6 selected studies - banco central do brasil · 6 selected studies the objective of this chapter is...

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Financial Stability Report November 2002

6 Selected studies

The objective of this chapter is to publish papers dealing with themes

related to those discussed in this Report. Three papers are presented:

a) Evaluating Methods of Calculating Capital Requirements forStock Portfolio Market Risk in BrazilThis paper examines four methods of determining capital

requirements for coverage of financial institution market risk

generated by exposure in stocks and their derivatives, except

options. Two theoretical portfolios were put together for

simulation purposes and are composed of Ibovespa assets. The

methods evaluated follow the orientation of the Basel Committee.

The first is termed the standardized approach and the others are

proprietary models that utilize the Value-at-Risk (VaR) concept.

Verification of the methods follows the methodology indicted by

Basel. At the same time, in the case of the methods based on

VaR, the Kupiec test for the proportion of failings is applied.

b) Risk-Neutral Probability DensitiesThis article discusses the extraction of market expectations

regarding the prices of financial assets in domestic options

markets. It argues that the implicit volatility extracted from

options has greater informational content than its econometric

counterparts and could be utilized in models for constructing

stress scenarios. Furthermore, it is possible to extract the risk-

neutral densities of these markets and this information can be

utilized with the same objective.

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c) Methodology for Evaluating the Capacity of FinancialInstitutions to Generate ResultsEvaluating the capacity of financial institutions to generate results

is an integral part of most banking supervision models and rating

systems used by supervisory authorities. This paper presents a

methodology for evaluating the capacity of financial institutions

to generate results and has the objective of aiding bank supervisors

in their pursuit of information on the level of results, their

composition and their sensitivity to changes in market conditions

and to formulate conclusion based on this information. The

methodology takes an essentially prospective view, utilizing

accounting and managerial information on past performance as

indicators of what can be expected in the future.

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Evaluating Methods of Calculating CapitalRequirements for Stock Portfolio MarketRisk in Brazil 1

Gustavo S. Araújo2

João Maurício S. Moreira2

Ricardo S. Maia Clemente2

1/ Our gratitude to our colleagues Alan Cosme R. da Silva and Flávio M. L. Traiano for their very

valuable collaboration in the initial stages of this paper. This is a preliminary version of the study

developed by Depep-RJ. Any criticisms and suggestions will be most welcome and can be sent to

[email protected], joã[email protected] or [email protected].

2/ Department of Research, Banco Central do Brasil.

3/ Models developed by the financial institutions themselves.

1 Introduction

The Basel Committee has elaborated a series of proposalsthat seek to provide national bank supervisory authorities with basicguidelines for regulating the capital requirements needed to covermarket risk. For purposes of capital allocation, the proposals suggesttwo calculation alternatives: the standard approach and the approachbased on internal risk management models3

.The standard approach has various technical

shortcomings in the methodology used to calculate capitalrequirements such as, for example, its static nature which makes itinappropriate for capturing alterations in volatility patterns and inthe correlations among risk factors. This is a worrisome limitation,particularly in dealing with markets in which asset prices are subjectto sharp fluctuations. On being calibrated to a specific scenario,capital requirements based on this method can rapidly becomeexcessive should a reduction in volatility levels occur, or insufficientshould the opposite come about.

The starting point for the approach based on internalmodels is the consensus that financial institutions would be capableof elaborating more precise models since they have better knowledgeof the portfolios they manage. Active management of these modelswould therefore lead to greater efficiency in the allocation of capital,

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compared to predefined percentages applied indiscriminately to allthe financial institutions. In order to ensure a minimum degree ofstandardization, transparency and consistency in the results obtainedfor different models, several qualitative and quantitative restrictionswere developed. In the latter case, one should highlight the holdingperiod of ten days, the 99% security level and the minimummultiplication factor of three4.

The purpose of this paper is to evaluate four methods ofcalculating capital requirements for covering market risk in stocksin Brazil following the guidelines defined by the Basel Committee.The first method follows the standard approach, while the othersare included among those considered as internal models. The latterare based on the VaR concept and have the advantage of adapting tovolatility fluctuations.

2. Methodology

2.1 Data

This study uses the Ibovespa5 series and the closing pricesof ten of the stocks included in that index (see Table 1) in the periodfrom 7.4.1994 to 7.31.2002, as well as the series of Telebrás closingprices in the period from 7.4.1994 to 10.14.19996. Some of thesestocks did not register the same liquidity at the start of the periodunder study and, consequently, prices were not registered for somedays included in the sample. In these cases, it was decided that thevalue of the most recently registered stock price would be used. Theuse of the Telebrás and Telemar series is a special case. The Telemarseries only began as of 9.29.1998. However, when one considersthat this stock has registered the highest percentage level7 of all stocksincluded in the index and that it has registered very high volumes oftrading since that time, it would not seem appropriate to exclude itfrom the analysis. Telebrás occupied an analogous position in the

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

5/ São Paulo Stock Market Index.

