6.3.2.1 Individual VaR

The individual VaR of the security (j ) within the portfolio is the VaR of all of these

securities; if their number is nj , we will have:

VaR j — = ’zq · σ (nj pj )

= ’zq · nj pj · σj

= ’pP · zq · Xj σj

As σij ¤ σi σj (the correlation coef¬cient is always 1 or less), we have:

N N

—

VaR P ≥ ’pP · zq · Xi Xj σi σj

i=1 j =1

N

= ’pP · zq · Xj σj

j =1

N

VaR j —

=

j =1

The right-hand member, which can be interpreted as the nondiversi¬ed VaR of the port-

folio, is therefore always lower (that is, more pessimistic) than the VaR diversi¬ed into

VaR — P .

6.3.2.2 Marginal VaR

The marginal VaR measures the alteration to the VaR of a portfolio following a minor

variation in its composition. More speci¬cally, it relates to the variation rate VaR P — =

’pP · zq · σP , explained by the variation in σP brought about by an in¬nitesimal increase

in the proportion Xj of the security (j ) while the other proportions remain constant. It is

therefore equal to:

VaR j — = ’pP · zq · (σP )Xj

Theory of VaR 195

As (σP )Xj = 2σP (σP )Xj and:

2

N

(σP )Xj = 2Xj σj2 +2 Xi σij

2

i=1

i=j

N

=2 Xi cov(Ri , Rj )

i=1

= 2 cov(RP , Rj )

= 2σj P

we ¬nally obtain:

σj P

VaR j — = ’pP · zq ·

σP

= ’pP · zq · σP βj P

= VaR P — · βj P

6.3.2.3 Components of VaR

We have seen that it is not possible to split the VaR on the basis of individual VaR values,

as these values do not ˜bene¬t™ from the diversi¬cation effect. The solution is to de¬ne

the VaR component that relates to the security (j ) through the marginal VaR affected by

a weight equal to the Xj proportion of (j ) within the portfolio:

CVaR j — = Xj · VaR j —

What we have, in fact, is:

N N

—

VaR j —

CVaR j = Xj ·

j =1 j =1

N

—

= VaR P · Xj βj P

j =1

N

VaR P —

= · Xj cov(Rj , RP )

σP

2

j =1

= VaR P —

6.3.3 Incremental VaR

6.3.3.1 De¬nition

Assume that we are in possession of a portfolio P and are anticipating the purchase of

an additional set of equities A, the values of P and A being expressed as pP and pA

196 Asset and Risk Management

respectively. It is considered that pA is positive or negative depending on whether an

investment or disinvestment is involved. The portfolio produced by this decision will be

expressed as P with value pP = pP + pA .

The incremental VaR following the acquisition of A is de¬ned as:

IVaR A — = VaR P — ’ VaR P —

Together with the use of the de¬nition, the practical use of the incremental VaR can

be simpli¬ed if the proportion of A is low. The proportions of A and P within P are

respectively:

pA pP

XA = XP = = 1 ’ XA

pP pP

The variance in the new portfolio, if the second-degree terms in XA can be ignored, is:

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

2 2 22

≈ (1 ’ 2XA )σP + 2XA σAP

2

With the same approximation, we have:

VaR —2 ’ VaR —2 ≈ 2VaR P — · IVaR A —

P P

In addition, we have:

VaR —2 ’ VaR —2 = ’pP · zq · σP + pP · zq · σP

2 2 2 2 2 2

P P

≈ ’pP · zq · (σP ’ σP )

2 2 2 2

σP

2

= VaR —2 ’1

P

σP2

σAP

≈ 2XA · VaR —2 ’1

P

σP2

= 2XA · VaR —2 (βAP ’ 1)

P

Therefore:

VaR —2 ’ VaR —2

P P

≈ XA · VaR P — (βAP ’ 1)

IVaR A ≈ —

2VaR P

6.3.3.2 Hedging investment

Now that the portfolio P and the composition of A have been de¬ned, we look for the

ideal amount of A to acquire. The need, therefore, is to determine pA in order for the

risk to be minimised, that is, on order for IVaR — A to be as high as possible or again, as

P is ¬xed, for VaR — to be as high as possible.

P

Theory of VaR 197

As:

σP = XP σP + XA σA + 2XP XA σAP

2 22 22

1

= 2 [pP σP + pA σA + 2pP pA σAP ]

22 22

pP

we have:

—

= ’pP · zq · σP

VaR P

= ’zq pP σP + pA σA + 2pP pA σAP

22 22

This quantity will be maximised if the expression under the root is minimised; as the

derivative of this last quantity with respect to pA is equal to 2pA σ 2 A + 2pP σAP . the total

of A to be acquired in order to minimise13 the risk is shown as

σAP

pA = ’pP

σA

2

6.3.3.3 Link to Sharp index

We saw in Section 3.3.3 that by using the Sharp index as an acquisition criterion for A

(SI p ≥ SIP ), we obtain the condition:

σP EP ’ RF

EA ≥ EP + ’1

σP XA

By replacing the standard deviation with the VaR in the Sharpe™s index (and by changing

the sign):

EP ’ RF

SIp = ’

VaR P —

the condition becomes:

VaR P — EP ’ RF

EA ≥ EP + — ’1

XA

VaR P

IVaR A — EP ’ RF

= EP + ·

VaR P — XA

that is, ¬nally,

EA ≥ EP + µ(EP ’ RF )

in which we have:

VaR P — ’ VaR P —

1 IVaR A — VaR P —

µ= · =

XP ’ XP

XA VaR P —

1

When XA is low (and XP therefore close to 1), this quantity is none other than the

elasticity of VaR with respect to XP .

13

The sign of the second derivative clearly shows that the total found constitutes a minimum.

