ńņš. 12 |

VaR's premium

calculation

Is the Conver-

no

gence acceptable?

yes

End of Picard's

process

END

Figure 10.2 Algorithm for taking account of VaR in the EGP method

The comparison of the portfolios obtained is based on the VaR premium for a VaR at

99 %, 95 % and 90 %. The historical period for the rates retained runs from 11 April 1995

to 11 April 2000. We have chosen as the risk-free rate the interbank rate ā˜Euribor 1 weekā™

for 11 April 2000, which was 3.676 %.23 See Tables 10.1ā“10.6.

10.2.4 Conclusion

It will be noted ļ¬rst of all that the two methods used propose portfolios higher than the

market index, regardless of the starting hypotheses applied.

Table 10.1 Composition of portfolio (VaR at 99 %)

EGP EGPā“VaR

Indra Systems 0.094135885 0.092850684

Tele Pizza 0.154240348 0.152162629

Amadeus Global Travel 0.110170100 0.108288303

NH Hotels (Ex-Coļ¬r) 0.131113608 0.130958267

Altadis 0.123080404 0.123684551

Union Fenosa 0.161211661 0.160615701

Terra Networks 0.041129011 0.039914293

Active de Constr. Y Serv. 0.063707852 0.064207223

Corp. Fin. Alba 0.084216091 0.086193217

Acciona 0.015113720 0.015959957

Aguas Barcelona 0.021991319 0.026165174

23

Data obtained from the ā˜Bank of Finlandā™ site.

Optimising the Global Portfolio via VaR 273

Table 10.2 Performance of portfolio (VaR at 99 %)

IBEX35 EGP EGPā“VaR

Weekly return on portfolio 0.004742222 0.011161026 0.011069952

Standard deviation for portfolio 0.028795127 0.033061249 0.032773963

VaR of portfolio at 99 % (weekly horizon) 0.062243883 0.065749358 0.065172119

Portfolio risk premium 0.140570337 0.316580440 0.316576644

Portfolio VaR premium 0.065030338 0.159188547 0.159201074

Table 10.3 Composition of portfolio (VaR at 99 %)

EGP EGPā“VaR

Indra Systems 0.094135885 0.092329323

Tele Pizza 0.154240348 0.151319771

Amadeus Global Travel 0.110170100 0.107524924

NH Hotels (Ex-Coļ¬r) 0.131113608 0.130895251

Altadis 0.123080404 0.123929632

Union Fenosa 0.161211661 0.160373940

Terra Networks 0.041129011 0.064409801

Active de Constr. Y Serv. 0.063707852 0.064409801

Corp. Fin. Alba 0.084216091 0.086995268

Acciona 0.015113720 0.016347870

Aguas Barcelona 0.021991319 0.026452697

Table 10.4 Performance of portfolio (VaR at 95 %)

IBEX35 EGP EGPā“VaR

Weekly return on portfolio 0.004742222 0.011161026 0.011033007

Standard deviation for portfolio 0.028795127 0.033061249 0.032657976

VaR of portfolio at 99 % (weekly horizon) 0.062243883 0.065749358 0.042751414

Portfolio risk premium 0.140570337 0.316580440 0.316569713

Portfolio VaR premium 0.065030338 0.159188547 0.241828870

Table 10.5 Composition of portfolio (VaR at 90 %)

EGP EGPā“VaR

Indra Systems 0.094135885 0.091788929

Tele Pizza 0.154240348 0.150446144

Amadeus Global Travel 0.110170100 0.106733677

NH Hotels (Ex-Coļ¬r) 0.131113608 0.130829934

Altadis 0.123080404 0.124183660

Union Fenosa 0.161211661 0.160123355

Terra Networks 0.041129011 0.038910766

Active de Constr. Y Serv. 0.063707852 0.064619774

Corp. Fin. Alba 0.084216091 0.087826598

Acciona 0.015113720 0.016749943

Aguas Barcelona 0.021991319 0.027787219

274 Asset and Risk Management

Table 10.6 Performance of portfolio (VaR at 90 %)

IBEX35 EGP EGPā“VaR

Weekly return on portfolio 0.004742222 0.011161026 0.010994713

Standard deviation for portfolio 0.028795127 0.033061249 0.032538097

VaR of portfolio at 99 % (weekly horizon) 0.062243883 0.065749358 0.030045589

Portfolio risk premium 0.140570337 0.316580440 0.316559138

Portfolio VaR premium 0.065030338 0.159188547 0.342820104

A comparison of the performances of the optimised portfolios shows that the results

obtained by EGPā“VaR in terms of ā˜VaR premiumā™ are higher than the classical EGP

method at all VaR conļ¬dence levels.24

Contrarily, however, it is evident that the EGP method provides a risk premium ā“ in

terms of variance ā“ higher than that of EGPā“VaR. This result clearly shows that optimis-

ing the risk in terms of variance is not consistent with optimising the risk in terms of

ļ¬nancial loss.

Next, it will be noted that the iterations in the Picard process, essential in the EGPā“VaR

method, do not really interfere with the algorithm as the convergence of the process is of

ļ¬ve signiļ¬cant ļ¬gures per iteration.

Finally, it must be stressed that the superiority of EGPā“VaR over EGP becomes more

marked as the level of conļ¬dence falls. This can be explained by the fact that with a

lower level of conļ¬dence in the calculation of the VaR, one is more optimistic about its

value and therefore about the VaR premium as well. This reduces interest in the method

to some extent, as the method is only really useful if the level of conļ¬dence that can be

shown in the portfolio is sufļ¬ciently high.

In order to assess the additional beneļ¬t of the EGPā“VaR method, we place ourselves

in the least favourable but most realistic case, in which we wish to optimise the VaR

premium for a 99 % level of conļ¬dence. In the example presented, the VaR premium

ratio is

EGPā’VaR

P

ā 1.00008

EGP

P

This implies that with an equal VaR, the ratio of weekly yields will also be 1.00008

and the ratio of annual yields will be 1.00416. As the weekly yields are in the region

of 1.1 %, the annual yield will be in the region of 77 %. If we start with two portfolios

of USD1 000 000 with identical VaR values, the portfolio proposed by the EGPā“VaR

method will propose an expected annual yield of USD3200 more than the portfolio with

the classical EGP method.

In conclusion, taking account of VaR in the EGP method supplies VaR premium port-

folios greater than the usual method (equal VaR) for a very small additional amount of

calculation. In addition, it is evident that the increase in value brings a signiļ¬cant increase

in annual yield.

10.3 OPTIMISING A GLOBAL PORTFOLIO VIA VaR

At present, a portfolio managerā™s work consists as much of choosing the keys for partition-

ing the various categories of assets (ā˜asset allocationā™) as of choosing the speciļ¬c assets that

24

Calculation of VaR with 90 %, 95 % or 99 % probability.

