Financial Stability Report – Chapter 6 – Item 2 · 2004-09-23 · 39/ Resolution no. 2,606, Circular no. 2,898, both of May 27, 1999, and Circular no. 3.229, of 03/25/2004 are

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This paper examines capital requirement determinationmodels in financial institutions in order to cover market riskstemming from exposure to foreign currencies and gold.The models examined belong to two groups according tothe approach involved: standardized and internal models. Inthe first group, we examine the Basel model and the modeladopted by the Brazilian legislation, in two versions: beforeand after the change which enabled the offsetting of oppositepositions in American Dollar, Euro, Sterling Pound, Yen,Swiss Franc and gold, implemented by Circular no. 3,217,of 12/19/2003,further modified by Circular no. 3,229, of 03/25/2004. In the second group, we consider the models basedon the concept of value at risk (VaR). We analyze the singleand double-window historical model and the exponentialsmoothing model (EWMA) and a hybrid approach thatcombines features of both models. The results suggest thatthe Basel model is inadequate to the Brazilian market,exhibiting a large number of exceptions. The model of theBrazilian legislation has no exceptions, though generatinghigher capital requirements than other internal models basedon VaR, even with the changes introduced by Circular no.3,229. In general, VaR-based models perform better andresult in less capital allocation than the standardized approachmodel applied in Brazil.

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33/ Research Departament, Central Bank of Brazil, [email protected].




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In 1996, the Basel Committee advanced several proposalsto provide bank supervision authorities with basic guidelinesin regulating capital requirements for the coverage of marketrisk37. These proposals suggest, for allocating regulatorycapital of a financial institution, two basic calculationalternatives: a standardized approach and an approach basedon internal models of risk management38. Banks that fulfillsome requisites may calculate their own capital requirementbased on certain parameters determined by the monetaryauthority. The remaining institutions would comply with therequirements posed by the standardized approach.

The structure put forward by the Basel Committee assumesthat capital requirement calculated by the standardizedapproach will be always higher than the requirementcalculated by an internal model, which is more adequate tothe risk profile and portfolios of each institution. This way,there would be a motivation, expressed as lower capitalrequirement, for financial institutions to develop more preciserisk management systems.

The standardized approach is characterized by an absenceof sensibility to changes in volatility patterns and correlationsof risk factors. This is a significant shortfall, especially whendealing with markets where asset prices changeconsiderably, as the Brazilian market.

The approach based on internal models enable financialinstitutions to use their own risk models. Qualitative andquantitative restrictions were imposed to ensure a minimumstandardization and consistency of results obtained indifferent proprietary systems, such as the calculation of thedaily value at risk (VaR), holding period of ten days, 99% ofconfidence level and a minimum multiplying factor of three.

The Central Bank of Brazil regulated the capitalrequirements to cover market risks resulting from exposureto foreign currencies- and gold-linked instruments using thestandardized approach, so that financial institutions are notallowed to use internal models to calculate capitalrequirements39.

37/ See Basel Committee on Banking Supervision (1996).

38/ Models developed by the financial institutions.

39/ Resolution no. 2,606, Circular no. 2,898, both of May 27, 1999, and Circular no. 3.229, of 03/25/2004 are the main documents on the

regulation on this matter.

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This paper aims at analyzing five capital requirementdetermination models covering market risk in foreigncurrencies and gold in Brazil. The two first models followthe standardized approach. We analyze the Basel modeland the model put forward by the Brazilian legislation intwo versions: before and after the change which enabledthe offsetting of opposite positions in American Dollar, Euro,Sterling Pound, Yen, Swiss Franc and gold, implemented byCircular 3,229, of 03/25/2004. The other models belong tothe internal model approach. We assess the one- and two-window historical model, the exponential smoothing and athird hybrid approach which presents characteristics of thetwo previous models.

The results suggest that the Basel model has a poorperformance, displaying a large number of exceptions dueto volatility fluctuations of the Brazilian market, coupled withreduced capital requirement. The model of the Brazilianlegislation does not exhibit exceptions, but it generates highercapital requirements than the VaR-based models, even iftaking into account the changes introduced by Circular no.3,229. As expected, models based on VaR performadequately and result in less capital allocation than thestandardized model effective in Brazil.

The remainder of this paper is organized as follows: thenext section defines the data used and methodologyemployed in building and assessing capital requirementcalculation models; the third section exhibits the resultsobtained; and the fourth and last one contains the conclusions.

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The series used in building the portfolios refer to theAmerican Dollar, Euro, Sterling Pound, Yen, Swiss Francand gold quotations from 1/4/2000 to 7/25/2003. The reasonfor selecting the first date was the starting of the Euro intocirculation. It is worth noticing that, though relatively short,the period under examination spans events that causedsignificant fluctuations in foreign exchange volatility40 inBrazil, enabling a more consistent assessment of the modelsexamined.

40/ Argentine crises, energy crises and presidential succession.

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Six portfolios were built based on different assetcompositions, with the purpose of reproducing situations thatmight promote an adequate gauging of the models underscrutiny. Such portfolios are depicted in Table 1.

Portfolio 1 comprises solely the most traded foreigncurrency in Brazil: the American Dollar. Portfolio 2 includesopposite positions in Dollar and Euro, enabling an assessmentof offsetting between currencies for the effect of capitalrequirement. Portfolio 3 combines positions in all currencies.Portfolio 4 includes currencies that featured the higher linearcorrelation value for the series examined, with the Euroand the Swiss Franc in opposite positions. Besides, a shortposition in gold was included. Portfolio 5 features again theDollar and Euro, this time in long positions, coupled with ashort position in gold. The sixth and last portfolio comprisessolely the Japanese Yen, which displayed the higheststandard deviation in the period analyzed. These portfoliosare considered, also, in their symmetrical versions.

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The return of a financial asset is given by

, where pt corresponds to the closing

price of the currency on date t.41.

For the exponential smoothing model we use its

approximation in logarithmical form, ,

where ln is the natural logarithm42.