7

VaR Estimation Techniques

7.1 GENERAL QUESTIONS IN ESTIMATING VaR

7.1.1 The problem of estimation

The de¬nition of the VaR parameter given in Section 6.2, Pr[ pt ¤ VaR] = 1 ’ q, or

more precisely, VaR q = min{V : Pr[ pt ¤ V ] ≥ 1 ’ q}, clearly shows that knowledge

of the distribution function F p (v) = Pr[ pt ¤ v] for the random variable ˜variation in

value™, pt = pt ’ p0 allows the VaR to be determined:

• either directly (by determining the quantile1 for the variable pt );

• or, assuming that the price of the assets follows a normal law, on the basis of the

expectation and the standard deviation for the variable: VaR q = E( pt ) ’ zq · σ ( pt ).

The aim of this chapter is therefore to present the standard methods for estimating the

distribution of the variable pt for a portfolio.

The possible inputs for this estimation are:

• the mathematical valuation models for the prices of the various assets; these are pre-

sented in brief in Section 3.3;

• the histories, that is, the observations of the various equity prices for a certain number

of past periods.

The problem of estimating the probability distribution for pt and the VaR may therefore

be summarised in the diagram shown in Figure 7.1.

The VaR can be calculated just as readily for an isolated position or single risk factor

as for a whole portfolio, and indeed for all the assets held by a business. Let us take the

case of a risk factor to lay down the ideas: if the value of this risk factor at a moment t

can be expressed as X(t), we will use the relative variation on X

X(t) ’ X(t ’ 1)

(t) = ,

X(t ’ 1)

for the period [t ’ 1; t]2 instead of the absolute variation X(t) ’ X(t ’ 1). Using pt

presents the twofold advantage of:

• making the magnitudes of the various factors likely to be involved in evaluating an

asset or portfolio relative;

• supplying a variable that has been shown to be capable of possessing certain distribu-

tional properties (normality or quasi-normality for returns on equities, for example).

1

Estimating quantiles is often a complex problem, especially for arguments close to 0 or 1. Interested readers should read

Gilchrist W. G., Statistical Modelling with Quantile Functions, Chapman & Hall/CRC, 2000.

If the risk factor X is a share price, we are looking at the return on that share (see Section 3.1.1).

2

200 Asset and Risk Management

Valuation models Historical data

Estimation

technique

VaR

Figure 7.1 Estimating VaR

Note

In most calculation methods, a different expression is taken into consideration:

X(t)

—

(t) = ln

X(t ’ 1)

As we saw in Section 3.1.1, this is in fact very similar to (t) and has the advantage

that it can take on any real value3 and that the logarithmic return for several consecutive

periods is the sum of the logarithmic return for each of those periods.

We will use the following notations:

• The risk factors are represented by X1 , X2 , . . . , Xn , and the price of the equity therefore

obeys a relation of the type p = f (X1 , X2 , . . . , Xn ) + µ.

• The combined duration of the histories is represented by T + 1. If the current moment is

expressed as 0, it is therefore assumed that the observations X(’T ),

X(’T + 1), . . . , X(’1), X(0) are obtainable for the risk factor X. On the basis of

these values, therefore, we can calculate ¬rst of all (’T + 1), . . . , (’1) or the

corresponding — values (we will refer to them hereafter as ).

• In addition, we suppose that the duration between two consecutive observation times

(a day, for example) is also the horizon for which the VaR is calculated. The current

value of the risk factor X is therefore X(0) and we wish to estimate X(1).

7.1.2 Typology of estimation methods

The VaR estimation techniques presented differ according to whether or not:

• distribution hypotheses have been formulated on the share prices;

• one is relying on the share price valuation models;

3

Something that is useful when relying on the hypothesis of normality.

VaR Estimation Techniques 201

• stationarity is assumed (the observed distribution of variations in price and/or the param-

eters estimated on the basis of the histories are still valid for the horizon for which the

VaR is being estimated).

Three methods are presented in detail, namely:

• Estimated variance“covariance matrix method (VC).

• Monte Carlo simulation (MC).

• Historical simulation (SH).

These techniques are described in the following three sections, being presented from the

point of view of the underlying ˜philosophy™ and giving a high-level view of the calcula-

tions, without going into detail about the re¬nements and speci¬c treatments introduced

by the institutions that have de¬ned them more precisely. The methodologies developed

are shown in diagrammatic form in Figure 7.2.4

Note

If we wish to estimate the weekly VaR parameter, for example, on the basis of daily

observations (¬ve working days), the loss variable will be calculated by:

(d) (d) (d) (d) (d)

p (w) = p1 + p2 + p3 + p4 + p5

Valuation models Historical data

Parameters

Simulation

Portfolio

MC

revaluation

VC

HS

VaR

Figure 7.2 The three estimation techniques

4

In fact, historical simulation uses evaluation models, as do the other two methods. However, it only uses them for certain

types of product (nonlinear optionals, for example) and only at the re-evaluation stage.

202 Asset and Risk Management

The distribution of the variation in the weekly value is therefore calculated from there,

because working on the hypothesis that the successive daily variations are independent,

we have:

(d) (d) (d) (d) (d)

E( p (w) ) = E( p1 ) + E( p2 ) + E( p3 ) + E( p4 ) + E( p5 )

(d) (d) (d) (d) (d)

var( p(w) ) = var( p1 ) + var( p2 ) + var( p3 ) + var( p4 ) + var( p5 )

The calculation of the VaR parameter based on the expectation and standard deviation for

the probability law will use the values:

E( p (w) ) = 5 · E( p (d) )

√

σ ( p (w) ) = 5 · σ ( p (d) )

The same reasoning can be applied to any other combination of durations, provided,

however, that the horizon considered is not too long, as any ¬‚aws in the hypothesis of

independence will then become too important.

7.2 ESTIMATED VARIANCE“COVARIANCE

MATRIX METHOD

The method of calculating the VaR using the estimated variance“covariance matrix system

is the method proposed and developed by J. P. Morgan5 using its RiskMetrics system.

This is currently the most complete form of VaR calculation technique, from an operational

point of view; it is also the system that has brought the VaR concept into prominence.

The method consists essentially of three stages, which are covered in the following

three paragraphs:

• The identi¬cation of primary risk factors into which the ¬nancial asset portfolio can be

split.

• The distribution of cash¬‚ows associated with these primary risk factors into simpler

cash¬‚ows, corresponding to standard maturity dates.6

• The effective calculation of the VaR.

