Exchange rate volatility and its impact on risk management with internal models in commercial banks

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1 Banks and Bank Systems, Volume, Issue 4, 007 Devjak Sreko (Slovena), Andraž Grum (Slovena) Exchange rate volatlty and ts mpact on rsk management wth nternal models n commercal banks Abstract Fnancal markets create a busness envronment of a commercal bank. Prce movement of an asset s an mportant attrbute of a fnancal market and s defned wth ts sze. Central banks adjust prce movements wth monetary polcy based on market actvty. The same holds for foregn exchange markets where central bank affects market actvty wth ts exchange rate. Due to the captal decree legslated by Bank of Slovena, Slovenan commercal banks can apply nternal models for captal requrements calculaton for currency rsk and selected market rsks (general poston rsk n lne wth debt and equty nstruments, prce change rsk for commodtes) as an alternatve or n combnaton wth standardzed methodology. If banks use nternal models for captal charge calculatons all features of fnancal markets should be embedded n the nternal model n order to assure proper accuracy of the model. The goal of ths paper s to dentfy reasons for a decay factor applcaton n nternal models n small fnancal markets, and to show a proper back testng procedure n case of applcaton of a decay factor. A proper back testng procedure shall be found usng lnear programmng. Keywords: monetary polcy, central bank, fnancal markets, rsk management, decay factor, nternal models, back testng. JEL Classfcaton: F3, G, G5, G, G3. Introducton In ths paper we shall understand a commercal bank as a fnancal nvestor. When calculatng captal charge for foregn exchange rsk, commercal banks can use standardzed prncple or nternal model. Internal rsk management model s an effectve approach for managng foregn exchange rsk as t assures prudental captal charge based on precse rsk measurement. When commercal banks use nternal models for rsk management and consequentally calculatng captal charge, value at rsk (VaR) as a rsk measure should be appled. Commercal banks commonly use two dfferent prncples for VaR calculaton. The frst prncple s hstorcal smulaton and the second one s delta normal approach. Hstorcal smulaton prncple of VaR calculaton s favourably used n practce because of ts ndependence of dstrbuton. The delta normal prncple s based on as assumpton of multnormal dstrbuton of returns, whch results n underestmatng the rsk n case dstrbutons dffer from normal dstrbuton. If necessary, commercal banks can apply tme weghtng of asset returns n order to obtan better rsk estmaton to whch they are beng exposed to. The goal of ths paper s to dentfy the reasons for applcaton of tme weghts n nternal model and to explan the modfcaton of backtestng approach n case when tme weghts are beng appled. Due to the drectves 000//EC, CAD 93/6/EEC and CAD3 drectves, commercal banks can apply nternal models for captal requrements calculaton for currency rsk and selected market rsks (general Devjak Sreko, Andraž Grum, 007. More about VaR can be found n Joron (00). poston rsk n lne wth debt and equty nstruments, prce change rsk for commodtes) as an alternatve or n combnaton wth standardzed methodology. When applyng for nternal VaR model, a commercal bank should, n lne wth drectves, use tme seres of data, whch s no shorter than one tradng year. Tme seres of data therefore can be longer, but t should never be shorter. Applcaton of tme weghtng and therefore a decay factor effectvely shortens tme seres of data. The condton n drectves refers to an effectve length of tme seres of data. Effectve observaton perod s calculated wth average tme lag of the ndvdual observatons, whch cannot be less than sx months (Basel Commttee on Bankng Supervson, 996).. Theoretcal background Holton (998) proposed a soluton to reweghtng hstorcal scenaros whch are able to adjust any moments (standard devaton, kurtoss, correlaton) or a varety of other parameters. Hull and Whte (998 and 998a) publshed a crude reweghtng methodology whch s able to match only one or two moments. Holton (999) ntroduced the methodology of weghted scenaros that can be used to enhance ether Monte Carlo VaR or hstorcal VaR. The technque solves the problem of phantom drft that arses when hstorcal VaR s based upon data from a perod that experenced a net ncrease or decrease n a rsk factor's value. Rchardson, Boudoukh, Whtelaw (998) presented a hbrd approach whch combnes two most popular methods of VaR estmaton: RskMetrcs and hstorcal smulaton. It estmates the VaR of a portfolo by applyng exponentally declnng weghts to 3