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Five models are assessed for determining capital requirementto face the market risk incurred by positions taken in foreigncurrency and gold. The first model uses the standardizedapproach put forward by the Basel Accord; the second

41/ Closing Ptax Quotation – the weighted average of trades in the interbank foreign exchange market, calculated by the Central Bank of Brazil

according to Communiqué no. 6,815/99.

42/ The model assumes that foreign exchange quotations are lognormal.

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follows the Brazilian regulations in two versions: before andafter the changes which enable offsetting of AmericanDollar, Euro, Sterling Pound, Yen, Swiss Franc and gold inopposite positions. The remaining models, the historical model,the model based on exponential smoothing and the hybridmodel, are under the internal models approach, the capitalrequirement (EC) of which is calculated by:43

(1)

where M is the multiplication factor of the 10-day VaRaverage ( ) of the last 60 days. The value of M isinitially set to 3 in the Basel Accord, and may reach 4depending on the performance of the model used by thefinancial institution. Considering the strong volatility patternsof the Brazilian market when compared to G-10 markets,and that this coupled to a 3=M may lead to exaggeratedcapital requirements in some periods, we additionallyevaluate EC with 2=M . Portfolios that are symmetricalin composition to the ones examined are also taken intoconsideration.

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The criteria used for allocation of capital to cover foreignexchange risk foresees a capital requirement equivalent to8% of the value of the net total open position of the institutionin foreign currency and gold, to be determined in two steps.In the first step, the institution calculates its net position ineach of the currencies it deals with (including gold), includingthe value of spot and forward positions and the value referredto option contracts, calculated by the Delta multiplied bythe quantity and by the size of the contract.

The second step deals with converting the net values, longor short positions, in different currencies, to the currencythe institution shall inform in its reports. Such net valuesshall be added separately, so as to obtain, at the end, twototal values: a long total value and a short total value. Thelarger of the two, in absolute value, is added to the absolutevalue of the net position in gold. The resulting valuecorresponds to the net total open position held by theinstitution in foreign currency.

43/ Basel Committee on Banking Supervision (1996).

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Resolution no. 2,606 and Circular no. 2,894, both of May27, 1999, are the basic rules used in Brazil up to 12.18.2003for determining capital requirements to cover the marketrisk in operations generating exposures in assets and liabilitiesreferred to foreign currencies and gold. Several laterregulations changed some of the factors used in calculatingcapital requirements. The last one, Circular no. 3,194, ofJuly 2, 2003, reduces the “F” factor (see definition below)from 1.0 to 0.5. Nevertheless, the calculation basis remainedthe same.

Circular no. 3.217, of 12/19/2003, introduced an importantchange in the calculation basis, which enabled the offsettingof opposite positions in American Dollar, Euro, SterlingPound, Yen and Swiss Franc. Circular no. 3.229, of 03/25/2004, substitutes for Circular no. 3.217, extending thispossibility to gold.

This paper examines the Model of Resolution no. 2,606 intwo versions: the first - from now referred as previous version- is based on the effective rule until 12/18/2003, and thesecond - from now referred as current version - is basedon Circular no. 3.229.

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According to this rule, total exposure in gold and assetsreferred in foreign currencies, including derivatives arelimited to 30% of the “Patrimômio de Referência” (PR)44.

Exposure is defined as the summation, in absolute value, ofdifference between long and short exposures, in eachcurrency, converted to the Brazilian Real. If such exposureis lower than 5% of PR, no capital is required. For anyexposure greater than 5% of PR, an additional capital isrequired, equivalent to 50% of total exposure (F’’=0,5).Analytically, the capital requirement (EC) is given by:

where:

=EC Capital requirement;the factor applicable to such operations;

44/ “Patrimômio de Referência” is a reference value defined in Resolution no. 2,837, of 05/30/2001.

Figure 1 - CR Backtest of the Standardized Approach Models - Portfolio 1

-60

-40

-20

0

20

40

60

4.22001

6.18 8.29 11.14 1.302002

4.17 7.2 9.12 11.26 2.102003

4.29 7.14

10-Days ReturnsCR Basle Standardized ModelCR Res. 2,606 - Previous Version CR Res. 2,606 - Modified Version

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= summation of the absolute values of net positions

in each currency and gold;

0%5%51

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kandPRofLifkn

i

i , otherwise;

PR = “Patrimômio de Referência”.

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This version is the result of the changed Articles 1 and 2 ofCircular no. 2,894/99. In practice, one may consider as onesingle currency exposures in Euro, United States Dollar, gold,Sterling Pounds, Yen, Swiss Franc and gold. When usingthis alternative, the lower of the following items, multipliedby a factor of 0.7 shall be added to the total exposure value:

(i) Summation, in absolute value, for gold and each of theforeign currencies mentioned above, of the excess longexposure over the short exposure;(ii) Summation, in absolute value, for gold and each of theforeign currencies mentioned above, of the excess shortexposure over the long exposure. Analytically, the capitalrequirement is given by:

where:

EC= Capital requirement;the factor applicable to such operations;

, corresponding to the total

exposure value in gold and foreign currencies;

, where iExpC is the long

exposure, iExpV is the short exposure on the i-th foreign currency and n1 relates to the group of selected currencies,including the Unites States Dollar, Euro, Yen, Sterling Pound,Swiss Franc and gold.

is the hedge factor related to the selectedcurrencies and gold;

Figure 2 - CR Backtest of the Standardized Approach Models - Portfolio 3

-250

-150

-50

50

150

250

4.22001

6.18 8.29 11.14 1.302002

4.17 7.2 9.12 11.26 2.102003

4.29 7.14

10-Days ReturnsCR Basle Standardized ModelCR Res. 2,606 - Previous Version CR Res. 2,606 - Modified Version

Figure 3 - CR Backtest of the Resolution 2.606 model and of the VaR-based models - Portfolio 1