6/ The series were obtained in the Economática information system.

7/ 13.58% on 7.30.2002. Petrobras PN occupies second position with 9.16%.

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previous period. Thus, in the constructed portfolios, Telebrás replacesTelemar in the period in which the latter stock was not yet beingtraded8.

Thus, each series of stock prices has 1,996 observations,with the same occurring in the Telebrás/Telemar series. Twoportfolios were constructed with ten assets each, in such a way thatthe value of each portfolio is constant over the period studied andthe proportion w

i of the financial value with which each stock

participates in a given portfolio is constant and equal, in absolutevalue, for all of the stocks. For both portfolios, an effort was madeto achieve an intermediate degree of diversification. It was consideredthat the greater the number of stocks, the lesser the concentration ofstocks in a single sector and the more significant the participation ofsuch stocks in the composition of Ibovespa, the more diversified is aportfolio9. At the same time, the composition of the portfolios variesexclusively in relation to the existence of long positions or long andshort positions. The two portfolios used are shown in Table 1. Inboth portfolios, the value of R$10,000.00 was attributed to eachposition. With this, portfolio I comes to R$100,000.00, since allpositions are long. Since portfolio II has six long positions and fourshort positions, its value comes to R$20,000.00.

It should be highlighted that, with the exception ofoptions, positions in stock derivatives can be treated, whenconsidering the stock price risk factor, as spot market positions.The interest rate risk must be dealt with separately in the frameworkof the interest rate risk factor.

8/ As a matter of fact, for methodological reasons that will be explained further on, Telebrás will be

included in the portfolios up to 9.29.1999, though Telemar was already being traded for a full

year. It should be noted that Telebrás was traded with relative liquidity up to mid-2000.

9/ According to Ceretta and Costa Jr. (2000), the major share of the benefits originating in the

diversification would be obtained with just 12 stocks. As of 18 stocks, the added benefits would be

practically negligible.

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2.2 Calculation of series of returns

The return of a f inancial asset is given by

, , in which pt is the closing price of the asset on date

t. For the parametric methods of calculating VaR, we use the

logarithmic returns, , in which 1n denotes the natural

logarithm. This approximation provides a better adjustment to thenormal distribution10.

2.3 Methods of determining capital requirements

Four methods were evaluated for determining the capitalrequirements needed to cope with the market risk incurred by stockpositions. The first method follows the standard approach and ischaracterized by a capital requirements level that is constant overtime. The others are based on the internal model approach asdetermined by the Basel Accord. In this case, capital requirementsare calculated daily by:

in which M = 3 is the multiplication factor of the mean VaR in thelast 60 days. In this paper, it was also implementted M = 2. VaR is

Portfolio Composition

I(+)ARCZ6, (+)BBDC4, (+)CMIG4, (+)CSNA3, (+)ELET6, (+)INEP4, (+)ITAU4, (+)PETR4, (+)RCTB41/TNLP4, (+)VALE5

II (-)ARCZ6, (+)BBDC4, (+)CMIG4, (-)CSNA3, (+)ELET6, (-)INEP4, (+)ITAU4, (-)PETR4, (+)RCTB41/TNLP4, (+)VALE5

Table 1: Composition of Portfolios Used on the Evaluation of Methods of Calculating Capital Requirement for Stocks Market Risk.

The operators (+) or (-) indicate that the position of the asset in the portfolio is long or short respectively. In both theportfolios the considered period ranges from August 10, 1995 to July 31, 2002. Telebrás stock will constitute bothportfolios up to September 20, 1999, and them will be replaced by Telemar. The assets above, identified by their codes inBOVESPA exchange, refer to the following enterprises: Aracruz, Bradesco, CEMIG, CSN, Eletrobrás, Inepar, Itaú,Petrobrás, Telebrás/Telemar e Vale do Rio Doce.

11

−

=−t

tt p

pR

=−1

lnt

tt p

pr

= ∑

=+− t

kktt VaRVaR

MmáxEC ,

60

60

11

10/ The VaR parametric methods analyzed here utilize the normal standardized distribution.

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calculated for a holding period of ten days. The required confidencelevel is 99% or, in other words, the pre-specified level for VaR is1%.

2.3.1 Standard approach method

Two types of risk are considered in this method: specificand general. The specific risk is associated to each individual stock,independently of what happens with the market. It is the so-calleddiversifiable or non-systematic risk. The general risk is associatedto the market context, when all stocks are to a greater or lesserextent subject to the influence of macroeconomic factors. This iswhat is called the nondiversifiable or systematic risk.

