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Market Risk Internal Models Approach (IMA):The process of calculation and validation of the

Value At Risk

10/21/2011 Isabelle Thomazeau

1st International Risk Management CongressFEBRABAN

Sao Paulo 19-21 October 2011


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Disclaimer

The views in this presentation are those of the authorand do not necessarily reflect those of the Autorit deContrle Prudentiel (ACP) or of the Banque de France.

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SummaryI. Introduction

II. Presentation of CCRM (Risk Modeling Control Unit)

III. Definition of the Value At Risk (VaR)

IV. Use of VaR for capital requirements

V. Computation of the VaR1. Risk Factors2. Modeling3. Importance of the risk factors time series in the VaR calculation4. Aggregation of the VaR

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. a pro uc on

6. Other remarks about the VaR

VI. Back-testing

VII. Stress-testing

VIII. Validation by supervisors

IX. Stressed VaR

X. Conclusion


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I. Introduction

The Basel Committee on Banking Supervision (BCBS) introduced in April1995 the notion of internal model to estimate the capital adequacy of bankstrading books. The risk indicator used is called Value-at-Risk (VaR).Documents are available on http://www.bis.org/list/bcbs/tid_21/index.htm.

Some references from the Basel Committee on Banking Supervision:

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n erna ona onvergence o ap a easuremen an ap aStandards, june 2006, [1]

which is a compilation of some documents among which the 1996Amendments to the Capital Accord to Incorporate Markets Risks;

Revisions to the Basel II market risk framework, june 2009, [2]

Enhancements to the Basel II framework, june 2009, [3]


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II Presentation of the CCRM

CCRM was created in 1995, in the wake of Brussels and Basel decisionsallowing banks to use for calculation of regulatory capital requirements theirinternal models developed to measure and monitor their market risks (thesemodels being, then, subject to prior examination and approval from the regulator)

Afterwards, a changing of name occurred in 1999 (Risk Modeling Control Unitinstead of Market Risk Control Unit, previously), to underline the larger scopeof risks (credit risk, operational risk,) to be taken into consideration, incoherence with the regulatory framework evolution (Basel II & EuropeanDirectives).

CCRM takes part in all on-site investigations involving modelling tools or riskquantification techniques (market risk, credit risk, ).

On-site investigation always covers various topics :

Of course, in-depth analysis of valuation models and of risk indicators

calculation,

But also, assessment of : quality of inputs, integrity of IT systems,reliability of internal controls, use-tests.

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II. Presentation of CCRM

So, the technical analysis of a model is just a part of a much largerframework and has to be completed by various assessments :

Accuracy of the perimeter covered by the model,

Reliability of the inputs,

Data integrity in the IT systems,

Soundness of the risk monitorin rocess

Effective use by operational management.

On-site validation missions will generally be made by a team of 4 to 7persons, and will take between 3 and 6 months.

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Regulatory approach, as well as its aims, is not exactly the same ineach case :

For VaR, IRB or AMA internal models : review must be made a priori, in

order to determine whether, or not, the institution can be allowed to use itsinternal models for prudential purposes ;

For economic capital models : no prior clearance is required. The objectiveis to fully understand the substance of the models the bank is using, with

,

between the bank and the supervisor regarding the final capitalrequirements (Pillar 2);

For valuation models : regulatory review is usually made a posteriori, inorder to audit the internal process developed by banks to validate thepricing models, and therefore to check the suitability of the valuation, as

recorded into the banks accounts ;

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A model is just a way to depict the reality, and the chosen representationmay be more or less accurate. Moreover, the quality of the representationmay change over time.

Some types of risks are not easy to quantify or, even, to be apprehended(reputational risk, strategic risk,).

Thus, how to take into account eventual shortcomings of a model, howto treat specific risks which are hard to quantify, are crucial issues.What adequate solutions can be offered ? As conservatism is needed,how to calibrate a satisfactory margin, as a precautionary cushion?

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III. Value At Risk definition

Value at Risk measures a confidence interval of the potential losses over a givenhorizon under standard market conditions.So VaR being a loss, it corresponds to a negative P&L.

The one-tailed left confidence interval of level p% for a variable Y, denoted

[ ; +[ is defined by :

Prob( Y [ ; +[ ) = p% (1)

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.

The Value-at-Risk for a time horizon h at a given confidence interval p% is thequantity VaR such that :

Prob(Vt+h - Vt [VaR ; +[ ) = p%.