-100

-80

-60

-40

-20

0

20

40

60

80

4.22001

6.18 8.29 11.14 1.302002

4.17 7.2 9.12 11.26 2.102003

4.29 7.14

10-days Returns CR Historical Model - 1 Window

CR Historical Model - 2 Windows CR EWMA

CR Res. 2,606 - Previous Version CR Res. 2,606 - Modified Version

CR Hybrid Model

Figure 4 - CR Backtest of the Resolution 2,606 model and of the VaR-based models - Portfolio 3

-250

-200

-150

-100

-50

0

50

100

150

200

250

4.22001

6.18 8.29 11.14 1.302002

4.17 7.2 9.12 11.26 2.102003

4.29 7.14

10-Days Returns CR Historical Model - 1 Window

CR Historical Model - 2 Windows CR EWMA

CR Res. 2,606 - Previous Version CR Res. 2,606 - Modified Version

CR Hybrid Model

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, where n2

relates to foreign currencies that are not in the group ofselected currencies;

0%5%5 �� kandPRofEXPifk , otherwise;

PR = “Patrimônio de Referência”.

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The historical model consists in using one given quantile ofthe series of portfolio returns as a VaR estimate. The portfolioreturns are simulated based on its current composition(financial proportion of each currency and gold in theportfolio) and on the observed returns of the assets duringthe last n days (n days moving window). The return of theportfolio is given by:

where ir is the daily return of asset i in the portfolio. Foreach date t, the weights w

i reflect the current proportions

of each currency value in the portfolio. Ten-day VaR, usedin calculating capital requirement, is given by the product ofthe one-day VaR by the square root of ten, as indicated bythe Basel Committee.

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This is the usual form of the non-parametric model ofempirical quantile, where a single moving window of 252observations is used, considering the 1% (left tail) and 99%(right tail) quantiles. This way, one-day VaR will be givenby:

252

%,1

1

, ti

d

compt QVVaR ��

252

%,99

1

, ti

d

vendt QVVaR ��

where Vi corresponds to the value of portfolio i.

Table 1 - Portfolios Composition (%)

Currency Portfolios 1/

1 2 3 4 5 6

Dolar 100.0 300.0 265.0 0.0 75.0 0.0

Euro 0.0 -200.0 -47.2 200.0 75.0 0.0

Pound 0.0 0.0 -3.4 0.0 0.0 0.0

Yen 0.0 0.0 -113.7 0.0 0.0 100.0

Swiss Franc 0.0 0.0 -0.8 -50.0 0.0 0.0

Gold 0.0 0.0 0.0 -50.0 -50.0 0.0

table can be considered in nominal terms or in percentual terms with respect

to the portfolio total value.

1/ The total value of each portfolio is $100 monetary units. The values in the

Table 2 - Total exposure and capital requirement

Standardized approach models

Models Portfolio 1 Portfolio 2

Expo- CR 1/ CR / Expo- CR 1/ CR /

sure Basle CR sure Basle CR

Basle 100.0 8.0 1.0 300.0 24.0 1.0

Version

Previous 100.0 50.0 6.3 500.0 250.0 10.4

Modified 100.0 50.0 6.3 240.0 120.0 5.0


Expo- CR CR / Expo- CR CR /


Basle 265.0 21.2 1.0 250.0 20.0 1.0

Version

Previous 430.0 215.0 10.1 300.0 150.0 7.5

Modified 215.5 107.7 5.1 170.0 85.0 4.3


Expo- CR CR / Expo- CR CR /


Basle 200.0 16.0 1.0 100.0 8.0 1.0

Version

Previous 200.0 100.0 6.3 100.0 50.0 6.3

Modified 135.0 67.5 4.2 100.0 50.0 6.3

1/ The total exposure and the capital requirement (CR) refers to $100

monetary units (portfolio value).

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This variant of the historical model considers simultaneouslytwo moving windows to obtain the empirical quantile.

The windows selected are of 126 and 252 days. Thequantiles of 1% (left tail) and 99% (right tail) are extractedfrom the two windows, and the most conservative (thehighest in module), for each tail, is used in calculating VaRof short and long positions according to the followingformulas:

252

%,1

126

%,1

1

, ; tti

d

compt QQMáxVVaR ��

� �252

%,99

126

%,99

1

, ; tti

d

vendt QQMáxVVaR ��

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This model follows the RiskMetrics (1996) methodology,45

in which each asset VaR in the portfolio is obtainedparametrically from its conditional volatility, estimated basedon exponential smoothing (EWMA)46. This way, one-dayVaR of each asset is given by:

where is the quantile of the standardized normaldistribution equivalent to the probability of VaR estimate, inthe case, 1% for the left tail and 99% for the right tail; and

tih , is the daily conditional volatility of logarithmic returnsfor asset i estimated for date t, calculated by exponentialsmoothing according to the expression:

where is the exponential decay factor, which is set to 0.94,the value suggested by RiskMetrics and widely used insuch models. One-day VaR for the asset portfolio is givenby:

45/ Developed by the U.S. bank J.P. Morgan.

46/ EWMA – Exponentially Weighted Moving Average

TM

TM

Table 3 - Evaluation of capital requirement (CR) exceptions

for standardized models and var-based models with

multiplication factor M = 2

Models Portfolio 1 Portfolio 2 Portfolio 3

Tail Tail Tail

Left Right Left Right Left Right

Basle 5 28 0 1 0 4

Historical

1 window 0 5 0 1 0 3

2 windows 0 1 0 1 0 3

EWMA 0 3 0 3 0 3

Hybrid 0 2 0 1 0 3


Tail Tail Tail


Basle 0 3 0 7 5 33

Historical

1 window 0 0 0 0 0 0

2 windows 0 0 0 0 0 0

EWMA 0 0 0 0 0 0

Hybrid 0 0 0 0 0 0

1/ There were no EC exceptions for the previous version and modified version

standardized models.

2/ Only one CR exception for the Historical model (in both versions) at the

right tail of Portfolio 3 was registered with respect to the VaR-Based Models

with multiplication factor M = 3.

