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Insurance products
Financia
l
Capital markets / Treasury risk
Market risk, Liquidity risk Credit
Analytics & modelling analytics
Integrated
Strategic Strategic risk
risk management
Strategic,
business,
COSO Control
Quality process
operations Self-
nal Engineering & cultural
Operatio compliance assessment Risk
management
COSO
financial
Financial
internal
ess control
Proc


Figure 2.6 Integrated Risk Management
Source: Deloitte & Touche


2.2.1.2 Back of¬ce vs. risk management
With regard to the back of¬ce vs. risk management debate, it is well worth remembering
that depending on the views of the regulator, the back of¬ce generally deals with the
administration of operations and as such must, like every other function in the institution,
carry out a number of control checks.
There are two types of back of¬ce control check:

• The daily control checks carried out by staff, for example each employee™s monitoring
of their suspense account.
• The ongoing continuous checks, such as monitoring of the accuracy and comprehen-
siveness of data communicated by persons responsible for business and operational
functions in order to oversee the operations administratively.

However, when mentioning checks to be made by risk management, one refers to excep-
tion process checks in accordance with the bank™s risk management policy, for example:

• Monitoring any limit breaches (limits, stop losses etc.).
• Monitoring (reconciliation of) any differences between positions (or results) taken (cal-
culated) within various entities (front, back, accounting etc.).

2.2.1.3 Internal audit vs. risk management
The role of audit in a ¬nancial group is based on four main aspects:

• Producing a coherent plan for the audit activities within the group.
Changes in Financial Risk Management 23

• Ensuring that the whole of the auditable activities, including the group™s subsidiaries
and holding company within the responsibilities of the parent company, are covered
through the conduct or review of audits.
• Applying a uniform audit method across all group entities.
• On the basis of a homogeneous style of reporting, providing the directors of the parent
company and of the subsidiaries maximum visibility on the quality of their internal
control systems.

Although risk management by its very nature is also involved with the ef¬ciency of the
internal control system, it must be remembered that this function is a tool designed to help
the management of the institution in its decision making. Risk management is therefore
part of the auditable domain of the institution.
We saw the various responsibilities of risk management in Section 2.1.


2.2.1.4 Position of legal risk
In practice, every banking transaction is covered by a contract (spoken or written) that
contains a certain degree of legal risk. This risk is more pronounced in transactions
involving complex securities such as derivative products or security lending.
From the regulator™s point of view, legal risk is the risk of contracts not being legally
enforceable. Legal risk must be limited and managed through policies developed by
the institution, and a procedure must be put in place for guaranteeing that the parties™
agreements will be honoured.
Before entering into transactions related to derivatives, the bank must ensure that its
counterparties have the legal authority to enter into these deals themselves. In addition,
the bank must verify that the conditions of any contract governing its activities in relation
to the counterpart are legally sound.
The legal risk linked to stock-market deals can in essence be subdivided into four types
of subrisk.

1. Product risk, which arises from the nature of the deal without taking into account
the counterparty involved; for example, failure to evaluate the legal risk when new
products are introduced or existing products are changed.
2. Counterparty risk. Here the main risk is that the counterparty does not have the legal
capacity to embark on the deal in question. For example, the counterparty may not
have the capacity to trade in derivative products or the regulatory authority for speci¬c
transactions, or indeed may not even have the authority to conclude a repo contract.
3. Transaction risk. This is certainly the most signi¬cant part of the legal risk and covers
actions undertaken in the conclusion of operations (namely, transaction and documen-
tation). When the deal is negotiated and entered into, problems may arise in connection
with regulatory or general legal requirements. For example: closing a spoken agree-
ment without listing the risks involved beforehand, compiling legal documentation or
contracts without involving the legal department, negotiating derivative product deals
without involving the legal department or without the legal department reviewing
the signed ISDA Schedules, signing Master Agreements with foreign counterparties
without obtaining an outside legal opinion as to the validity of default, and ¬nally
documentary errors such as inappropriate signatures, failure to sign the document or
24 Asset and Risk Management

failure to set up procedures aimed at ensuring that all contractual documentation sent
to counterparties is returned to the institution duly signed.
4. Process risk. In the event of litigation in connection with a deal or any other con-
sequence thereof, it will be necessary to undertake certain action to ensure that the
¬nancial consequences are minimised (protection of proof, coordination of litigation
etc.). Unfortunately, this aspect is all too often missing: records and proof of trans-
action are often insuf¬cient (failure to record telephone conversations, destruction of
emails etc.).

These four categories of risk are correlated. Fundamentally, the legal risk can arise at any
stage in the deal (pre-contractual operations, negotiation, conclusion and post-contractual
procedures).
In this context of risk, the position of the legal risk connected with the ¬nancial deals
within the risk management function presents certain advantages:

• Assessment of the way in which the legal risk will be managed and reduced.
• The function has a central position that gives an overall view of all the bank™s activities.
• Increased ef¬ciency in the implementation of legal risk management procedures in
¬nancial transactions, and involvement in all analytical aspects of the legal risk on the
capital market.


2.2.1.5 Integration
It is worrying to note the abundance of energy being channelled into the so-called problem
of the ˜fully integrated computerised risk-management system™. One and the same system
for market risks, credit risks and operational risks? Not possible!
The interesting problem with which we are confronted here is that of integrating
systems for monitoring different types of risk. We have to ask ourselves questions on
the real added value of getting everything communicated without including the unmen-
tionable “ the poorly secured accessories such as spreadsheets and other non-secured
relational databases.
Before getting involved with systems and expensive balance sheets relating to the
importance of developing ˜black boxes™ we think it wiser to ask a few questions on the
cultural integration of risk management within a business.
The regulator has clearly understood that the real risk management debate in the next 10
years will be on a qualitative, not a quantitative, level. Before moving onto the quantitative
models proposed by Basle II, should we not ¬rst of all pay attention to a series of
qualitative criteria by organising ourselves around them? Surely the ¬gures produced by
advanced operational risk methods are of a behavioural nature in that they show us the
˜score to beat™. To sum up, is it better to be well organised with professional human
resources who are aware of the risk culture, or to pride ourselves on being the owners of
the Rolls Royce of the Value at Risk calculation vehicles?
When one remembers that Moody™s5 is attaching ever-increasing importance to the
evaluation of operational risk as a criterion for awarding its ratings, and the impact of
these ratings on ¬nance costs, is it not worth the trouble of achieving compliance from

