that he manages. To do

this properly, he will compare the return on his portfolio with the return on the market

in which he is investing. From a practical point of view, this comparison will be made in

relation to a market representative index for the sector in question.

Note

The return on a real portfolio between moments s and t is calculated simply using the

V t ’ Vs

relation RP ,[s;t] = , provided there has been no movement within the portfolio

Vs

during the interval of time in question. However, there are general ¬‚ows (new securities

purchased, securities sold etc.). It is therefore advisable to evaluate the return with the

effect of these movements eliminated.

Note t1 < . . . < tn the periods in which these movements occur and propose that t0 = s

and tn+1 = t. The return to be taken into consideration is therefore given by:

n

RP ,]s;t] = (1 + RP ,]tk ;tk+1 [ ) ’ 1

k=0

Here, it is suggested that

Vt(’) ’ Vt(+)

RP ,[tk ;tk+1 ] = k+1 k

Vt(+)

k

Vt(’) and Vt(+) represent the value of the portfolio just before and just after the movement

j j

at moment tj respectively.

In Section 3.2, it has been clearly shown that the quality of a security or a portfolio is

not measured merely by its return. What should in fact be thought of those portfolios A

and B in which the returns for a given period are 6.2 % and 6.3 % respectively but the

attendant of B is twice that of A. The performance measurement indices presented below

take into account not just the return, but also the risk on the security or portfolio.

47

Management strategies, both active and passive, are dealt with in the following paragraph.

100 Asset and Risk Management

The indicators shown by us here are all based on relations produced by the ¬nan-

cial asset valuation model and more particularly on the CAPM equation. They therefore

assume that the hypotheses underlying this model are satis¬ed.

The ¬rst two indicators are based on the market straight-line equation and the CAPM

equation respectively; the third is a variation on the second.

3.3.3.2 Sharpe index

The market straight-line equation is:

EM ’ RF

EP = RF + · σP

σM

which can be rewritten as follows:

EP ’ RF EM ’ RF

=

σP σM

This relation expresses that the excess return (compared to the risk-free rate), standardised

by the standard deviation, is (in equilibrium) identical to a well-diversi¬ed portfolio and

for the market. The term Sharpe index is given to the expression

EP ’ RF

SIP =

σP

which in practice is compared to the equivalent expression calculated for a market repre-

sentative index.

Example

Let us take the data used for the simple Sharpe index model (Section 3.2.4):

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 consider the speci¬c portfolio relative to the value » = 0.010 for the risk

parameter. In this case, we will have X1 = 0.4387, X2 = 0.1118 and X3 = 0.4496, and

therefore EP = 0.0758 and σP = 0.0912. We will also have EI = 0.04 and σI = 0.0671,

and RF is taken, as in Section 3.2.5, as 0.03.

The Sharpe index for the portfolio is therefore given as:

0.0758 ’ 0.03

SIP = = 0.7982

0.0912

The Sharpe index relative to the index equals:

0.04 ’ 0.03

SII = = 0.1490

0.0671

This shows that the portfolio in question is performing better than the market.

Equities 101

Although Section 3.3.4 is given over to the portfolio management strategies for equities,

some thoughts are also given on the role of the Sharpe index in the taking of investment

(and disinvestment) decisions.

Suppose that we are in possession of a portfolio P and we are envisaging the purchase of

an additional total of equities A, the proportions of P and A being noted respectively as XP

and XA . Of course, XP + XA = 1 and XA is positive or negative depending on whether

an investment or a disinvestment is involved. The portfolio produced as a result of the

decision taken will be noted as P and its return will be given by RP = XP RP + XA RA .

The expected return and variance on return for the new portfolio are

EP = (1 ’ XA )EP + XA EA

σP = (1 ’ XA )2 σP + XA σA + 2XA (1 ’ XA )σP σA ρAP

2 2 22

We admit as the purchase criterion for A the fact that the Sharpe index for the new

portfolio is at least equal to that of the old one: SIP ≥ SIP , which is expressed as:

(1 ’ XA )EP + XA EA ’ RF EP ’ RF

≥

σP σP

By isolating the expected return on A, we obtain as the condition

σP EP ’ RF

EA ≥ EP + ’1

σP XA

It is worth noting that if A does not increase the risk of the portfolio (σP ¤ σP ), it is not

even necessary for EA ≥ EP to purchase A.

