<<

. 12
( 16)



>>


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 ˜Euribor 1 week™
for 11 April 2000, which was 3.676 %.23 See Tables 10.1“10.6.

10.2.4 Conclusion
It will be noted ¬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 EGP“VaR

Indra Systems 0.094135885 0.092850684
Tele Pizza 0.154240348 0.152162629
Amadeus Global Travel 0.110170100 0.108288303
NH Hotels (Ex-Co¬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 ˜Bank of Finland™ site.
Optimising the Global Portfolio via VaR 273
Table 10.2 Performance of portfolio (VaR at 99 %)

IBEX35 EGP EGP“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 EGP“VaR

Indra Systems 0.094135885 0.092329323
Tele Pizza 0.154240348 0.151319771
Amadeus Global Travel 0.110170100 0.107524924
NH Hotels (Ex-Co¬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 EGP“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 EGP“VaR

Indra Systems 0.094135885 0.091788929
Tele Pizza 0.154240348 0.150446144
Amadeus Global Travel 0.110170100 0.106733677
NH Hotels (Ex-Co¬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 EGP“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 EGP“VaR in terms of ˜VaR premium™ are higher than the classical EGP
method at all VaR con¬dence levels.24
Contrarily, however, it is evident that the EGP method provides a risk premium “ in
terms of variance “ higher than that of EGP“VaR. This result clearly shows that optimis-
ing the risk in terms of variance is not consistent with optimising the risk in terms of
¬nancial loss.
Next, it will be noted that the iterations in the Picard process, essential in the EGP“VaR
method, do not really interfere with the algorithm as the convergence of the process is of
¬ve signi¬cant ¬gures per iteration.
Finally, it must be stressed that the superiority of EGP“VaR over EGP becomes more
marked as the level of con¬dence falls. This can be explained by the fact that with a
lower level of con¬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 con¬dence that can be
shown in the portfolio is suf¬ciently high.
In order to assess the additional bene¬t of the EGP“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 con¬dence. In the example presented, the VaR premium
ratio is
EGP’VaR
P
≈ 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 EGP“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 signi¬cant increase
in annual yield.

10.3 OPTIMISING A GLOBAL PORTFOLIO VIA VaR
At present, a portfolio manager™s work consists as much of choosing the keys for partition-
ing the various categories of assets (˜asset allocation™) as of choosing the speci¬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 (˜stock picking™). 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
· 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 variance“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 ¬rst stage will therefore involve generalisation of the asset model used in
the EGP method, namely Ri = ai + bi RI + µi .
To start, we note that the introduction of the estimated variance“covariance matrix
method into our method involves more restrictive hypotheses than those used for equities:

• the hypothesis of stationarity of returns;
• the hypothesis of normality in asset returns;
• the hypothesis of linear dependency
” of prices on assets with respect to risk factor prices; and
” 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 + . . . + µi
2 2
k
k=1

r
Here, µi 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 speci¬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 ¬rst order:
n n n
Xk Xlr
r
r r r
Ri (X1 , . . . , Xn , t) = Ri (0) + (Ri )Xr Xk + (Ri )Xr ,Xr + · · · + µi
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 ¬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 + µi
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 + µi

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 + µi
k=1

Here, µI 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 EGP“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
±
relations:31
 cov(µi , µj ) = δij σµ2i
E(µi ) = 0

ai = constant
The expression of the variance and covariances according to those of the n risk factors
then becomes n n
r
σij = bik bj l σkl + δij σµ2i
k=1 l=1

Here, σkl 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
σP = Xi Xj σij = Xi Xj bik bj l σkl + Xi2 σµ2i
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) ’ RF ’ bik σkl blj Zj ’ Zi σµ2i = 0
k=1 l=1 j =1

As stated above, some assets do not have their own expression of risk: σµ2i = 0.
1
As zi depends on 2 , the equations will therefore have to be adapted.
σµ i
If we proceed as before, in an iterative way, we will with the same de¬nition of K
arrive at:
K n n
Ei (1 + ψP ) ’ RF ’ Zj bik σkl blj ’ Zi σµ2i = 0
j =1 k=1 l=1

By suggesting
n n
vij = bik σkl 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 δij .
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) ’ RF ’ vij Zj ’ Zi σµ2i = 0
j =1

K’1
Ei (1 + P) ’ RF ’ vij Zj ’ vii Zi ’ Zi σµ2i = 0
j =1

K’1
vij Zj
Ei (1 + P ) ’ RF j =1
’ = Zi
vii + σµ2i vii + σµ2i

That is:
θi ’ φK = 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:

• Classi¬cation of equities according to decreasing θI values.
• Iterative calculation of ZI rates according to the above equation in the order of classi-
¬cation of the equities.
• Stopping at ¬rst asset for which ZI < 0 (K of them will thus be selected).
Zi
• Deduction of the portfolio composition by Xi = K .
Zj
j =1
• Calculation of the VaR premium.
• Subsequent iteration of the Picard process.

