mathematical analysis of a hyperbolic … financial stability report may 2003 1/ studies and...

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Financial Stability Report May 2003

1/ Studies and Research Department, Central Bank of Brazil, [email protected]

Mathematical Analysis of aHyperbolic MultiplicationFactor and Performance of aPolynomial MultiplicationFactor

Solange Maria Guerra1

This paper studies the mathematical properties of a multiplicationfactor defined in Technical Note on Circular no. 2,692 and examineswhether the performance is improved by using a polynomial, insteadof a hyperbolic function to define the multiplication factor. Theresults suggest that while a polynomial multiplication factoreliminates the jumps that are likely to occur whenever there issignificant change in the value of percentile volatility, it fails tooperate as a curve smoothening.

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1 – Introduction

As the financial stability of a country, or even the world as a whole,may be threatened by the frailty of its bank system, there has been anincreased concern in establishing forms of fostering the soundnessof economic systems. Tangible results of these concerns may berecognized in recommendations, guidelines and rules issued by theBasle Committee.

In Brazil, up to June 1994, financial institutions earned large revenuesfrom inefficiencies made possible by high inflation rates, leavingbehind the financial results associated to activities typical of theindustry. With the implementation of the Real Plan, the bank industrywas compelled to undergo a restructuring, reducing spreads, curbingcosts and profiting from economies of scale. In line with thesechanges, the National Financial System underwent a process ofglobalization and internationalization.

Faced by this new environment, the Central Bank implementedmeasures to put its supervising structure in line with the newcontrolling and monitoring needs of the financial market, aiming atimproving the management of risks inherent to the activities of thebank industry and reaching internationally adopted standards.

Through Resolution no. 2,099, of August 17, 1994, Brazilincorporated the recommendations set by the 1998 Basel Accord.According to such recommendations, the supervisory approachchanged its focus from liabilities, largely referred to shareholders’equity, to the setting of limits identified by the level of credit riskresulting from operations performed by the institutions. Morespecifically, the financial institutions shall maintain, in addition tothe minimum paid-in capital and shareholders’ equity requirementsset by that Resolution, a shareholders’ equity adjusted to 8% of risk-weighted assets.

Furthering the improvement of prudential regulation in what relatesto risk-covering asset requirements, in 2000 the Central Bank issuedResolution no. 2,692, where a specific risk is considered, relatedto exposure of operations subject to changes in predetermined

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interest rates. This new rule incorporates dynamic aspects to theprocess of Required Stockholders’ Equity (PLE), contrasting tothe static approach then recommended by the Basle BankSupervision Committee, which may result in a requirement ofexcessive capital in an environment of normality and insufficientcapital in stress conditions.

In calculating the Required Stockholders’ Equity, Resolution no.2,692 considers, in addition to 11% of risk-weighted assets, the creditrisk from swap operations and a portion of PLE to cover the interestrates market risk of a certain remuneration currency/base. Thisportion of PLE is calculated according to the criteria and conditionsestablished in Circular no. 2,972, of March 2000, to wit, the largerof (i) the daily value at risk (VaR) and (ii) the daily VaR average ofthe past sixty days, multiplied by a factor, called multiplication factor,publicized by the Central Bank on a daily basis.

According to Araújo et al. (2002), the purpose of the multiplicationfactor is to smooth capital requirements in post-crises periods to createa protective cushion so that any further crisis may not fall on anunprepared institution. Despite fulfilling its role, depending on thevolatility dynamics of a one-year time window, the multiplication factormay increase abruptly, even in the absence of significant change in theprevailing volatility. In order to avoid a jolty multiplication factor,Araújo et al. (2002) suggest a change in the determination of volatilityof a certain percentile used in the multiplication factor calculation.

Pursuing a solution for this problem, this paper analyses themathematical aspect of the multiplication factor defined in Circularno. 2,972 and examines whether the use of some polynomial functionsin the multiplication factor definition eliminates the possibility ofjolts, while keeping its characteristic of smoothening the curve.