Example

Suppose that one has a portfolio for which EP = 0.08, that the risk-free total is RF = 0.03

and that one is envisaging a purchase of A at the rate XA = 0.02. The condition then

becomes

5 σP

EA ≥ 0.08 + ’1

2 σP

In the speci¬c case where the management of risks is such that σA = σP , the ratio of

the standard deviations is given by

σP

= (1 ’ XA )2 + XA + 2XA (1 ’ XA )ρAP

2

σP

= 0.9608 + 0.0392ρAP

This allows the conditions of investment to be determined according to the correlation

coef¬cient value: if ρAP = ’1, 0 or 1, the condition becomes EA ≥ 0.02, EA ≥ 0.0305

and EA ≥ 0.08 respectively.

102 Asset and Risk Management

3.3.3.3 Treynor index

The CAPM equation for the k th equity in the portfolio, Ek = RF + βk (EM ’ RF ), allows

the following to be written:

N N N

Xk Ek = Xk · RF + Xk βk · (EM ’ RF )

k=1 k=1 k=1

or, EP = RF + βP (EM ’ RF )

Taking account of the fact that βM = 1, this last relation can be written as:

EP ’ RF EM ’ RF

=

βP βM

The interpretation is similar to that of the Sharpe index. The Treynor index is therefore

de¬ned by:

EP ’ RF

TI P =

βP

which will be compared to the similar expression for an index.

Example

Let us take the data above, with the addition of (see Section 3.2.4): β1 = 0.60, β2 =

1.08, β3 = 1.32. This will give βP = 0.9774.

The Treynor index for this portfolio is therefore obtained by:

0.0758 ’ 0.03

TI P = = 0.0469

0.9774

meanwhile, the index relative to the index is

0.04 ’ 0.03

TI I = = 0.0100

1

This will lead to the same conclusion.

3.3.3.4 Jensen index

According to the reasoning in the Treynor index, we have EP ’ RF = βP (EM ’ RF ).

This relation being relative (in equilibrium) for a well-diversi¬ed portfolio, a portfolio

P will present an excess of return in relation to the market if there is a number ±P > 0

so that: EP ’ RF = ±P + βP (EM ’ RF ).

The Jensen index, JI P = ±, is the estimator for the constant term of the regression:

ˆ

EP ,t ’ RF,t = ± + β (EI,t ’ RF,t ).

For this, the variable to be explained (explanatory) is the excess of return of portfolio

in relation to the risk-free rate (excess of return of market representative index). Its value

is, of course, compared to 0.

Equities 103

Example

It is easy to verify that with the preceding data, we have

J IP = (0.0758 ’ 0.03) ’ 0.9774 · (0.04 ’ 0.03) = 0.0360, which is strictly positive.

3.3.4 Equity portfolio management strategies

3.3.4.1 Passive management

The aim of passive management is to obtain a return equal to that of the market. By the

de¬nition of the market, the gains (returns higher than market returns) realised by certain

investors will be compensated by losses (returns lower than market returns) suffered by

other investors:48 the average return obtained by all the investors is the market return.

The reality is a little different: because of transaction costs, the average return enjoyed

by investors is slightly less than the market return.

The passive strategy therefore consists of:

• Putting together a portfolio of identical (or very similar) composition to the market,

which corresponds to optimal diversi¬cation.

• Limiting the volume of transactions as far as is possible.

This method of operation poses a number of problems. For example, for the management

of some types of portfolio, regulations dictate that each security should only be present

to a ¬xed maximum extent, which is incompatible with passive management if a security

represents a particularly high level of stock-exchange capitalisation on the market. Another

problem is that the presence of some securities that not only have high rates but are

indivisible, and this may lead to the construction of portfolios with a value so high that

they become unusable in practice.

