ńņš. 2 |

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 ofļ¬ce vs. risk management

With regard to the back ofļ¬ce vs. risk management debate, it is well worth remembering

that depending on the views of the regulator, the back ofļ¬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 ofļ¬ce control check:

ā¢ The daily control checks carried out by staff, for example each employeeā™s monitoring

of their suspense account.

ā¢ 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 bankā™s risk management policy, for example:

ā¢ Monitoring any limit breaches (limits, stop losses etc.).

ā¢ 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 ļ¬nancial group is based on four main aspects:

ā¢ Producing a coherent plan for the audit activities within the group.

Changes in Financial Risk Management 23

ā¢ Ensuring that the whole of the auditable activities, including the groupā™s subsidiaries

and holding company within the responsibilities of the parent company, are covered

through the conduct or review of audits.

ā¢ Applying a uniform audit method across all group entities.

ā¢ 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 efļ¬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 regulatorā™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 partiesā™

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 speciļ¬c

transactions, or indeed may not even have the authority to conclude a repo contract.

3. Transaction risk. This is certainly the most signiļ¬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 ļ¬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

ļ¬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 insufļ¬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 ļ¬nancial deals

within the risk management function presents certain advantages:

ā¢ Assessment of the way in which the legal risk will be managed and reduced.

ā¢ The function has a central position that gives an overall view of all the bankā™s activities.

ā¢ Increased efļ¬ciency in the implementation of legal risk management procedures in

ļ¬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 ā˜fully integrated computerised risk-management systemā™. 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 ā“ 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 ā˜black boxesā™ 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 ļ¬rst of all pay attention to a series of

qualitative criteria by organising ourselves around them? Surely the ļ¬gures produced by

advanced operational risk methods are of a behavioural nature in that they show us the

ā˜score to beatā™. 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 Moodyā™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 ļ¬nance costs, is it not worth the trouble of achieving compliance from

5

Moodyā™s, Moodyā™s Analytical framework for Operational Risk Management of Banks, Moodyā™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:

ā¢ Report directly to executive management.

ā¢ Be independent of the front and back ofļ¬ce functions.

ā¢ Be located at a sufļ¬ciently senior hierarchical level to guarantee real independence, hav-

ing the authority and credibility it needs to fulļ¬l its function, both internally (especially

vis-` -vis the front and back ofļ¬ces) and externally (vis-` -vis the regulator, external audit

a a

and the ļ¬nancial community in general).

ā¢ Be a member of the asset and liability management committee.

ā¢ Where necessary, oversee all the decentralised risk-management entities in the sub-

sidiaries.

ā¢ 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.

ā¢ Have a clearly deļ¬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.

ā¢ Play a threefold role in the ļ¬eld of risks: advice, prevention and control.

But at what price?

2.2.2 The ā˜costā™ 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 ā˜hatsā™ for the front and back ofļ¬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 quantiļ¬able and/or non-quantiļ¬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-ofļ¬ce and accounting procedures.

At present, market and regulatory pressure is such that it is unthinkable for a respectable

ļ¬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 proļ¬t centre for the

institution, and concentrate on its added value.

We have seen that a well-thought-out risk management limits:

ā¢ Excessive control (large-scale savings, prevention of doubling-up).

26 Asset and Risk Management

ā¢ Indirect costs (every risk avoided is a potential loss avoided and therefore

money gained).

ā¢ 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 proļ¬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 investorā™s point of view produces a paradox:

ā¢ 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.

ā¢ On the other hand, however, although accepting the ā˜risk premiumā™ represents proļ¬t

ļ¬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 ļ¬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 ļ¬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 ļ¬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 ļ¬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

ā

The square root of the second, Ļ (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 ā˜idealā™ for an investor (that is, efļ¬cient), if, for a given expected return, it has

a minimal variance or if, for a ļ¬xed variance, it has a maximum expected return. All

the portfolios thus deļ¬ned make up what is termed the efļ¬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 investorā™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 efļ¬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 ļ¬nancial assets should take account of two fundamental aspects ā“ chance

and time.