2 Banks and Bank Systems, Volume, Issue 4, 007 past returns and then fndng the approprate percentle of ths tme-weghted emprcal dstrbuton. Emprcal tests show a sgnfcant mprovement n the precson of VaR forecasts usng the hybrd approach relatve to RskMetrcs and Hstorcal Smulaton. It s especally approprate for calculatng the VaR of fat-taled and hghly skewed data wth rapdly changng moments. A good revew of varous papers that proposed reweghtng schemes can be found n Dowd (005). Hsueh, Shyng, Ln (00) compared the accuracy of varous approaches to hstorcal smulaton (alternatve weghtng schemes for hstorcal exchange rate data). Hseh and Ln (003) employed alternatve exponentally weghted movng average estmator whch s based on the maxmum lkelhood estmator of the varance of generalzed error dstrbuton n conjuncton wth hstorcal smulaton when computng Value-at-Rsk of exchange rate. They suggest that ncorporatng generalzed error dstrbuton nto the hstorcal smulaton method s a substantal mprovement n exchange rate. Foregn exchange rsk management process n a commercal bank should nclude all rsk factors to whch a bank s beng exposed to. Each rsk factor s managed by a central bank and s a functon of macroeconomc varables. Stablty n foregn exchange markets s n general one of the basc goals of monetary polcy. Bank of Slovena snce ERM entrance mantans stable nomnal exchange rate wth monetary polcy. Its nterest rate polcy s subordnated to assure stablty of nomnal exchange rate. Monetary polcy goal of exchange rate stablty and potental structural adaptablty of nstrument set of central bank requre adaptablty of commercal banks n tradng on foregn exchange markets. Central bank wth a monetary and exchange rate polcy determnes busness envronment of a commercal bank. Ths should be consdered wthn nternal model for foregn exchange rsk management by a commercal bank and has a specal role n the stress testng programme whch s a part of an nternal rsk management model. Rsk factors defne busness envronment of a commercal bank and should be captured n a rsk measurng process and n a stress testng program of a bank. For captal requrements calculaton purpose commercal banks can use standardzed methodology or nternal models. When usng an nternal model a commercal bank has to take nto consderaton macroeconomc envronment as ts busness surroundng. Moreover, the exchange rate aganst home currency s determned by a central bank. Exchange rate can be more or less varable dependng on goals that central bank has. If a commercal bank manages 3 exchange rate rsk wth an nternal model t has to observe the varablty of an exchange rate. Foregn exchange markets can be dstngushed accordng to ther daly tradng volume. Usng ths crteron foregn exchange markets can be large or small sze foregn exchange markets. On small fnancal markets we can expect larger prce movements and therefore yeld clusterng more frequently. Hgher varablty can be shown wth a leptokurtoss of a yeld probablty dstrbuton functon. Busness envronment of a commercal bank defnes corporate and retal customers wth ther demand and a central bank wth ts monetary and exchange rate polcy. The latest s of a great mportance. Yeld ndependence s one among key assumptons of a temporally ndependently and dentcally dstrbuted or IDD model. For emprcal analyss EUR/SIT, EUR/HRK, EUR/CSD, EUR/CZK and EUR/PL currency pars were selected. Durbn-Watson test shall be used n order to show evdence of frst order autocorrelaton. In ths research Reuters exchange rates from June 8, 004 to December 30, 005 were consdered. Each currency has ts own holdays. These holdays are ncluded n the tme seres and as there are no market data avalable for these dates, the lengths of all tme seres are dfferent. Daly exchange rate values were used to calculate contnuously compounded daly yelds, whch are used n the analyss. When we wrte about exchange rates n ths artcle we wll refer to daly exchange rate yelds... Autocorrelaton test and statonarty test. In EUR/SIT exchange rate tme seres from ERM entrance onward evdence of a negatve frst order autocorrelaton has been detected usng Durbn- Watson test. The value of d statstc s 3,03. Autocorrelaton dagnostc for EUR/HRK exchange rate shows there s no frst order autocorrelaton present as Durbn-Watson d statstc s,989. Frst order autocorrelaton can be detected on EUR/CSD tme seres as the value of d statstc s,49. There was no frst order autocorrelaton detected ether for EUR/CZK ar for EUR/PL tme seres. The value of d statstc for EUR/CZK s,007, and the value of d statstc for EUR/PL s,9633. Both results show there s no evdence of frst order autocorrelaton. Therefore a null hypothess for Durbn-Watson test cannot be rejected. Autocorrelaton dagnostc for EUR/USD exchange rate tme seres shows IDD assumpton holds as Durbn-Watson d statstc for EUR/USD s,005. More about IDD model can be found n Campbell, Lo, MacKnlay (997).