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where , the correlation between assets i and j on date t,is given by such that is the conditional covariance betweenassets i and j on date t, resulting from the expression:

Extending the holding period to ten days, the portfolio VaRwill be given by:

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This model is an approach that combines the historicalmethodology and exponential smoothing. The main featureof the model is granting greater weight to more recentinformation in the portfolio return moving window. Theweight is calculated by the expression:

n

kWeight �

�

��

�

�� ]

)1(

)1([

where is the smoothing factor, set to 0.97, k is the size ofthe return historical window and n is the order number ofreturns, from the most recent to the remotest47. The mostrecent return, for instance, has n=0 and weight=0.03 in awindow of 252 business days.

The portfolio VaR is calculated by disposing returns inincreasing order and summing up the corresponding weightsuntil the result equals the desired VaR percentile. Thepercentile of 1% for the long position and 99% for the shortposition are obtained by linear interpolation between adjacentreturns48. Ten-day VaR is given by multiplying one-day VaRby the square root of 10.

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Both portfolio return distribution tails are assessed. Theperiod considered in the backtesting ranges from 4/2/2001to 7/25/2003, totaling 572 observations. Notice that the BaselAccord does not mention backtesting for capitalrequirements, but for 1-day VaR of internal models.

47/ The value of 0.97 for is based on the Historical Hybrid model original article, The Best of Both Worlds, authored by Boudoukh,

Richardson and Whitelaw (1998).

48/ In cases where the first weight represents a percentage greater than 1%, the criterion shall be selection of the first return.

Table 4 - 1-Day VaR Backtest

Basle Test - portolios 1 and 2

Model Exceptions - 1-Day VaR

Period Portfolio 1 Portfolio 2

Long Short Long Short

Historical

1 Windows

05/01/2001 to 04/01/2002 9 10 5 2

06/04/2001 to 08/04/2002 8 8 4 2

09/07/2001 to 08/07/2002 3 4 4 3

05/10/2001 to 03/10/2002 4 5 6 6

09/01/2002 to 03/01/2003 4 5 5 6

11/04/2002 to 04/04/2003 4 5 5 6

11/07/2002 to 08/07/2003 4 4 4 5

2 Windows

05/01/2001 to 04/01/2002 5 8 5 2

06/04/2001 to 08/04/2002 4 6 4 2

09/07/2001 to 08/07/2002 0 2 4 3

05/10/2001 to 03/10/2002 4 5 5 6

09/01/2002 to 03/01/2003 4 5 4 6

11/04/2002 to 04/04/2003 4 5 4 6

11/07/2002 to 08/07/2003 4 4 3 5

EWMA

05/01/2001 to 04/01/2002 5 9 2 3

06/04/2001 to 08/04/2002 5 5 3 3

09/07/2001 to 08/07/2002 3 3 4 6

05/10/2001 to 03/10/2002 4 5 4 8

09/01/2002 to 03/01/2003 2 5 3 7

11/04/2002 to 04/04/2003 2 5 4 8

11/07/2002 to 08/07/2003 3 3 5 6

Hybrid

05/01/2001 to 04/01/2002 5 6 6 5

06/04/2001 to 08/04/2002 4 3 3 4

09/07/2001 to 08/07/2002 2 2 2 5

05/10/2001 to 03/10/2002 4 4 4 5

09/01/2002 to 03/01/2003 3 4 4 4

11/04/2002 to 04/04/2003 3 4 4 5

11/07/2002 to 08/07/2003 2 3 3 6

Obs 1: Green zone - less than 5 exceptions in the sub-period;

yellow zone - between 5 and 9 exceptions; red zone - more than 9 exceptions

Obs. 2: The dates format is DD/MM/YYYY.

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However, this article intends also to assess the performanceof capital requirements calculated by different models.

The Basel Committee suggests a holding period of 10 days,during which losses could hypothetically accumulate. Thesubjacent idea is that in critical periods a financial institutionwould need more time to close its positions than in normalconditions. This way, the comparison is between ten daysreturns, for each portfolio assessed, and the respectivecapital requirement, registering the number of exceptions inthe period. An exception is a loss higher than the capitalrequirement previously set according to the model. In thiscase, considering daily observations of ten business dayreturns49 may generate clusters of exceptions, to the extentthat the effects of an extreme daily return remain for theten subsequent days.

Then, for standardized models, one observes only the numberof times in which the losses of the portfolio exceed the capitalrequirement in the backtest period. Assessment of VaR-based models follows the guidelines of the documentpublished by the Basel Committee specifically for thispurpose50. Each three months one checks the number oftimes in which, over the past 250 business days, 1-day VaRis exceeded by the loss of the day, for each portfolio. TheCommittee establishes three zones for the number ofexceptions51 verified (along 250 observations) in the backtestof daily VaR. Up to four exceptions, the assessed model isin the green zone, that is to say, approved. A number ofexceptions between five and nine puts the model on theyellow zone, which may lead, in the absence of strongreasons, to an increase of the multiplication factor up toM=4. Above nine exceptions, the model lays in the red zoneand the institution may be compelled to using the standardizedapproach. In addition to the Basel methodology, one alsoexamines the number of times in which the ten-dayaggregate losses exceed the capital requirement consideringa total of 572 observations.

Besides, one uses the Kupiec test52, with a 5% level ofsignificance, for the percentage of daily VaR exceptions.One tests the null hypothesis that the proportion of modelexceptions obtained empirically is equal to the predicted(theoretical) level for VaR.

49/ Each day, the daily returns of the last 10 days are aggregated.

50/ Supervisory Framework for the Use of Backtesting in Conjunction with the Internal Models Approach to Market Risk Capital Requirements

51/ In this article, the idea of exception applies both to capital requirements and VaR estimates.

52/ Kupiec (1995).