This paper is organized in five sections. In the next section, weintroduce the multiplication factor definition and background. Section3 deals with a mathematical analysis of the multiplication factor,while in Section 4 one examines whether polynomial functions mayimprove the multiplication factor performance. Finally, the lastsection lays down the conclusions of the paper.

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2 – The multiplication factor

The Basle Committee recommends that the precise value of themultiplication factor shall be set by local regulatory authorities at avalue not lower than 3. According to Jorion (1998), the InternationalSwap and Derivatives Association (ISDA) considers this lower limittoo high and has shown that a multiplication factor of 1 would providesufficient capital to cover periods of global unrest such as the onesbetween 1987 and 1992.

The Central Bank selected a variablemultiplication factor, currently ranging between3 and 1. This selection may have been madebecause of volatility discontinuities and peaksbetween 1997 and 1999, a period prior to thechange in the criteria for calculating therequirement of risk-covering assets, as one maysee in Graph 1 below. One pursued to smooththe capital requirement over a certain period oftime whenever there was a sudden change involatility, while keeping a protective cushion inperiods of low volatility to prevent financialinstitutions from being unexpectedly caught ina crisis.

The methodology for calculating the multiplication factor waspublicized by the Central Bank in a Technical Note on Circular no.2,972, of March 23, 2002, securing that the procedures to be usedwould be completely transparent to financial institutions.

The multiplication factor tM was defined as a decreasing functionof volatility, considering the frequency distribution of standardvolatility and the highest (M) and lowest (m)3 values. Thus,

2/ Standard volatility calculated according to the methodology established in the Technical Note onCircular no. 2,692, to wit, the maximum volatility among those estimated for each of the verticesused in calculating capital requirements.

3/ The current values in effect are M=3 and m=1

Graph 1 – Annualized daily standard volatility2

Standard deviation%

0

5

10

15

20

25

3.51997

1.21998

11.61998

9.91999

7.112000

5.152001

3.182002

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≥+

<==

%21

%

....

...........)(

Ptt

Pt

tt ifCCifM

MM σσσ

σσσ

(1)

where P% is the value related to a certain percentile4 associated tothe frequency distribution of volatility in a one-year window.Still according to the Technical Note, constants 1C and 2C weredetermined considering and , where

Picoσ is the larger volatility observed with an increasing window.Therefore, one has:

(2) (3)

The behavior of the standard volatility, publicized on a daily basisby the Central Bank, displays two sub-periods of largely dissimilarcharacteristics in 2001. In the first such sub-period, up to 03/14,standard volatility was low, with a yearly average of 0.0013. Fromthat date onwards, the statistics suffered chocks and had its averageincreased to 0.0056 in this second span of time, closing the yearwith an average of 0.0047.

Due to the smoothening effect of the multiplication factor and thefact that the percentile volatility used in its calculation had beendetermined in a one-year window, the dynamics of volatility in 2001was reflected in capital requirement of 2002.

On March 19, 2002, the window used to obtain the percentilevolatility came to exclude any data previous to 03/15/2001, whichbelonged to a period of low volatility, and came to contain only datafrom a period of greater volatility and increased instability. Thiscaused the percentile volatility to increase to 0.25%, from 0.13%,which would cause a raise of about 43% in the multiplication factorand, consequently, could lead to a strong increase in the capitalrequirement under a relatively stable environment. Given this

4/ The current percentile P% is 0, meaning that the percentile volatility is the minimum standardvolatility in a one-year window.

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possibility, the Central Bank decided to make an adjustment thatwould be spread along the next 40 days, through a transitory ruleestablished in Communiqué no. 9,329.

The fact mentioned above shows that, though themultiplication factor fulfils the role of smoothenthe curve, it may change in an abrupt way,depending on the dynamics of the volatility in a252 business day window. This behavior may beobserved in Graph 2, built based on data publishedby the Central Bank and the multiplication factorcalculated according to the Technical Note onCircular no. 2,692, in the period from 04/19/2001to 04/19/2002.