In its early versions,7 this method of determining the VAR was based essentially on

three hypotheses:

• The hypothesis of stationarity: the statistical parameters measured on the observed

distribution of the price variations (or of the returns) are good estimations of these

same (unknown) parameters for which the VaR is estimated.

• Returns on the various assets (or risk factors) obey a normal law.

• Asset prices depend on the risk factors in linear manner.

J. P. Morgan, RiskMetrics “ Technical Document, 4th Edition, Morgan Guaranty Trust Company, 1996. Also: Mina J.

5

and Yi Xiao J., Return to RiskMetrics: The Evolution of a Standard, RiskMetrics, 2001.

6

This distribution is known as mapping, and we will use this term in the sequel.

7

Up until the third edition of RiskMetrics.

VaR Estimation Techniques 203

The drawbacks of these hypotheses, which may be restrictive, led J. P. Morgan to

generalise the framework of the method. This means that the third hypothesis (linearity)

is no longer essential because of the way in which optional products are treated (see

Section 7.2.3). In the same way, the hypothesis of normality has been made more ¬‚exible,

as follows.

The hypothesis of normality allowed developments in the return on an asset to be

described using a random walk model, written as: Rt = µ + σ µt . In this model, the µt

random variables are assumed to be independent and identically distributed (iid), with

zero expectation and variance of 1. The µ and σ parameters are therefore given by:

E(Rt ) = µ

var(Rt ) = σ 2

The hypothesis of normality is therefore less of a problem if one assumes that:

• the returns have a volatility that varies over time Rt = µ + σt µt ;

• they are self-correlated, as in the GARCH8 models or even in the random volatil-

ity models.

More generally, for the N assets within a portfolio, the hypothesis becomes Rj t =

µj + σt µj t , where the µj t follows a multi-normal law with zero expectations and of

variance“covariance matrix Vt dependent on time. Returns that obey this condition are

said to verify the hypothesis of conditional normality.

To generalise further, it is also possible to choose a law other than the normal law for the

µt variable, such as the generalised law of errors that allows the leptokurtic distributions

to be taken into account.9

In addition, even by generalising the normality hypothesis, it is possible to calculate

VaR — = ’zq · σ ( pt ) in this methodology, and if one wishes, it is possible to come back

q

to the concept of VaR that corresponds to an absolute variation, via VaR q = VaR — + q

E( pt ).

In essence, this method of working10 simply brings the problem of determining VaR

down to estimating the variance in the portfolio loss.

7.2.1 Identifying cash ¬‚ows in ¬nancial assets

The ¬rst stage therefore consists of converting the various positions in the portfolio into

linear combinations of a certain number of risk factors that are easy to measure and have

a variance (and therefore a VaR) that can be easily calculated.

p = f (X1 , X2 , . . . , Xn ) + µ

= a1 X1 + a2 X2 + · · · + an Xn + µ

8

Interested readers will ¬nd further information on this subject in Gourieroux C., Mod´ les ARCH et applications ¬nanci` res,

e e

Economica, 1992 and in Appendix 7.

9

Readers are referred to Appendix 2 for developments on this law of probability.

In reality, RiskMetrics simply calculates VaR— . In other words, it assumes that the expectation loss is zero, or rather

10

that the subsequent movements in the portfolio prices show no trends; the parameter µ is merely equal to 0.

204 Asset and Risk Management

Each of the risk factors is a cash¬‚ow:

• characterised by a certain amount;

• expressed in a certain currency;

• paid on a certain date.

These cash¬‚ows must of course be discounted on the date on which VaR is calculated

(this date is termed 0).

The choice of risk factors is somewhat arbitrary and faces the following dilemma:

representation of an asset as a number of elementary risks ensures very accurate results,

but is costly in terms of calculation and data needed to supply the model. We now examine

a number of decomposition models to support our statement.

7.2.1.1 Fixed-income securities

Let us consider a security that brings in certain income in amounts X1 , X2 , . . . , XT at the

respective times t1 , t2 , . . . , tT . Some of these income amounts may be positive and others

negative. The full range of the risk factors on which the security depends is expressed as:

X1 (1 + r1 )’t1 X2 (1 + r2 )’t2 · · · XT (1 + rT )’tT

t1 t2 ··· tT

Here, rj represents the market interest rate for the corresponding period.

We are therefore looking at the breakdown of a ¬xed-income security into a total of

zero-coupon bonds and the statistical data required will therefore relate to these bonds.

This type of decomposition will therefore apply to most products with interest rates, such

as simple and variable-rate bonds, interest swaps, FRAs, futures with interest rates etc.

7.2.1.2 Exchange positions

For an exchange position at the current time, the two cash¬‚ows to be taken into consider-

ation are the two amounts that are the subject of the transaction. They are merely linked

by the exchange rate for the two currencies in question.

For an exchange position concluded for a future date T at a current (forward) rate F ,

we are required to consider not just the exchange rate on T (spot rate), but also S, the

interest rate on the two currencies. The principle of exchange rate parity in fact states

that the purchase of a currency X with a currency Y (the exchange rates being expressed

as ˜Y per X™) is the same as borrowing the currency Y at the present time at rate rY and

reselling it at T for X or exchanging Y for X at the present time and investing it at rate

rX : (1 + rY ) · S = F · (1 + rX ).

The purchase and sale of the two currencies at the time T can therefore be split into a

loan, an investment and an exchange deal at moment 0. The same type of reasoning can

also be applied to exchange swaps.

7.2.1.3 Other assets

For equities, the cash¬‚ows correspond simply to spot positions, which may be adapted

according to an exchange rate if the security is quoted on the foreign market.

VaR Estimation Techniques 205

Raw materials are treated in the same way as ¬xed-income securities, on the basis of

spot prices and the principle of discounting.

7.2.2 Mapping cash¬‚ows with standard maturity dates

Once the cash¬‚ows have been determined, application of the VaR calculation VaR q — =

’zq · σ ( pt ) requires the variances in each of the cash¬‚ows and of all the two-by-two

covariances to be known. If one looks at all the different possible dates, the task is quite

clearly impossibly large; such a set of data does not exist and even if it did, processing

the data would be far too unwieldy a process.