Optimising the Global Portfolio via VaR 275

will make up the portfolio (ā˜stock pickingā™). In consequence, most of the portfolios will

not consist exclusively of equities. In this context, generalisation of methods for optimising

equity portfolios in moving to global portfolios shows its true importance.

The current practice followed by portfolio managers consists of optimising different

asset portfolios separately and combining them subsequently. In many cases, the discre-

tionary clients are managed on the basis of a very restricted number of benchmarks

(for example, equity benchmarks, bond benchmarks, monetary fund benchmarks and

shareholder fund benchmarks). These are combined with each other linearly in order

to offer enough model portfolios for the clients needs; this is referred to as pooling.

In our example, we could combine the four portfolios two by two in sections of 10 %,

4

Ā· 11 = 66 model portfolios that represent the client portfolio

which would lead to

2

management methods.

Here we offer an alternative solution that allows a global optimal portfolio to be con-

structed by generalising the method shown in the preceding paragraph. In this case, the

optimisation process will involve correlations between assets of different natures estimated

for different maturity dates. The estimated varianceā“covariance matrix method will help

us to estimate VaR within our method.

10.3.1 Generalisation of the asset model

The generalisation proposed involves the inclusion of assets other than equities in the

portfolio. The ļ¬rst stage will therefore involve generalisation of the asset model used in

the EGP method, namely Ri = ai + bi RI + Īµi .

To start, we note that the introduction of the estimated varianceā“covariance matrix

method into our method involves more restrictive hypotheses than those used for equities:

ā¢ the hypothesis of stationarity of returns;

ā¢ the hypothesis of normality in asset returns;

ā¢ the hypothesis of linear dependency

ā” of prices on assets with respect to risk factor prices; and

ā” of returns on assets in relation to risk factor prices.25

These hypotheses will allow us to carry out the following developments.

A very general way of representing an asset according to its risk factors is to apply the

Taylor26 development to the asset according to the risk factors:

n n n

Xk Xlr

r

r r r

Ri (X1 , . . . , Xn , t) = Ri (0) + (Ri )Xr Xk + (Ri )t t + (Ri )Xr ,Xr

2

k k l

k=1 k=1 l=1

n r

Xk t t2

+ (Ri )Xr ,t + (Ri )t,t + . . . + Īµi

2 2

k

k=1

r

Here, Īµi represents the error in the development and Xk is the k th risk factor.

25

This is especially the case with shares (CAPM).

26

The speciļ¬c case in which the development is made in the area close to the reference point 0 is called the MacLaurin

development.

276 Asset and Risk Management

In the case of assets that are linear according to their risk factors and independent of

time, all the terms of order higher than 1 will disappear and it is possible to write:

n

r r r

Ri (X1 , . . . , Xn , t) = Ri (0) + (Ri )Xr Xk

k

k=1

In the case of nonlinear assets independent of time such as bonds, the time-dependent

terms will disappear but the Taylor development will not stop at the ļ¬rst order:

n n n

Xk Xlr

r

r r r

Ri (X1 , . . . , Xn , t) = Ri (0) + (Ri )Xr Xk + (Ri )Xr ,Xr + Ā· Ā· Ā· + Īµi

2

k k l

k=1 k=1 l=1

r

However, when the risk factors are small27 (Xk << 1), the higher-order terms will be

low in relation to the ļ¬rst-order term and they can be ignored. This therefore brings us

to the expression obtained for the purely linear assets:28

n

r r r

Ri (X1 , . . . , Xn , t) = Ri (0) + (Ri )Xr Xk + Īµi

k

k=1

In addition, some future linear assets, such as FRAs, can be linked to one or more risk

factors by construction or mapping.29 This is also the case for bonds when they are

expressed according to their coupons rather than the market rate. In this case, the asset

does not have its own risk, as this risk is always a function of the factor taken into

account, in which case we have once again:

n

r r r

Ri (X1 , . . . , Xn , t) = Ri (0) + (Ri )Xr Xk

k

k=1

Finally, this model will not be valid for time-dependent nonlinear assets, such as options,

as in this case the time-connected terms cannot be ignored in relation to the terms linked

to other risk factors and we cannot therefore reach an expression of the kind:

Ri = ai + bi RI + Īµi

To sum up, if we propose

bik = (Ri )Xr k = 1, . . . , n

k

ai = Ri (0)

27

In reality, this will be valid when the risk factor value is close to the reference point around which the development is

made. In our case (MacLaurin development) this reference point is 0 and the relative return on the risk factor must be close

to 0.

28

Bonds are nonlinear when expressed according to interest rates. They are linear when expressed according to the

corresponding zero coupon.

29

This is the case when the estimated variance and covariance matrix method is used for calculating VaR.

Optimising the Global Portfolio via VaR 277

We will obtain the generalised form for all the linear assets and assets that can be made

linear30 from the following equation:

n

r

Ri = ai + bik Xk + Īµi

k=1

Here, ĪµI will be zero in the case of assets that are linear by construction.

10.3.2 Construction of an optimal global portfolio

We will now adapt the EGPā“VaR method developed in Section 10.2 to the general

expression of a linear asset or asset that can be made linear according to its risk fac-

tors. This in fact corresponds to a multiple linear regression on the risk factors with the

ļ£±

relations:31

ļ£² cov(Īµi , Īµj ) = Ī“ij ĻĪµ2i

E(Īµi ) = 0

ļ£³

ai = constant

The expression of the variance and covariances according to those of the n risk factors

then becomes n n

r

Ļij = bik bj l Ļkl + Ī“ij ĻĪµ2i

k=1 l=1

Here, Ļkl represents the covariance between the risk factors Xk and Xlr .

r r

From the preceding equation, we deduce the expression of the variance for a portfolio

consisting of N assets:32

N N N N n n N

r

ĻP = Xi Xj Ļij = Xi Xj bik bj l Ļkl + Xi2 ĻĪµ2i

2

i=1 j =1 i=1 j =1 k=1 l=1 i=1

Developments similar to those made in Section 10.2.2 lead to the equation:

n n N

Ei (1 + P) ā’ RF ā’ bik Ļkl blj Zj ā’ Zi ĻĪµ2i = 0

k=1 l=1 j =1

As stated above, some assets do not have their own expression of risk: ĻĪµ2i = 0.

1

As zi depends on 2 , the equations will therefore have to be adapted.