3 Banks and Bank Systems, Volume, Issue 4, 007 The presence of frst order autocorrelaton s one reason why a bank could consder an applcaton of a decay factor n ts nternal rsk management model. Besdes ths feature of exchange rate behavor banks should also consder clusterng of returns. Assume there s no clusterng of hgh and/or low returns. Then there s no need for applcaton of a decay factor. In case of evdence of yeld clusterng, bank wll face tme subperods of hgh exchange rate varance, and tme subperods wth low exchange rate varance. We shall test clusterng of exchange rate yelds wth a concept of tme seres statonarty. If a tme seres s statonary, then ts mean, varance and autocovarance should be tme nvarant. Accordng to the fnancal theory, asset prces follow random walk and are therefore nonstatonary. But the frst dfferences of a random walk tme seres are statonary. The value of an asset should be equal to ts prce on the prevous day plus a random shock. If there s no random shock a tme seres has no unt root problem and s therefore statonary (Gujarat, 99). The research shows selected tme seres of exchange rates exhbt a frst order autocorrelaton. These tme seres are therefore also non-statonary as they do not fulfl a requrement of tme nvarant autocovarance. Stochastc process s statonary f there s no autocovarance n tme and f also ts mean and varance are tme nvarant. In order to see f mean and varance of these exchange rate tme seres are tme nvarant, we shall splt the observed tme horzon n two tme subperods and test the assumpton of dfference between means and the assumpton of equal varances between so defned tme seres. For the research purpose tme horzon shall be splt n two equal parts. The frst part shall nclude the frst half of exchange rate tme seres, and the second one shall nclude the second half of tme seres. Table. Independent samples test of equal means and equal varances EUR/SIT EUR/CSD Equal varances Equal varances not Equal varances Equal varances not Levene s test for equalty of varances F Sg. t df Sg. (-taled) t-test for equalty of means Mean dfference Std. error dfference 95% confdence Interval of the dfference Lower Upper,49,5,63 39,793,000003,0000 -,0000, ,63 37,687,793,000003,0000 -,0000, ,49,035,345 39,80,00035, ,00046, , ,99,80,00035, ,00046, Source: Reuters data and own calculaton wth SPSS. for Wndows. Before testng the assumpton of a dfference between means, analyss of varances should be performed. Testng the equalty of two varances n ths research wll be done wth Levene s test. The null hypothess has a general form of H 0 : and adequate alternatve hypothess has a general form of H :. For EUR/SIT t holds FL,49 F,05, m, m 39 3, ull hypothess cannot be rejected. The varances n two tme subperods are not dfferent and therefore no yeld clusterng exsts. For EUR/SIT statonarty was not rejected but t cannot be confrmed ether. We cannot be sure f EUR/SIT s statonary as ths research captures only two tme subperods and gnores the combnatons of all remanng subperods. There mght be volatlty clusterng present n other tme subperods whch were not tested n ths research. For statonary tme seres of asset prces the use of a decay factor n rsk management process s not grounded. For EUR/CSD Levene s test shows sgnfcant dfferences between varances for two tme subperods. Comparng expected yelds between two tme subperods for EUR/SIT and for EUR/CSD wth equalty of means test shows no sgnfcant dfferences. Therefore the assumpton of equal means H 0 : cannot be rejected but t cannot be confrmed ether. The assumpton holds only for selected tme subperods of data. For total confrmaton of the assumpton of equal means, the set of all samples of rme subperods should be tested. For all other currency pars, EUR/HRK, EUR/CZK, EUR/PL and EUR/USD, Dckey-Fuller test wll be appled n order to test statonarty of exchange rate tme seres. Let Y be an exchange rate tme se- 33