This is the reason for carrying out mapping “ redistributing the various cash¬‚ows

across a limited, previously determined range of standard maturity dates. In this way,

RiskMetrics supplies the data that allows the VaR to be calculated for 14 standard

maturity dates:11 1 month, 3 months, 6 months, 1 year, 2 years, 3 years, 4 years, 5 years,

7 years, 9 years, 10 years, 15 years, 20 years and 30 years.

For a cash¬‚ow that corresponds to one of the vertices, no mapping will be necessary.

On the other hand, a cash¬‚ow that occurs between two vertices should be divided between

the two neighbouring vertices, one directly above and the other directly below. Suppose

that a cash¬‚ow with current value V0 and maturity date t0 has to be divided between the

vertices t1 and t2 with t1 < t0 < t2 (see Figure 7.3).

The problem is that of determining the current values V1 and V2 of the cash¬‚ows

associated with the maturity dates t1 and t2 . A number of procedures, based on slightly

different principles, can be applied to make this division.

7.2.2.1 Elementary mapping

The ¬rst way of carrying out the mapping appears a priori to be very natural, as it relies

on the most frequently applied principles of ¬nancial calculations. In fact, it requires:

• Preservation of the current value, V0 = V1 + V2 .

t0 V0 t1 V1 + t2 V2

• Preservation of the duration = , or also: t0 V0 = t1 V1 + t2 V2 .

V0 V 1 + V2

The system put together by these two conditions is very easy to resolve:

t2 ’ t0 t0 ’ t1

V1 = V0 V2 = V0

t2 ’ t1 t2 ’ t1

(V0)

Real maturities

t0

(V1) (V2)

Standard maturities

t1 t2

Figure 7.3 Mapping

11

Better known as vertices.

206 Asset and Risk Management

Example

Let us consider a cash¬‚ow with a maturity date of t0 = 12 years and a total 1000 (at

that moment). The standard neighbouring maturity dates are t1 = 10 years and t2 = 15

years, and we assume that the interest rates for the two periods are r1 = 7 % and r2 =

8 %. From that, the rate of interest corresponding to the duration of the cash¬‚ow to

be processed (r0 = 7.4 %) is determined by linear interpolation and the current value is:

V0 = 1000 · (1 + 0.074)’12 = 424.57.

As the data have been speci¬ed thus, the divided current values can easily be calculated:

15 ’ 12

V1 = · 424.57 = 254.74

15 ’ 10

12 ’ 10

V2 = · 424.57 = 169.83

15 ’ 10

If one expresses the returns associated with each cash¬‚ow as R0 , R1 and R2 , we

clearly have:

R0 V0 = R1 V1 + R2 V2

hence the following expression for the variance:

σ0 V 2 0 = σ1 V12 + σ2 V22 + 2σ12 V1 V2

2 2 2

= σ1 V12 + σ2 V22 + 2σ1 σ2 ρ12 V1 V2

2 2

Because of the presence of the correlation coef¬cient ρ12 , which is less than 1, the variance

thus calculated may be underestimated:

σ0 V02 ¤ σ1 V12 + σ2 V22 + 2σ1 σ2 V1 V2

2 2 2

= (σ1 V1 + σ2 V2 )2

In addition, with the values found for V1 and V2 , the maximum value of the standard

deviation can be written as:

σ 0 V0 = σ 1 V1 + σ 2 V2

t2 ’ t0 t0 ’ t1

= σ1 V0 + σ 2 V0

t2 ’ t1 t2 ’ t1

This shows that σ0 is obtained by linear interpolation on the basis of σ1 and σ2 .

7.2.2.2 Mapping according to RiskMetrics

The problem of possibly underestimating the standard deviation (and therefore the VaR)

will be overcome if the standard deviation is determined by linear interpolation. The

mapping suggested by RiskMetrics is therefore de¬ned by:

• Preservation of the current value: V0 = V1 + V2 .

VaR Estimation Techniques 207

• Preservation of the risk σ0 V02 = σ1 V12 + σ2 V22 + 2σ1 σ2 ρ12 V1 V2 where σ1 , σ2 and ρ12

2 2 2

are data and σ0 is obtained by linear interpolation.

Because of the ¬rst relation, it is possible to introduce the parameter ±, for which

V1 V2

±= 1’± =

V0 V0

This gives, in the second relation: σ0 = σ1 ± 2 + σ2 (1 ’ ±)2 + 2σ1 σ2 ρ12 ±(1 ’ ±).

2 2 2

This can also be written: ± 2 (σ1 + σ2 ’ 2σ1 σ2 ρ12 )+2±(σ1 σ2 ρ12 ’ σ2 )+(σ2 ’ σ0 ) = 0.

2 2 2 2 2

This second-degree equation will allow two solution couples to be found (V1 , V2 ) and

the couple consisting of two numbers of the same sign as V0 will be preserved.

Example

Let us take the previous example, for which the cash¬‚ow to be divided was V0 = 424.57

and the respective maturity dates were t1 = 10, t0 = 12 and t2 = 15.

Let us assume that we have the following additional data: σ1 = 0.6 %, σ2 = 0.9 %,

ρ12 = 0.94.

We note ¬rst of all a slight underestimation12 of the risk caused by the diversi¬cation

effect when the mapping is applied, on the basis of preservation of duration. The variance

formula gives σ 2 0 V 2 0 = 9.0643.

Hence σ0 = 0.7091 % instead of the value σ0 = 0.72 % obtained by linear interpolation.

With this latter value for σ0 , the mapping equation based on preservation of risk allows

us to write:

1548± 2 ’ 6048± + 2916 = 0

We therefore obtain the two solutions ± = 3.3436 and ± = 0.5634. Only the second will

guarantee cash¬‚ows with the same sign as V0 , and allows the following to be calculated:

V1 = ±V0 = 239.19

V2 = (1 ’ ±)V0 = 185.37

7.2.2.3 Alternative mapping

The mapping suggested by RiskMetrics has the advantage over the previous form

of guaranteeing the preservation of risk (and it is exactly that parameter that interests

us here).