The base for calculating capital requirements for specificrisk is the sum of the absolute value of the long and short positions.In the case of general risk, the calculation base is given by the absolutevalue of the sum of the long and short positions. Percentages of 8%,12% and 15% were applied alternately to these bases. The capitalrequirement for coverage of market risk for stocks is given by thetotal sum of these two amounts.

2.3.2 Diagonal method

This method utilizes the parametric VaR, based on normalstandardized distribution and calculated for the period of one dayby11:

ttcd

t zVVaR σα ××= %,1

in which Vc,t

is the financial value of the portfolio on day t,%αz is the

quantile of the normal standardized distribution in relation to thepercentile α and tσ is the standard deviation of the portfolio on day

t. The Committee determines a percentile equal to 1%, so that%αz

11/ One should note that the use of this formula leads to a question. The value of the VaR was calculated

on the basis of the series of logarithmic returns, since these come closer to the normal distribution

than effective returns. However, this value will be used as an estimate of an empirical distribution

quantile of effective returns. The alternative would be to work only with the series of effective

returns. We opted to maintain the calculations based on the logarithmic returns.

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is approximately -2.33. The VaR for the ten-day interval on day t is

obtained by:

Calculation of tσ can become complex for a large number

of assets, since it involves the use of a covariance (or correlation)matrix of order n (number of assets in the portfolio). As a matter offact, for n assets it will be necessary to calculate n(n-1)/2 covariancesand n variances or, in other words, n(n+1)/2 parameters. An attemptto make this calculation more practical resides in the utilization ofthe stocks betas (ß

i) for calculating portfolio volatility. This method

is also known as the diagonal model in which the portfolio varianceis given by12:

∑∑∑ == =

+= N

i i

n

i

n

jtjtjtititmt i

www1

22

1 1,,,,

2,

2εσββσσ

in which 2,tmσ is the proportion of the stock value in the portfolio, is

the variance of Ibovespa calculated for day t13 and 2

tεσ is the variance

of the residuals related to the regressions of the returns of stock iagainst the returns of Ibovespa for obtaining the respective betas.With this method, the volume of calculation is sharply reduced, inrelation to the traditional covariance matrix, on presuming that thejoint movement of the stocks is caused by a single common factor,the market14. The ti,β were calculated daily based on a moving

window of 252 business days15. Thus, considering the lack ofcontinuity between the Telebrás and Telemar series, Telebrás shareswill be used in the portfolios up to 9.29.1999, though Telemar wasalready being traded for a full year.

We also implemented a simplified form that would havebeen suggested by the Basel Committee to reflect the market risk of

10110

10 ×= −d

td

t VaRVaR

12/ See Jorion (1998), pages 154-6.

13/Estimated daily by the sample variance , in which µr is the

average of the return in a sample of 252 business days.

14/The number of parameters estimated drops to 2n+1.

15/ One should recall that ß1 corresponds to the slope of the regression line of the series of returns of

stock i against the series of Ibovespa returns.

( )2

1

2

1

1 ∑=

−−

=n

trtt r

ns µ

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highly diversified portfolios16. In this approximation, the term

is ignored since it tends to become irrelevant as the

quantity of securities in the portfolio increases (by generatingsmall w

i). Though this tends to occur in the case of portfolios

with exclusively long positions, it is possible that it will not happenin portfolios with long and short positions, in which the overallvalue (net) of the portfolio is small in relation to the absolutevalues of the amounts of some positions, whether long or short,generating signif icant weights for the respective stocks.

2.3.3 Method based on exponential smoothing

This method follows the RiskMetrics17 methodology, inwhich the VaR of each component asset of the portfolio is obtainedon the basis of its specific volatility which, in turn, is calculatedthrough exponential smoothing (EWMA18). In this way, the VaR forone day of each asset would be obtained by19:

( )titidti hzVVaR ,%,

1, exp ××= α ,

in which hi,t is the daily conditional volatility of the returns of asset i

estimated for date t, calculated by exponential smoothing, on thebasis of the expression:

21,

21,, )1( −− −+= tititi rhh λλ ,

in λ which is the exponential decline factor for which two values

∑ =

N

i i iw

1

22εσ

16/ According to Jorion (1998), pg. 155.

17/ Methodology developed by the J.P. Morgan Bank of the United States.

18/ Exponentially Weighted Moving Average.

19/Here, one does not encounter the same problem found in the diagonal method. The use of the

exponential function in the formula makes it possible to make a direct comparison between the

VaR estimate and the effective returns, by transforming the estimate of the quantile of the series of

logarithmic returns in the corresponding estimate of the quantile of the series of effective returns.