4 Banks and Bank Systems, Volume, Issue 4, 007 res and let u t be an error term. In ths research Dckey-Fuller test wll be appled to the regresson n the followng form (Gujarat, 99): Y 34 t Model t Y t u Table. Regresson model coeffcents Unstandardzed coeffcents B t Std. error Standardzed coeffcents Beta t Sg. (Constant),00,00,0,7 t -3,38E-06,000 -,049 -,37,7 EURUSD (t-) -,004,050 -,7-9,909,000 ote: Dependent varable: deur/usdt. Source: Reuters data and own calculaton wth SPSS. for Wndows. Table 3. Regresson model summary Model R R Square Adjusted R Square Std. Error of the Estmate,70 a,504,50,00553 ote: a. Predctors: (constant), EUR/USD (t-), t. Source: Reuters data and own calculaton wth SPSS. for Wndows. As computed t 9, 909 for the EUR/USD ndcates null hypothess H 0 : 0 can be rejected and alternatve hypothess H : 0 therefore apples. For all remanng currency pars n the fnancal analyss the value of statstcs corresponds to nequalty 9, 53. 9, 53 holds for EUR/PL currency par. All tme seres of exchange rates n research were shown to be statonary. Consequently, homoscedastcty has been shown along wth the statonarty and therefore no volatlty clusterng has been detected based on selected tme horzon. In case of yeld clusterng detected a decay factor should be appled n order to mprove the rsk exposure assessment accuracy. The value of appled decay factor should be n a negatve correlaton wth the detected kurtoss of a yeld probablty dstrbuton functon. The exstence of kurtoss proves yelds of an asset are not statonary. The hgher the kurtoss of a yeld dstrbuton functon s, the lower should be the value of a decay factor. In case there s no excess kurtoss then the kurtoss of a yeld dstrbuton functon s 3 and a correspondng decay factor should be. Then the followng equaton therefore apples: lm 0 f. Table 4. Descrptve statstc for selected exchange rate yelds Curreney pars Std. Kurtoss Statstc Statstc Statstc Std. error EUR/PL 393,00536,454,46 EUR/CZK 394,009,975,45 EUR/SIT 394,000,000,45 EUR/HRK 394,0004,84,45 EUR/CSD 394,0033 4,80,45 EUR/USD 396,00553,30,45 Vald (lstwse) 393 Source: Reuters data and own calculaton wth SPSS. for Wndows. daly yeld EUR/SIT 0,000 0,0005 0,0000-0,0005-0,000 9-DEC DEC OV OCT-005 -OCT SEP SEP AUG AUG JUL JUL JU MAY MAY APR-005 -APR MAR MAR FEB FEB JA DEC DEC OV OV OCT OCT-004 -SEP SEP AUG AUG JUL JU-004 Datum Source: own calculaton. Fg.. EUR/SIT exchange rate movements from ERM entrance onward.. Modellng and back testng. Let us assume a commercal bank s usng an nternal model for managng foregn exchange rate rsk. When exchange rate yeld s more varable and when clusterng of varablty can be observed, a commercal bank has to properly model these specfcs. Clusterng can be at latest shown n back testng results as clusterng of excessons can prove an exstence of yeld varablty clusterng. Proper rsk management wth an nternal model would ndcate the need to use tme weghtenng and mplement the use of a decay factor. Of course there are several levels of foregn exchange market lqudty to observe and several levels of leptokurtoss can be assgned. The hgher the leptokurtoss of a yeld probablty dstrbuton of an exchange rate as an nstrument, the lower decay factor should be assgned.