It does, however, have an adverse effect: it compensates for the diversi¬cation effect

by allocating the greater part of the cash¬‚ow to the vertex with the greater volatility, even

though the actual maturity date may be closer to the vertex with the lower volatility. This

phenomenon is evident in the ¬gures in the example that we have just developed, but it

can be veri¬ed generally.13

12

The correlation coef¬cient here is in fact very close to 1.

13

Schaller P., On Cash¬‚ow Mapping in VaR Estimation, Creditanstalt-Bankverein, CA-RISC-19960227, 1996.

208 Asset and Risk Management

The phenomenon would not be of great import if mapping only had to be produced

for a single cash¬‚ow. It becomes much more problematic, however, if two cash¬‚ows are

considered simultaneously: one with a total X corresponding to a vertex t1 and another

with a total ’X with a maturity date close to t1 . P. Schaller13 , shows that by varying the

second maturity date continually on one side or another of t1 , the parameter ± shows a

discontinuity at t1 .

Schaller™s proposal is therefore to retain the condition of risk preservation, in order to

prevent underestimation of the risk through the diversi¬cation effect, and to replace the

V1

, allocated to the vertex t1 tends

other condition by expressing that the proportion

V 1 + V2

towards 100 % (resp. 0 %) when the actual maturity date t0 tends towards t1 (resp. t2 ).

More speci¬cally, he proposes that the proportion in question should be a linear function

f of the variable t0 so that f (t1 ) = 1 and f (t2 ) = 0.

V1 t2 ’ t0

= f (t0 ) = , and therefore that t2 V1 ’ t1 V1 =

It is easy to see that

V 1 + V2 t2 ’ t1

t2 V1 + t2 V2 ’ t0 (V1 + V2 ).

To sum up, Schaller™s mapping is de¬ned by

• Preservation of the quasi-duration: t0 (V1 + V2 ) = t1 V1 + t2 V2 .

• Preservation of the risk: σ 2 0 V 2 0 = σ 2 1 V 2 1 + σ 2 2 V 2 2 + 2σ1 σ2 ρ12 V1 V2 , where σ1 , σ2

and ρ12 are data and σ0 is obtained by linear interpolation.

V1 t2 ’ t0

By introducing β = = , which corresponds to the ¬rst condition, the

V1 + V 2 t2 ’ t1

second can be written:

V02

σ0 = σ1 β 2 + σ2 (1 ’ β)2 + 2σ1 σ2 ρ12 β(1 ’ β)

2 2 2

(V1 + V2 )2

This allows (V1 + V2 ) and therefore V1 and V2 to be calculated.

Example

With the data from the example already used for the ¬rst two mappings, we ¬nd succes-

sively that:

15 ’ 12

β= = 0.6

15 ’ 10

424.572

0.0072 · = 0.00005028

2

(V1 + V2 )2

V1 + V2 = 431.08

From this, it can be deduced that:

V1 = β(V1 + V2 ) = 258.65

V2 = (1 ’ β)(V1 + V2 ) = 172.43

VaR Estimation Techniques 209

Note

It may seem surprising that Schaller™s mapping is made without preserving the current

value. In reality, this is less serious than it may appear, because unlike normal ¬nancial

calculations, we are not looking at the value of a portfolio but at its risk. In addition, in

his paper, the author provides a thorough justi¬cation (through geometric logic) for his

method of work.

7.2.3 Calculating VaR

7.2.3.1 Data

One reason why the estimated variance“covariance matrix method leads to mapping

of elementary risk positions associated with a standard ¬nite set of maturity dates is

because it would be unthinkable to obtain this matrix for all the possible positions. We

are thinking about:

• the various families of available securities;

• the full range of securities in each group (number of quoted equities, for example), in

various stock-exchange locations;

• the consideration of various exchange rates between these ¬nancial locations;

• the abundance of derivatives, which have an in¬nite number of possible combinations;

• the various maturity dates of all the products.

The numerical values supplied for using this method are therefore the evaluations of

the expected values and variances of the various risk factors, as well as the covariances

relating to the corresponding couples.

It has therefore been agreed to use these parameters for the following risk factors:

• exchange rates (RiskMetrics uses 30 currencies);

• interest rates for the representation of ¬xed-rate instruments, such as bonds with map-

ping with a zero-coupon bond total (RiskMetrics uses 14 levels of maturity varying

from one month to 30 years);

• national stock-exchange indices for representing equities using the ˜beta™ model (Risk-

Metrics uses 30 national indices);

• a representative panel of raw materials.

Note 1

In Section 2.2.2 we introduce two concepts of volatility relative to one option: histor-

ical and implicit. Of course the same can be applied here for all the variances and

covariances. It has also been said that the implicit parameters are more reliable than

the historical parameters. However, because of the huge volume and frequency of data

obtained, it is the historical variances and covariances that will be adopted in prac-

tice.

210 Asset and Risk Management

Note 2

The estimates of the variances and covariances can be calculated using the usual descrip-

tive statistical formulae. If the estimation of such a parameter for the period t uses

observations relative to periods t ’ 1, t ’ 2, t ’ T , these formulae are:

T

1

σR,t = (Rt’j ’ R)2

2

T j =1

T

1

σ12,t = (R1,t’j ’ R 1 )(R2,t’j ’ R 2 )

T j =1

In actual fact, RiskMetrics does not use these formulae, but instead transforms the

equally weighted averages into averages whose weight decreases geometrically in order

to give greater weight to recent observations. For » positioned between 0 and 115 , we

calculated:

T

»j ’1 (Rt’j ’ R)2

σR,t = (1 ’ »)

2

j =1

T

»j ’1 (R1,t’j ’ R 1 )(R2,t’j ’ R 2 )

σ12,t = (1 ’ »)

j =1

These formulae are justi¬ed by the fact that if T is high, the weights will have a sum

equal to 1 as:14

∞

1

»j ’1 =

1’»

j =1

In addition, they present the advantage of not requiring all the data involved in the

formulae to be memorised, as they are included in the calculation by recurrence:

T

»j ’1 (Rt+1’j ’ R)2

σR,t+1 = (1 ’ »)

2

j =1

T ’1

»j ’1 (Rt’j ’ R)2

= »(1 ’ »)

j =0

®

T

1

= »(1 ’ ») ° (Rt ’ R)2 »

»j ’1 (Rt’j ’ R)2 ’ »T ’1 (Rt’T ’ R)2 +

»

j =1

= »σR,t + (1 ’ »)(Rt ’ R)2

2

14

This concept, known as a ˜geometric series™, is dealt with in Appendix 1.