These problems have led to the creation of ˜index funds™, collective investment organ-

isations that ˜imitate™ the market. After choosing an index that represents the market in

which one wishes to invest, one puts together a portfolio consisting of the same securities

as those in the index (or sometimes simply the highest ones), in the same proportions.

Of course, as and when the rates of the constituent equities change, the composition

of the portfolio will have to be adapted, and this presents a number of dif¬culties. The

reaction time inevitably causes differences between the return on the portfolio and the

market return; these are known as ˜tracking errors™. In addition, this type of management

incurs a number of transaction costs, for adapting the portfolio to the index, for reinvesting

dividends etc. For these reasons, the return on a certain portfolio will in general be slightly

lower than that of the index.

3.3.4.2 Active management

The aim of active management is to obtain a return higher than the market return.

A fully ef¬cient market can be beaten only temporarily and by chance: in the long

term, the return cannot exceed the market return. Active management therefore suggests

that the market is fully ef¬cient.

48

This type of situation is known in price theory as a zero total game. Refer for example to Binmore K., Jeux et th´ orie

e

des jeux, De Boeck & Larcier, 1999.

104 Asset and Risk Management

Two main principles allow the target set to be achieved.

1) Asset allocation, which evolves over time and is also known as market timing, consists

of putting together a portfolio consisting partly of the market portfolio or an index

portfolio and partly of a risk-free asset (or one that is signi¬cantly less risk than

equities, such as a bond). The respective proportions of these two components are

then changed as time passes, depending on whether a rise or a fall in the index is

anticipated.

2) Stock picking consists of putting together a portfolio of equities by choosing the

securities considered to be undervalued and likely to produce a return higher than the

market return in the near or more distant future (market reaction).

In practice, professionals use strategies based on one of the two approaches or a mixture

of the two.

In order to assess the quality of active management, the portfolio put together should

be compared with the market portfolio from the point of view of expected return and of

risk incurred. These portfolio performance indexes have been studied in Section 3.3.3.

Let us now examine some methods of market timing and a method of stock picking:

the application of the dividend discount model.

3.3.4.3 Market timing

This technique therefore consists of managing a portfolio consisting of the market portfolio

(M) for equities and a bond rate (O) in the respective proportions X and 1 ’ X, X being

adapted according to the expected performance of the two components.

These performances, which determine a market timing policy, may be assessed using

different criteria:

• The price-earning ratio, introduced in Section 3.1.3: PER = rate/pro¬t.

• The yield gap, which is the ratio between the return on the bond and the return on the

equities (dividend/rate).

• The earning yield, which is the product of the PER by the bond rate.

• The risk premium, which is the difference between the return on the market portfolio

and the return on the bond: RP = EM ’ EO . It may be estimated using a history, but

it is preferable to use an estimation produced beforehand by a ¬nancial analyst, for

example using the DDM (see below).

Of course, small values for the ¬rst three criteria are favourable to investment in equities;

the situation is reversed for the risk premium.

The ¬rst method for implementing a market timing policy is recourse to decision

channels. If one refers to one of the four criteria mentioned above as c, for which historical

observations are available (and therefore an estimation c for its average and sc for its

standard deviation), we choose, somewhat arbitrarily, to invest a certain percentage of

equities depending on the observed value of c compared to c, the difference between the

two being modulated by sc . We may choose for example to invest 70 %, 60 %, 50 %, 40 %

Equities 105

c

30 %

40 %

c 50 %

t

60 %

70 %

Figure 3.29 Fixed decision channels

Figure 3.30 Moving decision channels

or 30 % in equities depending on the position of c in relation to the limits:49 c ’ 3 sc , c ’

2

sc , c + 1 sc and c + 3 sc (Figure 3.29).

1

2 2 2

This method does not take account of the change of the c parameter over time. The

c and sc parameters can therefore be calculated over a sliding history (for example, one

year) (Figure 3.30).

Another, more rigorous method can be used with the risk premium only. In the search

for the ef¬cient frontier, we have looked each time for the minimum with respect to the

proportions of the expression σP ’ »EP in which the » parameter corresponds to the

2

risk (» = 0 for a cautious portfolio, » = +∞ for a speculative portfolio). This parameter

is equal to the slope of the straight line in the plane (E, σ 2 ) tangential to the ef¬cient

frontier and coming from the point (RF , 0). According to the separation theorem (see

Section 3.3.1), the contact point for this tangent corresponds to the market portfolio (see

σM2

Figure 3.31) and in consequence we have: » = .