5 Banks and Bank Systems, Volume, Issue 4, 007 Back testng shows the accuracy of an nternal model wth a number of excessons. It uses VaR as a crtera number along wth actual and hypothetcal portfolo loss. Clusterng of loss excessons accordng to VaR could mply hgh autocorrelaton n rsk. If the nternal model n a commercal bank s used for foregn exchange rsk management, the autocorrelaton refers to an exchange rate autocorrelaton. An optmal back testng result s an even dstrbuton of excessons n dfferent volatlty regmes, whch shows that the VaR model s responsve to a varety of market condtons. Market condtons are a busness envronment of a commercal bank and should therefore also been captured wthn VaR nternal model. The clusterng of loss excessons therefore supports selecton of a decay factor. If VaR s not responsve to the ncreased revenue volatlty, ths ndcates poor parameterzaton and lower decay factor should be selected..3. Model. Here we shall defne what an effcent tme seres s to be used wthn nternal models when a decay factor has been appled. When a decay factor (smaller than ) was appled, tme seres of data should be extended as a condton for nternal model use requres an effcent tme seres of data, whch should be no shorter than daly data. Let us assume that a commercal bank s usng an nternal model for managng foregn exchange rate rsk and no tme weghtenng has been appled. In ths case a commercal bank has to use daly rsk factor values. Let be a weghted average and an effectve number of data n tme seres. Let be a number of all data n observed tme seres, and let n be a successve number of a data n a tme seres, where stands for ts place n the range queue. When there s no tme weghtenng n general holds the followng equaton: n 5,5. Effectve number of data n tme seres should not be shorter than 5,5. Ths corresponds to the requrement that effectve observaton perod cannot be less than sx months (Basel Commttee on Bankng Supervson, 996). The dea of an effectve tme seres s dentcal to the average weghted maturty concept of a securty. If we assume that uncondtonal returns are not IDD, then t can be that the data on returns from the nearer past are better representatve of future rsk than other. As possble soluton to the problem Boudoukh, Rchardson and Whtelaw (998) suggested generalzed hstorcal smulaton method known as BRW model. BRW model assgns dfferent weghts to returns, dependng on tme of ther orgn. Last hstorcal return r t has assgned weght a, the return before that r t- has assgned weght a, where a a and analogcally. represents exponental decay factor wth values on the nterval between 0 and. The largest weghts are assgned to the yelds from nearer past. If a commercal bank apples tme weghtenng wth a weghtenng scheme, then the general equaton holds: a n a a a. Let be a decay factor. Then the followng apples: a n aa a a a a a a a a 3 a a a 5,5. If the last equaton does not hold, a commercal bank has to extend ts tme seres of data. Tme seres of data should be extended that the followng condton would be met: a n a a a a 5,5. The bank would seek for a mnmal so that the last nequalty would hold. Ths s a soluton of a mnmzaton problem for whch a bank wll seek a soluton. The length of tme seres can be longer therefore can be 5,5. When a bank apples a decay factor, the value of a decay factor determnes relatve mportance of a daly tradng result n a tme seres of tradng results on a daly bass. In case there s no tme weghtenng appled, the sum of all weghts s determned wth equalty a. As a commercal bank can use a longer tme seres of data and as tradng days s a mnmum length of a tme seres, equalty 35

