A Probabilistic Approach to Worst Case Scenarios

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1 A Probabilisic Approach o Wors Case Scenarios A Probabilisic Approach o Wors Case Scenarios By Giovanni Barone-Adesi Universiy of Albera, Canada and Ciy Universiy Business School, London Frederick Bourgoin Millennium Global Invesmen, London & Kosas Giannopoulos Universiy of Wesminser, London March

2 A Probabilisic Approach o Wors Case Scenarios Hisorical simulaion is a naural seing for scenario analysis, bu i mus pay aenion o curren marke condiions Value a Risk (VaR) is increasingly popular as a managemen and regulaory ool. To furher is accepance i is necessary o assess is reliabiliy under condiions likely o be encounered in financial markes. A logical venue o invesigae his issue is hrough he use of hisorical simulaion. Hisorical simulaion relies on a uniform disribuion o selec innovaions from he pas. These innovaions are applied o curren asse prices o simulae heir fuure evoluion. Once a sufficien number of differen pahs has been explored i is possible o deermine a porfolio VaR wihou making arbirary assumpions on he disribuion of porfolio reurns. This is especially useful in he presence of abnormally large porfolio reurns. From he early days of modern finance large reurns are known o cluser in ime. The resuling flucuaions in daily volailiy make he confidence levels of VaR compuaions ha ignore clusering unreliable. This is he case wih VaR measuremens based on he variance-covariance marix and Mone-Carlo mehods, ha ypically ignore curren marke condiions o produce fla volailiy forecass for fuure days. Moreover he use of he covariance marix of securiy reurns or he choice of an arbirary disribuion in he Mone-Carlo mehod usually desroys valuable informaion abou he disribuion of porfolio reurns. To make our hisorical simulaion consisen wih he clusering of large reurns we model he volailiy of our porfolio as a GARCH process. Pas daily porfolio 1

3 Barone-Adesi, Bourgoin & Giannopoulos reurns are divided by he GARCH volailiy esimaed for he same dae o obain sandardised residuals. A simulaed porfolio reurn for omorrow is obained muliplying a randomly seleced sandardised residual by he GARCH forecas of omorrow volailiy. This simulaed reurn is used o updae he GARCH forecas for he following day, ha is hen muliplied by a newly seleced sandardised residual o simulae he reurn for he second day. Our recursive procedure is repeaed unil he VaR horizon (i.e.,10 days) is reached, generaing a sample pah of porfolio volailiies and reurns. We repea our procedure o obain a bach of sample pahs of porfolio reurns. A confidence band for he corresponding porfolio values is buil by aking he Kernel (empirical) frequency disribuion of values a each ime. The lower 1% area idenifies he wors case over he nex en days. To illusrae our procedure we consruced a hypoheical porfolio, diversified across all hireen naional equiy markes of our daa sample. To form our porfolio each equiy marke is weighed proporionally o is capialisaion in he world index as on December The porfolio weighs are repored in he able below: Table 1 : Porfolio weighs counry our porfolio world index (dec 95) Denmark France Germany Hong Kong Ialy Japan Neherlands Singapore Spain Sweden Swizerland UK US

4 A Probabilisic Approach o Wors Case Scenarios Hence hese weighs are held consan for he enire 10 year period and muliplied by he hireen local index reurns. Since marke risk needs o quanify evenual porfolio losses in one currency all local porfolio reurns are measured in US dollars. The descripive saisics ogeher wih he Jarque-Bera 1 normaliy es are repored on able 2. Figure 1 displays he empirical disribuion of porfolio's reurns. Table 2 : Descripive saisics of he equally weighed porfolio mean (p.a.) sd. dev (p.a.) skewness kurosis normaliy p value 10.92% 12.34% The las column is he probabiliy ha our porfolio reurns are generaed from a normal disribuion. The rejecion of normaliy in able 1 and he paern of clusering visible in figure 1 led us o model our porfolio reurns, r, as a GARCH process wih asymmeries, wih daily volailiy h given by eq.1: r = 100 Ln P P 1 + ε ε ~ N( 0, h ) (1.a) h ( ) = ω+ α ε + γ + βh (1.b) 1 The es for normaliy is he Jarque-Bera es, 2 2 (( 6) ( 3) 24) JB skewness Kurosis law = / + / χ 2 2 df 3