Basle Test - portfolios 3 and 4

Model Exceptions - 1-Day VaR

Portfolio 3 Portfolio 4


Historical

1 Windows

05/01/2001 to 04/01/2002 8 10 6 6

06/04/2001 to 08/04/2002 7 4 6 5

09/07/2001 to 08/07/2002 4 2 5 3

05/10/2001 to 03/10/2002 4 7 8 5

09/01/2002 to 03/01/2003 4 7 6 6

11/04/2002 to 04/04/2003 4 7 6 6

11/07/2002 to 08/07/2003 4 6 5 5

2 Windows

05/01/2001 to 04/01/2002 6 8 6 4

06/04/2001 to 08/04/2002 5 2 6 3

09/07/2001 to 08/07/2002 3 2 5 2

05/10/2001 to 03/10/2002 3 7 6 5

09/01/2002 to 03/01/2003 3 7 4 6

11/04/2002 to 04/04/2003 3 7 4 6

11/07/2002 to 08/07/2003 3 6 3 5

EWMA

05/01/2001 to 04/01/2002 3 3 1 5

06/04/2001 to 08/04/2002 5 2 1 4

09/07/2001 to 08/07/2002 5 3 2 3

05/10/2001 to 03/10/2002 5 5 2 3

09/01/2002 to 03/01/2003 4 5 2 3

11/04/2002 to 04/04/2003 4 6 2 4

11/07/2002 to 08/07/2003 4 5 2 5

Hybrid

05/01/2001 to 04/01/2002 5 4 4 5

06/04/2001 to 08/04/2002 4 3 3 5

09/07/2001 to 08/07/2002 5 5 5 6

05/10/2001 to 03/10/2002 5 7 7 6

09/01/2002 to 03/01/2003 4 6 6 6

11/04/2002 to 04/04/2003 5 7 7 4

11/07/2002 to 08/07/2003 6 7 7 4




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To make a cost-benefit appraisal of the models, onecontrasts the average capital requirement by portfolio withthe respective numbers of exceptions observed. There is atrade-off between these indicators, since that for a givenmodel, larger capital requirement tends to generate smallernumber of exceptions. In this case, the lower both valuesare, the more efficient the model is. In addition, one considersthe difference between capital requirement and therespective loss (EC – loss) as a way of assessing theconflicting aspects between the protection given by a certainmodel and the efficiency in capital allocation for riskcoverage. For instance, in cases where EC is free fromexceptions, the larger the difference between EC and theloss, the higher the protection, though less efficient is thecapital allocation. Besides, one considers the differencebetween the respective loss and capital requirement (loss –EC) since, in case EC exception have occurred, one mayassess for each model how larger is the loss than the EC.

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Exceptions of one-day VaR, assessed by the Kupiec test,and exceptions of EC regarding ten-day returns are shownfor each type of portfolio for the purpose of comparing themethodologies. In addition, indicators are introduced withthe aim of assessing, whenever there are exceptions, byhow much losses exceed EC (named [Losses – EC]), andby how much losses are short of EC (named [EC – losses]),otherwise.53 Backtesting graphs are shown for portfolios 1and 3, since the conclusions extracted from the interpretationof such results are applicable to the remaining portfolios.

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Among standardized approach models, one may notice thatthe model implemented under Brazilian legislation in itsprevious version is more conservative than the Basel model.The current version lessens the situation by allowing, undercertain conditions, the offsetting of opposing positions indifferent currencies. All these models share the samecharacteristics of lacking response to changes in volatility,which generates graphs featuring straight lines parallel tothe abscissa axis, as the value of the portfolios is constant(Graphs 1 and 2).

53/ Values of [Losses – EC] and [EC – losses] are presented as a proportion of the portfolio’s value.


Basle Test - portfolios 5 and 6

Model Exceções - VaR diário

Period Portfolio 5 Portfolio 6


Historical

1 Windows

05/01/2001 to 04/01/2002 9 7 6 7

06/04/2001 to 08/04/2002 9 6 6 8

09/07/2001 to 08/07/2002 4 2 2 4

05/10/2001 to 03/10/2002 7 4 5 9

09/01/2002 to 03/01/2003 5 5 5 10

11/04/2002 to 04/04/2003 5 5 5 9

11/07/2002 to 08/07/2003 5 5 5 7

2 Windows

05/01/2001 to 04/01/2002 6 6 6 6

06/04/2001 to 08/04/2002 6 5 6 7

09/07/2001 to 08/07/2002 2 1 2 3

05/10/2001 to 03/10/2002 3 4 4 8

09/01/2002 to 03/01/2003 3 5 4 9

11/04/2002 to 04/04/2003 3 5 4 8

11/07/2002 to 08/07/2003 3 5 4 6

EWMA

05/01/2001 to 04/01/2002 1 4 3 5

06/04/2001 to 08/04/2002 1 3 3 6

09/07/2001 to 08/07/2002 1 2 1 6

05/10/2001 to 03/10/2002 1 3 2 8

09/01/2002 to 03/01/2003 1 4 2 8

11/04/2002 to 04/04/2003 2 4 4 7

11/07/2002 to 08/07/2003 3 5 4 4

Hybrid

05/01/2001 to 04/01/2002 5 6 6 6

06/04/2001 to 08/04/2002 5 1 1 6

09/07/2001 to 08/07/2002 6 4 4 4

05/10/2001 to 03/10/2002 6 6 6 3

09/01/2002 to 03/01/2003 6 6 6 2

11/04/2002 to 04/04/2003 4 6 6 2

11/07/2002 to 08/07/2003 4 5 5 2




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This is the only model among those based on standardizedapproach that feature exceptions. The most relevantoccurrences relate to portfolios 1 and 6, as shown in Table3, made up of just one asset (Dollar positions in portfolio 1and Yen positions in portfolio 6). Five exceptions arerecorded on the left tail of both portfolios, while for the righttail 28 exceptions on portfolio 1 and 33 for portfolio 6 areshown. The remaining portfolios feature few exceptions,and always in the right tail of returns distribution.