EM ’ RF

In addition, the return on portfolio consisting of a proportion X of the market portfolio

and a proportion 1 ’ X of the bond rate is given by RP = XRM + (1 ’ X)RO , which

49

The order of the channels must be reversed for the risk premium.

106 Asset and Risk Management

σP

2

σM

2

RF EM EP

Figure 3.31 Separation theorem

allows the following to be determined:

EP = XEM + (1 ’ X)EO

σP = X2 σM + 2X(1 ’ X)σMO + (1 ’ X)2 σO

2 2 2

The problem therefore consists of determining the value of X, which minimises the

expression:

Z(X) = σP ’ »EP = X2 σM + 2X(1 ’ X)σMO + (1 ’ X)2 σO ’ »[XEM + (1 ’ X)EO ].

2 2 2

The derivative of this function:

Z (X) = 2XσM + 2(1 ’ 2X)σMO ’ 2(1 ’ X)σO ’ »(EM ’ EO )

2 2

= 2X(σM ’ 2σMO + σO ) + 2σMO ’ 2σO ’ » · RP

2 2 2

provides the proportion sought:

» · RP ’ 2(σMO ’ σO )

2

X=

2(σM ’ 2σMO + σO )

2 2

or, in the same way, replacing » and RP by their value:

EM ’ E0

· σM ’ 2(σMO ’ σO )

2 2

E ’ RF

X= M

2(σM ’ 2σMO + σO )

2 2

Example

If we have the following data:

EM = 0.08 σM = 0.10

EO = 0.06 σO = 0.02

RF = 0.04 ρMO = 0.6

Equities 107

we can calculate successively:

σMO = 0.10 · 0.02 · 0.6 = 0.0012

0.102

»= = 0.25

0.08 ’ 0.04

PR = 0.08 ’ 0.06 = 0.02

and therefore:

0.25 · 0.02 ’ 2 · (0.0012 ’ 0.022 )

X= = 0.2125

2 · (0.102 ’ 2 · 0.0012 + 0.022 )

Under these conditions, therefore, it is advisable to invest 21.25 % in equities (market

portfolio) and 78.75 % in bonds.

3.3.4.4 Dividend discount model

The aim of the dividend discount model, or DDM, is to compare the expected return

of an equity and its equilibrium return, which will allow us to determine whether it is

overvalued or undervalued.

˜

The expected return, Rk , is determined using a model for updating future dividends.

A similar reasoning to the type used in the Gordon“Shapiro formula (Section 3.1.3),

or a generalisation of that reasoning, can be applied. While the Gordon“Shapiro relation

suggests a constant rate of growth for dividends, more developed models (two-rate model)

use, for example, a rate of growth constant over several years followed by another, lower

rate for subsequent years. Alternatively, a three-rate model may be used with a period of

a few years between the two constant-rate periods in which the increasing rate reduces

linearly in order to make a continuous connection.

The return to equilibrium Ek is determined using the CAPM equation (Section 3.3.1).

This equation is written Ek = RF + βk (EM ’ RF ).

If one considers that it expresses Ek as a function of βk , we are looking at a straight-line

equation; the line passes through the point (0, RF ) and since βM = 1, through the point

(1, EM ). This straight line is known as the ¬nancial asset evaluation line or the security

market line.

˜

If the expected return Rk for each security is equal to its return on equilibrium Ek , all

˜

the points (βk , Rk ) will be located on the security market line. In practice, this is not the

case because of certain inef¬ciencies in the market (see Figure 3.32).

˜

Rk

•

• • •

EM

• •

•

• •

RF

• •

bk

1

Figure 3.32 Security market line

108 Asset and Risk Management

˜

This technique considers that the Rk evaluation made by the analysts is correct and that

the differences noted are due to market inef¬ciency. Therefore, the securities whose repre-

sentative point is located above the security market line are considered to be undervalued,

and the market should sooner or later rectify the situation and produce an additional return

for the investor who purchased the securities.