The random aspect

It is obvious that the changes in value of a ļ¬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 ļ¬‚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 deļ¬ned as

ā˜risk-freeā™, 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, ļ¬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 ļ¬nancial asset valuation models are termed monoperiodic, such as Markowitzā™s

portfolio theory. These models examine the ā˜photographā™ 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 ļ¬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 ā“ that of the perfect mar-

ket ā“ will be put forward. Despite its reductive nature, it deļ¬nes a context in which

ļ¬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:

ā¢ Information is available in equal measure to all investors.

ā¢ There are no transactional or issue costs associated with the ļ¬nancial assets.

ā¢ There is no tax deduction on the income produced by the ļ¬nancial assets (where

increases in value or dividends are involved, for example).

ā¢ Short sales are authorised without restriction.

Absence of arbitrage opportunity

An arbitrage opportunity is a portfolio deļ¬ned in a context in which:

ā¢ No ļ¬nancial movement occurs within the portfolio during the period in question.

ā¢ The risk-free interest rate does not alter during the period in question and is valid for

any maturity date (a ļ¬‚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 speciļ¬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 ā„ 0 or V0 ā¤ 0 and VT > 0.

Generally speaking, the absence of arbitrage opportunity hypothesis is constructed in

the ļ¬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 proļ¬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 ļ¬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 (ļ¬‚at, constant rate curve), then Vt = 0 for any t ā [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 ā˜valuation modelā™

A valuation model for a ļ¬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 ) + Īµ.

The residual Īµ 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 ļ¬rst approximation), and is therefore deļ¬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:

ā¢ The imperfect nature of valuation models.

ā¢ The imperfect knowledge of the rules and limitations particular to the institution.

ā¢ 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 ā˜event-basedā™ risks. However, this does not mean that the careful risk manager

should not include ā˜catastrophe scenariosā™, 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 speciļ¬c model4 that will be analysed in further

detail in this second part.

Linear models

We will look ļ¬rst at the simple case in which the function f of the valuation model

is linear, or more speciļ¬cally, the case in which the price variation p = pt ā’ p0 is a

ļ¬rst-degree function of the variations X1 , . . . , Xn of the various explanatory variables

and of that ( Īµ) of the residue:

p = a0 + a1 X1 + . . . + an Xn + Īµ.

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 ļ¬rst-degree function of the variation in a general index of the market (of course,

the coefļ¬cients of this ļ¬rst-degree function vary from one security to another: p =

Ī± + Ī² I + Īµ.

In practice, the coefļ¬cients Ī± and Ī² 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.

Ā“

Devolder P., Finance Stochastique, Editions de lā™ULB, 1993.

Ā“

Roger P., Lā™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 ) + Īµ 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 + . . . + Īµ

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

pā fXk (X1 , . . . , Xn ) Xk + Īµ

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 speciļ¬c models that take

account of this additional ingredient. We no longer have a stationary random model, such

as Sharpeā™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 Blackā“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 ļ¬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 ā’ 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 deļ¬ned as:

Ct ā’ Ctā’1 + Dt

Rt =

Ctā’1

We are therefore looking at a value without dimension, which can easily be broken down

into the total of two terms:

Ct ā’ Ctā’1 Dt

Rt = +

Ctā’1 Ctā’1

ā¢ The ļ¬rst of these is the increase in value, which is ļ¬ctitious in that the holder of the

equity does not proļ¬t from it unless the equity is sold at the moment t.

ā¢ 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 inļ¬‚ation when deļ¬ning the return parameter,

the nominal return Rt(n) (excluding inļ¬‚ation), the real return Rt(r) (with inļ¬‚ation) and

the rate of inļ¬‚ation Ļ„ are all introduced. They are linked by the relation 1 + Rt(n) =

(1 + Rt(r) ) Ā· (1 + Ļ„ ).