ĻĪµ i

If we proceed as before, in an iterative way, we will with the same deļ¬nition of K

arrive at:

K n n

Ei (1 + ĻP ) ā’ RF ā’ Zj bik Ļkl blj ā’ Zi ĻĪµ2i = 0

j =1 k=1 l=1

By suggesting

n n

vij = bik Ļkl blj

k=1 l=1

30

The case of nonlinear time-independent assets.

The quantity that is worth 1 if i = j and 0 if i = j is termed Ī“ij .

31

r

The Xj proportions of the securities in the portfolio should not be confused with the risk factors Xk .

32

278 Asset and Risk Management

We arrive successively at:

K

Ei (1 + P) ā’ RF ā’ vij Zj ā’ Zi ĻĪµ2i = 0

j =1

Kā’1

Ei (1 + P) ā’ RF ā’ vij Zj ā’ vii Zi ā’ Zi ĻĪµ2i = 0

j =1

Kā’1

vij Zj

Ei (1 + P ) ā’ RF j =1

ā’ = Zi

vii + ĻĪµ2i vii + ĻĪµ2i

That is:

Īøi ā’ ĻK = Zi

This equation is equivalent to the relation in Section 10.2 that gives the composition of

the portfolio.

The process will therefore be equivalent to the case of the equities:

ā¢ Classiļ¬cation of equities according to decreasing ĪøI values.

ā¢ Iterative calculation of ZI rates according to the above equation in the order of classi-

ļ¬cation of the equities.

ā¢ Stopping at ļ¬rst asset for which ZI < 0 (K of them will thus be selected).

Zi

ā¢ Deduction of the portfolio composition by Xi = K .

Zj

j =1

ā¢ Calculation of the VaR premium.

ā¢ Subsequent iteration of the Picard process.

10.3.3 Method of optimisation of global portfolio

The main difļ¬culty of moving from a portfolio of equities to a global portfolio comes

from the fact that the global portfolio introduces a range of ļ¬nancial products that are very

different in terms of their behaviour when faced with risk factors and of their characteristic

future structure.

The ļ¬rst effect of this is that we are not envisaging a case in which all the ļ¬nancial

instruments included contain optional products. The second is that it is now necessary to

choose a global risk calculation method, in our case that of VaR.

The VaR calculation method used here is that of the estimated varianceā“covariance

matrix. This choice is based on the fact that this method is ā˜the most complete VaR

calculation technique from the operational point of viewā™,33 but predominantly because it

is the only method that allows VaR to be expressed explicitly according to the composition

of the equities. In this regard, it is the one best adapted to our method.

In practice, the estimated varianceā“covariance matrix method expresses a position in

terms of cashļ¬‚ows and maps these ļ¬‚ows for the full range of risk factors available.

33

Esch L., Kieffer R. and Lopez T., Value at Risk ā“ Vers un risk management moderne, De Boeck, 1997, p. 111.

Optimising the Global Portfolio via VaR 279

We then obtain the expression of the discounted price of the asset according to the risk

factors:34

n

r

pi = Aik Xk

k=1

This will lead to the expression in terms of the equivalent returns:

n

r

Ri = bik Rk

k=1

Where: ļ£± p ā’ pi,tā’1

ļ£“ Ri = i,t

ļ£“

ļ£“ pi,tā’1

ļ£“

ļ£“

ļ£“

ļ£“

ļ£“ Xr ā’ Xk,tā’1r

ļ£“r

ļ£“ R = k,t

ļ£²k r

Xk,tā’1

ļ£“ r

ļ£“

ļ£“ b ā Aik Xk

ļ£“ ik

ļ£“

ļ£“ n

ļ£“

ļ£“

ļ£“ r

Aij Xj

ļ£“

ļ£³

j =1

It must be stressed that in our case, if mapping can be avoided (either because it is

possible to apply linear regression to the risk factor as is the case for equities, or because

it is possible to express the return on the asset directly as a function of the risk-factor

return), we will skip this stage.

In the speciļ¬c case of non-optional derivative products, for which the discounted price

(or current value) is initially zero, the return on them cannot be deduced. In this case, the

derived product will be separated into two terms with a non-zero price. For example, a FRA

6ā“9 long will be split into a nine-month loan and a six-month deposit. The expression of

the expected return on the equity according to the risk factor returns will follow as:

n

r

Ei = bik Ek

k=1

Note

The expression above supposes that the terms Aik are constant. In addition, the equities

do not require mapping over a full range of products as they are expressed according to

spot positions on market indices and we will therefore retain, in this case:

Ri = ai + bi RI + Īµi

We will now summarise the stages of this method.

1. The ļ¬rst stage consists of choosing the characteristics of the population of assets from

which we will choose those that will optimise the portfolio. It will be particularly

important to choose the maturity dates, due dates etc. from the assets in question as

these will condition the value of the VaR and therefore the optimisation of the portfolio.

34

Note that this mapping can be applied to prices, to each term or directly to yields. In this situation, calculation of the

zj values will sufļ¬ce.

280 Asset and Risk Management

2. The second stage of the optimisation process involves a search for all the coefļ¬cients

involved in the equations:

n

r

Ei = bik Ek

k=1

n n

r

Ļij = bik bj l Ļkl + Ī“ij ĻĪµ2i

k=1 l=1

This will be supplied by the historical periods of the assets and the estimated vari-

anceā“covariance matrix method. This stage will also condition the temporal horizon

for optimising the VaR premium. In addition, these positions do not require any map-

ping, which means that the equities may be understood in the classical way through

linear regression on the corresponding market index.

3. The third stage consists of optimising the VaR premium in the true sense, the premium

being arrived at according to the scheme described in Section 10.3.2.

In its general outlay, the algorithm is similar to that described in Section 10.2.2 and

speciļ¬cally involves the Picard iteration. The ļ¬‚ow chart shown in Figure 10.3 sets out

the main lines to be followed.

Example

In order not to complicate the exposition unnecessarily, we will deal with the case of a

basket of assets containing (see Tables 10.7ā“10.11):

START

Asset's choice

Calculation of

coefficients

Picard's

EGPā’VaR

process

END

Figure 10.3 Algorithm for optimising a global portfolio using VaR

Table 10.7 Daily volatility and yield of assets

Variance Return

Equity 1 0.001779239 0.006871935

Equity 2 0.000710298 0.003602820

Equity 3 0.000595885 0.003357344

Equity 4 0.002085924 0.004637212

Equity 5 0.000383731 0.003891459

Bond 1 7.11199E-05 0.000114949

Bond 2 4.01909E-05 0.000125582

Table 10.8 Expression of assets in terms of risk factors

Assets Market Rate, Rate, Rate, Rate, Rate, Rate, Rate, Rate,

index 1 year 2 years 3 years 4 years 5 years 7 years 9 years 10 years

Equity 1 0.907016781 0 0 0 0 0 0 0 0

Equity 2 0.573084264 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

Equity 3 ā’0.524903478

Equity 4 0.982081231 0 0 0 0 0 0 0 0

Equity 5 0.421222706 0 0 0 0 0 0 0 0

Bond 1 0 0.047260003 0.045412546 0.043420969 0.041501147 0.822405334 0 0 0

Bond 2 0 0.044719088 0.042970958 0.041086458 0.039269854 0.058192226 0.0793941 0.046557894 0.647809442

282 Asset and Risk Management

ā¢ One ļ¬ve-year bond in euros, coupon 5 % repayment at par.