5
Moody™s, Moody™s Analytical framework for Operational Risk Management of Banks, Moody™s, January 2003.
Changes in Financial Risk Management 25

the qualitative viewpoint (notwithstanding the savings made on capital through bringing
the equity fund into line)?
A risk management function should ideally:

• Report directly to executive management.
• Be independent of the front and back of¬ce functions.
• Be located at a suf¬ciently senior hierarchical level to guarantee real independence, hav-
ing the authority and credibility it needs to ful¬l its function, both internally (especially
vis-` -vis the front and back of¬ces) and externally (vis-` -vis the regulator, external audit
a a
and the ¬nancial community in general).
• Be a member of the asset and liability management committee.
• Where necessary, oversee all the decentralised risk-management entities in the sub-
sidiaries.
• Have as its main task the proposal of an institution-wide policy for monitoring risks
and ensuring that the decisions taken by the competent bodies are properly applied,
relying on the methodologies, tools and systems that it is responsible for managing.
• Have a clearly de¬ned scope of competence, which must not be limited to market
and credit risks but extend to operational risks (including insurance and BCP), the
concentration risk and the risks linked to asset management activity in particular.
• Play a threefold role in the ¬eld of risks: advice, prevention and control.

But at what price?

2.2.2 The ˜cost™ of risk management
A number of businesses believed that they could make substantial savings by spending
a bare minimum on the risk management function. It is this serious lack of foresight,
however, that has led to collapse and bankruptcy in many respectable institutions. The
commonest faults are:

1. One person wearing two ˜hats™ for the front and back of¬ce, a situation that is, to say
the least, conducive to fraud.
2. Non-existence of a risk management function.
3. Inability of management or persons delegated by management to understand the activ-
ities of the market and the products used therein.
4. Lack of regular and detailed reporting.
5. Lack of awareness of employees at all levels, of quanti¬able and/or non-quanti¬able
risks likely to be generated, albeit unwittingly, by those employees.
6. Incompatibility of volumes and products processed both with the business and with
back-of¬ce and accounting procedures.

At present, market and regulatory pressure is such that it is unthinkable for a respectable
¬nancial institution not to have a risk management function. Instead of complaining about
its cost, however, it is better to make it into a direct and indirect pro¬t centre for the
institution, and concentrate on its added value.
We have seen that a well-thought-out risk management limits:

• Excessive control (large-scale savings, prevention of doubling-up).
26 Asset and Risk Management

• Indirect costs (every risk avoided is a potential loss avoided and therefore
money gained).
• Direct costs (the capital needed to be exposed to the threefold surface of market, credit
and operational risk is reduced).

The promotion of a real risk culture increases the stability and quality of pro¬ts, and
therefore improves the competitive quality of the institution and ensures that it will last.


2.3 A NEW RISK-RETURN WORLD
2.3.1 Towards a minimisation of risk for an anticipated return
Assessing the risk from the investor™s point of view produces a paradox:

• On one hand, taking the risk is the only way of making the money. In other terms,
the investor is looking for the risk premium that corresponds to his degree of aversion
to risk.
• On the other hand, however, although accepting the ˜risk premium™ represents pro¬t
¬rst and foremost, it also unfortunately represents potential loss.

We believe that we are now moving from an era in which investors continually looked
to maximise return for a given level of risk (or without thinking about risk at all), into
a new era in which the investor, for an anticipated level of return, will not rest until the
attendant risk has been minimised.
We believe that this attitude will prevail for two different reasons:

1. Institutions that offer ¬nancial services, especially banks, know the levels of return that
their shareholders demand. For these levels of return, their attitude will be that they
must ¬nd the route that allows them to achieve their objective by taking the smallest
possible risk.
2. The individual persons and legal entities that make up the clientele of these institutions,
faced with an economic future that is less certain, will look for a level of return that at
least allows them to preserve their buying and investing power. This level is therefore
known, and they will naturally choose the ¬nancial solution that presents the lowest
level of risk for that level.


2.3.2 Theoretical formalisation
As will be explained in detail in Section 3.1.16 in the section on equities, the return R is a
random factor for which the probability distribution is described partly by two parameters:
a location index, the expected value of which is termed E(R), and a dispersion index,
the variance which is noted var(R). The ¬rst quantity corresponds to the expected return.

6
Readers are referred to this section and to Appendix 2 for the elements of probability theory needed to understand the
considerations that follow.
Changes in Financial Risk Management 27

E(R)


P




var(R)

Figure 2.7 Selecting a portfolio


E(R)



P



Q
E




var(R)


Figure 2.8 Selecting a portfolio


The square root of the second, σ (R) = var(R), is the standard deviation, which is a
measurement of risk.
A portfolio, like any isolated security, will therefore be represented by a mean-variance
couple. This couple depends on the expected return level and variance on return for the
various assets in the portfolio, but also on the correlations between those assets. A portfolio
will be ˜ideal™ for an investor (that is, ef¬cient), if, for a given expected return, it has
a minimal variance or if, for a ¬xed variance, it has a maximum expected return. All
the portfolios thus de¬ned make up what is termed the ef¬cient frontier, which can be
represented graphically in the Figure 2.7.
In addition, in the same plane the indifference curves represent the portfolios with an
equivalent mean-variance combination in the investor™s eyes (that is, they have for him
the same level of utility7 ). The selection is therefore made theoretically by choosing the
portfolio P from the ef¬ciency frontier located on the indifference curve located furthest
away (that is, with the highest level of utility), as shown in Figure 2.7.
In a situation in which an investor no longer acts on the basis of a classic utility
structure, but instead wishes for a given return E and then tries to minimise the variance,
the indifference curves will be cut off at the ordinate E and the portfolio selected will be
Q, which clearly presents a lower expected return than that of P but also carries a lower
risk that P . See Figure 2.8.

7
Readers are referred to Section 3.2.7.
Part II
Evaluating Financial Assets




Introduction
3 Equities
4 Bonds
5 Options
30 Asset and Risk Management

Introduction
Two fundamental elements
Evaluation of ¬nancial assets should take account of two fundamental aspects “ chance
and time.