VaR Estimation Techniques 211

since »T ’1 is negligible if T is suf¬ciently high. In the same way, we have:

σ12,t+1 = »σ12,t + (1 ’ »)(R1t ’ R 1 )(R2t ’ R 2 )

7.2.3.2 Calculations for a portfolio of linear shares

Let us consider a portfolio consisting of N assets in respective numbers n1 , . . . , nN , the

value of each asset being expressed on the basis of risk factors X1 , X2 , . . . , Xn through

a linear relation:

pj = aj 1 X1 + · · · + aj n Xn j = 1, . . . , N

This is the case with securities such as bonds, equities, exchange positions, raw materials,

interest and exchange swaps, FRAs etc.

The value of the portfolio can therefore be written as:

N

pP = n j pj

j =1

N n

= nj aj k Xk

j =1 k=1

«

n N

nj aj k Xk

=

k=1 j =1

Here, if one wishes to take account of time:15

«

n N

nj aj k Xk (t)

pP (t) =

k=1 j =1

The variation in the portfolio™s value can therefore be written:

pP (t) = pP (t) ’ pP (t ’ 1)

«

n N

nj aj k (Xk (t) ’ Xk (t ’ 1))

=

k=1 j =1

«

n N

Xk (t) ’ Xk (t ’ 1)

nj aj k Xk (t ’ 1)

=

Xk (t ’ 1)

k=1 j =1

n

= xk (t ’ 1) k (t)

k=1

It is assumed that the in¬‚uence of the risk factors (measured by aj k ) and the composition of the portfolio (nj ) does not

15

change at the VaR calculation horizon.

212 Asset and Risk Management

Here, k has the standard interpretation, and xk (t ’ 1) = N=1 nj aj k Xk (t ’ 1) can be

j

interpreted as the total invested in (t ’ 1) in the k risk factor.

th

We are interested in the changes to the portfolio between time 0 (now) and time 1 (VaR

calculation horizon). It is therefore assumed that we have the following observations:

• The current positions in the various risk factors xk (0) (k = 1, . . . , n), which we will

subsequently express simply as xk .

• The estimation, based on historical periods, of the expected relative variations for the

various risk factors E( k (1)) = Ek (k = 1, . . . , n).

• The estimation, also based on historical periods, of the variances and covariances for

the risk factors:

var( k (1)) = σk2 = σkk k = 1, . . . , n

k (1), l (1)) = σkl k, l = 1, . . . , n

cov(

These data can be expressed in matrix form:

« « «2

σ1 σ12 ··· σ1n

x1 E1

¬ x2 · ¬ E2 · ¬ σ21 σ2n ·

σ2 ···

2

¬· ¬· ¬ ·

x=¬ . · E=¬ . · V =¬ . .·

. ..

. . . . .

.

. . . . .

xn En σn1 σn2 ··· σn2

The ¬‚uctuations in the variation in value can be written as (see Section 3.1.1):

σ 2 ( pP (1)) = var( pP (1))

n n

= xk xl σkl

k=1 l=1

= xt V x

This allows the following calculation:

√

VaR — = ’zq σ ( pP (1)) = ’zq x t V x.

q

And therefore:

VaR q = VaR — + E( pP (1))

q

√

= x t E ’ zq x t V x

Note 1

In actual fact, the RiskMetrics system does not produce the matrix V for the variances

and covariances, but two equivalent elements.

• The value zq σk (k = 1, . . . , n) of the VaR parameter for each risk factor, where q is

chosen as equalling 0.95 and therefore zq = 1.6465. On the basis of these data, we

construct the matrix:

S = diag(’zq σ1 , . . . , ’zq σn ) = ’zq diag(σ1 , . . . , σn ).

VaR Estimation Techniques 213

σkl

• The matrix for the correlations ρkl = :

σk σl

«

ρ12 · · · ρ1n

1

¬ ρ21 · · · ρ2n ·

1

¬ ·

C=¬ . .·

. ..

. . .

.

. . .

ρn1 ρn2 ··· 1

Here, we clearly have: V = [diag(σ1 , . . . , σn )]t Cdiag(σ1 , . . . , σn ).

We can therefore calculate the VaR by:

VaR — = ’ zq x t [diag(σ1 , . . . , σn )]t Cdiag(σ1 , . . . , σn )x

2

q

√

= ’ x t S t CS x

= ’ (S x)t C(S x)

Note 2

This method shows the effect of diversi¬cation on risk with particular clarity; if there

were no diversi¬cation, the elements in the matrix C would all be worth 1. In reality,

however, they are worth 1 or less, which improves the value of the VaR. In addition,

without the effects of diversi¬cation, all the components of C(Sx) would be equal to

n

k=1 (S x)k , and the VaR parameter would be given by:

n

—

V aRq = (S x)k

k=1

n

= ’zq xk σk

k=1

7.2.3.3 The speci¬c case of equities

If one is considering a portfolio of equities, it is no longer possible to use the vari-

ance“covariance matrix, which would be too bulky. For this reason, equities are dealt

with on an approximation basis “ a diagonal model. This model is comparable to Sharpe™s

simple index model presented in Section 3.2.4, but its index must be representative of the

market.16

The only statistical elements needed in this case are the expected return EI and the

standard deviation σI for the index, the coef¬cients aj and bj of the model for the j th

security

pj I

= aj + bj + µj

pj I

and the variances for the residuals from the various regressions:

σµ2j (j = 1, . . . , N )

16

Sharpe™s model, depending on the type of asset in question, may involve a sectorial index.