With Y= Vt+h - Vt and =VaR in (1)

i.e. in p% of the cases, the losses of the portfolio for the time horizon h may notexceed VaR.


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Banks which use internal models to estimate the capital adequacy of theirtrading books have to calculate the VaR:

For a time horizon h equal to10 days,

For a confidence interval p of 99%.

Denoted VaR(10 days, 99%)

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If an institution computes the VaR for a time horizon of 1 day, the 10 daysVaR will be approximated by :

day)(1VaR10days)(10VaR


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The capital adequacy for a trading book due to its market risk is calculated,for the day t, as :

Capital = Max ( VaR(t-1) ; ( x VaRAvg))

+ additional requirements (cf infra)

where VaR = VaR ( 10 days ; 99% ), global VaR daily computed,

and VaRAvg is the averaged of the daily VaR measures of the preceding sixty

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- .

is determined by the ACP :

its minimum value is equal to 3,

There is often a first add-on x which depends on the quality of the model orof the general risk management framework,

there is an add-on y which depends on the result of the back-testing.

So = 3 + x + y


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Additional requirements for specific (credit or equity) risk, which is the riskof variation in a price due to factors specific to the issuer :

VaRSpec(t-1) or average, calculated daily.

The bank has two different options to measure the specific risk:

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Either it is able to calculate both a VaR on the general risk (which is the riskof variation in a price due to market data level fluctuations) and a VaR on thespecific risk,

Or the bank defines the portfolios which it considers bear a specific risk andcalculates a global VaR on them (combining the general risk and the specific

risk). The scaling factor used for the specific risk is then applied to the VaRwhich is obtained on these portfolios,

Note that the second option is conservative since you apply a greater scalingfactor to a VaR of which a part is imputable to the general risk.


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Since the financial crisis began in mid-2007, an important source oflosses occurred in the trading book. A main contributing factor was thatthe capital framework for market risk, based on the 1996 amendment tothe Capital Accord to incorporate market risks didnt capture some keyrisks.

So in 2009, the Basel Committee on Banking Supervision has completed themarket risk framework (Basel 2.5) :

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Add-on for stressed VaR (sVaR), computed using a stressed referenceperiod ,

Max (latest sVaR available ; (s x sVaRAvg))

sVaR = sVaR (10 days ; 99% ), at least weekly computed,

sVaRAvg is the averaged of the sVaR numbers calculated overthe preceding sixty business days,

s is dtermined by the ACP (same way than for the VaR) :

s = 3 + xs + ys


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Deletion of the Add-on for specific risk, replaced by 2 new add-ons,calculated at least weekly , for a time horizon of 1 year, and at a confidenceinterval of 99,9%. These add-on will not be detailed in this presentation.

IRC (Incremental Risk Charge) add-on, to better capture default risk as

well as rating migration risk for credit trading portfolios, except forcorrelation trading products (CDO : Collaterized Debt Obligations):

Max ( last IRC calculation, average on the 12 last weeks).

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-

migration and default risk for correlation trading products:

Max ( last CRM calculation, average on the 12 last weeks), with afloor in the requirement.

In European Union, these models should be used for 12/31/2011 capitaladequacy computations for banks whose VaR has already beenapproved by the supervisors, otherwise :

Use of the standard approach if stressed VaR not approved; Use of the standard method for specific risk if IRC/CRM not approved.


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V. Computation of the VaR

1 Risk factors

Risk factors are generally market data whose variations affect thevaluation of the positions. Sometimes they are deduced from marketdata.

They have to be identified precisely for each type of instruments beforecomputing the VaR,

Their choice is specific to each bank and varies during the time, The same parameters are often used for P&L and VaR purposes.

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At first, choice of the market risk factors, denoted Xi hereafter, which maybe correlated :

Interest rates, Equity /indices prices,

Foreign Exchange rates, Credit spreads, Commodity prices, Option implied volatilities,


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The daily variations of the price of the trading book (ie the daily P&L) issupposed to be entirely explained by the variation of these risk factors.But it happens that some market parameters are not risk factors, forexample when these data :

have a very small impact on the VaR : the bank has to justify its choice,

are difficult to observe on a daily basis (dividend forecasts or repos rates,implicit correlations between stocks and indices used to price some equitystructured products, ) : specific risk indicators must be calculated to

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.