ā¢ One 10-year bond in euros, coupon 6 % reimbursement at par.

ā¢ Five equities based on the same market index.

The VaR is calculated at 99 % probability and with a horizon of one day. The risk-free

rate is 3.75 %.

The data relative to these shares and the associated risk factors are purely ļ¬ctitious.35

We can see ļ¬rst of all that the composite portfolio beats each of the assets in isolation,

thus clearly showing the diversiļ¬cation effect sought.

In addition, it is best to stress that when applied to a portfolio of bonds, this method

offers real optimisation of the yield/VaR ratio, unlike the known portfolio immunisation

methods. In the case of bonds, the method proposed is therefore not an alternative, but

an addition to the immunisation methods.

Table 10.9 Classiļ¬cation of assets

Classiļ¬cation

Equity 5 1.238051233

Equity 2 1.180947722

Equity 3 0.921654797

Equity 1 0.764163854

Equity 4 0.620600304

Bond 2 0.614979773

Bond 1 0.19802287

Table 10.10 Composition of portfolio

Classiļ¬cation

Equity 1 0.070416113

Equity 2 0.114350746

Equity 3 0.068504735

Equity 4 0.036360401

Equity 5 0.146993068

Bond 1 0.041847479

Bond 2 0.521527457

Table 10.11 VaR premiums on the various assets

Classiļ¬cation VaR premium

Equity 1 0.031922988

Equity 2 0.028453485

Equity 3 0.024136102

Equity 4 0.023352806

Equity 5 0.030366115

Bond 1 0.000722099

Bond 2 0.001690335

Portfolio 0.0576611529

35

See the CD-ROM attached to this book, ā˜Global optimisation of VaR premium.xlsā™.

Optimising the Global Portfolio via VaR 283

Finally, the major advantage of this method is that it allows a portfolio to be optimised

in terms of asset allocation as well as stock picking, which is not the case with the

pooling methods. In pooling, the combinations of benchmarks do not take account of the

correlation between these and still less take account of the correlation between each asset

making up the benchmarks. This is the great advantage of the method, as asset allocation

accounts for the greater part of a portfolio managerā™s work.

11

Institutional Management:

APT Applied to Investment Funds

The APT1 model described in Section 3.3.2 allows the behaviour of investment funds to

be analysed in seven points.

11.1 ABSOLUTE GLOBAL RISK

Normal volatility (the ā˜standard deviationā™ of statisticians) is a measurement of the impact

of all market conditions observed during a year on the behaviour of an asset. It sum-

marises recent history. If other market conditions had prevailed, the same calculation

would have given another volatility. As risk has to be calculated a priori, the probable

average volatility for the period ahead must be determined in advance.

Absolute risk, calculated within the APT framework, is the most reliable forecast of

the probable average volatility of an asset. The APT system calculates the global risk for

the fund, which it breaks down into:

ā¢ systematic risk and APT factor-sensitivity proļ¬le;

ā¢ residual or speciļ¬c risk.

The systematic proļ¬le consists of a series of ā˜rodsā™, each of which represents the

sensitivity of the fund to the corresponding systematic risk factors. Systematic risk is the

quadratic mean of all the sensitivities to the various systematic factors. The observation

of the proļ¬le suggests a genetic similarity; systematic risk is thus the product of a genetic

code, a kind of ā˜DNAā™ of the risk of each instrument. This will allow portfolio adjustments

to be measured with respect to its objective and the risk of performance divergence to be

ascertained:

ā¢ Two very different systematic risk proļ¬les will lead to a signiļ¬cant probability of

divergence (Figure 11.1).

ā¢ In contrast, two very similar systematic risk proļ¬les will have a correspondingly sig-

niļ¬cant probability of similar behaviour (Figure 11.2).

11.2 RELATIVE GLOBAL RISK/TRACKING ERROR

The relative risk of a portfolio with respect to a target calculated using APT is a very

reliable a priori risk estimator.

When the global risk for a portfolio has been calculated with respect to a target (ex ante

tracking error), a series of risk tranches can be traced. These ranges are a very reliable

1

Interested readers should read Ross S. A., The arbitrage theory of capital asset pricing, Journal of Economic Theory,

1976, pp. 343ā“62. Dhrymes P. J., Friend I. and Gultekin N. B., A critical re-examination of the empirical evidence on the

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

of the Stock Market, Journal of Business, No. 59, 1986, pp. 383ā“403.

286 Asset and Risk Management

Absolute global risk

European equities portfolio

Systematic

APT profile ā“ systematic risk

0.10

100 %

0.08

90 %

0.06

80 %

0.04

70 %

0.02

60 %

0.00

50 %

ā“0.02

40 %

ā“0.04

30 %

ā“0.06

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

20 %

Real European equities Target

10 %

European equities portfolio

FT EURO 0%

Global risk 16.82 19.64 Portolio/Cash FT Euro/Cash

Target - portfolio stock overlap 0.00 0.00

Number of securities 723.00 10.00

Specific

Systematic 16.75 17.61

Specific 1.53 8.69

Specific/systematic ratio 1/99 20/80

Figure 11.1 Stock picking fund

Absolute global risk

European equities portfolio

APT profile ā“ systematic risk Systematic

0.10

100 %

0.08

90 %

0.06

80 %

0.04

70 %

0.02

0.00 60 %

ā“0.02 50 %

ā“0.04

40 %

ā“0.06

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 30 %

20 %

Real European equities Target

10 %

FT EURO European equities portfolio

Global risk 16.82 17.27 0%

Portfolio/Cash FT Euro/Cash

Target - portfolio stock overlap 0.00 3.14

Number of securities 723.00 72.00

Systematic 16.75 17.04

Specific

Specific 1.53 2.83

Specific/systematic ratio 1/99 3/97

Figure 11.2 Index fund

indicator of the a priori risk, and can be used as an indicator of overperformance or

underperformance.

When the portfolio reaches the upper range, 96 % (Figure 11.3), the probability of rising

above the range will have dropped to 2 %. Conversely, the probability of arbitrage of the

strategy by the market, in accordance with the theory of APT, is close to certainty (98 %).