214 Asset and Risk Management

In fact, it is known that the expected return and the portfolio variance are given in this

case by:

E( pP ) = aP + bP EI

N

var( pP ) = bP σI2 + n2 pj σµ2j

2

j

j =1

where we have:

N

aP = nj aj pj

j =1

N

bP = nj bj pj

j =1

Note that this model, in which correlations between securities do not appear, nevertheless

takes account of them as the various equities are linked to the same index. It can also be

shown that:

pj

pi

, = bi bj σI2

cov

pi pj

To make things easier, the RiskMetrics system actually uses a variance on this diag-

onal model, which differs only from the model we have just presented by the absence of

N

n2 pj σµ2j from the second member of the formula that gives the variance for

the term j

j =1

the portfolio.

This term takes account of the differences between the returns observed on the various

securities and those shown by the model. This method of working therefore gives a

slightly underestimated value for the VaR parameter, which according to the previous

section, is:

VaR — = ’zq bP σI

q

7.2.3.4 Calculations for a portfolio of nonlinear values

For nonlinear products (essentially options17 ), the situation is much more complicated

because of their nonlinear nature. Linearity is in fact essential for applying the estimated

variance“covariance matrix method, as a linear combination of multinormal variables

(correlated or otherwise) is itself distributed according to a normal law. We will assume

only that the variances in underlying prices are distributed according to a normal law

with zero expectation and standard deviation σt . As for the distribution of the variation

in option price, we will show how this is determined.

Let us approach the case of an isolated option ¬rst. The method involves Taylor™s

development of the valuation model. The importance of the underlying equity price over

the option price is such that for this variable, we develop up to order 2 and the variation

17

For other methods of approach to VaR for options products, read Rouvinez C., Going Greek with VaR, Risk Magazine,

February 1997, pp. 57“65.

VaR Estimation Techniques 215

in the premium is therefore obtained according to the option™s various sensitivity coef¬-

cients:18

pt = fS (St , K, „, σt , RF ) St + 1 fSS (St , K, „, σt , RF )( St )2

2

+ fK (St , K, „, σt , RF ) K + f„ (St , K, „, σt , RF ) „

+ fσ (St , K, „, σt , RF ) σt + fR (St , K, „, σt , RF ) RF

= · St + · ( St )2 + fK · K’ · „ +V · σt + ρ · RF

1

2

As in RiskMetrics, we will limit our analysis to the , and parameters only:

pt = · St + · ( St )2 ’ · „

1

2

It can be demonstrated on that basis that the expectation, variance, skewness coef¬cient

γ1 and kurtosis coef¬cient γ2 of pt are obtained by:

E( pt ) = σt2 ’ ( „)

1

2

var( pt ) = σt2 + σt4

2 2

1

2

σt (3 + σt2 )

2 2

γ1 ( pt ) =

( 2+ 2 σ 2 )3/2

1

t

2

σt2 (4 + 2 σt2 )

2 2

3

γ2 ( pt ) =

( 2+ 2 σ 2 )2

1

t

2

To evaluate the parameters involved in the formulae, , and are determined using

the Black and Scholes valuation model through suitably adapted pricing software and

σt is calculated on the basis of historical observations, as explained in the previous

section.

The distribution of pt is then determined as follows: from among a series of prob-

ability laws, known as the Johnson Distributions,19 we look for the one for which the

above four parameters correspond best, More speci¬cally, from among the following

relations:

pt ’ c

St = a + b

d

pt ’ c

St = a + b · ln

d

pt ’ c

St = a + b · arcsh

d

pt ’ c

St = a + b · ln

c + d ’ pt

which de¬ne respectively normal, log-normal, bounded and unbounded laws, the law and

parameters a, b, c and d can be determined20 so that for the four parameters mentioned

above, St follows a normal law with zero expectation and standard deviation σt .

In fact, t = 1 when the VaR calculation horizon is 1 day.

18

19

Johnson N. L., Systems of frequency curves generated by methods of translation, Biometrika, Vol. 36., 1949, pp. 149“75.

20

Using the algorithm described in Hill I. D., Hill R. and Holder R. L., Fitting Johnson curves by moments (Algorithm

AS 99), Applied Statistics, Vol. 25 No. 2, 1976, pp. 180“9.

216 Asset and Risk Management

As the aim here was to take account of nonlinearity (and therefore skewness and kurtosis

among other properties), we are no longer looking at calculating VaR using the formula

VaR — = ’zq · σ ( pt ).

q

In this case, take once again the initial de¬nition, based on the quantile

Pr[ pt ¤ VaR] = 1 ’ q and therefore, for the law determined, write the equation St =

zq and resolve it with respect to pt .

In the case of a portfolio of optional products, the method shown above is generalised

using the matrix calculation. Thus, for a portfolio consisting of N optional assets in

respective numbers n1 , . . ., nN , the ¬rst two distribution parameters for the variance

pP ,t in the portfolio price are written as:

N

1

E( pP ,t ) = tr( V ) ’ nj j( „)

2 j =1

1

t

var( pP ,t ) = V + tr( V V )

2

where V is the variance“covariance matrix for the returns of the various underlying

equities, and:

« «

n1 1 ··· n1 1

0 0

¬0 0· ¬ n2 2 ·

n2 2 · · ·

¬ · ¬ ·

=¬ . =¬ . ·

.·

. ..

. . . .

.

. . . .

· · · nN N nN N

0 0

Note 1

For optional products, RiskMetrics also proposes the use of the Monte Carlo methodol-

ogy, as described in Section 7.3, alongside the approach based on Taylor™s development

of the valuation model.

Note 2

This VaR evaluation method is a local method, as it ¬rst determines the VaR parameter

for each of the risk factors separately (the matrix S from Note for linear assets), and then

aggregates them through the estimated variance“covariance method.

Conversely, the Monte Carlo and historical simulation methods introduced in

Sections 7.3 and 7.4 initially determine the future value of the portfolio, and then deduce

the VaR value compared to the current value of the portfolio.

7.3 MONTE CARLO SIMULATION

7.3.1 The Monte Carlo method and probability theory

7.3.1.1 Generation of probability distribution

Let us examine ¬rst of all the problem of generating values for a random variable X

distributed according to a probability law whose distribution function FX is known.21

21

Vose D., Quantitative Risk Analysis, John Wiley & Sons, Ltd, 1996.