The random aspect
It is obvious that the changes in value of a ¬nancial asset cannot be predicted in a
deterministic manner purely by looking at what happened in the past. It is quite clear
that for equities, whose rates ¬‚uctuate according to the law of supply and demand, these
rates are themselves dictated by the perception that market participants have of the value
of the business in question. The same applies to products that are sometimes de¬ned as
˜risk-free™, such as bonds; here, for example, there is the risk of bankruptcy, the risk of
possible change and the risk posed by changes in interest rates.
For this reason, ¬nancial assets can only be evaluated in a random context and the
models that we will be putting together cannot work without the tool of probability (see
Appendix 2 for the essential rules).

The temporal aspect
Some ¬nancial asset valuation models are termed monoperiodic, such as Markowitz™s
portfolio theory. These models examine the ˜photograph™ of a situation at a given moment
and use historical observations to analyse that situation.
On the other hand, there may be a wish to take account of development over time,
with the possible decision for any moment according to the information available at
that moment. The random variables mentioned in the previous paragraph then turn into
stochastic processes and the associated theories become much more complex.
For this reason, the following chapters (3, 4 and 5) will feature both valuation models
(from the static viewpoint) and development models (from the dynamic viewpoint). In
addition, for the valuation of options only, the development models for the underlying
asset are essential because of the intrinsic link between this product and the time variable.
The dynamic models can be further divided into discrete models (where development
is observed at a number of points spaced out over time) and continuous models (where
the time variable takes its values within a continuous range such as an interval). The
mathematical tools used for this second model are considerably more complex.

Two basic principles
The evaluation (or development) models, like all models, are based on a certain number
of hypotheses. Some of these are purely technical and have the aim of guaranteeing the
meaning of the mathematical expressions that represent them; they vary considerably
according to the model used (static or dynamic, discrete or continuous) and may take the
form of integrability conditions, restrictions on probability laws, stochastic processes, and
so on.
Other hypotheses are dictated by economic reality and the behaviour of investors,1
and we will be covering the two economic principles generally accepted in ¬nancial
models here.
1
We will be touching on this last aspect in Section 3.2.6
Evaluating Financial Assets 31

The perfect market
Often, a hypothesis that is so simplistic as to be unrealistic “ that of the perfect mar-
ket “ will be put forward. Despite its reductive nature, it de¬nes a context in which
¬nancial assets can be modelled and many studies have been conducted with the aim of
weakening the various elements in this hypothesis.
The perfect market2 is a market governed by the law of supply and demand, on which:

• Information is available in equal measure to all investors.
• There are no transactional or issue costs associated with the ¬nancial assets.
• There is no tax deduction on the income produced by the ¬nancial assets (where
increases in value or dividends are involved, for example).
• Short sales are authorised without restriction.

Absence of arbitrage opportunity
An arbitrage opportunity is a portfolio de¬ned in a context in which:

• No ¬nancial movement occurs within the portfolio during the period in question.
• The risk-free interest rate does not alter during the period in question and is valid for
any maturity date (a ¬‚at, constant rate curve).

This is a portfolio with an initial value (value at the point of constitution) that is
negative but presents a certain positive value at a subsequent time. More speci¬cally, if
the value of the portfolio at the moment t is termed Vt , we are looking at a portfolio for
which: V0 < 0 and VT ≥ 0 or V0 ¤ 0 and VT > 0.
Generally speaking, the absence of arbitrage opportunity hypothesis is constructed in
the ¬nancial modelling process. In fact, if it is possible to construct such portfolios,
there will be considerable interest in putting together a large number of them. However,
the numerous market operations (purchases/sales) that this process would require would
lead, through the effect of supply and demand, to alterations to the prices of the various
portfolio components until the pro¬ts obtained through the position of arbitrage would all
be lost.
Under this hypothesis, it can therefore be said that for a portfolio of value V put
together at moment 0, if VT = 0, no ¬nancial movement occurs in that portfolio between
0 and T and the interest rate does not vary during that period and is valid for any maturity
date (¬‚at, constant rate curve), then Vt = 0 for any t ∈ [0; T ].
This hypothesis of absence of arbitrage can be expressed as follows: in the context
mentioned above, a portfolio which has been put together so as not to contain any random
element will always present a return equal to the risk-free rate of interest.

The concept of ˜valuation model™
A valuation model for a ¬nancial asset is a relation that expresses quite generally the
price p (or the return) for the asset according to a number of explanatory variables3

2
See for example Miller and Modigliani, Dividend policy, growth and the valuation of shares, Journal of Business, 1961.
3
In these circumstances it is basically the risk of the security that is covered by the study; these explanatory variables are
known as risk factors.
32 Asset and Risk Management

X1 , X2 , . . . , Xn that represent the element(s) of the market likely to affect the price:
p = f (X1 , X2 , . . . , Xn ) + µ.
The residual µ corresponds to the difference between reality (the effective price p) and
the valuation model (the function f ).
Where the price valuation model is a linear model (as for equities), the risk factors
combine together to give, through the Central Limit Theorem, a distribution for the vari-
able p that is normal (at least in the ¬rst approximation), and is therefore de¬ned by the
two mean-variance parameters only.
On the other hand, for some types of assets such as options, the valuation model ceases
to be linear. The previous reasoning is no longer valid and neither are its conclusions.
We should state that alongside the risk factors that we will be mentioning, the explana-
tory elements of the market risk can also include:

• The imperfect nature of valuation models.
• The imperfect knowledge of the rules and limitations particular to the institution.
• The impossibility of anticipating changes to legal regulations.

We should also point out that alongside this market risk, the investor will be confronted
with other types of risk that correspond to the occurrence of exceptional events such as
wars, oil crises etc. This group of risks cannot of course be evaluated using techniques
designed for the risk market. The technique presented here will not therefore be including
these ˜event-based™ risks. However, this does not mean that the careful risk manager
should not include ˜catastrophe scenarios™, in order to take account of the exceptional
risks, alongside the methods designed to deal with the market risks.
In this section we will be covering a number of general principles relative to valuation
models, and mentioning one or another speci¬c model4 that will be analysed in further
detail in this second part.

Linear models
We will look ¬rst at the simple case in which the function f of the valuation model
is linear, or more speci¬cally, the case in which the price variation p = pt ’ p0 is a
¬rst-degree function of the variations X1 , . . . , Xn of the various explanatory variables
and of that ( µ) of the residue:

p = a0 + a1 X1 + . . . + an Xn + µ.