The opposite to this is underperformance, which pulls the relative performance of the

portfolio down towards the lower range. It should lead the manager, after analysis of his

portfolio and validation of his initial strategy, to wait for his portfolio to rebound; the

brief period of underperformance should in principle be arbitrated by the market.

Thus, by taking care to track the relative performance of the portfolio regularly with

respect to its target within the risk ranges, we will have a very reliable indicator of when

Institutional Management: APT Applied to Investment Funds 287

103.00

102.00

101.00

67 %

100.00

96 %

99.00

98.00

97.00

16 July 1991 6 April 2002

Figure 11.3 Ex ante tracking error

to revise the portfolio in cases of overperformance or to maintain the strategy in cases of

underperformance.

The 98 %-VaR is the maximum loss with 98 % probability for a portfolio in relation to

a risk-free asset, on a clearly deļ¬ned horizon.

If the ex ante tracking error is calculated in relation to the risk-free asset, 98 %-VaR

will be represented according to time by the second lower envelope.

11.3 RELATIVE FUND RISK VS. BENCHMARK ABACUS

The ā˜bestā™ (minimum risk) curve gives the minimum tracking error for a number of

securities (x-axis) held in the portfolio ā“ the deviation of the fund with respect to the

benchmark (y-axis), as shown in Figure 11.4.

For each portfolio studied one can, with knowledge of its number of securities, locate

oneā™s relative risk on the graph and note the vertical distance that separates it from the

minimum risk curve.

12

Minimum risk

Minimum risk Ć— 2

Minimum risk Ć— 3

10

Real European equities

OPT European equities

8

6

4

2

0

10 20 30 40 50 60 70 80 90 100 110

Figure 11.4 The best

288 Asset and Risk Management

For any structured portfolio, the optimum weightings for the securities held can be

identiļ¬ed in order to minimise the relative global risk.

In Figure 11.4, the black square shows the relative global risk of the initial portfolio and

the grey square that of the optimised portfolio (securities the same as the initial portfolio,

but weightings altered and possibly down to zero). For a given preselection of securities,

the grey square indicates the minimum relative global risk that can be reached with this

choice of securities.

11.4 ALLOCATION OF SYSTEMATIC RISK

The allocation of systematic risk allows the distribution of the risk to be displayed accord-

ing to key variables: sector, country, economic variables, ļ¬nancial variables etc. The

allocation can be made taking account of statistical dependencies (joint allocation) or

otherwise (independent allocation). For example, in a European portfolio, the ā˜ļ¬nancial

sectorā™ variable can be linked to MSCI Europe Finance index, and the ā˜long-term France

interest rateā™ variable can be linked to the 10-year Matif index.

11.4.1 Independent allocation

As the portfolioā™s systematic risk is represented by its APT factor-sensitivity vector, it is

possible to calculate the portfolioā™s sensitivity to each of the independent variables taken

individually.

The portfolioā™s sensitivity to each of the variables is calculated by projecting the port-

folioā™s systematic risk vector onto the vector that represents the systematic risk of the

variable. The length of the projection shows the sensitivity of the portfolio to the variable.

Figure 11.5 shows that the portfolio has hardly any sensitivity to the performance of

variable D as the length of the projection of its systematic risk onto that of variable D is

close to zero.

The various sensitivities thus calculated with respect to several different variables allow

the individual contribution made by each variable to the portfolioā™s systematic behaviour

to be measured. They can be compared to each other but not added together.

Factor 3

Systematic risk

of the portfolio

Variable A

Variable C

Variable B

Factor 2

Variable D

Factor 1

Figure 11.5 Independent allocation

Institutional Management: APT Applied to Investment Funds 289

APT ā“ factor 3

Systematic risk

of the portfolio

Growth

Not explained

Value

APT ā“ factor 2

APT ā“ factor 1

Figure 11.6 Joint allocation

11.4.2 Joint allocation: ā˜valueā™ and ā˜growthā™ example

As the systematic risk of the portfolio is expressed by its APT factor-sensitivity vector, it

can be broken down into the explicative variables ā˜growthā™ and ā˜valueā™, representing the

S&P Value and the S&P Growth (Figure 11.6).

One cannot, however, be content with projecting the portfolio risk vector onto each of

the variables. In fact, the ā˜growthā™ and ā˜valueā™ variables are not necessarily independent

statistically. They cannot therefore be represented by geometrically orthogonal variables.

It is in fact essential to project the portfolio risk vector perpendicularly onto the space

of the vectors of the variables. In the present example, it is a matter of projection onto

the ā˜growthā™ and ā˜valueā™ variables plan.

Once this projection is calculated, the trace thus produced needs to be again projected

onto the two axes to ļ¬nd out the contribution made by each variable to the systematic

risk of the portfolio.

It is clear from this example that the two variables retained only explain part of the

systematic risk of the portfolio. The component of the portfolioā™s systematic risk not

explained by the explicative variables chosen is represented by the perpendicular vector.

This perpendicular component allows allocation of the systematic risk to be completed.

This component is not a residual.

Whatever the extent of the explicative variables retained, it is essential to calculate this

additional component very accurately. If it is not calculated, its contribution will appear

as a residual, neutral by deļ¬nition in statistical terms, while it is in fact systematic.

This is the exact problem encountered by all the models constructed on the basis

of regression on extents of prespeciļ¬ed variables. These variables do not explain the

systematic component, as it is orthogonal to them (that is, statistically independent).

11.5 ALLOCATION OF PERFORMANCE LEVEL

In the same way as the APT model allows a portfolioā™s risk to be broken down ex ante

over the range of independent variables, it also allows the portfolioā™s performance to be

broken down ex post over the same independent variables for variable periods between

1991 and the present.

290 Asset and Risk Management

30 June 98

PricewaterhouseCoopers Performance allocation APTimum counsel

Europe portfolio

15

10

5

0

ā“5

ā“10

ā“15

ā“20

Origin 15/07/98 05/08/98 26/08/98 16/09/98 07/10/98 28/10/98 18/11/98 09/12/98 30/12/98 20/01/99

ā“25

Automobile Financial services ActiveInput: Level deviations

Figure 11.7 Allocation of performance level

The technique used is termed ā˜factor-mimicking portfoliosā™; each risk factor can be

mimicked by an arbitrage portfolio. For example, it is possible to construct a portfolio

that presents a unitary sensitivity to factor 8 and zero sensitivity to the other factors. This

is called a factor-8 mimicking portfolio.

Being aware of the composition of 20 portfolios that mimic 20 statistical factors, we

can also ļ¬nd out the proļ¬tability of the securities making up the portfolios and therefore

ļ¬nd out the factors. Being aware of the various sensitivities of the funds to the various

factors and their yields, we can allocate the performance levels.