The real return can then be easily calculated:

1 + Rt(n)

Rt(r) = ā’1

1+Ļ„

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 ā’ 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 inļ¬‚ation for that year

is 5 %, the real return will be:

1.13

(r)

Rj une = ā’ 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 deļ¬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 Ā· Ct ā’ Ctā’1 + Dt X Ā· Ct ā’ Ctā’1 + X Ā· Dt

Rt = or

Ctā’1 Ctā’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, ā˜averageā™ 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 ) Ā· (1 + R2 ) Ā· . . . Ā· (1 + R12 )

Therefore:

R1 month = (1 + R1 ) Ā· . . . Ā· (1 + R12 ) ā’ 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 ā’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 ļ¬‚uctuations in partial returns are signiļ¬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 difļ¬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 ā“ that of the logarithmic

return.

Ct

Rtā— = ln .

Ctā’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 ā’ Ctā’1

Rtā— = ln 1 +

Ctā’1

= ln(1 + Rt )

ā Rt

The advantage of Rtā— compared to Rt is that:

ā¢ Only Rtā— can take values as small as one wishes: if Ctā’1 > 0, we have:

Ct

= ā’ā

lim ln

Ctā’1

Ct ā’ā’0+

Ct ā’ Ctā’1

ā„ ā’1

which is compatible with statistical assumption about return, though

Ctā’1

ā¢ Rtā— allows the variation to be calculated simply over several consecutive periods:

Ct Ct Ctā’1 Ct Ctā’1

= ln Ā· = ln + ln

ln

Ctā’2 Ctā’1 Ctā’2 Ctā’1 Ctā’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 Rtā— for a few values of Ct .

The differences observed are small, and in addition, we have:

11 100

= 0.0039 + 0.0271 ā’ 0.0794 ā’ 0.0907 = ā’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

Rtā—

Ct Rt

12 750

12 800 0.0039 0.0039

13 150 0.0273 0.0271

ā’0.0760 ā’0.0794

12 150

ā’0.0864 ā’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 ā’ Vtā’1 + Dt

RP ,t =

Vtā’1

N N N

nj Cj t ā’ nj Cj,tā’1 + nj Dj t

j =1 j =1 j =1

= N

nk Ck,tā’1

k=1

N

nj (Cj t ā’ Cj,tā’1 + Dj t )

j =1

= N

nk Ck,tā’1

k=1

N

nj Cj,tā’1

= Rj t

N

j =1

nk Ck,tā’1

k=1

nj Cj,tā’1

The quantity Xj = represents the portion of the equity (j ) invested in the

N

nk Ck,tā’1

k=1

portfolio at the moment t ā’ 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-

isļ¬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 difļ¬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 deļ¬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 ļ¬gures are often difļ¬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 ā’ Itā’1

RI,t = .

Itā’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:

ā¢ Some relate to prices and others to returns.

ā¢ Some use arithmetic means for prices, others use equity market capitalisation.

ā¢ Some take account of dividends paid, others do not.

ā¢ 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:

ļ£« ļ£¶ ļ£« ļ£¶

X1 E1

ļ£¬ X2 ļ£· ļ£¬ E2 ļ£·

ļ£¬ ļ£· ļ£¬ ļ£·

X=ļ£¬ . ļ£· E=ļ£¬ . ļ£·

ļ£.ļ£ø ļ£.ļ£ø

. .

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 ļ¬‚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 Ļj2 and the standard

deviation Ļj of the return:

Ļj2 = var(Rj ) = E[(Rj ā’ Ej )2 ] = E(Rj ) ā’ Ej

2 2

In practice, this is evaluated using its ergodic estimator:

T T

1 1 2

sj = (Rj t ā’ R j ) = Rj t ā’ 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 ā˜smallā™

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 ā˜expected return-riskā™ couple, which is sufļ¬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 ā’ Ej |) or semi-variance, which is deļ¬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 deļ¬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. 419ā“33.