An example of the linear valuation model is the Sharpe simple index model used for
equities (see Section 3.2.4). This model suggests that the variation5 in price of an equity
is a ¬rst-degree function of the variation in a general index of the market (of course,
the coef¬cients of this ¬rst-degree function vary from one security to another: p =
± + β I + µ.
In practice, the coef¬cients ± and β are evaluated using a regression technique.6
4
Brearley R. A. and Myers S. C., Principles of Corporate Finance, McGraw-Hill, 1991.
Broquet C., Cobbaut R., Gillet R. and Vandenberg A., Gestion de Portefeuille, De Boeck, 1997.
Copeland T. E. and Weston J. F., Financial Theory and Corporate Policy, Addison-Wesley, 1988.
´
Devolder P., Finance Stochastique, Editions de l™ULB, 1993.
´
Roger P., L™Evalation Des Actifs Financiers, De Boeck, 1996.
5
This is a relative variation in price, namely a return. The same applies to the index.
6
Appendix 3 contains the statistical base elements needed to understand this concept.
Evaluating Financial Assets 33

Nonlinear models independent of time
A more complex case is that in which the function f of the relation p = f (X1 , X2 , . . . ,
Xn ) + µ is not linear.
When time is not taken into consideration, p is evaluated using a Taylor development,
as follows:
n n n
1
p= fXk (X1 , . . . , Xn ) Xk + fXk Xl (X1 , . . . , Xn ) Xk Xl + . . . + µ
2!
k=1 k=1 l=1

For as long as the Xk variations in the explanatory variables are low, the terms of the
second order and above can be disregarded and it is possible to write:
n
p≈ fXk (X1 , . . . , Xn ) Xk + µ
k=1

This brings us back to a linear model, which will then be processed as in the previ-
ous paragraph.
For example, for bonds, when the price of the security is expressed according to the
interest rate, we are looking at a nonlinear model. If one is content to approximate using
only the duration parameter (see Section 4.2.2), a linear approximation will be used. If,
however one wishes to introduce the concept of convexity (see Section 4.2.3), the Taylor
development used shall take account of the second-degree term.

Nonlinear models dependent on time
For some types of asset, duration is of fundamental importance and time is one of the
arguments of the function f .
This is the case, for example, with conditional assets; here, the life span of the contract
is an essential element. In this case, there is a need to construct speci¬c models that take
account of this additional ingredient. We no longer have a stationary random model, such
as Sharpe™s example, but a model that combines the random and temporal elements; this
is known as a stochastic process.
An example of this type of model is the Black“Scholes model for equity options (see
Section 5.3.2), where the price p is a function of various variables (price of underlying
asset, realisation price, maturity, volatility of underlying asset, risk-free interest rate). In
this model, the price of the underlying asset is itself modelled by a stochastic process
(standard Brownian motion).
3
Equities

3.1 THE BASICS
An equity is a ¬nancial asset that corresponds to part of the ownership of a company,
its value being indicative of the health of the company in question. It may be the subject
of a sale and purchase, either by private agreement or on an organised market. The law
of supply and demand on this market determines the price of the equity. The equity can
also give rise to the periodic payment of dividends.


3.1.1 Return and risk
3.1.1.1 Return on an equity
Let us consider an equity over a period of time [t ’ 1; t] the duration of which may be
one day, one week, one month or one year. The value of this equity at the end of the
period, and the dividend paid during the said period, are random variables1 referred to
respectively as Ct and Dt .
The return on the equity during the period in question is de¬ned as:

Ct ’ Ct’1 + Dt
Rt =
Ct’1

We are therefore looking at a value without dimension, which can easily be broken down
into the total of two terms:
Ct ’ Ct’1 Dt
Rt = +
Ct’1 Ct’1

• The ¬rst of these is the increase in value, which is ¬ctitious in that the holder of the
equity does not pro¬t from it unless the equity is sold at the moment t.
• The second is the rate of return, which is real as it represents an income.

If one wishes to take account of the rate of in¬‚ation when de¬ning the return parameter,
the nominal return Rt(n) (excluding in¬‚ation), the real return Rt(r) (with in¬‚ation) and
the rate of in¬‚ation „ are all introduced. They are linked by the relation 1 + Rt(n) =
(1 + Rt(r) ) · (1 + „ ).
The real return can then be easily calculated:

1 + Rt(n)
Rt(r) = ’1
1+„

1
Appendix 2 contains the basic elements of probability theory needed to understand these concepts.
36 Asset and Risk Management

Example
An equity is quoted at 1000 at the end of May and 1050 at the end of June; it paid a
dividend of 80 on 12 June. Its (monthly) return for this period is therefore:

1050 ’ 1000 + 80
Rj une = = 0.13 = 13 %
1000

This consists of an increase in value of 5 % and a rate of return of 8 %.
We are looking here at the nominal return. If the annual rate of in¬‚ation for that year
is 5 %, the real return will be:

1.13
(r)
Rj une = ’ 1 = 0.1254 = 12.54 %
(1.05)1/12

For certain operations carried out during the return calculation period, such as division
or merging of equities, free issue or increase in capital, the principle of de¬nition of return
is retained, but care is taken to include comparable values only in the formula. Therefore,
when an equity is split into X new equities, the return will be determined by:

X · Ct ’ Ct’1 + Dt X · Ct ’ Ct’1 + X · Dt
Rt = or
Ct’1 Ct’1

This will depend on whether the dividends are paid before or after the date of the split.
If a return is estimated on the basis of several returns relating to the same duration but
for different periods (for example, ˜average™ monthly return estimated on the basis of 12
monthly returns for the year in question), then mathematical common sense dictates that
the following logic should be applied:

1 + R1 year = (1 + R1 ) · (1 + R2 ) · . . . · (1 + R12 )

Therefore:
R1 month = (1 + R1 ) · . . . · (1 + R12 ) ’ 1
12




The expression (1 + R1 month ) is the geometric mean of the corresponding expressions for
the different months. We therefore arrive at, and generally use in practice, the arithmetic
mean.
R1 + . . . + R12
R1 month =
12

This last relation is not in fact correct, as is shown by the example of a security quoted
at 1000, 1100 and 1000 at moments 0, 1 and 2, respectively. The average return on this
security is obviously zero. The returns on the two subperiods total 10 % and ’9.09 %,
respectively, which gives the following values for the average return: 0 % for the geometric
mean and 0.45 % for the arithmetical mean.
Generally speaking, the arithmetic mean always overestimates the return, all the more
so if ¬‚uctuations in partial returns are signi¬cant. We are, however, more inclined to use
Equities 37

the arithmetic mean because of its simplicity2 and because this type of mean is generally
used for statistical estimations,3 and it would be dif¬cult to work with variances and
covariances (see below) estimated in any other way.

