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σj2 = βj σM + σµ2j
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

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,

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