42 Asset and Risk Management

of diversiļ¬cation. The linear dependence between the return of the security (i) and its

security (j ) is measured by the covariance:

Ļij = cov(Ri , Rj ) = E (Ri ā’ Ei )(Rj ā’ Ej ) = E(Ri Rj ) ā’ Ei Ej

This is evaluated by the ergodic estimator

T T

1 1

sij = (Rit ā’ R i )(Rj t ā’ R j ) = (Rit Rj t ) ā’ 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

difļ¬cult to express. To avoid this problem, we use the correlation coefļ¬cient

Ļij

Ļij = corr(Ri , Rj ) =

Ļi Ā· Ļj

For this coefļ¬cient, the ergodic estimator is of course given by

sij

rij =

si Ā· sj

Remember that this last parameter is a pure number located between ā’1 and 1, of

which the sign indicates the way of dependency between the two variables and the values

close to Ā±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 Ļij = Xt V X

ĻP = var(RP ) =

2

i=1 j =1

Here: Ļii = Ļi2 and one has determined

ļ£«

ļ£¶ ļ£« ļ£¶

Ļ1 Ļ12 Ā· Ā· Ā· Ļ1N

2

X1

ļ£¬ X2 ļ£· ļ£¬ Ļ21 Ā· Ā· Ā· Ļ2N ļ£·

Ļ22

ļ£¬ ļ£· ļ£¬ ļ£·

X=ļ£¬ . ļ£· V =ļ£¬ . .ļ£·

. ..

ļ£.ļ£ø ļ£. . .ļ£ø

.

. . . .

XN ĻN1 ĻN2 Ā· Ā· Ā· ĻN2

If one wishes to show the correlation coefļ¬cients, the above formula becomes:

N N

ĻP = Xi Xj Ļi Ļj Ļij

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 Ļ1 = 0.03, Ļ2 = 0.02, Ļ12 = 0.01, is calculated regardless by:

2 2

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

2

0.03 0.01 0.3

ĻP = 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 ā˜expected return-riskā™ 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

Markowitzā™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:

ļ£« ļ£¶

N N N N

Xi ļ£ Xj Ļij ļ£ø

ĻP = Xi Xj Ļij =

2

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

The total risk for the security (i) within the portfolio therefore depends on Ļi2 but also

on the covariances with other securities in the portfolio. It can be developed as follows:

N N

Xj Ļij = Xj cov(Ri , Rj )

j =1 j =1

ļ£« ļ£¶

N

= cov ļ£Ri , Xj Rj ļ£ø

j =1

= cov(Ri , RP )

= ĻiP

The relative importance of the total risk for the security (i) in the portfolio risk is

therefore measured by:

N

Xj Ļij

ĻiP j =1

= .

ĻP ĻP

2 2

These relative risks are such as:

N

ĻiP

Xi = 1.

ĻP2

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:

Ļ1P = 0.3 Ā· 0.03 + 0.7 Ā· 0.01 = 0.016

Ļ2P = 0.3 Ā· 0.01 + 0.7 Ā· 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 Ā· 0.958 + 0.7 Ā· 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 :

ĻiM

Ī²i =

ĻM2

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 = Ī±i + Ī²i 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 efļ¬ciency

Here follows a brief summary of the concept of market efļ¬ciency,8 which is a necessary

hypothesis (or one that must be at least veriļ¬ed approximately) for the validity of the

various models of ļ¬nancial analysis and is closely linked to the concept of the ā˜perfect

marketā™.

Ri

ā¢

ā¢

ā¢ ā¢ ā¢

ā¢ ā¢

ā¢

ā¢

RM

Figure 3.1 Systematic risk

8

A fuller treatment of this subject is found in Gillet P., Lā™Efļ¬cience Des MarchĀ“ s Financiers, Economica, 1999.

e

Equities 45

3.1.2.1 General principles

It was Eugene Fama9 who explicitly introduced the concept of ā˜efļ¬ciencyā™. The deļ¬nition

that he gave to the concept was as follows: ā˜A ļ¬nancial market is said to be efļ¬cient if,

and only if, all the available information on each ļ¬nancial asset quoted on the market is

immediately included in the price of that assetā™.