VaR Estimation Techniques 217

It is relatively easy to obtain the values for a uniform random variable within the

interval [0; 1]. One can, for example, choose the list for a particular ¬gure (the units

¬gure or the last informed decimal) in a sequence of numbers taken within a table: a

table of mathematical or statistical values, mortality table, telephone directory etc. One

can also use a table of random numbers, available in most formula collections.22 It is also

possible to use software that generates pseudo-random numbers.23

As the full range of values for the distribution function FX is precisely the interval [0;

1], it is suf¬cient for the purpose of generating the values of X to transform the values

obtained for a uniform random variable U within [0; 1], using the reciprocal function of

FX . This function is the quantile QX for the variable X, and we will calculate QX (u) for

each value u generated within [0; 1].

The variable QX (U ) (random through the channel of U ) is properly distributed accord-

ing to the law of X, as, through de¬nition of the uniform random variable,

Pr[QX (U ) ¤ t] = Pr[FX (QX (U )) ¤ FX (t)]

= Pr[U ¤ FX (t)]

= FX (t)

This property is also illustrated in Figure 7.4.

7.3.1.2 Using the variables generated

The Monte Carlo simulation is a statistical method used when it is very dif¬cult to

determine a probability distribution by mathematical reasoning alone. It consists of ˜fab-

ricating™ this distribution through the generation of a large number of pseudo-random

samples extracted from the distribution, whether it is:

• known theoretically with parameters of a speci¬ed numerical value (by estimation,

for example);

• the result of observations or calculations on observations.

Fx(t)

x

0

Figure 7.4 Monte Carlo simulation

22

For example, Abramowitz M. and Stegun A., Handbook of Mathematical Functions, Dover, 1972.

23

A method for generating pseudo-random numbers uniformly distributed within the interval [0; 1] is developed in

Appendix 2.

218 Asset and Risk Management

Suppose for example that one wishes to ¬nd the distribution of samples through a

random variable X for which the law of probability is known but not normal. A sample

is extracted from this law by the procedure explained above, and the average of this

procedure is a value that we will call x1 . If one then generates not one but many samples

M in the same way, a distribution will be built up: (x 1 , x 2 , . . ., x M ). This, with a good

approximation if the size of the samples and M are suf¬ciently large, is the distribution

of sampling for the expectation of X.

7.3.2 Estimation method

For the problem of constructing the distribution of the variation in value and estimating

the VaR, the Monte Carlo simulation method is recommended by Bankers Trust in its

RaRoc 2020 system.

7.3.2.1 Risk factor case

Let us take ¬rst of all the very simple case in which the value for which the VaR is

required is the risk factor X itself (the price for an equity, for example). The random

variable

X(1) ’ X(0)

=

X(0)

is distributed according to a certain law of probability. This law can of course be evaluated

on the basis of observations:

X(t) ’ X(t ’ 1)

(t) = t = ’T + 1, . . . , ’1, 0

X(t ’ 1)

However, as this variable will have to be generated using the Monte Carlo technique,

it is essential for the distribution used not to present any irregularities corresponding

to differences with respect to the real distribution. The Monte Carlo method is in fact

very sensitive to these irregularities and they are especially likely to show themselves for

extreme values, which are precisely the values of interest to us.

The solution adopted is therefore a valuation model that is adapted to the risk factor

X. Thus, for the return on an equity, a normal or more leptokurtic (see Section 6.1.2)

distribution will be chosen; if one wishes to take account of a development over time,

an adapted stochastic process known as geometric Brownian motion (see Section 4.3.2)

may be used. It expresses a relative variance in the price St at time t according to the

expected return ER and the volatility σR of the security in a two-term sum:

St+dt ’ St dSt

= = ER · dt + σR · dwt

St St

Only the second term is random, dwt being distributed according to a normal law of zero

√

expectation and standard deviation dt.

If the general form of the valuation model is the result of theoretical reasoning,24 the

parameters (expectation and standard deviation in the cases mentioned above) for this

model shall be evaluated directly on the basis of observations.

24

Reasoning for which the suitability needs to be veri¬ed using observations.

VaR Estimation Techniques 219

Through Monte Carlo simulation of the underlying random variable and use of the

adapted valuation model, we therefore generate a large number M of pseudo-observations25

(t)

(t = 1, . . ., M) for the random variable .

The relation X(1) ’ X(0) = . X(0) allows the future value of the risk factor X to be

estimated by:

X(t) (1) = X(0) + (t) · X(0) t = 1, 2, . . . , M

The distribution of the variation in value can therefore be estimated by:

p (t) = X(t) (1) ’ X(0) = (t)

· X(0). t = 1, 2, . . . , M

7.3.2.2 Isolated asset case

Let us now consider the case in which the asset for which one wishes to determine the

VaR depends on several risk factors X1 , X2 , . . . , Xn . The value of this asset is therefore

expressed by a relation of the type p = f (X1 , X2 , . . . , Xn ).

The observations of these risk factors Xk (t), (k = 1, . . . , n, t = ’T + 1, . . . , ’1, 0),

allow the valuation model parameters and the stochastic processes relative to these risk

factors to be estimated.

If the k variables (k = 1, . . . , n) are generated independently of each other, the

correlation that links the various risk factors exerting an in¬‚uence on the price of the equity

will be lost completely. Also, the Monte Carlo simulation method has to be adapted in

order for various variables to be generated simultaneously while respecting the associated

variance and covariance matrix. This matrix will of course also be calculated on the basis

of available observations.

Let us begin by reasoning over the case of two risk factors. We wish to generate 1

and 2 according to a given law, so that:

E( 1) = µ1 E( 2) = µ2

1) = σ1 2) = σ2

2 2

var( var(

1, 2) = σ12

cov(

If we generate two other variables δ1 and δ2 that obey the same theory, but so that

E(δ1 ) = 0 E(δ2 ) = 0

var(δ1 ) = 1 var(δ2 ) = 1

δ1 and δ2 are independent

It is suf¬cient to de¬ne:

= σ1 δ1 + µ1