3.4 EQUITY DYNAMIC MODELS

The above paragraphs deal with static aspects, considering merely a ˜photograph™ of the

situation at a given moment. We will now touch on the creation of models for develop-

ments in equity returns or rates over time.

The notation used here is a little different: the value of the equity at moment t is noted

as St . This is a classic notation (indicating ˜stock™), and in addition, the present models

are used among other things to support the development of option valuation models

for equities (see Section 5.3), for which the notation Ct is reserved for equity options

(indicating ˜call™).

Finally, we should point out that the following, unless speci¬ed otherwise, is valid only

for equities that do not give rise to the distribution of dividends.

3.4.1 Deterministic models

3.4.1.1 Discrete model

Here, the equity is evaluated at moments t = 0, 1, etc. If it is assumed that the return on

St+1 ’ St

the equity between moments t and t + 1 is i, we can write: i = , which leads

St

to the evolution equation St+1 = St · (1 + i).

If the rate of return i is constant and the initial value S0 is taken into account, the

equation (with differences) above will have the solution: St = S0 · (1 + i)t .

If the rate varies from period to period (ik for the period] k ’ 1; k]), the previous

relation becomes St = S0 . (1 + i1 ) (1 + i2 ) . . . (1 + it ).

3.4.1.2 Continuous model

We are looking here at an in¬nitesimal development in the value of the security. If it is

assumed that the return between moments t and t + t (with ˜small™ t) is proportional

to the duration t with a proportionality factor δ:

St+ ’ St

t

δ· t=

St

the evolution equation is a differential equation50 St = St · δ.

The solution to this equation is given by St = S0 · eδt .

The link will be noted between this relation and the relation corresponding to it for the

discrete case, provided δ = ln (1 + i).

t tend towards 0.

50

Obtained by making

Equities 109

If the rate of return δ is not constant, the differential development equation will take

t

δ(t) dt

the form S t = St · δ (t), thus leading to the more complex solution St = S0 · e .

0

Note

The parameters appear in the above models (the constant rates i and δ, or the variable

rates i1 , i2 , . . . and δ(t)) should of course for practical use be estimated on the basis of

historical observations.

3.4.1.3 Generalisation

These two aspects, discrete and continuous, can of course be superimposed. We there-

fore consider:

• A continuous evolution of the rate of return, represented by the function δ(t). On top

of this:

• A set of discrete variations occurring at periods „1 , „2 , . . . , „n so that the rate of return

between „k’1 and „k is equal to ik .

If n is the greatest integer so that „n ¤ t, the change in the value is given by

t

δ(t) dt

„1 „2 ’„1 „n ’„n’1 t’„n

St = S0 · (1 + i1 ) (1 + i2 ) . . . (1 + in ) (1 + in+1 ) ·e .

0

This presentation will allow the process of dividend payment, for example, to be taken

into consideration in a discrete or continuous model. Therefore, where the model includes

only the continuous section represented by δ(t), the above relation represents the change

in the value of an equity that pays dividends at periods „1 , „2 etc. with a total Dk paid in

„k and linked to ik by the relation

Dk

ik = ’ (’)

Sk

Here, Sk (’) is the value of the security just before payment of the k th dividend.

3.4.2 Stochastic models

3.4.2.1 Discrete model

It is assumed that the development from one period to another occurs as follows: equity

at moment t has the (random) value St and will at the following moment t + 1 have

one of the two values St .u (higher than St ) or St .d (lower than St ) with the respective

probabilities of ± and (1 ’ ±).

We therefore have d ¤ 1 ¤ u, but it is also supposed that d ¤ 1 < 1 + RF ¤ u, without

which the arbitrage opportunity will clearly be possible. In practice, the parameters u, d

and ± should be estimated on the basis of observations.

110 Asset and Risk Management

Generally speaking, the following graphic representation is used for evolutions in equity

prices:

S = St · u (±)

’ ’ t+1

St ’ ’

’

’’

’ S = S · d (1 ’ ±)

t+1 t

It is assumed that the parameters u, d and ± remain constant over time and we will no

longer clearly show the probability ± in the following graphs; the rising branches, for

example, will always correspond to the increase (at the rate u) in the value of the security

with the probability ±.