5 Barone-Adesi, Bourgoin & Giannopoulos Fig 1 : World Capialisaion weighed Porfolio Reurns /01/85 22/05/85 09/10/85 26/02/86 16/07/86 03/12/86 22/04/87 09/09/87 27/01/88 15/06/88 02/11/88 22/03/89 09/08/89 27/12/89 16/05/90 03/10/90 20/02/91 10/07/91 27/11/91 15/04/92 02/09/92 20/01/93 09/06/93 27/10/93 16/03/94 03/08/94 21/12/94 10/05/95 27/09/95 14/02/96 Therefore our porfolio volailiy is modelled o depend on he mos recenly observed porfolio reurn. The combinaion of GARCH volailiy and porfolio hisorical reurns offers us a fas and accurae measure of he pas, curren (and fuure) volailiy of he curren porfolio. No esimaion of he correlaion marix of securiy reurns is required. Furhermore our VaR mehod conains fewer "bad surprises", since GARCH models allow for fa ails on he uncondiional disribuion of he daa. The effecs of our choice become apparen if he reurns in figure 1 are compared wih figure 2, where he reurns have been scaled by heir daily volailiy, as in equaion 2: z = r h (2) Clusering of reurns in ime is reduced by volailiy scaling and he disribuion of reurns now appears o be more uniform. However he large number of reurns sill exceeding 3 sandard deviaions suggess ha our scaling does no 4

6 A Probabilisic Approach o Wors Case Scenarios make reurns normal. Our annualised porfolio volailiy, in figure 3, varied from 7 o 21 % over 10 years. 10 Fig 2 : Porfolio Sress Analysis (Sandardised Residuals) /01/86 21/05/86 08/10/86 25/02/87 15/07/87 02/12/87 20/04/88 07/09/88 25/01/89 14/06/89 01/11/89 21/03/90 08/08/90 26/12/90 15/05/91 02/10/91 19/02/92 08/07/92 25/11/92 14/04/93 01/09/93 19/01/94 08/06/94 26/10/94 15/03/95 02/08/95 20/12/95 Fig 3 : Annualized volailiy of he porfolio /01/86 29/05/86 16/10/86 05/03/87 23/07/87 10/12/87 28/04/88 15/09/88 02/02/89 22/06/89 09/11/89 29/03/90 16/08/90 03/01/91 23/05/91 10/10/91 27/02/92 16/07/92 03/12/92 22/04/93 09/09/93 27/01/94 16/06/94 03/11/94 23/03/95 10/08/95 28/12/95 5

7 Barone-Adesi, Bourgoin & Giannopoulos The scaled reurns are he foundaion of our simulaion. To simulae porfolio reurns over nex 10 days we selec randomly 10 reurns from figure 2. We hen consruc ieraively he daily porfolio volailiy ha hese reurns imply according o equaion 1. We use his volailiy o rescale our reurns. The resuling reurns reflec herefore curren marke condiions raher han marke condiions associaed wih reurns in figure 1. To obain he disribuion of our porfolio reurns we replicaed he above procedure 10,000 imes. The resuling -normalised- disribuion is shown in figure 4. The normal disribuion is shown in dos in he same figure for ease of comparison. Fig 4 : Normalized Esimaed Disribuion of Reurns in 10 days versus he normal densiy (10,000 Simulaions) % Smoohed Densiy Normal Densiy 6

8 A Probabilisic Approach o Wors Case Scenarios No surprisingly, simulaed reurns on our well-diversified porfolio are almos normal, excep for heir seeper peaking around 0 and some clusering in he ails. The general shape of he disribuion suppors he validiy of he usual measure of VaR for our porfolio. However a closer examinaion of our simulaion resuls shows how even our well-diversified porfolio may depar from normaliy under wors case scenarios. There are in fac several occurrences of very large negaive reurns, reaching a maximum loss of 9.52%. Our empirical disribuion implies losses of 3.38% and 2.24% a confidence levels of 1% and 5% respecively. 180 Fig 5 : Esimaed Disribuion of Porfolio VaR in 10 days (10,000 Simulaions) mean = 3.25% The reason for his deparure is he changing porfolio volailiy and hus porfolio VaR, shown in figure 5. Porfolio VaR over nex 10 days depends on he random reurns seleced in each simulaion run. Is paern is skewed o he 7

9 Barone-Adesi, Bourgoin & Giannopoulos righ, showing how large reurns end o cluser in ime. These clusers provide realisic wors case scenarios consisen wih hisorical experience. Of course our mehodology may produce more exreme deparures from normaliy for less diversified porfolios. In conclusion, our simulaion mehodology allows for a fas evaluaion of VaR and wors case scenarios for large porfolios. I akes ino accoun curren marke condiions and does no rely on he knowledge of he correlaion marix of securiy reurns. Our compuaions were performed using he RiskClock sofware. A full descripion of i is available from he auhors. 8

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