3 – Mathematical analysisof the multiplicationfactor

As the purpose of this paper is assessing whether the problemreported in the section above may be eliminated when defining themultiplication factor through a polynomial function with the samemathematical properties of the hyperbolic function that is currentlyused in the multiplication factor definition, we shall conduct amathematical analysis of (1).

Mt was built in such a way to be continuous, as shown by the sidelimits (4) and (6) when tends to and to .

(4)

(5)

and

Graph 2 – Multiplication factor calculated by the formula given in the Technical Note about the Circular 2,692 and by the transitory formula and standard volatilityMultiplication factor

0.0

0.7

1.4

2.1

2.8

3.5

10.12001

10.232001

11.142001

12.62001

12.282001

1.212002

2.132002

3.62002

3.272002

4.182002

Volatility

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Mult. factor Circular 2,692 Mult. factor transit. formula

Volatility

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(6)

In addition, one has that:

>−

<=

%21

%

......

............0)('

Ptt

Pt

ifCif

tM σσσ

σσ

(7)

As 01 >C , Then M(t) is decreasing, as mentioned in the

Technical Note.

>

<=

%31

%

......2............0

)(''Pt

t

Pt

ifCif

tM σσσ

σσ

(8)

Since ,we have that the graph M(t) has its concavity turnedupwards.

Besides, M(t) reaches the minimum and the maximum global m and ,M respectively, since, by construction it is limited by such values.

Using the values in effect for M and m, 3 and 1, respectively, percentilevolatility and maximum volatility ,

we build the graph for M(t) (Graph 3).

As the volatility values are close to zero, we willanalyze the behavior of M(t) when tends tozero.

Although , it is worthwhileto examine the expression below:

(9)

Graph 3 – Hiperbolic multiplication factor of the Circular 2,692Multiplication factor

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.000 0.006 0.012 0.018 0.024 0.030 0.036 0.042 0.048 0.054 0.060

Standard volatility

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Though a situation in which the limit tends to infinity does not happen,the expression indicates that the multiplication factor is very sensitivewhen the volatility is significantly low.

In practice, determination of the multiplication factor is achieved bya family of functions, since expression (1) uses parameters and , which is changeable along time. The lower , the

higher the multiplication factor sensitivity. Then, a change in value may cause a “large” variation in the multiplication factor, a“leap”, as the one that happened on March 19, 2002.

In order to illustrate the fact, we display besidegraphs of multiplication factors 1 and 2 (Graph 4),where and ,respectively.

4 – Polynomialmultiplication factor

This section analyses whether a polynomialmultiplication factor of degree three is a feasiblealternative to avoid excessive sensitivity to very

low volatility in the hyperbolic multiplication factor defined in theTechnical Note on Circular no. 2,972. The choice of a polynomialfunction was made in an attempt to find a downward slope smootherthan that of the hyperbolic function.

In order to build the polynomial , let

us consider the same characteristics of hyperbolical M, that is tosay, being a continuous, decreasing function, with concavity

turned upwards and, in addition, and .

Thus, we have to determine and such as inequalities and equations(10) to (13) below are maintained:

Graph 4 – Behaviour of the hyperbolic multiplication factor when the percentile volatility changesMultiplication factor

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0000 0.0071 0.0142 0.0213 0.0284 0.0355 0.0426 0.0497 0.0568

Volatility

Multiplication factor 1 Multiplication factor 2

Leap

Perc. volat. 0,001118 Perc. volat. 0,002475

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(10)

(11)

(12)

(13)

As we have two equations and four unknowns, we shall determinean incomplete polynomial.

For D equal to zero, graph of P(x) would pass through the originand, for a decreasing P(x) D would be negative, which is notdesirable. We shall, therefore, make D different from zero.

We make B=0. Then, in order to satisfy (11), A shall be positive.

From equations (12) and (13) we have:

(14)

From conditions A>0 and , we have

that .

As , we define

(15)

Hence

(16)

and

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(17)

Since condition (10) was not used to obtain A, C and D, we shall seewhether it remains true for the values determined in equations from(15) to (17).

in the interval

For M = 3 and m = 1, we have that the above intervalcontains . We may, therefore remain assured that P(x) isdecreasing over the interval .