Note that the return of the equity between the period t and (t + 1) is given by

St+1 ’ St u ’ 1 (±)

=

d ’ 1 (1 ’ ±)

St

Between the moments t + 1 and t + 2, we will have, in the same way and according to

the branch obtained at the end of the previous period:

’ S = St+1 · u = St · u

2

’ ’ t+2

’

St+1’

’’’ S = S · d = S · ud

t+2 t+1 t

or

St+2 = St+1 · u = St · ud

’’

’’

St+1’

’’’ S = S · d = S · d2

t+2 t+1 t

It is therefore noted that a rise followed by a fall leads to the same result as a fall followed

by a rise. Generally speaking, a graph known as a binomial trees can be constructed (see

Figure 3.33), rising from period 0 (when the equity has a certain value S0 ) to the period t.

It is therefore evident that the (random) value of the equity at moment t is given

by St = S0 · uN d t’N , in which the number N of rises is of course a random binomial

variable51 with parameters (t; ±):

t

± k (1 ’ ±)t’k

Pr[N = k] =

k

The following property can be demonstrated:

E(St ) = S0 · (±u + (1 ’ ±)d)t

S0 . u3 ¦

S 0 . u2

S0 . u S0 . u2d ¦

S0 . ud

S0

S0 . d S0 . ud2 ¦

S 0 . d2

S0 . d2 ¦

Figure 3.33 Binomial tree

51

See Appendix 2 for the development of this concept and for the properties of the random variable.

Equities 111

In fact, what we have is:

t

t

S0 · uk d t’k · ± k (1 ’ ±)t’k

E(St ) =

k

k=0

t

t

(±u)k ((1 ’ ±)d)t’k

= S0 ·

k

k=0

This leads to the relation declared through the Newton binomial formula.

Note that this property is a generalisation for the random case of the determinist formula

St = S0 · (1 + i)t .

3.4.2.2 Continuous model

The method of equity value change shown in the binomial model is of the random

walk type. At each transition, two movements are possible (rise or fall) with unchanged

probability. When the period between each transaction tends towards 0, this type of

random sequence converges towards a standard Brownian motion or SBM.52 Remember

that we are looking at a stochastic process wt (a random variable that is a function of

time), which obeys the following processes:

• w0 = 0.

• wt is a process with independent increments : if s < t < u, then wu ’ wt is independent

of wt ’ ws .

• wt is a process with stationary increments : the random variables wt+h ’ wt and wh

are identically distributed.

• Regardless of what t may be, the random variable wt is distributed according to a

√

normal law of zero mean and standard deviation t:

1

e’x /2t

2

fwt (x) = √

2πt

The ¬rst use of this process for modelling the development in the value of a ¬nancial asset

was produced by L. Bachelier.53 He assumed that the value of a security at a moment t

is a ¬rst-degree function of the SBM: St = a + bwt . According to the above de¬nition, a

is the value of the security at t = 0 and b is a measure of the volatility σ of the security

for each unit of time. The relation used was therefore St = S0 + σ · wt .

The shortcomings of this approach are of two types:

• The same absolute variation (¤10 for example) corresponds to variations in return that

are very different depending on the level of price (20 % for a quotation of ¤50 and 5 %

for a value of ¤200).

• The random variable St follows a normal law with mean S0 and standard deviation

√

σ t; this model therefore allows for negative prices.

52

Appendix 2 provides details of the results, reasoning and properties of these stochastic processes.

53

Bachelier L., Th´ orie de la sp´ culation, Gauthier-Villars, 1900. Several more decades were to pass before this reasoning

e e

was ¬nally accepted and improved upon.

112 Asset and Risk Management

For this reason, P. Samuelson54 proposed the following model. During the short interval

of time [t; t + dt], the return (and not the price) alters according to an Itˆ process :

o

St+dt ’ St dSt

= = ER · dt + σR · dwt

St St

Here, the non-random term (the trend) is proportional to the expected return and the

stochastic term involves the volatility for each unit of time in this return. This model is

termed a geometric Brownian motion.