Table 2 shows that exposure in the Basel model is alwayslower or equal to the exposure calculated by the previousmodel and higher or equal to the exposure calculated by thecurrent version model. The capital requirement, however,is invariably lower for the Basel model, because of the lowerpercentage used.

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The capital requirement determined based on the previousversion does not exhibit exceptions in any of the portfolios(see remark on Table 3). This is partially due to thecalculation of the foreign exchange exposure, whichconsiders the summation, in absolute value, of all netexposure, both long and short, in foreign currency and gold.The value will be identical to the one calculated by the Baselmodel when the net positions are all short or all long. Thismethodology does not encourage foreign exchange hedgingby opposing positions for strongly correlated currencies.

However, the main cause of differences between capitalrequirements calculated by this model, as opposed to theBasel model, is the percentage to be applied. The F” factorcurrently in use is 50%, contrasted to the 8% of the Baselmodel. Table 2 shows that, despite the calculated exposureis the same for models in portfolios 1, 5 and 6, capitalrequirement calculated by the previous version reached 50%of the portfolio value and was 6.25 times larger than thecapital requirement calculated according to the parametersof the Basel model. In portfolios 2 and 3, on the other hand,the aggregate effect of a broader calculation basis and ahigher percentage applied results in capital requirementsover tenfold the ones suggested by the Basel model. Graphs1 and 2 show the disparity between capital requirementsdetermined by the different models. If, on the one hand,there were no exceptions under the current legislation, onthe other, the capital required by the foreign exchange marketrisk was quite high.


KupiecTest


Tail Tail Tail


Historical

1 window 13 15 10 8 12 17

2 windows 9 13 9 8 9 15

EWMA 9 14 9 12 10 10

Hybrid 8 11 13 13 13 14


Tail Tail Tail


Historical

1 window 12 12 14 12 14 12

2 windows 10 10 9 11 10 15

EWMA 4 11 4 11 7 14

Hybrid 9 12 9 9 7 10

1/ Obs: The test non-rejection interval ranges between two and ten exceptions.

The rejection events are in red.

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As with the previous version, the current version of theBrazilian rule does not show exceptions. Regarding theprevious version, we may say that this one tends towardsthe Basel model. Indeed, the Basel model calculation baseis identical to the summation of T

1, T

2 components when

there are not weak currencies or gold. In this case, thecalculation base of the current version will be lower thanthe corresponding base of the Basel model to the extentthat T

2 is multiplied by an hedge factor lower than 1. That is

to say, all positions not reached by T1 will be part of T

2 with

just 70% of their value. Therefore, the selected currencieswith opposite net positions will feature a reduction of 30%in their offsetting positions in calculating exposure. This way,what keeps the requirement well above the Basel model isthe F” factor that, as in the previous version, was definedat 50%, contrasting with the 8% of the Basel model. Graphs1 and 2 make clear that the difference of capitalrequirements between this and the Basel model is stillsubstantial. However, one may notice in portfolios 2, 3, 4and 5 the effect of encouragement to adopt opposite positionsin the selected currencies (Table 2). In this case, the capitalrequirement resulted practically half of what wouldotherwise be required in the previous version.

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The VaR based models present average EC lower than bothversions of Resolution no. 2,606 models and higher than theBasel model. However, with a multiplication factor of 3,there are few situations in which they require higher capitalallocation than the current version. The smaller differencebetween the average capital requirements of these models(VaR-based versus current version) happens in portfolioscomprising one single asset (Tables 6, 7 and 8).

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As observed in Section 2, there are two versions of thehistorical model: with a single and with a double window.According to the Kupiec test, the results of VaR are betterfor the double-window historical model (Table 5). The single-window historical model is not rejected just for portfolio 2in both tails.

Regarding the seven sub-periods analyzed (Basel test), theresults are not satisfactory (Table 4). The single-windowhistorical model is the only VaR-based model that displaysoccurrences (three occurrences in portfolios 1, 3 and 6) in

Table 6 - Mean Capital Requirement

Standardized approach models

Models Portfolios

1 2 3 4 5 6

Basle Long 8.0 24.0 21.2 20.0 16.0 8.0

Short 8.0 24.0 21.2 20.0 16.0 8.0

Version

Previous Long 50.0 250.0 215.0 150.0 100.0 50.0

Short 50.0 250.0 215.0 150.0 100.0 50.0

Modified Long 50.0 120.0 107.7 85.0 67.5 50.0

Short 50.0 120.0 107.7 85.0 67.5 50.0

Table 7 - Mean capital requirement

VaR-based models - multiplicator factor M = 2

Models Portfolios

1 2 3 4 5 6

Historical

1 Window

Long 18.57 25.50 21.57 28.33 26.12 26.12

Short 18.58 28.74 21.79 32.27 28.40 28.40

2 Windows

Long 20.69 27.21 22.95 30.25 28.20 23.41

Short 19.37 29.88 23.47 33.99 30.65 22.80

EWMA

Long 17.30 25.06 21.30 27.78 25.00 19.91

Short 17.92 25.93 22.00 28.88 25.97 20.68

Hybrid

Long 20.55 28.28 24.57 29.83 28.49 22.82

Short 17.76 27.42 21.35 32.18 28.68 21.41

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the red zone. The double-windowed historical model displaysproportionally equal occurrences in the green and yellowzones.

Regarding EC calculated as M = 2, the double-windowhistorical model shows the lowest number of exceptionsamong VaR-based models (five, all in the right tail). Thesingle-window historical model is the one displaying thelargest number of exceptions (nine, all again in the right tail)among VaR-based models (Table 3). However, when oneassesses [loss – EC], it becomes clear that thesemeasurements are not significant (Table 9).

When using M = 3, there was only one exception in portfolio3 in both versions, for the right tail (see remark on Table 3).However, the value of the loss (loss – EC] is irrelevant (seeremark on Table 9).

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Regarding VaR estimates, as shown in Table 5, theExponential Smoothing Model, as the double-windowedhistorical model, is not rejected for any portfolio for the lefttail by the Kupiec test. However, for the right tail, the modelis not rejected just for portfolio 3 where the upper limit forexceptions is reached (10 exceptions). Portfolios recordingthe highest number of exceptions (14) were the onescomprising one single currency (portfolios 1 and 6).