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were used. The first, 0.94, corresponds to the value adopted byRiskMetrics and has been widely utilized in practice. The second, 0.90,was estimated by maximum likelihood based on Ibovespa returns20 inorder to capture the Brazilian stock market volatility with greaterprecision. Therefore, the one-day VaR for each stock is:

∑∑= =

××=n

itij

n

j

dtj

dti

dt VaRVaRVaR

1,

1

1,

1,

1 ρ,

in which the correlation between assets i and j on date t, tji ),,(ρ , is

obtained by tjti

tjitji hh

h

,,

),,(),,( =ρ , in such tjih ),,( a way that it denotes the

conditional covariance between assets i and j on date t, obtainedthrough the formula:

1,1,1),,(),,( )1( −−− −+= tjtitjitji rrhh λλ

Extending the holding period to ten days, the VaR forasset i will be given by:

( )10exp 10,%10,10, ×××= −− titi

dti hzVVaR α

The portfolio’s VaR is calculated by the expression:

∑∑= =

××=n

itij

n

j

dtj

dti

dt VaRVaRVaR

1,

1

10,

10,

10 ρ

20/ The returns encountered between 12.1.1994 and 7.31.2002 were utilized.

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2.3.4 Method based on historic VaR

This method consists simply in using a given quantile (inthis case, referring to the percentile 1%) of the empirical distributionof the portfolio’s returns, in a moving window of 252 business days,as the VaR estimate that will be used in calculating capitalrequirements. The portfolio’s return is given by ∑ ×=

iiic rwR , in

which Rc refers to one day or ten days, depending on whether the r

i

are daily or accumulated over ten days. For each date t, the weights(w

i) used for the last 252 returns are those found in t, so that the

series of returns will reflect the composition of the portfolio on thatdate21. It should be noted that the VaR used in calculating capitalrequirements for this method is given by a quantile of the empiricaldistribution of accumulated returns over ten days and not by themultiplication of the VaR of one day by the square root of ten.

2.4 Verification of methods

The argument put forward by the Basel Committee isthat, in times of crisis, the liquidity of many assets may diminishsharply, making it difficult to change positions over a short periodof time. In this light, it suggests a period of ten days over whichlosses could be hypothetically accumulated. Thus, the comparisonis made between returns accumulated over ten days by the portfolioand the respective capital requirements. The verif icationencompasses the period from 10.19.1995 to 7.31.2002, totaling1,675 observations.

In the case of the standardized method, the number oftimes in which the losses of a given portfolio surpass the capitalrequirement is observed. Verification of the methods based on VaRfollows the guidelines stated in the document published for thispurpose by the Basel Committee22. The period considered heregenerated 23 subperiods of one year. Basel determines an upperlimit of four failures in 250 observations23. A number of failures

21/ In this paper, since portfolio composition does not change over time, this aspect is found

automatically.

22/ Supervisory Framework of the Use of “Backtesting” in Conjunction with the Internal Models

Approach to Market Risk Capital Requirements (1996).

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between 5 and 10 can lead to an increase in the multiplication factorto as much as four, unless there is a very good reason for thedeviation. Above 10 failures, the institution evaluated may be calledupon to adopt the standard approach. Finally, one observes thenumber of times in which the loss accumulated over ten daysexceeds the capital requirement.

It should be noted that the procedure for evaluating themodels (backtesting) recommended by the Committee generatesinertial behavior of the failure pattern. Since the samples have 250observations and are verified every three months, the intersectionbetween adjacent periods are approximately three quarters of thesample. At the same time, in the case of the capital requirement, thefact that one considers returns accumulated over a holding period often business days can generate inertia in the number of failuresoccurring in a given period, since the effects of an extreme dailyreturn are perpetuated for the ten subsequent business days24.

The methods analyzed were also evaluated by theKupiec test25 for the percentage of failures, applied to the samesamples of 250 days and to the complete period. In this paper, theKupiec test for the proportion of failures was applied to the seriesof VaR estimates, with a significance level of 5%. Intervals for thenonrejection of the null hypothesis (H0) were constructed for theempirical proportion of times in which the VaR was surpassed (seeTable 2). When the verified proportion was kept within theseintervals, H0 was not rejected. A p-value was calculated for theproportions found. Given the significance level of the test, thegreater the p-value in relation to the 5%, the greater the margin ofsafety for nonrejection of H0.

23/ Failure is understood as the occurrence of a loss that exceeds the estimate of VaR for the day in

question. In this paper, this concept will also be used for an accumulated ten day loss which surpasses

the respective capital requirements.

24/ For each day, a return of ten days is calculated accumulating the last ten daily returns.

25/ Kupiec (1995).

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3. Results

3.1 Standard method

The standard method was shown to be inefficient for acapital requirement calculated as 8% of the amount of portfolio Ifor each risk (total of 16%), since it accumulated 35 capitalrequirement failures in the entire period analyzed (Table 4). Withrespect to the inertial behavior of the number of failures, it shouldbe noted that the vast majority of occurrences were registered in 9contiguous subperiods of 250 days, encompassing the internationalcrises that broke out in Asia and Russia. As a matter of fact, 11failures were registered in the short period of just over two weeks(10.27.1997 to 11.12.1997), in which the São Paulo Stock Exchangeregistered sharp declines26. In much the same way, 17 failuresoccurred in less than one month (8.12.1998 to 9.10.1998)27. Giventhe weak result, capital requirements were calculated with thealternative percentages of 12% and 15% of the portfolio value28.Three and zero failures were registered, respectively29.