Modeling risk factors correlation for a long only portfolio is completelydifferent than for a long/short portfolio :

For a long only portfolio, overestimating correlations may generally lead toan overestimation of the risks through an underestimation of the

diversification effect,

For a long / short portfolio, overestimating correlations may lead to asignificant underestimation of the risks, as it may overestimate somecompensation (hedging) effects between instruments.


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2 Modeling

The computation of the VaR is not straightforward and needs some

modeling techniques.

Then, 3 main approaches can be developed to compute the VaR :

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Parametric approach (a)

Historical simulation (b)

Monte Carlo simulation (c)

Each approach has its advantages.


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2.a Parametric VaR

V(t) : Valuation of the portfolio at t, function of the values of the riskfactors Xi

Two main additional hypothesis :

( ) ( ) ( ) ( )[ ]tXtXtXFtV n,...,, 21=

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first order derivatives :

The variation of the risk factors Xi is assumed to follow a

Gaussian law.Consequently V(t,t+dt) is distributed following a Gaussian lawN(0 ; (V)), because its mean is neglected.

( ) ( ) ( )i

n

i i

dXX

FtVdttVdtttV

=

+=+

1

,


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Its standard deviation (V) is given by :

( )

=

nn

nn

F

X

F

FFFV

...

...

,...,,

1

22

2

22121

112112

2

1

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Where the i and ij are the standard deviation and correlationparameters of the daily variations of the risk factors.

n

nnnnn

n

X

F

...

...

............2

2211

21


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The 1-day VaR will be given by :

VaR (1 day) = kx (V)

Where kx is the x% percentile of a Gaussian distribution (for x=99%, thiscoefficient will be equal to 2,33).

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VaR(99%, 1 day) = 2,33 x(V)

This methodology will be adapted only to linearportfolios.


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2.b Historical simulation

The daily variations of the risk factors are supposed to be stationary.

Step 1 : compute the 260 scenarios corresponding to the daily variations ofthe risk factors observed during the last year,

Step 2 : apply these variations to today prices : one obtains 260 possible

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,

Step 3 : price the portfolio for each of these 260 scenarios.

The VaR will be the 1% percentile of these scenarios if sorting them fromthe worst to the best.

So for a time series of 1 year for the risk factors, the VaR is between the2nd and the 3rd worst cases.

Sometimes banks take 300 days time series for the VaR to be the 3rd

worst case.


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Some approximations may be used to compute the price of the tradingbook for each of the scenarios because there are so many valuations asdays in the time series.

In theory, estimating a 1% percentile with only 260 scenarios should notgive a precise estimation.

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n prac ce, as e a y var a ons o mar e pr ces are no n epen en ,

the convergence of the VaR estimator is generally quite acceptable.

When historical data are not available for a given risk factor, some proxyhave to be used.

There are no assumptions on the statistical distribution of the risk factorsor on their correlations.


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2.c Monte Carlo simulation

The daily variations of the risk factors follow a given statistical distribution

(not necessarily Gaussian).

Step 1 : estimate the parameters of the distribution of the risk factors,

Step 2 : simulate randomly N (typically 10 thousands or more) possible

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var a ons o e r s ac ors, a ng n o accoun e corre a ons e ween

the risk factors, Step 3 : price the portfolio for each of these Nscenarios.

The VaR will be the 1% percentile of the value of the portfolio on the N

scenarios.


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Monte Carlo simulation could appear as the most sophisticated (one caneasily add risk factors,).

But this technique is time consuming. Some institutions may then usesome simplifications :

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They may choose a reduced number of risk factors, or simplify their

correlation structure,

They may use simplified valuation techniques or even approximate thevariation of the price of the portfolio by its first or second order sensitivities,

They may reduce the number of simulations.


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3. Importance of the risk factors time series in the VaR calculation

The observation period of the market data and their variations has asignificant impact on the results of the VaR calculation :

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observed variations on the risk factors ;

For parametric calculation and Monte Carlo simulations : the dynamics ofthe risk factors and the way they are linked (represented by the variance-covariance or the correlation matrix) depend on their past evolutions.


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Main components :

The length of the time series and the characteristics of the marketduring this period : when a time series corresponds to a calmperiod, for a given position the VaR decreases.

The update frequency of the time series of risk factors (and so of

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shocks or correlation structure, depending on the methodology

chosen),

Between 2 updates of the risk factors, the new shocks are not takeninto account in the VaR, so the amount of the VaR may be under orover estimated ;

The P&L is calculated with real market data. So in case of importantmoves of the markets (high volatilities of the markets), the back-testingmay fail.