Note
We also use another calculation formula when no dividend is paid “ that of the logarithmic
return.
Ct
Rt— = ln .
Ct’1

This formula differs only slightly from the formula shown above, as it can be developed
using the Taylor formula as follows, if the second-degree and higher terms, which are
almost always negligible, are not taken into consideration:

Ct ’ Ct’1
Rt— = ln 1 +
Ct’1
= ln(1 + Rt )
≈ Rt

The advantage of Rt— compared to Rt is that:

• Only Rt— can take values as small as one wishes: if Ct’1 > 0, we have:

Ct
= ’∞
lim ln
Ct’1
Ct ’’0+


Ct ’ Ct’1
≥ ’1
which is compatible with statistical assumption about return, though
Ct’1
• Rt— allows the variation to be calculated simply over several consecutive periods:

Ct Ct Ct’1 Ct Ct’1
= ln · = ln + ln
ln
Ct’2 Ct’1 Ct’2 Ct’1 Ct’2

which is not possible with Rt . We will, however, be using Rt in our subsequent reasoning.

Example
Let us calculate in Table 3.1 the quantities Rt and Rt— for a few values of Ct .
The differences observed are small, and in addition, we have:

11 100
= 0.0039 + 0.0271 ’ 0.0794 ’ 0.0907 = ’0.1391
ln
12 750

2
An argument that no longer makes sense with the advent of the computer age.
3
See, for example, the portfolio return shown below.
38 Asset and Risk Management
Table 3.1 Classic and logarithmic returns

Rt—
Ct Rt

12 750
12 800 0.0039 0.0039
13 150 0.0273 0.0271
’0.0760 ’0.0794
12 150
’0.0864 ’0.0907
11 100


3.1.1.2 Return on a portfolio
Let us consider a portfolio consisting of a number N of equities, and note nj , Cj t and
Rj t , respectively the number of equities (j ), the price for those equities at the end of
period t and the dividend paid on the equity during that period.
The total value Vt of the portfolio at the moment t, and the total value Dt of the
dividends paid during period t, are therefore given by:
N
Vt = nj Cj t
j =1
N
Dt = nj Dj t
j =1

The return of the portfolio will therefore be given by:
Vt ’ Vt’1 + Dt
RP ,t =
Vt’1
N N N
nj Cj t ’ nj Cj,t’1 + nj Dj t
j =1 j =1 j =1
= N
nk Ck,t’1
k=1
N
nj (Cj t ’ Cj,t’1 + Dj t )
j =1
= N
nk Ck,t’1
k=1
N
nj Cj,t’1
= Rj t
N
j =1
nk Ck,t’1
k=1

nj Cj,t’1
The quantity Xj = represents the portion of the equity (j ) invested in the
N
nk Ck,t’1
k=1
portfolio at the moment t ’ 1, expressed in terms of equity market capitalisation, and one
Equities 39

Xj = 1. With this notation, the return on the portfolio takes the following
thus arrives at
form:
N
RP ,t = Xj Rj t
j =1


Note
The relations set out above assume, of course, that the number of each of the securities in
the portfolio remains unchanged during the period in question. Even if this condition is sat-
is¬ed, the proportions Xj will be dependent on t through the prices. If therefore one wishes
to consider a portfolio that has identical proportions at two given different moments, the
nj must be altered in consequence. This is very dif¬cult to imagine in practice, because
of transaction costs and other factors, and we will not take account of it in future. Instead,
our reasoning shall be followed as though the proportions remained unchanged.

As for an isolated security, when one considers a return estimated on the basis of several
returns relating to the same duration but from different periods, one uses the arithmetical
mean instead of the geometric mean, which gives:

12
1
RP ,1 = RP ,t
month
12 t=1
N
12
1
= Xj Rj t
12 t=1 j =1

N 12
1
= Xj Rj t
12
j =1 t=1


Therefore, according to what was stated above:4

N
RP ,1 = Xj Rj,1 month .
month
j =1



3.1.1.3 Market return
From a theoretical point of view, the market can be considered as a portfolio consisting
of all the securities in circulation. The market return is therefore de¬ned as: RM,t =
N
j =1 Xj Rj t where Xj represents the ratio of global equity market capitalisation of the
security (j ) and that of all securities.
These ¬gures are often dif¬cult to process, and in practice, the concept is usually
replaced by the concept of a stock exchange index that represents the market in question:
It ’ It’1
RI,t = .
It’1

4
Note that this relationship could not have existed if the arithmetical mean was not used.
40 Asset and Risk Management

A statistical index is a parameter that allows a magnitude X between the basic period
X(s)
t and the calculation period s to be described as: It (s) = .
X(t)
When X is composite, as for the value of a stock exchange market, several methods
of evaluation can be envisaged. It is enough to say that:

• Some relate to prices and others to returns.
• Some use arithmetic means for prices, others use equity market capitalisation.
• Some take account of dividends paid, others do not.
• Others relate to all quoted securities, others are sectorial in nature.

The best known stock exchanges indexes are the Dow Jones (USA), the S&P 500 (USA),
the Nikkei (Japan) and the Eurostoxx 50 (Europe).

3.1.1.4 Expected return and ergodic estimator
As we indicated above, the return of an equity is a random variable, the distribution
of which is usually not fully known. The essential element of this probability law is of
course its expectation:5 the expected return Ej = E(Rj ).
This is an ex ante mean, which as such is inaccessible. For this reason, it is estimated
on the basis of available historical observations, calculated for the last T periods. Such
an ex post estimator, which relates to historical data, is termed ergodic. The estimator for
the expected return on the security (j ) is therefore:
T
1
Rj = Rj t
T t=1

In the same way, for a portfolio, the expected return equals:
EP = E(RP ) = N=1 Xj Ej = Xt E, introducing the X and E vectors for the propor-
j
tions and expected returns on N securities:
«  « 
X1 E1
¬ X2 · ¬ E2 ·
¬ · ¬ ·
X=¬ . · E=¬ . ·
. .
. .
XN EN

The associated ergodic estimator is thus given by:
T N
1
RP = RP t = Xj R j .
T t=1 j =1

In the following theoretical developments, we will use the probability terms (expec-
tation) although it is acknowledged that for practical calculations, the statistical terms
(ergodic estimator) should be used.

