The Ten Commandments for Managing Value-at-Risk Under the Basel II Accord*

Transcription

1 The Ten Commandmens for Managing Value-a-Risk Under he Basel II Accord* Juan-Ángel Jiménez-Marín Deparmen of Quaniaive Economics Compluense Universiy of Madrid Michael McAleer Deparmen of Quaniaive Economics Compluense Universiy of Madrid Teodosio Pérez-Amaral Deparmen of Quaniaive Economics Compluense Universiy of Madrid March 2009 * For financial suppor, he firs auhor wishes o hank he Secrearía de Esado de Universidades of Spain hrough projec SEJ and Compluense Universiy hrough projec UCM , he second auhor is graeful o he Ausralian Research Council, and he hird auhor wishes o hank he Secrearía de Esado de Universidades of Spain hrough projec ECO /ECON. Corresponding auhor: Juan-Ángel Jiménez-Marín (juanangel@ccee.ucm.es). 1

2 Absrac Under he Basel II Accord, banks and oher Auhorized Deposi-aking Insiuions (ADIs) are required o communicae heir daily marke risk esimaes o he relevan naional moneary auhoriy a he beginning of each rading day, using one of a variey of Value-a-Risk (VaR) models o measure risk. The purpose of his paper is o provide a simple explanaion and a se of prescripions for managing VaR under he Basel II Accord. The commandmens deal wih undersanding he Basel II colours, undersanding he risk model before choosing, varying he choice of risk model, avoiding he green zone and being willing o violae, incurring large violaions, sopping before he red zone, avoiding frequen violaions, avoiding he esimaion of large porfolios, aggregaing porfolios ino a single index, and inerpreing commandmens sensibly as guidelines. Key words and phrases: Financial porfolios, daily capial charges, frequency of violaions, magniude of violaions, opimizing sraegy, risk forecass, value-a-risk, green zone, red zone. JEL Classificaions: G32, G11, G17. 2

3 Jack Sparrow: The only rules ha really maer are hese: wha a man can do and wha a man can do. Piraes of he Caribbean: Curse of he Black Pearl Elizabeh: You re piraes. Hang he code, and hang he rules. They re more like guidelines anyway. Piraes of he Caribbean: Curse of he Black Pearl Jack Sparrow: I hough you were supposed o keep o he code. Mr. Gibbs: We figured hey were more acual guidelines. Piraes of he Caribbean: Curse of he Black Pearl 1. Inroducion Value-a-Risk (VaR) may be defined as a wors case scenario on a ypical day, and hence is concerned wih unlikely (or possibly exreme) evens. Under he Basel II Accord for marke risk involving socks and bonds (see Basel Commiee on Banking Supervision (1988), (1995), (1996) for furher deails), banks and oher Auhorized Deposi-aking Insiuions (ADIs) are required o communicae heir daily marke risk esimaes o he relevan naional moneary auhoriy a he beginning of each rading day, using one of a variey of VaR models o measure risk (see Jorion (2000)). A violaion is said o occur when he negaive reurns exceed he VaR forecas a ime, such ha acual losses on a porfolio are worse han had been prediced. In he conex of he Basel II Accord, he opimizaion problem facing ADIs, wih he number of violaions and accurae forecass of VaR as decision variables (see, for example, McAleer (2009) and McAleer, Jiménez-Marin and Peréz-Amaral (2009) for alernaive sraegies o endogenize he number of violaions), is as follows: Minimize sup ( 3 + k) { k,var} DCC = VaR 60, VaR 1, (1) where he minimizaion problem is defined in erms of DCC = daily capial charges, which is he higher of ( 3 + ) VaR 60 and VaR 1 VAR = Value-a-Risk for day, k, 3