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Regulatory requirements

Minimum length of the observation period : 1 calendar year (= 262 days or261 daily excepting week-ends and 1st of january )

Update at least every month, more often in case of high volatilities of the

markets

In France, most banks use one year time series for risk factors.

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,

If it corresponds to a quiet period, the VaR amounts will be lower.

The market data are generally updated :

Daily for historical simulations : every day the oldest data are replaced bythe most recent one.

For a VaR calculated on End Of Day D market data, the latest shocksare the risk factors variations between D-1 and D-2.

Monthly or twice a month for parametric calculations and Monte Carlosimulations, to determine the variance-covariance matrix or the historicalvolatilities.


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4. Aggregation of the VaR

VaR calculations : At aggregated level At portfolio level, business line, trader level, , to identify the most risky ones.

The sum of the VaR from different activities is not equal to the VaRcalculated on the whole activity.

The difference between these 2 calculations represent the netting effectbetween different activities.

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The computation should be made at the trading book level. ComputingVaR-like indicators for different risk factors / portfolio and sum themshould not be considered as acceptable.

When different methodologies are used to calculate the VaR, depending

on the activity, the global VaR is the sum of the VaR calculated with eachof these methodologies, and diversification effect are not taken intoaccount.


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5. VaR production

Institutions which have been authorized to use their internal modelhave to calculate a VaR daily .

It is time consuming process :

Feed the trading positions in the calculation engine,

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ee an up a e o mar e a a an or s or ca s mu a ons, o e

shocks), Valuations of the portfolio (which take a long time for historical and Monte

Carlo simulations).


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6 Other remarks about the VaR

VaR does not provide accurate risk measurements for activities whosepotential losses have very high amplitudes and occurrence probabilityless than 1%.

VaR generally doesnt provide satisfying measure of risk for some typesof positions, for which the capital requirement is to be estimated using thestandard method if no specific processing has been developed, forexample :

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Event risk (mergers,) which may affect significantly risk arbitrage desks :VaR is based on time series of equity prices and generally doesnt take intoaccount sudden spreads increases between the prices of the two stocks,which induce losses in case of failure of the transaction;

Hedge funds (liquidation values mostly not available on a daily basis) :

stress scenarios may be useful.


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The VaR is a synthetic indicator which measures a confidence interval ofthe potential losses. It does not give any information about the maximalloss or at any other percentile.Without further information, it is not possible to determine the part of the

various risk factors and the influence of the size of the positions in theevolution of the VaR amounts.

Basel requires the VaR to be used for daily risk-monitoring purposes(use-tests).

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VaR limits are generally defined for different aggregation levels.

But VaR is not an operational indicator to monitor positions of the traders(these are monitored by sensitivities indicators and various limits definedfor finest levels).

The coherence between operational and VaR limits has to be checked.


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VI. Back-testing

Back-testing : Institutions have to compare daily Profit and Losses (P&L)to VaR figures (not in absolute real values, negative) :

It is required to be done at least quarterly on the 250 last business days.

Two P&L : Hypothetical P&L : changes in the portfolio value calculated on end of day

positions unchanged;

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ea : actua tra ng outcomes, .e a y c anges n port o o va ue

(excluding fees, commissions and net interest incomes),

Supervisors require banks to perform back-testing on one of these P&Lor both.

In France, since 2011, both have to be back-tested.

A back-testing is required on the VaR and on the VaR for specific risk.


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VI. Back-testing

The add-on for capital requirements depends on the number of failures ofthe back-testing.

Number of failures = number of days for which :

P&L


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VI. Back-testing

DAILY P/L (M tM ) / VaR 99 1 DAY 99%

.

1 0 0 0 0 .0

2 0 0 0 0 .0

3 0 0 0 0 .0

(USD) R ea l / C lean M tM R IS K T heo ret ic a l P /L

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In theory, losses should exceed the 1 day VaR around 1% of the cases.

- 3 0 0 0 0 .0

- 2 0 0 0 0 .0

- 1 0 0 0 0 .0

0. 0

S


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VII. Stress-tests

Three sets of stress-tests are required in order for an internal model to bevalidated:

Historical stress-tests;

Hypothetical stress-tests;

Adverse stress-tests.