6 Banks and Bank Systems, Volume, Issue 4, 007 a can be generalzed to a. The last nequalty s very mportant. The sum of all weghts should be at least, what makes sense. But ths s only one condton n a constrant set when the requred length of tme seres s beng determned. The other constrant refers to an effectve observaton perod. When determnng requred length of a tme seres, the larger value of all constrant wll be selected as all constrants should be met. Therefore the value of corresponds to the soluton of the followng optmzaton problem: Mn subject to a a a n a a a 3 5, 5 a a 0,,, 0. Suppose 0, 999. If a commercal bank consders only data n exchange rate tme seres, we get: a, a, 0, a. Calculaton result shows that the length of exchange rate tme seres data s too short and should therefore be extended. Searchng the soluton only for the condton a would gve a soluton of 88. The calculaton shows that average effectve length of tme seres wth 88 data s 9,6049. Therefore the frst condton s fulflled, but the second one s not fulflled. We shall now search the mnmal length of tme seres so that the second condton n predefned optmzaton problem would be fulflled. The followng therefore apples: a a mn a a 5,5, a a a a mn 5, We have calculated that the soluton of optmzaton problem mn subject to a a n a a a a 3 5, 5 a a 0,,, 0 when 0, 999 s 306. Concluson The goal of ths paper was to search for reasons when and why a commercal bank should apply a decay factor n nternal model for foregn exchange rsk management. We showed the autocorrelaton or statonarty s the reason to mplement a decay factor n an nternal model for foregn exchange rsk management purpose n a commercal bank. Autocorrelaton has been shown for EUR/SIT and EUR/CSD currency pars. EUR/SIT and EUR/CSD currency pars were shown to have a unt root due to exstng frst order autocorrelaton. Presence of a volatlty clusterng or mean varablty n tme cannot be rejected as the research was not based on the set of all samples. All remanng currency pars n the research were shown to be statonary. If a tme seres s non-statonary, the reason for a decay factor mplementaton s supported. When a decay factor s appled, the length of a tme seres should be extended. The length of a tme seres can be calculated wth an optmzaton mathematcal model, whch has frst been explaned and then appled wth a selected decay factor. References. Basel Commttee on Bankng Supervson: Amendment to the captal accord to ncorporate market rsks, Boudoukh J., Rchardson M.P., Whtelaw R. The best of both worlds: A hybrd approach to calculatng value at rsk // Rsk, 998. o. 5. pp

7 Banks and Bank Systems, Volume, Issue 4, Campbell J.Y., Lo A.W., MacKnlay C.A. The econometrcs of fnancal markets. Prnceton: Prnceton Unversty Press, pp. 4. Dowd K. Measurng Market Rsk. ew York: John Wley & Sons, nd edton, Gujarat. D. Basc econometrcs. ew York: McGraw-Hll, pp. 6. Holton G.A. Smulatng value-at-rsk // Rsk, 998. o. 5. pp Holton G.A. Smulatng Value-at-Rsk wth Weghted Scenaros // Fnancal Engneerng ews, January Hseh C.-C., Ln C.-H. Applyng Generalzed Error Dstrbuton to the hstorcal smulaton method for Value-at- Rsk to mprove performance of rsk management of exchange rate // atonal Kaohsung Frst Unversty of Scence and Technology, Hsueh P.-H., Shyng W.-T., Ln C.-H. A Study on Estmatng Value-at-Rsk Model for US Dollars Aganst T Dollars Exchange Rate by Hstorcal Smulaton Approach // atonal Kaohsung Frst Unversty of Scence and Technology, Hull J., Whte A. Value at rsk when daly changes n market varables are not normally dstrbuted // Journal of Dervatves 5, 998. o. 3. pp Hull J., Whte A. Incorporatng volatlty updatng nto the hstorcal smulaton method for value-at-rsk // Journal of Rsk, Fall 998, pp Joron P. Value at rsk: The new benchmark for controllng market rsk. ew York: McGraw-Hll, pp. 37