4 VAR = Yˆ z σˆ, 60 VaR = mean VaR over he previous 60 working days, Yˆ = esimaed reurn a ime, z = 1% (negaive) criical value of he disribuion of reurns a ime, σˆ = esimaed risk (or square roo of volailiy) a ime, 0 k 1 is he Basel II violaion penaly (see Table 1). Invesors can choose he financial asses in consrucing a porfolio, bu hey canno conrol he daily reurn of he porfolio. However, hey can conrol daily capial charges by a judicious choice of violaion penaly. σˆ, which will affec VaR and he value of k arising from he According o Table 1, he value of k can be deermined exogenously or endogenously according o he penaly zones. The penalies arising from being colour coded are imporan because higher (lower) absolue esimaes of VaR will end o increase (decrease) daily capial charges, wih a resuling lower (higher) value of k. ADIs ha sray ino he red zone can be given he ulimae penaly, namely a emporary or permanen suspension from rading aciviies. 2. The Ten Commandmens This secion presens en commandmens ha are inended o assis in managing VaR for marke risk when he number of violaions, and hence he value of k, can be exogenous or endogenous. 4

5 Table 1: Basel Accord Penaly Zones Zone Number of Violaions k Green 0 o Yellow Red Noe: The number of violaions is given for 250 business days. Noe: The penaly srucure under he Basel II Accord is specified for he number of penalies and no heir magniude, eiher individually or cumulaively. 5

6 (C1) Undersand he Basel II Colours Green means go (away), red means sop (now), and yellow is lef o he discreion of he invesor. The penalies arising from individual and accumulaed violaions should be undersood in order o make informed decisions regarding wheher or no o violae on any given day. The penaly srucure is given, bu he invesor can choose which penaly is opimal on any given day. (C2) Undersand he Risk Model before Choosing There are many risk models, depending on he daa frequency used. Condiional volailiy is mos ofen used o esimae marke risk, especially for daily daa, bu his choice probably arises from hisorical expediency raher han model superioriy. Riskmerics (1996) seems o be he indusry sandard. Helpful reviews of he condiional volailiy lieraure are given in Li, Ling and McAleer (2002) and McAleer (2005), while recen reviews of sochasic volailiy and realized volailiy models are given in Asai, McAleer and Yu (2006) and McAleer and Medeiros (2008), respecively. Undersanding he choice of model, especially by appreciaing wha each model can and canno do, will make invesmen decisions more meaningful. (C3) Vary he Choice of Risk Model Be willing o change he risk model ha is used o forecas VaR on a daily basis. No risk model dominaes all ohers a all imes, and hence no single risk model will minimize daily capial charges on a daily basis. A judicious choice among alernaive risk models can and will lead o lower daily capial charges, bearing in mind he penalies imposed for an excessive number of violaions. (C4) Avoid he Green Zone and Be Willing o Violae Green may be necessary for a healhy plane, bu i is far from opimal for managing VaR. The green zone is boh a reward (in erms of k) for few violaions, bu here will be few violaions when VaR forecass are oo conservaive (meaning oo high in 6

7 absolue erms). Say away from he green zone because he daily capial charges will be oo high (for furher deails, see McAleer and da Veiga (2008a, 2008b)). (C5) Incur Large Violaions The Basel II Accord does no penalize excessively large violaions, as he penaly srucure is inended only for oo many violaions. If an invesor is going o violae, hen he violaion should be serious (meaning huge). There is no purpose in increasing k if he associaed VaR is no being decreased more han proporionaely, given he Basel II penaly srucure. (C6) Sop Before he Red Zone Avoid he red zone as his can lead o serious penalies, including emporary or permanen suspension. Enering he red zone arises hrough having accumulaed penalies for an excessive number of violaions. Alhough his may mean ha VaR is relaively low, he size of violaions can also be small, in which case he daily capial charges may no be small. (C7) Avoid Frequen Violaions The Basel II Accord penaly srucure is inended o punish an excessively frequen number of violaions. Therefore, a combinaion of one or more risk models ha do no accumulae an excessively large number of violaions should be considered. In seeking o minimize daily capial charges, i is preferable o combine a few violaions of large magniude han a large number of violaions of small magniude. (C8) Avoid Esimaing Large Porfolios Large porfolios may be unavoidable in erms of balancing specializaion and hedging sraegies, bu hey have oo many covariances and/or correlaions for purposes of esimaion. For example, a 10 asse porfolio has 10 variances and 45 covariances, whereas a 100 asse porfolio has 100 variances and 4950 covariances (see Caporin and 7