From here on, we will use the index t not for the random return variable relative to period t, but for referencing a
5

historical observation (the realised value of the random variable).
Equities 41

3.1.1.5 Risk of one equity
The performance of an equity cannot be measured on the basis of its expected return only.
Account should also be taken of the magnitude of ¬‚uctuations of this return around its
mean value, as this magnitude is a measurement of the risk associated with the security
in question. The magnitude of variations in a variable around its average is measured
using dispersion indices. Those that are adopted here are the variance σj2 and the standard
deviation σj of the return:

σj2 = var(Rj ) = E[(Rj ’ Ej )2 ] = E(Rj ) ’ Ej
2 2


In practice, this is evaluated using its ergodic estimator:

T T
1 1 2
sj = (Rj t ’ R j ) = Rj t ’ R j
2 2 2
T T
t=1 t=1

Note
Two typical values are currently known for the return on an equity: its (expected) return
and its risk. With regard to the distribution of this random variable, if it is possible
to accept a normal distribution, then no other parameter will be needed as the law of
probability is characterised by its average and its standard deviation.
The reason for the omnipresence of this distribution is the central limit theorem (CLT),
which requires the variable in question to be the sum of a very large number of ˜small™
independent effects.
This is probably the reason why (number of transactions) it is being noted empirically
that returns relating to long periods (a month or a year) are often normally distributed,
while this is not necessarily the case for daily returns, for example. In these cases, we
generally observe distributions with fatter tails6 than those under the normal law. We will
examine this phenomenon further in Part III, as value at risk is particularly interested in
these distribution tails.
However, we will consider in this part that the distribution of the return is characterised
by the ˜expected return-risk™ couple, which is suf¬cient for the Markowitz portfolio the-
ory.7 In other cases (dynamic models), it will be supposed in addition that this is normal.
Other dispersion indices could be used for measuring risk, as mean deviation
E(|Rj ’ Ej |) or semi-variance, which is de¬ned as the variance but takes account only
of those return values that are less than the expected return. It is nevertheless the vari-
ance (and its equivalent, the standard deviation) that is almost always used, because
of its probability-related and statistical properties, as will be seen in the de¬nition of
portfolio risk.

3.1.1.6 Covariance and correlation
The risk of a portfolio depends of course on the risk of the securities of which it is
composed, but also on the links present between the various securities, through the effect

6
This is referred to as leptokurtic distribution.
7
Markowitz H., Portfolio selection, Journal of Finance, Vol. 7, No. 1, 1952, pp. 419“33.
42 Asset and Risk Management

of diversi¬cation. The linear dependence between the return of the security (i) and its
security (j ) is measured by the covariance:

σij = cov(Ri , Rj ) = E (Ri ’ Ei )(Rj ’ Ej ) = E(Ri Rj ) ’ Ei Ej

This is evaluated by the ergodic estimator

T T
1 1
sij = (Rit ’ R i )(Rj t ’ R j ) = (Rit Rj t ) ’ R i R j
T T
t=1 t=1

The interpretation of the covariance sign is well known, but its order of magnitude is
dif¬cult to express. To avoid this problem, we use the correlation coef¬cient
σij
ρij = corr(Ri , Rj ) =
σi · σj

For this coef¬cient, the ergodic estimator is of course given by
sij
rij =
si · sj

Remember that this last parameter is a pure number located between ’1 and 1, of
which the sign indicates the way of dependency between the two variables and the values
close to ±1 correspond to near-perfect linear relations between the variables.

3.1.1.7 Portfolio risk
If one remembers that RP ,t = N=1 Xj Rj t , and given that the formula for the variance
j
of a linear combination of random variables, the variance of the return on the portfolio
takes the following form:

N N
Xi Xj σij = Xt V X
σP = var(RP ) =
2

i=1 j =1


Here: σii = σi2 and one has determined
«
 « 
σ1 σ12 · · · σ1N
2
X1
¬ X2 · ¬ σ21 · · · σ2N ·
σ22
¬ · ¬ ·
X=¬ . · V =¬ . .·
. ..
. . . .
.
. . . .
XN σN1 σN2 · · · σN2


If one wishes to show the correlation coef¬cients, the above formula becomes:
N N
σP = Xi Xj σi σj ρij
2

i=1 j =1
Equities 43

Example
The risk of a portfolio consisting of two equities in respective proportions, 30 % and
70 %, and such that σ1 = 0.03, σ2 = 0.02, σ12 = 0.01, is calculated regardless by:
2 2


σP = 0.32 · 0.03 + 0.72 · 0.02 + 2 · 0.3 · 0.7 · 0.01 = 0.0167, or by:
2


0.03 0.01 0.3
σP = 0.3 0.7 = 0.0167.
2
0.01 0.02 0.7

It is interesting to compare the portfolio risk with the individual security risk.
The ˜expected return-risk™ approach to the portfolio therefore requires a knowledge of
the expected returns and individual variances as well as all the covariances two by two.
Remember that the multi-normal distribution is characterised by these elements, but that
Markowitz™s portfolio theory does not require this law of probability.

3.1.1.8 Security risk within a portfolio
The portfolio risk can also be written as:
« 
N N N N
Xi  Xj σij 
σP = Xi Xj σij =
2

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


The total risk for the security (i) within the portfolio therefore depends on σi2 but also
on the covariances with other securities in the portfolio. It can be developed as follows:
N N
Xj σij = Xj cov(Ri , Rj )
j =1 j =1
« 
N
= cov Ri , Xj Rj 
j =1

= cov(Ri , RP )
= σiP

The relative importance of the total risk for the security (i) in the portfolio risk is
therefore measured by:
N
Xj σij
σiP j =1
= .
σP σP
2 2


These relative risks are such as:
N
σiP
Xi = 1.
σP2
i=1
44 Asset and Risk Management

Example

Using the data in the previous example, the total risks for the two securities within the
portfolio are given as:

σ1P = 0.3 · 0.03 + 0.7 · 0.01 = 0.016
σ2P = 0.3 · 0.01 + 0.7 · 0.02 = 0.017

The corresponding relative risks therefore total 0.958 and 1.018 respectively. Note that
what we actually have is: 0.3 · 0.958 + 0.7 · 1.018 = 1.
The concept of the relative risk applied to the market as a whole or to a particular
portfolio leads us to the concept of systematic risk :

σiM
βi =
σM2


It therefore represents the relative importance of the total security risk (i) in the market
risk, that is, the volatility of Ri in relation to RM , as the quotient in question is the slope
of the regression line in which the return on the security (i) is explained by the return of
the market (see Figure 3.1):
Ri = ±i + βi RM

It can be accepted, in conclusion, that the risk of a particular security should never be
envisaged in isolation from the rest of the portfolio in which it is included.