Example

Figure 3.34 shows a simulated trajectory (development over time) for 1000 very short

periods with the values ER = 0.1 and σR = 0.02, based on a starting value of S0 = 100.

We can therefore establish the ¬rst property in the context of this model: the stochastic

process St showing the changes in the value of the equity can be written as

σR2

St = S0 · exp ER ’ · t + σR · wt

2

This shows that St follows a log-normal distribution (it can only take on positive values).

In fact, application of the Itˆ formula55 to the function f (x, t) = ln ·x where x = St ,

o

we obtain:

1 12 1

d(ln St ) = 0 + ER St ’ 2 σR St2 · dt + σR St · dwt

St St

2St

σR2

= ER ’ · dt + σR · dwt

2

σR2

—

This equation resolves into: ln St = C + ER ’ · t + σR · wt

2

101.2

101

100.8

100.6

100.4

100.2

100

99.8

99.6

99.4

99.2

1 101 201 301 401 501 601 701 801 901 1001

Figure 3.34 Geometric Brownian motion

54

Samuelson P., Mathematics on speculative price, SIAM Review, Vol. 15, No. 1, 1973.

55

See Appendix 2.

Equities 113

The integration constant C — is of course equal to ln S0 and the passage to the exponential

gives the formula declared.

It is then easy to deduce the moments of the random variable St :

E(St ) = S0 · eER t

var(St ) = S0 · e2ER t (eσR t ’ 1)

2

2

The ¬rst of these relations shows that the average return E(St /S0 ) on this equity over the

interval [0; t] is equivalent to a capitalisation at the instant rate, ER .

A second property can be established, relative to the instant return on the security

over the interval [0; t]. This return obeys a normal distribution with mean and standard

deviation, shown by

σR σR

2

ER ’ ;√

t

2

This result may appear paradoxical, as the average of the return is not equal to ER .

This is because of the structure of the stochastic process and is not incompatible with the

intuitive solution, as we have E(St ) = S0 · eER t .

To establish this property, expressing the stochastic instant return process as δt , we can

write St = S0 · eδt ·t, that is, according to the preceding property,

St

1

δt = · ln

t S0

σR wt

2

= ER ’ + σR ·

t

2

This establishes the property.

4

Bonds

4.1 CHARACTERISTICS AND VALUATION

To an investor, a bond is a ¬nancial asset issued by a public institution or private company

corresponding to a loan that confers the right to interest payments (known as coupons)

and repayment of the loan upon maturity. It is a negotiable security and its issue price,

redemption value, coupon total and life span are generally known and ¬xed beforehand.

4.1.1 De¬nitions

A bond is characterised by various elements:

1. The nominal value or NV of a bond is the amount printed on the security, which,

along with the nominal rate of the bond, allows the coupon total to be determined.

2. The bond price is shown as P . This may be the price at issue (t = 0) or at any

subsequent moment t. The maturity price is of course identical to the redemption

value or R mentioned above.

3. The coupons Ct constitute the interest paid by the issuer. These are paid at various

periods, which are assumed to be both regular and annual (t = 1, 2, . . . , T ).

4. The maturity T represents the period of time that separates the moment of issue and

the time of reimbursement of the security.

The ¬nancial ¬‚ows associated with a bond are therefore:

• From the purchaser, the payment of its price; this may be either the issue price paid to

the issuer or the rate of the bond paid to any seller at a time subsequent to the issue.

• From the issuer, the payment of coupons from the time of acquisition onwards and the

repayment on maturity.

The issue price, nominal value and repayment value are not necessarily equal. There may

be premiums (positive or negative) on issue and/or on repayment.

The bonds described above are those that we will be studying in this chapter; they are

known as ¬xed-rate bonds. There are many variations on this simple bond model.

It is therefore possible for no coupons to be paid during the bond™s life span, the

return thus being only the difference between the issue price and the redemption value.

This is referred to as a zero-coupon bond .1 This kind of security is equivalent to a

¬xed-rate investment.