8 McAleer (2009a) for possible remedies). In his sense, i may be useful o impose parameric resricions and o calibrae he model (as in Riskmerics (1996)). (C9) Aggregae Porfolios ino a Single Index Sock marke indexes summarize he capializaion of leading financial asses, and are also easy o undersand as a single daa series. Moreover, a single index requires he specificaion and esimaion of a single risk model, and hence does no have any covariances o esimae. For hese reasons, i is sensible o aggregae any given porfolio ino a single index and o calculae he VaR accordingly (see, for example, McAleer and da Veiga (2008a, 2008b)). (C10) Inerpre Commandmens Sensibly as Guidelines The concep of VaR is inended o capure possibly bad oucomes on a ypical day, and no when marke panic ses in. Invesors should be especially careful o inerpre hese recommendaions judiciously when financial markes are in free fall. Someimes cash is he opimal financial asse, especially when markes are diving and volailiy is excessively high. 3. Conclusion Jack Sparrow: [afer Will draws his sword] Pu i away, son. I s no worh you geing bea again. Will Turner: You didn bea me. You ignored he rules of engagemen. In a fair figh, I d kill you. Jack Sparrow: Tha s no much incenive for me o figh fair, hen, is i? Piraes of he Caribbean: Curse of he Black Pearl Alhough commandmens are widely read, as confirmed by reference o SSRN views and download saisics, ogeher wih laws, rules, regulaions, agreemens, direcives, prescripions, recommendaions, guidelines and codes, commandmens are made o be broken, if no alogeher ignored. In any maers relaed o dealing wih financial markes, his can be a cosly omission, regardless of wheher markes are fair or no (for furher deails, see McAleer (2009) and Caporin and McAleer (2009b)). 8

9 References Asai, M., M. McAleer and J. Yu (2006), Mulivariae sochasic volailiy: a review, Economeric Reviews, 25, Basel Commiee on Banking Supervision, (1988), Inernaional Convergence of Capial Measuremen and Capial Sandards, BIS, Basel, Swizerland. Basel Commiee on Banking Supervision, (1995), An Inernal Model-Based Approach o Marke Risk Capial Requiremens, BIS, Basel, Swizerland. Basel Commiee on Banking Supervision, (1996), Supervisory Framework for he Use of Backesing in Conjuncion wih he Inernal Model-Based Approach o Marke Risk Capial Requiremens, BIS, Basel, Swizerland. Caporin, M. and M. McAleer (2009a), Do we really need boh BEKK and DCC? A ale of wo covariance models, Deparmen of Economics, Universiy of Padova (Available a SSRN: hp://ssrn.com/absrac= ). Caporin, M. and M. McAleer (2009b), The Ten Commandmens for managing invesmens, o appear in Journal of Economic Surveys (Available a SSRN: hp://ssrn.com/absrac= ). Jorion, P. (2000), Value a Risk: The New Benchmark for Managing Financial Risk, McGraw-Hill, New York. Li, W.K., S. Ling and M. McAleer (2002), Recen heoreical resuls for ime series models wih GARCH errors, Journal of Economic Surveys, 16, Reprined in M. McAleer and L. Oxley (eds.), Conribuions o Financial Economerics: Theoreical and Pracical Issues, Blackwell, Oxford, 2002, pp McAleer, M. (2005), Auomaed inference and learning in modeling financial volailiy, Economeric Theory, 21, McAleer, M. (2009), The Ten Commandmens for opimizing value-a-risk and daily capial charges, o appear in Journal of Economic Surveys (Available a SSRN: hp://ssrn.com/absrac= ). McAleer, M., J.-Á. Jiménez-Marin and T. Pérez-Amaral (2009), A decision rule o minimize daily capial charges in forecasing value-a-risk, Deparmen of Quaniaive Economics, Compluense Universiy of Madrid, Spain (Available a SSRN: hp://ssrn.com/absrac= ). 9

10 McAleer. M. and M.C. Medeiros (2008a), Realized volailiy: a review, Economeric Reviews, 27, McAleer, M. and B. da Veiga (2008a), Forecasing value-a-risk wih a parsimonious porfolio spillover GARCH (PS-GARCH) model, Journal of Forecasing, 27, McAleer, M. and B. da Veiga (2008b), Single index and porfolio models for forecasing value-a-risk hresholds, Journal of Forecasing, 27, Riskmerics TM (1996), J.P. Morgan Technical Documen, 4 h Ediion, New York, J.P. Morgan. 10