In Figure 11.7, the performance level is explained by the contributions made by all the

variables from the automobile and ļ¬nancial sectors.

11.6 GROSS PERFORMANCE LEVEL

AND RISK WITHDRAWAL

The adjusted APT performance for the risk is equal to the historical performance over a

given period divided by the APT beta for the fund, compared to a reference strategy that

can be the average for the fund within a category or an index.

The APT beta measures the overall sensitivity of the fund to this reference strategy.

The APT beta is not identical to the traditional beta in the ļ¬nancial assets equilibrium

model, which only measures the sensitivity of the funds to certain market measurements

such as an index. Graphically, it is the orthogonal projection of the systematic risk vector

for the fund onto the vector for the average in the category (Figure 11.8).

However, like the traditional beta, it is a number for which the average for a given

homogeneous group, like European equities, is 1. A number in excess of 1 indicates that

the fund is more sensitive to the systematic risk factors that its pairs. A number less than

1 indicates that the fund is less sensitive.

Let us consider a fund A invested in European equities with a gross performance level

that over one year has reached 16.58 %. On the basis of this gross performance level, this

Institutional Management: APT Applied to Investment Funds 291

APT ā“ factor 3

Fund A

Average for

Length = the category

APT ā“ systematic risk

cos (Ī±) = APT ā“ correlation

APT ā“ factor 2

APT ā“ beta = lever effect

gross performance

Corrected performance =

APT ā“ beta

APT ā“ factor 1

Figure 11.8 Gross performance level and risk withdrawal

fund A will be classiļ¬ed 63rd out of the 314 in its category. Let us take as a reference

within this category the virtual fund equal to the average for the funds.

The vector that represents the systematic risk for fund A differs from the systematic

risk for the average virtual fund, taken as a reference in length and in position within the

area. This difference between the two vectors indicates the probability that the behaviour

patterns will diverge. Fund A follows a strategy identical to that of the reference fund for

a proportion equal to its projection onto the reference fund. This is the beta APT.

The beta APT measures the proportion of the fundā™s performance covered by the

reference strategy. As the beta APT is less than 1 (0.867), this indicates that fund A is

less ā˜investedā™ than the reference fund within the reference strategy. If Fund A had been

as much ā˜investedā™ as the reference fund in the reference strategy, its performance level

would have reached 16.58/0.867 = 19.12, that is, its gross performance multiplied by its

withdrawal coefļ¬cient of 1/Ī². This performance level is the withdrawal performance of

fund A within the average strategy. If the withdrawal coefļ¬cient 1/Ī² is very different

from 1 (for example 0.8 or 1.2), this indicates that the strategy of A is very different

compared to the reference for the category. This may be caused by:

ā¢ either the angle between vector A (absolute systematic risk) and the average vector for

the category, which is very open;

ā¢ or by vector A being too long or too short.

11.7 ANALYSIS OF STYLE

The APT model calculates an absolute systematic risk vector for each fund. As these

calculations are carried out for a very wide range of funds, it will be noticed that the

vectors for all the funds dealt with group together naturally in clearly demarcated cones.

Each cone represents a group of funds, such as, for example, ā˜European equity fundsā™.

Once the map graph (Figure 11.9) has been completed, it will allow:

ā¢ the actual homogeneous management categories to be delimited;

ā¢ all funds to be located within one of these categories;

ā¢ its level of typicality to be measured;

ā¢ its withdrawal performance to be calculated in relation to the average strategy for

the category.

292 Asset and Risk Management

Factor 3

Homogeneous Group 1

Homogeneous Group 3

Factor 2

Homogeneous Group 2

Factor 1

Figure 11.9 Map graph

In practice, it will be noted that the angle of opening of each cone is less than 24ā—¦ . In

general, only funds with E values between 0.8 and 12 will be included.

Part V

From Risk Management to Asset

and Liability Management

Introduction

12 Techniques for measuring structural risks in balance sheets

294 Asset and Risk Management

Introduction

In addition to the traditional functions allocated to risk management, such as manage-

ment of credit risks and market risks in certain banking products, the discipline has also

expanded into the study of operational risks and random insurance risks.

The aim of asset and liability management, or ALM, is to arrive at an understanding

of the problems of risk (rates, exchange etc.) across the whole balance sheet.

12

Techniques for Measuring Structural Risks

in Balance Sheets

The aim of the developments that follow is to make the analytical tools used in ALM

easily understandable.

Understanding of the concepts must not be treated separately from understanding of

the accounting, banking and ļ¬nancial data needed to use the concepts.

Naturally, ALM is normally managed through management software packages. The

interface between computer processes and software applications allows essential elements

such as the type of product and its characteristics in terms of ļ¬‚ows and currency type to

be captured.

First, we will introduce the traditional ALM tools and models, together with the possible

uses for VaR in ALM.

On the other hand, the interface between ALM software and the bankā™s computer net-

work cannot be achieved effectively without addressing the contractual hazards particular

to certain products. In fact, how can liquidity risk and therefore interest rate risk be known

on contracts that do not have a contractual maturity, such as a current account? How can

a ļ¬‚oating-rate contract be modelled? These two elements present real practical problems

for the use of analytical tools in traditional risks such as liquidity or interest rate gaps or

even in calculating a duration for the balance sheet.

Next, we will propose techniques for modelling ļ¬‚oating-rate products and calculate

maturity and the liquidity risk proļ¬les for products that have no maturity dates.

12.1 TOOLS FOR STRUCTURAL RISK ANALYSIS IN ASSET

AND LIABILITY MANAGEMENT

Let us take the balance sheet of a bank in order to illustrate the asset and liability

management tools (Table 12.1).

The equity portfolio consists of a negotiable treasury bond with a nominal rate of 4.5 %

and ļ¬ve-year maturity. The variable-risk equity portfolio is held long term (15 years). It

is not a trading portfolio.

The property loan book has a 20-year term at rates that are adjusted every six months

based on Euribor six-month (3.1 % at t0 ) and a 10-year maturity. Interest is paid on

15 June and 15 December in each year. The principal is reimbursed on maturity.

The bonds are issued at a ļ¬xed rate of 5.5 % over ļ¬ve years, with a single bullet

payment on maturity.

The demand deposits are held in a replicating portfolio, a concept that we will explain

in more detail later. Of the current account balances 20 % vary over the month and

are reļ¬nanced through the interbank market at one month. Of the deposits 30 % have a

maturity date of two years and 50 % have ļ¬ve years.

Interbank: the nostri and lori have a reporting period of one day and maturity of 50 %

one month and 50 % three months.