As far as sub-periods are concerned (Basel test), the modeldisplays, in general, better result than others, with more sub-periods in the green zone. There are not occurrences in thered zone.

Regarding EC, the exponential smoothing model has a loweraverage capital requirement than others that follow theinternal approach, as shown in Tables 7 and 8.

For the multiplication factor 3, the model does not recordexceptions for EC for any portfolio in both tails (see remarkon Table 3). For the multiplication factor 2, there is a totalof 9 EC exceptions, all for the short position, which shows aperformance slightly below that of double-window and hybridhistorical models (Table 3). However, in Table 9 one noticesthat the average and the upper value of [loss – EC] are notrelevant. Tables 10 and 11 show that this model displays,among VaR-based models, the lowest [EC loss] averageand often one of the highest minimum values of the

Table 8 - Mean capital requirement

VaR-based models - multiplicator factor M = 3

Models Portfolios

1 2 3 4 5 6

Historical

1 Window

Long 27.85 38.25 32.35 42.49 39.18 39.18

Short 27.87 43.11 32.69 48.41 42.60 42.60

2 Windows

Long 31.04 40.81 34.43 45.38 42.30 35.11

Short 29.05 44.83 35.20 50.98 45.97 34.20

EWMA

Long 25.81 37.54 31.87 41.63 37.38 29.80

Short 26.71 38.81 32.88 43.25 38.79 30.93

Hybrid

Long 30.46 42.24 36.64 44.39 42.32 33.87

Short 26.64 41.13 32.03 48.27 43.02 32.11

Table 9 - Differences between the losses larger than

CR and the CR (% of portfolio value)

Short position - multiplicator factor M = 2

Models Portfolios

1 2 3 4 5 6

Basle

Mean 4.54 1.95 1.88 5.07 5.15 3.72

Max 14.33 1.95 3.54 9.26 11.97 14.78

Historical

1 Window

Mean 0.96 2.65 5.29 0.00 0.00 0.00

Max 2.85 2.65 8.29 0.00 0.00 0.00

2 Windows

Mean 2.85 2.65 5.29 0.00 0.00 0.00

Max 2.85 2.65 8.29 0.00 0.00 0.00

EWMA

Mean 2.96 2.37 4.38 0.00 0.00 0.00

Max 5.81 5.63 7.38 0.00 0.00 0.00

Hybrid

Mean 2.74 3.41 3.91 0.00 0.00 0.00

Max 4.56 3.41 6.92 0.00 0.00 0.00

1/ There were no EC exceptions for the previous version and modified version

standardized models .

2/ Only one CR exception for the Historical model (in both versions) at the right

tail of Portfolio 3 was registered with respect to the VaR-Based Models with

multiplication factor M = 3. The difference between the loss and the CR is 0,07%.

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measurement, which means better protection withoutexcessive capital allocation to cover the market risk.

One may see by Graphs 3 and 4 that the EC series calculatedby the exponential smoothing model follows a closer pathto the changes in volatility of returns than the two versionsof the historical model.

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Regarding VaR results, the hybrid model shows the secondworst performance according to the Kupiec test (Table 5),better only than the one-windowed historical model.

However, regarding the Basel test (Table 4), the model doesnot display any sub-period with exceptions higher than seven.Nevertheless, the model shows less sub-periods within thegreen zone than the Exponential Smoothing Model andDouble-Window Historical Model.

Regarding the EC backtesting for multiplication factor 2,the model shows a total of six exceptions for EC, all in theright tail (Table 3). The maximum value of [loss – EC] (Table9), in such cases, is not relevant. The EC average (Table9) and [EC – loss] average for the multiplication factor 2(Tables 10 and 11) is close to the single-window historicalmodel and higher than the one of exponential smoothing.For the multiplication factor 3, the backtesting does not showexceptions of EC. Similarly to what happens in theexponential smoothing model, the EC series calculated bythe hybrid model responds more quickly to volatilityfluctuations, compared to both versions of the historicalmodel (Graphs 3 and 4).

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This paper assesses capital requirement determinationmodels to cover the risk market resulting from exposure inforeign exchange. The examined models were groupedaccording to standardized approach models or internalmodels (VaR-based). The first group comprises the Baselmodel and the model adopted by the Brazilian legislation, inthe versions before and after the changes implemented byCircular no. 3,217, of 12/19/2003,further modified by Circularno. 3,229, of 03/25/2004. The second group comprises single-and double-window historical models, exponential smoothing(EWMA) and the hybrid model.

Table 10 - Differences between the CR and the losses

smaller than CR (% of portfolio value)