Since all the correlations of the assets considered in thispaper are positive, portfolios with long and short positions tend tohave a reduced risk level. The standard model only partially reflectsthis risk reduction, since the specific risk considers the sum of theposition absolute values as the basis of calculation. This, to a great

Sample Failures Proportion

Period with 250 observations 0 a 6 0% a 2,46%

Period with 1,675 observations 10 a 25 0,56% a 1,51%

Table 2: Intervals of Nonrejection of the Null Hypothesis for the Empirical Proportion and Number of Failures Based on Kupiec’s Test with 5% Significance Level

Note.: The null hypothesis corresponds to the acceptance that the real failures proportion of the model equals the pre-specified level for the Var (1%). If the empirical proportion is within the specified interval , H0 can not be rejected at the5% significance level.

26/ For example, on 10.27.1997, the Ibovespa dropped by 14.98%, while Bradesco, Eletrobrás and

Petrobras shares dropped by respective rates of 13.15%, 17.36% and 18.68%.

27/ For example, on 8.12.1998, Eletrobrás and CSN shares dropped by 10.20% and 14.35%,

respectively. On 8.27.1998, Cemig, Eletrobrás and Petrobras shares fell by respective rates of

10.93%, 13.64% and 13.10%.

28/ For each risk or, in other words, 24% and 30% in the total, respectively.

29/ 15% was the first rounded percentage for which there were no capital requirements failures.

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extent, explains why there are no failures with 8% in the case ofportfolio II (Table 5).

3.2 Diagonal method

3.2.1 Simplified diagonal method

The daily VaR calculated on the basis of the simplifieddiagonal method for portfolio I has more than four failures in ninesubperiods. The VaR failures are concentrated in the period from7.14.1997 to 10.30.1997 (11 failures) and the Kupiec test rejectsthe null hypothesis for eight adjacent subperiods. The worst resultsencompass those dates. For the whole sample, the Kupiec test alsorejects the null hypothesis with 31 failures, i.e. six more than themaximum defined ceiling (Table 3). The capital requirement failureswere two and zero for M equal to 2 and 3, respectively (Table 4).

For portfolio II, the simplified diagonal model indicatesa very poor performance, with more than four VaR failures in 21subperiods. The same thing occurs in relation to the Kupiec test.For the complete period, 108 failures were registered (6.45%),sharply exceeding the maximum limit of the Kupiec test (Table 3).This result indicates that the simplified method has problems whenapplied to portfolios with long and short positions, since significantweights can occur for some stocks.

Failures % p-value Failures % p-value

Simplified31 1.85 0.00176 108 6.45 0.00000

Complete27 1.61 0.02078 25 1.49 0.05937

λλλλ = 0,94 36 2.15 0.00004 29 1.73 0.00643

λλλλ = 0,90 41 2.45 0.00000 35 2.09 0.00009

23 1.37 0.14634 24 1.43 0.09455

Table 3: Summary of the Kupiec’s Test Results for Diagonal, Exponential Smoothing Based and Historical Models – Complete Sample

Portfolio I Portfolio II

Methods

Diagonal

Exponential Smoothing

Historical

Note. The "p-value" column refers to Kupiec’s test with 5% significance level , applied only to the 1% VaR. Therefore, p-values equal or higher than 0.05 indicate that the VaR method has being approved, i.e. the null hypothesis considering thereal failures proportion equal to the pre-specified level for the VaR (1%) can not be rejected at the 5% significance level.The failures proportion is calculated for the complete sample with 1,675 days.

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Capital requirement has seven failures for M=2 and noneat all for M=3, indicating that the deficiencies of a especific VaRmodel can be hidden by a high multiplication factor, when only thecapital requirement failures are considered (Table 5).

Failures Mean Min. Max. Mean Min. Max.

EC = 8% 35 4.14% 0.00% 13.71% 11.71% 0.14% 15.98%

EC = 12% 3 3.03% 1.34% 5.71% 19.03% 0.16% 23.98%

EC = 15% 0 0.00% 0.00% 0.00% 24.94% 0.29% 29.98%

M = 2 5 2.32% 0.15% 5.51% 26.36% 0.47% 52.33%

M = 3 0 0.00% 0.00% 0.00% 41.76% 4.77% 78.53%

M = 2 2 3.47% 3.08% 3.86% 28.16% 0.86% 54.53%

M = 3 0 0.00% 0.00% 0.00% 44.64% 7.24% 81.84%

M = 2 0 0.00% 0.00% 0.00% 20.81% 0.83% 54.68%

M = 3 0 0.00% 0.00% 0.00% 33.74% 11.43% 82.11%

M = 2 0 0.00% 0.00% 0.00% 20.26% 1.04% 55.96%

M = 3 0 0.00% 0.00% 0.00% 32.89% 11.27% 84.03%

M = 2 8 3.25% 0.07% 9.10% 26.16% 0.23% 48.12%

M = 3 1 0.62% 0.62% 0.62% 41.31% 0.74% 72.19%


Historical

Note.: The difference between CR and the respective loss is calculated as a percentage of the Portfolio I amount of R$100.000,00.