Taking C=0 instead of B=0, we obtain that the concavity of thepolynomial graph in the interval is turned downwards,failing to keep the characteristics of the function currently definingthe multiplication factor.

Therefore, we define the polynomial multiplication factor by:

≥++<

==%

3%

.....................................

)(Pttt

Pttt ifDCA

ifMPP

σσσσσσ

σ (18)

where A,C and D are determined by (15), (16)and (17).

Graph 5 enables comparing the performance of ahyperbolic multiplication factor defined by (1) witha polynomial multiplication factor defined by (18).

The polynomial multiplication factor Pt eliminatesan excessively sensitive response of themultiplication factor to low volatility, as we may

Graph 5 – Comparison between hyperbolic and polynomial multiplication factorsMultiplication factor

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0000 0.0071 0.0142 0.0213 0.0284 0.0355 0.0426 0.0497 0.0568

Volatility

Hiperbolic multiplication factorPolynomial multiplication factor

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see in Graphs 5 and 6, where in the latter we havegraphs of polynomial multiplication factors 1 and 2that were generated considering and , respectively.

However, notice that Pt in the interval is almost linear, being therefore uninterestingsince it will not fulfill our objective of smoothenthe curve5.

Tests performed with a second degree polynomialand some incomplete polynomials of degree

higher than 3 display similar results. These results are possibly dueto the fact that we are trying to make an approximation of a hyperbolicfunction by a polynomial function over a very small interval.

5 – Conclusion

This paper examines whether a change from a hyperbolic functionto a polynomial function in the definition of a multiplication factorestablished in Technical Note to Circular no. 2,972 is a feasiblesolution to avoid sudden increases in the multiplication factorwhenever there is a significant change in the value of percentilevolatility, while current volatility keeps relatively stable.

The results suggest that a multiplication factor defined by a thirddegree polynomial function, given by (18) avoids the problems of a“jerky” multiplication factor, though failing to smoothen the curve.Indeed, its behavior is very similar to the one of a linear multiplicationfactor. A test performed with data from April 2000 to April 2002shows that a linear multiplication factor ranges from 2.7 to 3 and apolynomial multiplication factor, from 2.5 to 3 over the same period,while the hyperbolic multiplication factor ranges from 1.18 to 3.Similar results were obtained using incomplete polynomials of degreetwo and higher than three in the multiplication factor’s definition. It

Graph 6 – Behaviour of the polynomial multiplication factor when the percentile volatility changesMultiplication factor

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0000 0.0071 0.0142 0.0213 0.0284 0.0355 0.0426 0.0497 0.0568

Volatility

Polynomial multipl. factor 1 Polynomial multipl. factor 2

Perc.Volat 0,001118 Perc.Volat 0,002475

5/ A test performed using data from April 2000 to April 2002 shows that the linear multiplicationfactor ranges from 2.7 to 3.

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is, therefore, uninteresting to use a polynomial multiplication factor,since it will carry a cost to be calculated and the effect will be veryclose to that of a fixed multiplication factor of value three, dependingon the dynamics of volatility.

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Bibliography

ARAÚJO, G.S; MOREIRA, J.M.S.; CLEMENTE, R.S.M., 2002.Multiplicador da Circular 2,972/00 – Uma Proposta de MetodologiaAlternativa para a Determinação da Volatilidade Percentil. (mimeo)

CENTRAL BANK OF BRAZIL. Resolution 2,099, August 17, 1994.

CENTRAL BANK OF BRAZIL. Resolution 2,692, February 24,2000.

CENTRAL BANK OF BRAZIL. Circular 2,972, March 23, 2000.

CENTRAL BANK OF BRAZIL. Technical Note on Circular 2,972,March 23, 2000.

CENTRAL BANK OF BRAZIL. Relatório de EstabilidadeFinanceira, November 2002 – Volume 1.

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

JORION, P. Value at Risk: a Nova Fonte de Referência para oControle de Risco de Mercado. Bolsa de Mercadorias e Futuros –São Paulo, 1998.

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