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 justiļ¬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 efļ¬ciency consists of assuming that

the prices will follow a random walk, that is, that the sequence Ct ā’ Ctā’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 deļ¬ne an efļ¬cient market are:

ā¢ 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.

ā¢ The information is available simultaneously to all investors and the reaction of the

investors to the information is instantaneous.

ā¢ The information is available free of charge.

ā¢ There are no transaction costs or taxes on the market.

ā¢ The market in question is completely liquid.

It is obvious that these conditions can never be all strictly satisļ¬ed in a real market.

This therefore raises the question of knowing whether the differences are signiļ¬cant

and whether they will have the effect of invalidating the efļ¬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 efļ¬ciency are inconclusive and

should not be used as a basis for forming clear and deļ¬nitive ideas.

9

Fama E. F., Behaviour of Stock Market Prices, Journal of Business, Vol. 38, 1965, pp. 34ā“105. Fama E. F., Random

Walks in Stock Market Prices, Financial Analysis Journal, 1965. Fama E. F., Efļ¬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., Lā™Analyse Technique, Economica, 1998.

46 Asset and Risk Management

3.1.2.2 Weak form

The weak form of the efļ¬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:

ā¢ The autocorrelation test. Is there a correlation (positive or negative) between the suc-

cessive return on security values that allows forecasts to be made?

ā¢ The run test. Is the distribution of the sequence lengths for positive returns and negative

returns normal?

ā¢ Statistical tests for random walk.

ā¢ 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 efļ¬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 efļ¬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:

ā¢ Determining the theoretical return on a security Rit = Ī±i + Ī²i RMt on the basis of

historical observations relating to a period that does not include such events.

ā¢ When such an event occurs, comparing the difference between the theoretical return

and the real return.

ā¢ Measuring the reaction time in order for the values to be altered again.

3.1.2.4 Strong form

The strong form of the efļ¬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 speciļ¬c circumstances:

ā¢ In recognised cases of misdemeanour by an initiated person.

Equities 47

ā¢ In cases of intensive trading on a market without the public being informed.

ā¢ In cases of intensive trading on the part of initiated persons.

ā¢ In cases of portfolios managed by professionals likely to have speciļ¬c information

before the general public has it, as in collective investment organisations.

3.1.2.5 Observed case of systematic inefļ¬ciency

Although the above analyses suggest that the efļ¬ciency hypothesis can be globally accep-

ted, cases of systematic inefļ¬ciency have been discovered. In these cases, the following

have sometimes been observed:

ā¢ Higher than average proļ¬tability at the end of the week, month or year.

ā¢ Higher proļ¬tability for low equity market capitalisation businesses than for high capi-

talisation companies.

Alongside these differences, pockets of inefļ¬ciency allowing arbitrage may present

themselves. Their origin may be:

ā¢ Speculative bubbles, in which the rate of a security differs signiļ¬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.

ā¢ Irrational behaviour by certain investors.

These various elements, although removed from the efļ¬ciency hypothesis, do not, how-

ever, bring it into question. In addition, the proļ¬t to investors wishing to beneļ¬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 efļ¬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 ļ¬nancial market efļ¬ciency

calls into question a number of practices currently used by the ļ¬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 ļ¬nancial theory. (. . .).

The myth. All the hypotheses necessary for accepting the theory of efļ¬ciency are accepted by

the theoryā™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 efļ¬ciency make the

market healthy and more credible, and therefore attract investors and businesses. To make a

48 Asset and Risk Management

market more efļ¬cient is to reduce the risk of the speculation bubble. (. . .). The aim of the

authorities is therefore to improve the efļ¬ciency of the ļ¬nancial markets (. . .).