10.3.3 Method of optimisation of global portfolio
The main dif¬culty of moving from a portfolio of equities to a global portfolio comes
from the fact that the global portfolio introduces a range of ¬nancial products that are very
different in terms of their behaviour when faced with risk factors and of their characteristic
future structure.
The ¬rst effect of this is that we are not envisaging a case in which all the ¬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 variance“covariance
matrix. This choice is based on the fact that this method is ˜the most complete VaR
calculation technique from the operational point of view™,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 variance“covariance matrix method expresses a position in
terms of cash¬‚ows and maps these ¬‚ows for the full range of risk factors available.
33
Esch L., Kieffer R. and Lopez T., Value at Risk “ 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: ± p ’ pi,t’1
 Ri = i,t

 pi,t’1




 Xr ’ Xk,t’1r
r
 R = k,t
k r
Xk,t’1
 r

 b ≈ Aik Xk
 ik

 n


 r
Aij Xj


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 speci¬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
6“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 + µi
We will now summarise the stages of this method.
1. The ¬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 suf¬ce.
280 Asset and Risk Management

2. The second stage of the optimisation process involves a search for all the coef¬cients
involved in the equations:
n
r
Ei = bik Ek
k=1
n n
r
σij = bik bj l σkl + δij σµ2i
k=1 l=1

This will be supplied by the historical periods of the assets and the estimated vari-
ance“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
speci¬cally involves the Picard iteration. The ¬‚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 10.7“10.11):

START



Asset's choice




Calculation of
coefficients



Picard's
EGP’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 ’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

• One ¬ve-year bond in euros, coupon 5 % repayment at par.
• One 10-year bond in euros, coupon 6 % reimbursement at par.
• 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 ¬ctitious.35
We can see ¬rst of all that the composite portfolio beats each of the assets in isolation,
thus clearly showing the diversi¬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 Classi¬cation of assets

Classi¬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

Classi¬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

Classi¬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, ˜Global optimisation of VaR premium.xls™.
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 manager™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 ˜standard deviation™ 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:

• systematic risk and APT factor-sensitivity pro¬le;
• residual or speci¬c risk.

The systematic pro¬le consists of a series of ˜rods™, 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 pro¬le suggests a genetic similarity; systematic risk is thus the product of a genetic
code, a kind of ˜DNA™ 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:

• Two very different systematic risk pro¬les will lead to a signi¬cant probability of
divergence (Figure 11.1).
• In contrast, two very similar systematic risk pro¬les will have a correspondingly sig-
ni¬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. 343“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. 323“46. Chen N. F., Roll R. and Ross S. A., Economic Forces
of the Stock Market, Journal of Business, No. 59, 1986, pp. 383“403.
286 Asset and Risk Management
Absolute global risk
European equities portfolio
Systematic
APT profile “ systematic risk
0.10
100 %
0.08
90 %
0.06
80 %
0.04
70 %
0.02
60 %
0.00
50 %
“0.02
40 %
“0.04
30 %
“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 “ systematic risk Systematic

0.10
100 %
0.08
90 %
0.06
80 %
0.04
70 %
0.02
0.00 60 %
“0.02 50 %
“0.04
40 %
“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 de¬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 ˜best™ (minimum risk) curve gives the minimum tracking error for a number of
securities (x-axis) held in the portfolio “ 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
one™s relative risk on the graph and note the vertical distance that separates it from the
minimum risk curve.

12
Minimum risk
Minimum risk — 2
Minimum risk — 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
identi¬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, ¬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 ˜¬nancial
sector™ variable can be linked to MSCI Europe Finance index, and the ˜long-term France
interest rate™ variable can be linked to the 10-year Matif index.

11.4.1 Independent allocation
As the portfolio™s systematic risk is represented by its APT factor-sensitivity vector, it is
possible to calculate the portfolio™s sensitivity to each of the independent variables taken
individually.
The portfolio™s sensitivity to each of the variables is calculated by projecting the port-
folio™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 portfolio™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 “ factor 3

Systematic risk
of the portfolio
Growth


Not explained


Value

APT “ factor 2



APT “ factor 1

Figure 11.6 Joint allocation


11.4.2 Joint allocation: ˜value™ and ˜growth™ 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 ˜growth™ and ˜value™, 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 ˜growth™ and ˜value™ 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 ˜growth™ and ˜value™ variables plan.
Once this projection is calculated, the trace thus produced needs to be again projected
onto the two axes to ¬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 portfolio™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 de¬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 prespeci¬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 portfolio™s risk to be broken down ex ante
over the range of independent variables, it also allows the portfolio™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


“5


“10


“15


“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

“25

Automobile Financial services ActiveInput: Level deviations


Figure 11.7 Allocation of performance level


The technique used is termed ˜factor-mimicking portfolios™; 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 ¬nd out the pro¬tability of the securities making up the portfolios and therefore
¬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 ¬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 ¬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 “ factor 3