3.1.2 Market ef¬ciency
Here follows a brief summary of the concept of market ef¬ciency,8 which is a necessary
hypothesis (or one that must be at least veri¬ed approximately) for the validity of the
various models of ¬nancial analysis and is closely linked to the concept of the ˜perfect
market™.


Ri




• • •
• •





RM


Figure 3.1 Systematic risk

8
A fuller treatment of this subject is found in Gillet P., L™Ef¬cience Des March´ s Financiers, Economica, 1999.
e
Equities 45

3.1.2.1 General principles
It was Eugene Fama9 who explicitly introduced the concept of ˜ef¬ciency™. The de¬nition
that he gave to the concept was as follows: ˜A ¬nancial market is said to be ef¬cient if,
and only if, all the available information on each ¬nancial asset quoted on the market is
immediately included in the price of that asset™.
Indeed, he goes so far as to say that there is no overvaluation or undervaluation of
securities, and also that no asset can produce a return greater than that which corresponds
to its own characteristics.
This hypothesis therefore guarantees equality of treatment of various investors: no
category of investor has any informational advantage. The information available on this
type of market therefore allows optimum allocation of resources.
The economic justi¬cation for this concept is that the various investors, in competition
and possessing the same information, will, through their involvement and because of the
law of supply and demand, make the price of a security coincide with its intrinsic value.
We are of course looking at a hypothesis that divides the supporters of fundamental
analysis from the supporters of technical analysis. The former accept the hypothesis and
indeed make it the entire basis for their reasoning; they assume that returns on securities
are unpredictable variables and propose portfolio management techniques that involve
minimising the risks linked to these variables.10 The latter propose methods11 that involve
predicting courses on the basis of historically observed movements.
From a more mathematical point of view, market ef¬ciency consists of assuming that
the prices will follow a random walk, that is, that the sequence Ct ’ Ct’1 (t = 1, 2, . . .)
consists of random variables that are independent and identically distributed. In these cir-
cumstances, such a variation can only be predicted on the basis of available observations.
The economic conditions that de¬ne an ef¬cient market are:

• The economic agents involved on the market behave rationally; they use the available
information coherently and aim to maximise the expected utility of their wealth.
• The information is available simultaneously to all investors and the reaction of the
investors to the information is instantaneous.
• The information is available free of charge.
• There are no transaction costs or taxes on the market.
• The market in question is completely liquid.

It is obvious that these conditions can never be all strictly satis¬ed in a real market.
This therefore raises the question of knowing whether the differences are signi¬cant
and whether they will have the effect of invalidating the ef¬ciency hypothesis. This
question is addressed in the following paragraphs, and the analysis is carried out at three
levels according to the accessibility of information. The least that can be said is that the
conclusions of the searches carried out in order to test ef¬ciency are inconclusive and
should not be used as a basis for forming clear and de¬nitive ideas.

9
Fama E. F., Behaviour of Stock Market Prices, Journal of Business, Vol. 38, 1965, pp. 34“105. Fama E. F., Random
Walks in Stock Market Prices, Financial Analysis Journal, 1965. Fama E. F., Ef¬cient Capital Markets: A Review of Theory
and Empirical Work, Journal of Finance, Vol. 25, 1970.
10
This approach is adopted in this work.
11
Refer for example to Bechu T. and Bertrand E., L™Analyse Technique, Economica, 1998.
46 Asset and Risk Management

3.1.2.2 Weak form
The weak form of the ef¬ciency hypothesis postulates that it is not possible to gain a
particular advantage from the range of historical observations; the rates therefore purely
and simply include the previous rate values.
The tests applied in order to verify this hypothesis relate to the possibility of predicting
rates on the basis of their history. Here are a few analyses carried out:

• The autocorrelation test. Is there a correlation (positive or negative) between the suc-
cessive return on security values that allows forecasts to be made?
• The run test. Is the distribution of the sequence lengths for positive returns and negative
returns normal?
• Statistical tests for random walk.
• Simulation tests for technical analysis methods. Do the speculation techniques give
better results than passive management?

Generally speaking, most of these tests lead to acceptance of the weak ef¬ciency hypoth-
esis, even though the most demanding tests from the statistical viewpoint sometimes
invalidate it.

3.1.2.3 Semi-strong form
The semi-strong form of the ef¬ciency hypothesis postulates that it is not possible to gain
a particular advantage from information made public in relation to securities; the rates
therefore change instantly and correctly when an event such as an increase in capital,
division of securities, change of dividend policy, balance sheet publication or take-over
bid is announced publicly.
The tests carried out to verify this hypothesis therefore relate to the effects of the events
announced. They consist successively of:

• Determining the theoretical return on a security Rit = ±i + βi RMt on the basis of
historical observations relating to a period that does not include such events.
• When such an event occurs, comparing the difference between the theoretical return
and the real return.
• Measuring the reaction time in order for the values to be altered again.

3.1.2.4 Strong form
The strong form of the ef¬ciency hypothesis postulates that it is not possible to gain a
particular advantage from nonpublic information relating to securities; the rates there-
fore change instantly and correctly when an event that is not public, that is, an insider
event, occurs.
The tests carried out to verify this hypothesis therefore relate to the existence of priv-
ileged information. They follow a method similar to that used for the semi-strong form,
but in speci¬c circumstances:

• In recognised cases of misdemeanour by an initiated person.
Equities 47

• In cases of intensive trading on a market without the public being informed.
• In cases of intensive trading on the part of initiated persons.
• In cases of portfolios managed by professionals likely to have speci¬c information
before the general public has it, as in collective investment organisations.

3.1.2.5 Observed case of systematic inef¬ciency
Although the above analyses suggest that the ef¬ciency hypothesis can be globally accep-
ted, cases of systematic inef¬ciency have been discovered. In these cases, the following
have sometimes been observed:

• Higher than average pro¬tability at the end of the week, month or year.
• Higher pro¬tability for low equity market capitalisation businesses than for high capi-
talisation companies.

Alongside these differences, pockets of inef¬ciency allowing arbitrage may present
themselves. Their origin may be:

• Speculative bubbles, in which the rate of a security differs signi¬cantly and for a
long time from its intrinsic value before eventually coming back to its intrinsic value,
without movements of the market economic variables as an explanation for the differ-
ence.
• Irrational behaviour by certain investors.