Table 4: Evaluation of Capital Requirement (CR) Failures for Standard, Diagonal, Exponential Smoothing Based and Historical Methods for Portfolio I.

Differences between CR and losses larger than CR (%)

Differences between CR and losses smaller than CR (%)

Methods

Standard

Diagonal Simplified

λλλλ = 0,94

λλλλ = 0,90

Complete

Failures Mean Min. Max. Mean Min. Max.

EC = 8% 0 0.00% 0.00% 0.00% 37.64% 2.44% 47.97%

EC = 12% 0 0.00% 0.00% 0.00% 61.64% 26.44% 71.97%

EC = 15% 0 0.00% 0.00% 0.00% 79.64% 44.44% 89.97%

M = 2 8 5.44% 0.71% 9.60% 41.56% 0.37% 89.12%

M = 3 0 0.00% 0.00% 0.00% 66.81% 4.47% 133.82%

M = 2 0 0.00% 0.00% 0.00% 65.15% 11.28% 118.19%

M = 3 0 0.00% 0.00% 0.00% 102.90% 37.28% 177.37%

M = 2 0 0.00% 0.00% 0.00% 51.56% 15.59% 114.37%

M = 3 0 0.00% 0.00% 0.00% 82.52% 39.71% 173.06%

M = 2 0 0.00% 0.00% 0.00% 51.19% 13.48% 115.91%

M = 3 0 0.00% 0.00% 0.00% 81.94% 41.20% 176.68%

M = 2 0 0.00% 0.00% 0.00% 47.07% 5.13% 100.23%

M = 3 0 0.00% 0.00% 0.00% 75.79% 23.68% 150.35%


λλλλ = 0,90

Historical

Table 5: Evaluation of Capital Requirement (CR) Failures for Standard, Diagonal, Exponential Smoothing Based and Historical Methods for Portfolio II

Differences between CR and losses larger than CR (%)

Differences between CR and losses smaller than CR (%)

Note.: The difference between CR and the respective loss is calculated as a percentage of the Portfolio II amount of R$20.000,00.

Methods

Standard

Diagonal Simplified

Complete

λλλλ = 0,94

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3.2.2 Complete diagonal method

The performance of the complete diagonal method wassignificantly better than in the case of the simplified form. In portfolioI, the number of subperiods in which occurred more than four failuresdropped from nine to eight and the total number of failures droppedfrom 31 to 27, i.e. just two more than the maximum determined bythe Kupiec test30. With regard to the subperiods, the Kupiec testrejected H0 eight times. In relation to capital requirements, theperformance was identical to that of the simplified method, withtwo failures for M=2 and zero failures for M=3.

In the case of portfolio II, there was a strong drop in thenumber of failures. In seven subperiods there were more than fourfailures. There were 25 for the whole sample31, in such a way thatH0 could not be rejected by the Kupiec test. For the subperiods, themethod was not approved on the Kupiec test only four times. Withregard to capital requirements, the performance was clearly betterthan for the simplified method, with zero failures for bothmultiplication factors.

3.3 Method based on exponential smoothing

3.3.1 Smoothing factor λ = 0.94

For portfolio I, there were more than four failures in 14of the 23 subperiods. In many cases, the number of exceptions wasclose to four. This is reflected on the Kupiec test which rejected H0nine times. For the entire sample, the Kupiec test also rejected H0.The failures are distributed over the period studied with no strongconcentrations in short periods of time, as occurred in the previousmethods. There were no capital requirement failures for M=2 andM=3, which could indicate that the method permits a more effectiveadjustment to volatility changes.

30/ Ten VaR failures are concentrated in the period from 7.14.1997 to 10.30.1997.

31/ Ten failures occurred in the period from 7.15.1997 to 10.30.1997.

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The method produced better results for portfolio IIthan for portfolio I. Though it failed more than four times in 12of the 23 subperiods, the numbers were always close to that limit.This is reflected on the Kupiec test, which rejected H0 in onlythree subperiods, all with seven failures. For the whole sample,H0 was rejected on surpassing the upper bound by four failures.There were no failures in capital requirements for both values ofmultiplication factor.