296 Asset and Risk Management

Table 12.1 Simpliļ¬ed balance sheet

Assets Liabilities

Tangible ļ¬xed assets 10 Equity fund 15

Bond portfolio 30 Bonds issued 20

Share portfolio 10 Non-interest-bearing 25

current accounts

Property Loans 20 Interbank 10

Total 70 Total 70

Table 12.2 Rate curve

[1 dayā“3 months] 2.5 %

[3 monthsā“1 year] 3.0 %

[1 yearā“3 years] 3.5 %

[3 yearsā“6 years] 4.0 %

[6 yearsā“10 years] 4.5 %

Market conditions: Euribor six-month rate 3.1 %; rate curve noted at actuarial rate (see

Table 12.2).

12.1.1 Gap or liquidity risk

Liquidity risk appears in a bank when the withdrawal of funds exceeds the incoming

deposits over a deļ¬ned period. The liquidity mismatch measures the foreseeable differ-

ences on various future dates between the full totals of liabilities and assets.

The projections represent the provisional needs for liquidity and reļ¬nancing and are a

basic management tool. Gaps may be calculated in terms of cashļ¬‚ows or in stocks.

Mismatches in cashļ¬‚ow terms are the differences between the variation in assets and

the variation in liabilities (funds coming in and funds going out) during a given period.

They determine the need for new ļ¬nance during the period through calculations of future

ļ¬‚ow maturities (Table 12.3).

Gaps in positions are the differences between the liability totals in assets and in liabilities

on a given date. They determine the total cumulative liquidity need at a given date. The

gaps in cashļ¬‚ow represent the variations in gaps for positions from one period to another.

The stocks gap must be identical in terms of absolute value to the cumulative cashļ¬‚ow

mismatches from the very beginning (see Table 12.4).

Table 12.3 Liquidity gaps (in ļ¬‚ows)

Mismatches 1 dayā“ 1 monthā“ 3 monthsā“ 1 yearā“ 3 yearsā“ 6 yearsā“ Over 10

in cashļ¬‚ow 1 month 3 months 1 year 3 years 6 years 10 years years

Falls in assets 0 30 20 5

ā’10 ā’5 ā’7.5 ā’32.5

Falls in

liabilities

ā’10 ā’5 ā’7.5 ā’2.5

Gap 0 20 5

ā’15 ā’15 ā’22.5 ā’25 ā’5

Cumulative gap 0

Techniques for Measuring Structural Risks in Balance Sheets 297

Table 12.4 Liquidity gaps (in stocks)

Position gaps: 1 dayā“ 1 monthā“ 3 monthsā“ 1 yearā“ 3 yearsā“ 6 yearsā“ Over 10

liability 1 month 3 months 1 year 3 years 6 years 10 years years

and asset

movements

Tangible ļ¬xed 10

assets

Portfolio 30

converted into

ļ¬xed assets

Variable risk share 10

portfolio

Credits on ļ¬xed 20

assets

Total assets

Equity funds 15

Issue of bonds 20

Current account 5 7.5 12.5

deposits not

paid

Interbank 5 5

Total liabilities

ā’10 ā’15 ā’15 ā’22.5 ā’25 ā’5

Gap 0

Position gaps are negative as the assets depreciate more slowly than the liabilities,

leading to a treasury deļ¬cit over the period as a whole (a positive gap represents an

excess of resources).

12.1.2 Rate mismatches

The rate mismatch is linked to the liquidity gap, as all forms of liquidity necessitate

ļ¬nancing. The interest rate gap is the difference between ļ¬‚oating rate assets and liabilities

over a certain period. Interest rate gaps can be calculated in stocks or in ļ¬‚ows, on a balance

sheet in a state of equilibrium.

To construct a gap analysis, we begin by compiling the balance sheet for operations in

progress, specifying their maturity date and rate. Each future maturity date gives rise to

a cashļ¬‚ow. This ļ¬‚ow will be positive for the assets as it corresponds to the repayment of

a loan or payment of a supplier. The ļ¬‚ows are shown in the repayment schedule on the

date corresponding to their maturity. The difference between the cumulative ļ¬‚ow of assets

and the cumulative ļ¬‚ow of liabilities represents the capital invested at an unknown rate.

The gap schedule summarises the simple information that shows the cashļ¬‚ow manager

the future development in his rate position; that is, his exposure to the rate risk. This

information, however, does not give him any information on the price that will be payable

if he decides to rebalance his balance sheet in terms of totals or duration. Neither does the

cashļ¬‚ow manager know the sensitivity of the repayment schedule to ļ¬‚uctuations in rates.

The ļ¬neness of the breakdown of repayment schedules varies greatly from one institution

to another and is a function of the number and total value of the positions in play. The

short term must be less than one year, in accordance with the budgetary and accounting

horizon, which is annual (Table 12.5).

298 Asset and Risk Management

Table 12.5 Gap report

Gap: liability and 1 dayā“ 1 monthā“ 3 monthsā“ 1 yearā“ 3 yearsā“ 6 yearsā“ Over 10

asset movements 1 month 3 months 1 year 3 years 6 years 10 years years

Tangible ļ¬xed assets 10

Portfolio converted 30

into ļ¬xed assets

Variable risk share 10

portfolio

Credits on ļ¬xed 20

assets

Assets 0 0 20 0 30 0 20

Equity fund 15

Issue of bonds 20

Current account

deposits not paid

Interbank 10

Liabilities 10 0 0 0 20 0 15

ā’10

Rate mismatches 0 20 0 10 0 5

(margins or ļ¬‚ow)

ā’10 ā’10

Cumulative rate or 10 10 20 0 5

asset mismatches

The bankā™s interest rate margin will beneļ¬t from a rise in rates in three months. After

three months, the position is long as there are more assets sensitive to variations in rates

than liabilities. Contrarily, up to three months the bank will have a short position in

rates. Naturally, this gap report will be used as a basis for the simulations that we will

show later.

A reading of the gap proļ¬les gives an overall idea of the ļ¬nance needs for the period.

The creation of liquidity and rate gap reports is a common method as the procedure is

easy to understand, calculate and display. In addition, it allows the impact of a change in

rates on the interest margin to be estimated. The method does, however, have a number

of drawbacks:

ā¢ The aggregation into periods masks the risks inherent within the method.

ā¢ It is not possible to calculate a single indicator.

12.1.3 Net present value (NPV) of equity funds and sensitivity

In ALM, return and risk are generally determined using two key indicators:

ā¢ The net interest income (or interest rate margin).

ā¢ NPV (or market value).