These various elements, although removed from the ef¬ciency hypothesis, do not, how-
ever, bring it into question. In addition, the pro¬t to investors wishing to bene¬t from
them will frequently be lost in transaction costs.

3.1.2.6 Conclusion
We quote P. Gillet in conclusion of this analysis.

Financial market ef¬ciency appears to be all of the following: an intellectual abstraction, a myth
and an objective.
The intellectual abstraction. Revealed by researchers, the theory of ¬nancial market ef¬ciency
calls into question a number of practices currently used by the ¬nancial market professionals,
such as technical analysis. (. . .) It suggests a passive management, while technical analysis
points towards an active management. (. . .) In addition, it is one of the basic principles of
modern ¬nancial theory. (. . .).
The myth. All the hypotheses necessary for accepting the theory of ef¬ciency are accepted by
the theory™s supporters. In addition to the classic hypotheses on circulation of information or
absence of transaction costs, which have been addressed, other underlying hypotheses have as
yet been little explored, especially those linked to the behaviour of investments and to liquidity.
(. . .).
An objective. The market authorities are aware that the characteristics of ef¬ciency make the
market healthy and more credible, and therefore attract investors and businesses. To make a
48 Asset and Risk Management
market more ef¬cient is to reduce the risk of the speculation bubble. (. . .). The aim of the
authorities is therefore to improve the ef¬ciency of the ¬nancial markets (. . .).


3.1.3 Equity valuation models
The principle of equivalence, the basis of ¬nancial mathematics, allows the expression
that the intrinsic value V0 of an equity at the moment 0 is equal to the discounted
values of the future ¬nancial ¬‚ows that the security will trigger. Put more simply, if one
assumes that the dividends (future ¬nancial ¬‚ows) are paid for periods 1, 2 etc. and have
a respective total of D1 , D2 etc., and if the discount rate k is included, we will obtain the
relation:

Dt (1 + k)’t
V0 =
t=1


Note 1
The direct use of this relation can be sensitive. In fact:

• The value of all future dividends is not generally known.
• This formula assumes a constant discount rate (ad in¬nitum).
• It does not allow account to be taken of speci¬c operations such as division or regroup-
ing of equities, free issues or increases in capital.

The formula does, however, provide a number of services and later we will introduce a
simpli¬ed formula that can be obtained from it.

Note 2
This formula, which links V0 and k, can be used in two ways:

• If V0 is known (intrinsic value on an ef¬cient market), the value of k can be deduced
from it and will then represent the expected return rate for the security in ques-
tion.
• If k is given, the formula provides an assessment of the security™s value, which can
then be compared to the real rate C0 , thus allowing overevaluation or underestimation
of the security to be determined.


3.1.3.1 The Gordon“Shapiro formula
This relation12 is based on the following hypotheses:

• The growth of the ¬rm is self-¬nancing.
• The rate of return r of the investments, and the rate of distribution d of the pro¬ts, are
constant from one period to the next.

12
See Gordon M. and Shapiro E., Capital equipment analysis: the required rate pro¬t, Management Science, Vol. 3,
October 1956.
Equities 49

Under these hypotheses, if Bt is ¬xed as the pro¬t for each action sold during the period
t and Et is the accounting value per equity at the moment t (capital divided by number
of equities), we have: ±
 Dt = d
Bt

Bt = r · Et’1

And therefore:

Bt+1 = Bt + r · (Bt ’ Dt )
= Bt [1 + r(1 ’ d)]

The pro¬ts therefore increase at a constant rate g = r(1 ’ d), which is the rate of prof-
itability of the investments less the proportion distributed. The dividends also increase at
this constant rate and it is possible to write Dt+1 = g.Dt , hence: Dt = D1 (1 + g)t’1 .
The present value can therefore be worked out as follows:

D1 (1 + g)t’1 (1 + k)’t
V0 =
t=1
∞ t
D1 1+g
=
1+k 1+k
t=0

D1
= 1+k
1+g
1’
1+k

This is provided the discount rate k is greater than the rate of growth g. This leads to the
Gordon“Shapiro formula:

D1 dB1 drE0
V0 = = =
k’g k’g k’g

Example
The capital of a company consists of 50 000 equities, for a total value of 10 000 000. The
investment pro¬tability rate is 15 %, the pro¬t distribution rate 40 %, and the discount
rate 12 %.
The pro¬t per equity will be:

10 000 000
B = 0.15 · = 30
50 000

The dividend per equity will therefore be D = 0.4 — 30 = 12. In addition, the rate of
growth is given as follows: g = 0.15 — (1 ’ 0.4) = 0.09.
50 Asset and Risk Management

The Gordon“Shapiro formula therefore leads to:

12 12
V0 = = = 400
0.12 ’ 0.09 0.03

The market value of this company is therefore 50 000 — 400 = 20 000 000, while its
accounting value is a mere 10 000 000.

D1
The Gordon“Shapiro formula produces the equation k = g + , which shows that
V0
the return k can be broken down into the dividend growth rate and the rate of payment
per security.

3.1.3.2 The price-earning ratio
One of the most commonly used evaluation indicators is the PER. It equals the ratio of
the equity rate to the expected net pro¬t for each equity:
C0
PER 0 =
B1
Its interpretation is quite clear: when purchasing an equity, one pays PER 0 — ¤1 for a
pro¬t of ¤1. Its inverse (pro¬t over price) is often considered as a measurement of returns
on securities, and securities whose PER is below the market average are considered to be
undervalued and therefore of interest.
This indicator can be interpreted using the Gordon“Shapiro formula, if the hypotheses
relative to the formula are satis¬ed. In fact, by replacing the rate with the V0 value given
for this formula:
D1 dB1
C0 = =
k’g k ’ r(1 ’ d)
we arrive directly at:
d
PER 0 =
k ’ r(1 ’ d)
This allows the following expression to be obtained for the rate of return k:
d
k = r(1 ’ d) +
PER 0
1’d
1
= r(1 ’ d) + ’
PER 0 PER 0
As PER 0 = C0 /rE0 , we ¬nd that:
r(1 ’ d)(C0 ’ E0 )
1
k= +
C0
PER 0

Example
If one takes the same ¬gures as in the previous paragraph:
Equities 51

r = 15 %
d = 40 %
10 000 000
E0 = = 200

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