3.3.2 Smoothing factor λ = 0.90

Despite being calculated for Ibovespa, the smoothingfactor λ = 0.90 caused deterioration in the results in comparison to

the previous case. For portfolio I, there were more than four failuresin 16 of the 23 subperiods. The Kupiec test rejected H0 in tensubperiods. For the entire sample, the Kupiec test also rejected H0with 41 failures (2.45%). There were no capital requirements failuresfor the two multiplication factors.

In a similar way to what is described in section 3.3.1, themethod produced better results for portfolio II than for portfolio I,with more than four failures in 15 subperiods. The numbers wereclose to this bound and this is reflected on the Kupiec test, whichrejected H0 in only five subperiods. For the whole sample, the methodregistered 35 failures (2.09%), rejecting H0. There were no capitalrequirements failures for the two values of the multiplication factor.

3.4 Historic method

The historic method turned in the best performance forthe VaR estimates, with just four subperiods in which there weremore than four failures for portfolio I. However, the distance fromthe bound of 4 failures was considerably greater (11 failures intwo subperiods) than that observed on the method based onexponential smoothing. For the same four subperiods, the Kupiectest rejected H0, as the failures were concentrated in the periodfrom 7.14.1997 to 10.30.1997 (ten fai lures). The betterperformance of this method is shown in the p-value for the completeperiod (0.1463), indicating that the null hypothesis of the Kupiec

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test cannot be rejected for 23 failures, corresponding to an empiricalproportion of failures of 1.37%. However, there was one capitalrequirement failure for M=3, when the loss that occurred on10.30.1997 was greater than the capital requirement by 0.62% ofthe portfol io amount. For M=2, there were eight capitalrequirements failures.

For portfolio II, the historic method once again turnedin the best performance for VaR, surpassing the bound in sevensubperiods, with H0 rejected in just three. For the whole sample,there were 24 failures (1.43%), being this number within the intervalof nonrejection of the Kupiec test. Seven failures occurred between7.15.1997 and 10.30.1997. There were no capital requirement failuresfor both multiplication factors.

4. Conclusion

The results obtained are shown in the appended figures1 to 6 and can be summarized as follows:

a) The standard method, whose capital requirement is equal to the

percentage suggested by Basel (8% of portfolio value for each

specific and general risk), was shown to be inefficient in dealing

with the volatility of the Brazilian stock market. The lowest

percentage for which the method does not register capital

requirement failures is 15%. Since this method generates a constant

level of capital requirements, potentially there will be moments in

which the risk will be underestimated, while there will be other

times in which the allocation of capital to cover risks will be much

higher than needed. Nonrecognition of the changes in volatility

patterns and correlations among assets is the principal criticism to

this method.

b) The complete diagonal method registers a reasonable performance

for both portfolios. However, the adjustment to volatility

fluctuations occurs only gradually, as evident in the long flat

segments in the VaR lines. On the other hand, the simplified

version was shown to be inadequate for portfolios with long and

short positions in which there is a expressive participation of a

single asset.

127


c) The method based on exponential smoothing had its best result

with a smoothing factor equal to 0.94. The 0.90 factor estimated

on the basis of the series of Ibovespa returns generated a larger

number of failures for both of the portfolios analyzed. It was

recognized that the basic characteristic of the method is its ability

to adjust rapidly to volatility, as demonstrated in the movements

of the VaR line. This important aspect is reflected on the inexistence

of capital requirement failures for both portfolios, even when a

multiplication factor equal to two is implemented (result achieved

only with this method) and even with an insufficient performance

of the one-day VaR. It is also reflected on the lowest average of

the differences between the capital requirements calculated and

the losses effectively registered (capital requirements – losses)

among all of the methods, when we consider the situations in

which there are no capital requirements failures. Even with a low

average of [capital requirements – losses], the protection provided

is more effective in comparison to the other methods in the sense

that the minimum value of this indicator is almost always higher

for this method.

d) The historic method has important contrasts. If, on the one hand,

it was the only method – in the approach based on internal models

– that was not rejected by the Kupiec test for whole sample in both

portfolios, on the other hand, it was the method that registered the

worst results for capital requirements: there were eight failures for

M=2 and one failure for M=3 (it was the only method that failed

for M=3, but only by 0.62% of the value of the portfolio) in the

case of portfolio I. This fact can be explained by the low level of

adaptability of the method to volatility fluctuations, which is

evident in the long plateaus that exist in the VaR lines. This

characteristic is a consequence of the methodology employed,

which uses a specific percentile of the sample of returns as a

estimate for VaR. Thus, the VaR can remain unaltered for relatively

long periods. One important advantage of this method, however,

is its simplicity and implementation ease. Despite being limited,

this explains why it is frequently used in practice.