3.1.3 Equity valuation models

The principle of equivalence, the basis of ļ¬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 ļ¬nancial ļ¬‚ows that the security will trigger. Put more simply, if one

assumes that the dividends (future ļ¬nancial ļ¬‚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:

ā

Dt (1 + k)ā’t

V0 =

t=1

Note 1

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

ā¢ The value of all future dividends is not generally known.

ā¢ This formula assumes a constant discount rate (ad inļ¬nitum).

ā¢ It does not allow account to be taken of speciļ¬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

simpliļ¬ed formula that can be obtained from it.

Note 2

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

ā¢ If V0 is known (intrinsic value on an efļ¬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.

ā¢ If k is given, the formula provides an assessment of the securityā™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 Gordonā“Shapiro formula

This relation12 is based on the following hypotheses:

ā¢ The growth of the ļ¬rm is self-ļ¬nancing.

ā¢ The rate of return r of the investments, and the rate of distribution d of the proļ¬ts, are

constant from one period to the next.

12

See Gordon M. and Shapiro E., Capital equipment analysis: the required rate proļ¬t, Management Science, Vol. 3,

October 1956.

Equities 49

Under these hypotheses, if Bt is ļ¬xed as the proļ¬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: ļ£±

ļ£² Dt = d

Bt

ļ£³

Bt = r Ā· Etā’1

And therefore:

Bt+1 = Bt + r Ā· (Bt ā’ Dt )

= Bt [1 + r(1 ā’ d)]

The proļ¬ts therefore increase at a constant rate g = r(1 ā’ 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)tā’1 .

The present value can therefore be worked out as follows:

ā

D1 (1 + g)tā’1 (1 + k)ā’t

V0 =

t=1

ā t

D1 1+g

=

1+k 1+k

t=0

D1

= 1+k

1+g

1ā’

1+k

This is provided the discount rate k is greater than the rate of growth g. This leads to the

Gordonā“Shapiro formula:

D1 dB1 drE0

V0 = = =

kā’g kā’g kā’g

Example

The capital of a company consists of 50 000 equities, for a total value of 10 000 000. The

investment proļ¬tability rate is 15 %, the proļ¬t distribution rate 40 %, and the discount

rate 12 %.

The proļ¬t per equity will be:

10 000 000

B = 0.15 Ā· = 30

50 000

The dividend per equity will therefore be D = 0.4 Ć— 30 = 12. In addition, the rate of

growth is given as follows: g = 0.15 Ć— (1 ā’ 0.4) = 0.09.

50 Asset and Risk Management

The Gordonā“Shapiro formula therefore leads to:

12 12

V0 = = = 400

0.12 ā’ 0.09 0.03

The market value of this company is therefore 50 000 Ć— 400 = 20 000 000, while its

accounting value is a mere 10 000 000.

D1

The Gordonā“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 proļ¬t for each equity:

C0

PER 0 =

B1

Its interpretation is quite clear: when purchasing an equity, one pays PER 0 Ć— Ā¤1 for a

proļ¬t of Ā¤1. Its inverse (proļ¬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 Gordonā“Shapiro formula, if the hypotheses

relative to the formula are satisļ¬ed. In fact, by replacing the rate with the V0 value given

for this formula:

D1 dB1

C0 = =

kā’g k ā’ r(1 ā’ d)

we arrive directly at:

d

PER 0 =

k ā’ r(1 ā’ d)

This allows the following expression to be obtained for the rate of return k:

d

k = r(1 ā’ d) +

PER 0

1ā’d

1

= r(1 ā’ d) + ā’

PER 0 PER 0

As PER 0 = C0 /rE0 , we ļ¬nd that:

r(1 ā’ d)(C0 ā’ E0 )

1

k= +

C0

PER 0

Example

If one takes the same ļ¬gures as in the previous paragraph:

Equities 51

r = 15 %

d = 40 %

10 000 000

E0 = = 200

ńņš. 2 |