Institutions frequently use the same systems for VaR calculation and stress-tests.

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But sometimes two independent processes exist, one for the VaR, and onefor the stress-tests :

P&L calculations with sensitivities suppose the risk factors variations to be small.So they shouldnt not be used for stress-tests, which assume the shocks to be

important. So in case of VaR calculation with a sensitivity approach, stress-tests could for

example be estimated with full valuations.



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Calculation engine :

Input :

Market Data :

Identification of those which are risk factors

Historical data time series

Variance/covariance matrix

For historical simulations, update of the shocks

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,

Valuations of the portfolio and VaR calculation

Output :

VaR

Stress-tests

Back-testing

Risk-monitoring



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The whole calculation engine has to be examined

The quality of the inputs and the robustness of the VaR architecturesystem are essential.

Because of the important number of steps in the VaR computation, risksof failure somewhere in the process are significant.

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So procedures to solve production incidents have to be set up.

Impacts of the incidents on the VaR measurements have to be analyzed,for the supervisor to be able to have an opinion on the quality of the VaRproduction process.

VIII V lid ti b i


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Modeling choices must be appreciated depending on the portfolio forwhich they will be used.

The validity of these choices may vary over time :

For example, if an institution develops new trading activities, or optimizesthe hedging of its main risks, it may have to improve its VaR methodology;

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e s a s ca s r u on o mar e pr ces a y var a on may c ange over

time : a market parameter that remains unchanged may become volatile,two market parameters that were historically perfectly correlated maybecome independent.

A too conservative approach should not be recommended :overestimation of a risk could hide the fact that an other risk may beunderestimated.

VIII Validation by supervisors


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Risk factors

Historical data used to compute the VaR should be verified, they shouldbe coherent with data used to compute daily P&L.

Some points to review :

Are the selected risk factors representative of the activities?

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Is there a good granularity?

Sometimes the number of points is reduced to reduce the computingtime (for example on interest rates).Use of data sources to estimate the scenarios of the VaR different fromthe one used for the daily valuation.

For market whose feed in the system is automatic (Bloomberg,Reuters,), which market makers, time window, price, tochoose?



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How to replace missing data in the time series ?

Rules to complete these data when no historical prices can be found mustbe defined. This is particularly important for risk factors specific to aninstitution. (e.g. : use of a proprietary model to represent the smile effect).

How to deal with a new risk-factor (e.g. a newly issued stock) with less than1 year historical or with illiquid parameters for which daily quotations are notavailable?

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Which frequency do we choose for the updates ?

When no time series are available, is the proxy used a goodone ?

Adequacy between risk factors and positions



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Portfolio valuation techniques

Monte Carlo and historical simulations being time consuming, if approximated valuationtechniques are used, one should check that the approximations are acceptable, forexample :

compare sensitivities (delta, vega,) computed using exact and approximatedvaluation models on the different portfolios;

Compare P&L calculated by full valuations and by sensitivities;

For sensitivities based approaches :

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Are they coherent across products ? Do they explain the risks of the portfolio ?

More the models are simplified, more the quality of the VaR is unreliable.

Control of the exhaustiveness of the positions : For positions for which there is no VaR computation, capital requirements arecalculated using the standard approach;

The supervisor checks that for each entity of the institution, a capital iscalculated (VaR or standard method) for every trading position.



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Back-testing and P&L attribution

If it is made at different portfolio level, back-testing is a very useful tool toassess the quality of the VaR computation.

An other very useful technique is to verify that the risk factors of the VaRand the representation of portfolio chosen (sensitivities, approximated

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va ua on mo e s exp a n correc y e a y ro oss.

Such decomposition of the P&L by risk factors is generally called P&Lattribution and may be computed by some institutions.


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X Conclusion


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X. Conclusion

Generally, more portfolio are hedged or complex,

Containing structured (exotic) and highly non linear transactions,

Making some thin arbitrage (deformation of a yield curve, arbitrage betweencorrelated products,),

Trading on illiquid markets,

more computing the VaR may be difficult and may need sophisticated tools.

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The whole calculation engine has to be examined by the supervisor for him to

determine whether, or not, an institution can be allowed to use its internalmodels for prudential purposes.

Since the financial crisis began in mid-2007, the Basel Committee on BankingSupervision has completed the market risk framework with IRC, CRM and

Stressed VaR which also have to be approved by the supervisors.

The BCBS has also started a fundamental review of trading activities.