Many of the failures registered by the one day VaR in allof the methods occurred at moments of severe international crises.Since all of the methods felt the impact of these events, one must be

128


very careful in analyzing them. Even the Basel Committee considersthe possibility of not automatically penalizing the internal modelsthat do not meet the minimum performance level (less than fourfailures of the daily VaR in 250 business days). Quite to the contrary,it is the task of the regulatory authority to evaluate the circumstancesin which the weak performance occurred, seeking to detect whetherthe model has truly demonstrated conceptual problems.

The VaR methods with the worst performances do notregister capital requirements failures for M=3 (except on failing forthe historic method in portfolio I), even in times of crisis. This couldmean that the multiplication factor suggested by the Committee forcalculating capital requirements may be excessive when applied tothe Brazilian stock market.

In fact, the good performance of the method based onexponential smoothing with respect to capital requirements indicatesthat a method that rapidly adapts to volatility changes would probablyallow the use of a multiplication factor smaller than 3. In this way, itwould be possible to maintain an adequate level of protection at thesame time in which one would achieve greater efficiency in capitalallocation.

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References

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

BASEL COMMITTEE ON BANKING SUPERVISION. Overviewof the Amendment to the Capital Accord to Incorporate Market Risks.January 1996.

BASEL COMMITTEE ON BANKING SUPERVISION.Supervisory Framework for the Use of "Backtesting" in Conjunctionwith the Internal Models Approach to Market Risk CapitalRequirements. January 1996.

CERETTA, P. S., COSTA JR., N.C.A. Quantas Ações Tornam umPortfólio Diversificado no Mercado de Capitais Brasileiro? (HowMany Stocks Make a Portfolio Diversified on the Brazilian CapitalMarket?) Mercado de Capitais - Análise Empírica no Brasil - ColeçãoCoppead de Administração, 2000.

DEPEP/RJ. Relatório sobre Alocação de Capital para Cobertura deRiscos de Mercado (Report on Capital Allocation for Market RiskCoverage). Banco Central do Brasil - December 1999.

EDERINGTON, L.H., GUAN, W., Forecasting Volatility. FinanceDivision, Michael F. Price College of Business, University ofOklahoma, Working Paper, April 1999.

JORION, P. Value at Risk: A Nova Fonte de Referência para oControle de Risco de Mercado (Value at Risk: the New Source ofReference for Controling Market Risk). Commodities and ExchangeMarket – São Paulo, 1998.

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

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Figure 1. Estimated Daily VaR and Daily Returns

- Portfolio I1/

-20

-15

-10

-5

0

5

10

15

20

25

30

10.191995

6.51996

1.141997

8.26 4.31998

11.16 7.11999

2.102000

9.19 5.42001

12.12 7.292002

R$ 1,000.00

Daily returns Complete Diagonal VaREWMA VaR Historical VaR

1/ Portfolio I value: R$100 000 00

-100

-80

-60

-40

-20

0

20

40

60

80

10.191995

6.51996

1.141997

8.26 4.31998

11.16 7.11999

2.102000

9.19 5.42001

12.12 7.292002

R$ 1,000.00

10 Days Returns Standard CR (15%) Complete Diagonal CR

EWMA CR Historical CR1/ Portfolio I value: R$100 000 00 CR = Capital Requirements

Figure 2. Estimated CR with M=2 and 10 Days Returns -

Portfolio I1/

-100

-80

-60

-40

-20

0

20

40

60

80

10.191995

6.51996

1.141997

8.26 4.31998

11.16 7.11999

2.102000

9.19 5.42001

12.12 7.292002

R$ 1,000.00

10 Days Returns Complete Diagonal CR EWMA CR

Historical CR Standard CR (15%)1/ Portfolio I value: R$100,000.00. CR = Capital Requirements.


Portfolio I1/

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Figure 4. Estimated Daily VaR and Daily Returns -

Portfolio II1/

-8

-6

-4

-2

0

2

4

6

8

10

10.191995

6.51996

1.141997

8.26 4.31998

11.16 7.11999

2.102000

9.19 5.42001

12.12 7.292002

R$ 1,000.00

Daily Returns Complete Diagonal VaR

EWMA VaR Historical VaR1/ Portfolio II value: R$20,000.00.

-40

-30

-20

-10

0

10

20

10.191995

6.51996

1.141997

8.26 4.31998

11.16 7.11999

2.102000

9.19 5.42001

12.12 7.292002

R$ 1,000.00


Historical CR Standard CR (15%)1/ Portfolio II value: R$20,000.00. CR = Capital Requirements.


Portfolio II1/

-40

-30

-20

-10

0

10

20

30

10.191995

6.51996

1.141997

8.26 4.31998

11.16 7.11999

2.102000

9.19 5.42001

12.12 7.292002

R$ 1,000.00


Historical CR Standard CR (15%)


Portfolio II1/

1/ Portfolio II value: R$20 000 00 CR = Capital Requirements

6 selected studies - banco central do brasil · 6 selected studies the objective of this chapter is...

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