There are also bonds more complex than those described above, for example:2

• Variable rate bonds, for which the value of each coupon is determined periodically

according to a parameter such as an index.

1

A debenture may therefore, in a sense, be considered to constitute a superimposition of zero-coupon debentures.

2

Read for example Colmant B., Delfosse V. and Esch L., Obligations, Les notions ¬nanci` res essentielles, Larcier, 2002.

e

Also: Fabozzi J. F., Bond Markets, Analysis and Strategies, Prentice-Hall, 2000.

116 Asset and Risk Management

• Transition bonds, which authorise repayment before the maturity date.

• Lottery bonds, in which the (public) issuer repays certain bonds each year in a draw.

• Convertible bonds (convertible into equities) etc.

4.1.2 Return on bonds

The return on a bond can of course be calculated by the nominal rate (or coupon rate)

rn , which is de¬ned as the relation between the total value of the coupon and the nomi-

nal value

C

rn =

NV

This de¬nition, however, will only make sense if all the different coupons have the same

value. It can be adapted by replacing the denominator with the price of the bond at a

given moment. The nominal rate is of limited interest, as it does not include the life span

of the bond at any point; using it to describe two bonds is therefore rather pointless.

For a ¬xed period of time (such as one year), it is possible to use a rate of return

equivalent to the return on one equity:

Pt + Ct ’ Pt’1

Pt’1

This concept is, however, very little used in practice.

4.1.2.1 Actuarial rate on issue

The actuarial rate on issue, or more simply the actuarial rate (r) of a bond is the rate for

which there is equality between the discounted value of the coupons and the repayment

value on one hand and the issue price on the other hand:

T

Ct (1 + r)’t + R(1 + r)’T

P=

t=1

Example

Consider for example a bond with a period of six years and nominal value 100, issued

at 98 and repaid at 105 (issue and reimbursement premiums 2 and 5 respectively) and a

nominal rate of 10 %. The equation that de¬nes its actuarial rate is therefore:

10 + 105

10 10 10 10 10

98 = + + + + +

1+r (1 + r)2 (1 + r)3 (1 + r)4 (1 + r)5 (1 + r)6

This equation (sixth degree for unknown r) can be resolved numerically and gives r =

0.111044, that is, r = approximately 11.1 %.

The actuarial rate for a zero-coupon bond is of course the rate for a risk-free investment,

and is de¬ned by

P = R(1 + r)’T

Bonds 117

The rate for a bond issued and reimbursable at par (P = N V = R), with coupons that

are equal (Ct = C for all t) is equal to the nominal rate: r = rn . In fact, for this particular

type of bond, we have:

T

C(1 + r)’t + P (1 + r)’T

P=

t=1

(1 + r)’1 ’ (1 + r)’T ’1

+ P (1 + r)’T

=C ’1

1 ’ (1 + r)

1 ’ (1 + r)’T

+ P (1 + r)’T

=C

r

From this, it can be deduced that r = C/P = rn .

4.1.2.2 Actuarial return rate at given moment

The actuarial rate as de¬ned above is calculated when the bond is issued, and is sometimes

referred to as the ex ante rate. It is therefore assumed that this rate will remain constant

throughout the life of the security (and regardless of its maturity date).

A major principle of ¬nancial mathematics (the principle of equivalence) states that

this rate does not depend on the moment at which the various ¬nancial movements are

˜gathered in™.

Example

If, for the example of the preceding paragraph (bond with nominal value of 100 issued at

98 and repaid at 105), paying an annual coupon of 10 at the end of each of the security™s

six years of life) and with an actuarial rate of 11.1 %, one examines the value acquired

for example on the maturity date, we have:

• for the investment, 98 · (l + r)6 ;

• for the generated ¬nancial ¬‚ows: 10 · [(l + r)5 + (l + r)4 + (l + r)3 + (l + r)3 + (l +

r)2 + (l + r)1 + l] + 105.

The equality of these two quantities is also realised for r = 11.1 %.

If we now place at a given moment t anywhere between 0 and T , and are aware of the

change in the market rate between 0 and t, the actuarial rate of return at the moment3 t,