Long position - multiplicator factor M = 2 and M = 3

%

Models Portfolios

1 2 3 4 5 6

Standardized

Basel

Mean 4.95 19.39 17.25 16.67 12.92 4.72

Min 0.01 8.05 7.49 7.60 5.36 0.05

Version

Previous

Mean 46.83 245.39 211.03 146.67 96.92 46.62

Min 40.07 234.05 201.27 137.60 89.36 41.38

Modified

Mean 46.83 115.39 103.80 81.67 64.42 46.62

Min 40.07 104.05 94.04 72.60 56.86 41.38

VaR-based

M = 2

Historical

1 window

Mean 17.60 21.29 18.29 26.23 24.36 24.36

Min 3.26 12.70 4.13 9.67 8.48 8.48

2 windows

Mean 19.97 23.05 19.76 28.32 26.77 21.95

Min 4.11 12.73 5.28 10.93 9.28 6.54

EWMA

Mean 15.00 19.88 17.11 24.62 22.17 17.33

Min 6.34 5.64 5.91 15.54 11.50 8.89

Hybrid

Mean 19.01 23.61 20.79 27.09 26.63 21.09

Min 5.23 7.99 7.07 12.14 9.25 6.14

M = 3

Historical

1 window

Mean 27.98 34.23 29.42 41.00 38.06 38.06

Min 7.61 24.51 10.10 20.70 16.44 16.44

2 windows

Mean 31.54 36.87 31.61 44.13 41.68 34.62

Min 8.89 24.56 11.82 22.60 17.03 13.23

EWMA

Mean 23.96 32.07 27.58 38.57 34.65 27.63

Min 10.42 16.44 13.59 25.77 19.01 16.67

Hybrid

Mean 29.73 37.52 32.92 42.10 41.09 32.96

Min 10.99 16.16 14.73 21.95 15.63 12.57

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The Brazilian legislation model, in both its versions, displaysa EC total average considerably larger than the VaR-basedmodels with the multiplication factor equal to 3. However,there are moments in which it requires less capital allocation.In its current version, despite partially considering thecorrelation among the main currencies and gold, the modelbehaves exactly as in the previous version in cases wherethe portfolio is comprised by one single currency, by differentcurrencies in the same position or currencies other than theselected ones. However, it means an advance from theprevious version by permitting some offsetting betweenselected currencies in opposite positions.

As expected, VaR-based models display less capitalallocation than standardized models, except for the Baselmodel that presents a large number of exceptions,evidencing its inadequacy to the high volatility of the Brazilianforeign exchange market.

The use of a single-window in the historical methodologyyields the poorest results when considering VaR-basedmodels. The other VaR-based models, two-windowhistorical, exponential smoothing and hybrid models, displaysimilar results. By the Basel test, the exponential smoothingmodels performed best, as well as for the capital requirementwith the multiplication factor 3, since it couples lower averagewith smaller number of exceptions. However, for themultiplication factor 2, this is the model displaying the largernumber of EC exceptions. In this case, the bestperformance belongs to the two-window historical model.It shall be noticed that EC for the hybrid and exponentialsmoothing models follows more closely the behavior ofreturns when compared to other models.

The period studied included moments of high volatility inthe Brazilian foreign exchange market. This way, part ofthe exceptions exhibited by 1-day VaR in all models is dueto these events. The Basel Committee advances thepossibility that internal models that fail to reach the minimumrequired performance are not automatically penalized.Therefore, one shall evaluate the circumstances when thepoor performance takes place in an attempt to detectwhether the model presents intrinsic flaws in its conception.

Table 11 - Differences between the CR and the losses

smaller than CR (% of portfolio value)

Short position - multiplicator factor M = 2 and M = 3

%

Models Portfolios

1 2 3 4 5 6

Standardized

Basel

Mean 5.20 19.69 17.10 15.34 11.92 4.73

Min 0.15 2.43 0.82 0.27 0.67 0.17

Version

Previous

Mean 46.34 245.61 210.62 145.14 95.50 45.83

Min 27.67 224.05 190.24 120.74 72.03 27.22

Modified

Mean 46.34 115.61 103.38 80.14 63.00 45.83

Min 27.67 94.05 83.00 55.74 39.53 27.22

VaR-based

M = 2

Historical

1 window

Mean 13.69 24.06 17.00 26.32 22.36 22.36

Min 0.45 1.76 0.73 2.77 0.62 0.62

2 windows

Mean 14.27 25.12 18.67 27.71 24.22 17.53

Min 0.14 1.76 1.27 5.34 2.87 1.36

EWMA

Mean 13.68 22.49 18.18 23.84 21.21 15.82

Min 0.37 4.51 1.88 6.67 1.44 0.79

Hybrid

Mean 13.02 23.44 17.46 26.17 22.91 16.94

Min 0.35 1.14 2.18 3.79 3.77 3.50

M = 3

Historical

1 window

Mean 22.01 38.17 27.40 41.93 35.81 35.81

Min 3.61 9.00 4.36 18.79 11.92 11.92

2 windows

Mean 23.15 39.75 29.90 44.02 38.60 28.38

Min 4.75 9.00 4.36 19.42 12.69 11.93

EWMA

Mean 21.94 35.44 28.94 38.11 33.91 25.72

Min 0.85 4.53 1.30 17.75 11.51 7.80

Hybrid

Mean 21.19 37.25 28.02 41.70 36.63 27.49

Min 2.72 7.86 2.00 18.14 15.01 10.34

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BASEL COMMITTEE ON BANKING SUPERVISION.Amendment to the Capital Accord to IncorporateMarket Risks. January 1996.

BASEL COMMITTEE ON BANKING SUPERVISION.Overview of the Amendment to the Capital Accord toIncorporate Market Risks. January 1996.

BASEL COMMITTEE ON BANKING SUPERVISION.Supervisory Framework for the Use of “Backtesting”in Conjunction with the Internal Models Approach toMarket Risk Capital Requirements. January 1996.

BOUDOUKH, J.; RICHARDSON, M.; WHITELAW, R.The Best of Both Worlds, Risk 11 (May), 64-67,1998.

JORION, P. Value at Risk: A Nova Fonte de Referênciapara o Controle de Risco de Mercado. Bolsa deMercadorias e Futuros – São Paulo, 1998.

KUPIEC, P. Techniques for Verifying the Accuracy of RiskMeasurement Models. Journal of Derivatives, 2, 73-84 -1995.

LEMGRUBER, E.F. & OHANIAN, G. O Modelo deProjeção de Volatilidade do RiskMetrics e a Hipótese deDistribuição Normal Condicional para alguns Fatores deRisco do Brasil. Gestão de Risco e Derivativos:Aplicações no Brasil, 2001.

RISKMETRICS GROUP. RiskMetrics - TechnicalDocument. New York - J. P. Morgan, 1996.

RISKMETRICS GROUP. RiskMetrics - TechnicalDocument. New York - J. P. Morgan, 1996.

Documents

Financial Stability Report – Chapter 6 – Item 2 · 2004-09-23 · 39/ Resolution no. 2,606, Circular no. 2,898, both of May 27, 1999, and Circular no. 3.229, of 03/25/2004 are