In the interest rate gap, we saw the concept of interest margin, which represents the

budgetary point of view and was short term in nature. Now, we will look at the market

value or NPV, which will represent the point of view of supply and will be long term

in nature. The NPV for the equity fund is obtained by the difference between the assets

Techniques for Measuring Structural Risks in Balance Sheets 299

and liabilities as evaluated under market conditions using a mark-to-market approach.

The NPV must be calculated on the basis of ļ¬‚ows in capital and interest, while the gap

schedule is only compiled in relation to the liabilities on elements of assets and liabilities.

The NPV for the equity fund is an approach to the bankā™s value. In this case, it is

considered that the market value of the bank depends on the value of the asset portfolio

and the cost of the debt. This approach also suggests that all the entries in the balance

sheet are negotiable. Unfortunately, however, measuring the NPV is not sufļ¬cient as it is

static, hence the interest in a study of sensitivity, the concept presented before together

with duration (see Section 4.2). It does not take account of the potential rate ļ¬‚uctuation

risks. On the other hand, the interest and capital ļ¬‚ows are often actualized on the basis of

a single rate curve without taking account of the spread in credit risk inherent in certain

assets (retail credits).

12.1.4 Duration of equity funds

The NPV of equity funds is static and sensitive to changes in interest rates. To measure

this sensitivity, the duration of equity funds can be calculated. We have shown before

that the market value of an asset is a function of its duration. By deļ¬nition:

ā¢ MVA = market value of the asset.

ā¢ MVL = market value of the liability.

ā¢ Def = equity funds duration.

ā¢ Da = duration of the asset.

ā¢ Dl = duration of the liability.

NPV NPV

= ā’Def Ā·

r 1+r

The duration of the equity fund is the algebraic sum of the durations of the elements

making up the portfolio

NPV 1 + r

Def = ā’ Ā·

r NPV

(MVA ā’ MVL) 1 + r

=ā’ Ā·

r NPV

ā’Da Ā· MVA Dl Ā· MVL 1+r

=ā’ + Ā·

1+r 1+r NPV

Da Ā· MVA ā’ Dl Ā· MVL

=

NPV

We have demonstrated the conditions for which the market value of the equity fund is

immunised. The NPV of the equity fund is not sensitive to variations in interest rates

when Da . MVA = Dl . MVA, which corresponds to a zero equity fund duration. Def can

assume any value. High values correspond to high sensitivity of economic equity fund

value to rate changes in one direction or the other.

300 Asset and Risk Management

12.2 SIMULATIONS

The aim of simulations is to explore the future conļ¬gurations of rate curves and their

impact on interest margins or value-measuring instruments (current value of equity fund

or economic reserve, which is the difference between the NPV and the accounting value

of the asset).

There are four main analytical categories of simulation, from the simplest to the

most complex.

ā¢ The ļ¬rst category of simulation relates to the rate curves with balance sheet volumes

for each ALM product renewed in the same way on transactional bases. The effect of

rate changes will be felt in existing contracts and in new production, according to the

transactional rollover hypothesis. A contract with accounting value of 100 maturing

within two months will be renewed in the same way every two months for the same

accounting value.

ā¢ The second category of simulation relates to the rate curves with balance sheet volumes

for each ALM product renewed in the same way according to the initial terms of each

contract. The effect of rate changes will be felt in existing contracts and in new produc-

tion, according to the contractual rollover hypothesis. A contract with accounting value

of 100 maturing within two months will be renewed in the same way within two months

for the same accounting value, with a term renewed as before (for example, one year).

ā¢ The third category of simulation relates both to rates and to balance sheet volumes of

ALM products. This simulation is both more complex and more realistic, as volumes

can be sensitive to changes in interest rates. In this case a contract with 100 maturing

within two months will be renewed on a contractual basis with a total of 100 + a %

or 100 ā’ a %.

ā¢ The fourth category of simulation also uses the previous simulation by introducing the

commercial and balance sheet development strategy. In this case a contract with 100

maturing within two months will be renewed on new or identical contractual bases with

a total of 100 + a % or 100 ā’ a %.

The aim of the simulations is to optimise the liquidity and rate covers. It is easily

understandable that this optimisation process will become more complex when the sen-

sitivities of the balance sheet volumes to rates are used and the effects of commercial

policies and development strategies are included in the balance sheet.

For each type of simulation, one or more rate scenarios can be used. When the banking

institutions use a single scenario rate, the most frequently used method is a parallel

shift of rates per currency per 100 basis points. This single scenario is in fact a stress

scenario. It can be usefully added to by truly macroeconomic bank forecasts of rate

curves, produced using the forward rates based on market forecasts and on the theory of

anticipation, or through a series of scenarios that makes use of stochastic rate generators

(Hull White, Vasicek, etc). Naturally, these scenarios are closer to a true margin forecast

and to short-term and medium-term value indicators.

Once the rate scenario has been drawn up, the methods for changing the rate conditions

on existing contracts and/or on new production will depend on the type of rate. There are

three types of interest rate:

ā¢ Variable rates, with recurring revision periods. These contracts are altered according to

market parameters (ļ¬‚oating rate) or other indicators (such as bank resource costs). In

Techniques for Measuring Structural Risks in Balance Sheets 301

this case, the ALM information system must allow the revision periods for each old or

new contract to be identiļ¬ed and the methods of revision to be known.

ā¢ The variable or ļ¬‚oating-rate contracts, with non-recurring revision periods, depend on

indicators that are not a contractual function of market indicators. In this case, the rates

will be revised on the bankā™s initiative. For ļ¬‚oating rates, the model must allow the

probable period of revision to be known for each rate scenario. This revision is applied

to all the outstanding amounts on old or new contracts. Floating-rate contracts can

be integrated into ALM by using a theoretical model founded on NPV or by using a

behavioural approach founded on canonical correlations and logistic regression. These

methods will be introduced later.

ā¢ Fixed rates require knowledge of the historical correlations between the market rates

and the contract rates. These correlations, based on the market reference rate, will apply

to new productions.

12.3 USING VaR IN ALM

VaR is used in a very speciļ¬c ways in asset and liability management. This market risk-

measuring instrument was initially used for dealing-room or treasury activities. Of course

the time horizon used is that of one day in order to assess the market risk for that day

and to satisfy the control requirements of the middle ofļ¬ce. ALM is not concerned with

this very short-term trading logic. If VaR is used, the VaR will be calculated on a longer

period, generally one month. VaR assesses the market risk for the bankā™s balance sheet as

a whole and not just for market activities in the narrow sense. The month is also linked

to the organisational process adopted for ALM, as the asset and liability management

committee usually meets on a monthly basis.

This instrument is a useful addition to the instruments introduced earlier as being

based essentially on interest rate risk. Finally, for practical reasons, the method used is

ńņš. 12 |