Fund A
Average for
Length = the category
APT “ systematic risk

cos (±) = APT “ correlation


APT “ factor 2
APT “ beta = lever effect


gross performance
Corrected performance =
APT “ beta
APT “ factor 1

Figure 11.8 Gross performance level and risk withdrawal


fund A will be classi¬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 fund™s performance covered by the
reference strategy. As the beta APT is less than 1 (0.867), this indicates that fund A is
less ˜invested™ than the reference fund within the reference strategy. If Fund A had been
as much ˜invested™ 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 coef¬cient of 1/β. This performance level is the withdrawal performance of
fund A within the average strategy. If the withdrawal coef¬cient 1/β 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:

• either the angle between vector A (absolute systematic risk) and the average vector for
the category, which is very open;
• 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, ˜European equity funds™.
Once the map graph (Figure 11.9) has been completed, it will allow:

• the actual homogeneous management categories to be delimited;
• all funds to be located within one of these categories;
• its level of typicality to be measured;
• 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 24—¦ . 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 ¬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 ¬‚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 bank™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 ¬‚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 ¬‚oating-rate products and calculate
maturity and the liquidity risk pro¬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 ¬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 ¬xed rate of 5.5 % over ¬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 re¬nanced through the interbank market at one month. Of the deposits 30 % have a
maturity date of two years and 50 % have ¬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 Simpli¬ed balance sheet

Assets Liabilities

Tangible ¬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 day“3 months] 2.5 %
[3 months“1 year] 3.0 %
[1 year“3 years] 3.5 %
[3 years“6 years] 4.0 %
[6 years“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 de¬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 re¬nancing and are a
basic management tool. Gaps may be calculated in terms of cash¬‚ows or in stocks.
Mismatches in cash¬‚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 ¬nance during the period through calculations of future
¬‚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 cash¬‚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 cash¬‚ow
mismatches from the very beginning (see Table 12.4).

Table 12.3 Liquidity gaps (in ¬‚ows)

Mismatches 1 day“ 1 month“ 3 months“ 1 year“ 3 years“ 6 years“ Over 10
in cash¬‚ow 1 month 3 months 1 year 3 years 6 years 10 years years

Falls in assets 0 30 20 5
’10 ’5 ’7.5 ’32.5
Falls in
liabilities
’10 ’5 ’7.5 ’2.5
Gap 0 20 5
’15 ’15 ’22.5 ’25 ’5
Cumulative gap 0
Techniques for Measuring Structural Risks in Balance Sheets 297
Table 12.4 Liquidity gaps (in stocks)

Position gaps: 1 day“ 1 month“ 3 months“ 1 year“ 3 years“ 6 years“ Over 10
liability 1 month 3 months 1 year 3 years 6 years 10 years years
and asset
movements

Tangible ¬xed 10
assets
Portfolio 30
converted into
¬xed assets
Variable risk share 10
portfolio
Credits on ¬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
’10 ’15 ’15 ’22.5 ’25 ’5
Gap 0


Position gaps are negative as the assets depreciate more slowly than the liabilities,
leading to a treasury de¬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
¬nancing. The interest rate gap is the difference between ¬‚oating rate assets and liabilities
over a certain period. Interest rate gaps can be calculated in stocks or in ¬‚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 cash¬‚ow. This ¬‚ow will be positive for the assets as it corresponds to the repayment of
a loan or payment of a supplier. The ¬‚ows are shown in the repayment schedule on the
date corresponding to their maturity. The difference between the cumulative ¬‚ow of assets
and the cumulative ¬‚ow of liabilities represents the capital invested at an unknown rate.
The gap schedule summarises the simple information that shows the cash¬‚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
cash¬‚ow manager know the sensitivity of the repayment schedule to ¬‚uctuations in rates.
The ¬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 day“ 1 month“ 3 months“ 1 year“ 3 years“ 6 years“ Over 10
asset movements 1 month 3 months 1 year 3 years 6 years 10 years years

Tangible ¬xed assets 10
Portfolio converted 30
into ¬xed assets
Variable risk share 10
portfolio
Credits on ¬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
’10
Rate mismatches 0 20 0 10 0 5
(margins or ¬‚ow)
’10 ’10
Cumulative rate or 10 10 20 0 5
asset mismatches



The bank™s interest rate margin will bene¬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 pro¬les gives an overall idea of the ¬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:

• The aggregation into periods masks the risks inherent within the method.
• 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:

• The net interest income (or interest rate margin).
• 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 ¬‚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 bank™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 suf¬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 ¬‚uctuation
risks. On the other hand, the interest and capital ¬‚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 de¬nition:

• MVA = market value of the asset.
• MVL = market value of the liability.
• Def = equity funds duration.
• Da = duration of the asset.
• Dl = duration of the liability.
NPV NPV
= ’Def ·
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 = ’ ·
r NPV
(MVA ’ MVL) 1 + r
=’ ·
r NPV
’Da · MVA Dl · MVL 1+r
=’ + ·
1+r 1+r NPV
Da · MVA ’ Dl · 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 con¬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.
• The ¬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.
• 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).
• 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 ’ a %.
• 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 ’ 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:
• Variable rates, with recurring revision periods. These contracts are altered according to
market parameters (¬‚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 identi¬ed and the methods of revision to be known.
• The variable or ¬‚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 bank™s initiative. For ¬‚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.
• 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 speci¬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 of¬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 bank™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

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