Value at Risk part II. Weighted Historical Simulation. BRW Approach. HW Approach

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1 Value a Risk par II Weighed Hisorical Simulaion Chuang I - Yuan Deparmen of Finance, NCCU Weighed Hisorical Simulaion 3 BRW Approach 4 Boudoukh, Richardson and Whielaw (Risk, 998) Hull and Whie (JR, 998) Duffie and Pan (JD, 997) Holon (Risk, 998) Barone-Adesi, Giannopoulos and Vosper (JFM, 999) Age-weighed hisorical simulaion: assigns exponenial weigh o each observaion Reflecs relaive imporance of he daa Combines he srengh of EWMA and hisorical simulaion Reduces he effecive sample size Sep Sep 2 Periods ago Reurns Weighs Sored Reurns Weighs Cumulaive Weighs Sep 3: use inerpolaion o find HW Approach Volailiy-weighed hisorical simulaion: updaes reurn informaion o ake accoun of recen changes in volailiy Le T = mos recen forecas of he volailiy, = he hisorical GARCH (or EWMA) forecas of volailiy a day T r * r 6

2 General Approaches o Weighed Hisorical Simulaion Duffie and Pan sugges a weighing scheme ha akes accoun of correlaion as well as volailiies Holon chooses weighs o reflec curren marke esimaes, no only of means, volailiies and correlaions, bu also of skewness, kurosis and higher momens 7 Filered Hisorical Simulaion Sep #: Fi GARCH model o he reurns Sep #2: Forecas volailiy in a sample period Sep #3: Obain sandardized reurns Sep #4: Boosrap he sandardized reurns and muliply each random drawing by he mos recen volailiy forecas Sep #5: find 8 Exreme Value Theory Exreme Value Theory Exreme value heory (EVT) focuses on exreme evens and heir associaed ail probabiliies I offers precisely he mehodology for characerizing he rare, bu no impossible, occurrences of losses EVT offers a parameric form for he ail of a disribuion To dae he EVT lieraure describes a univariae heory Two Principal Disribuions () Two Principal Disribuions (2) 2 GEV: The generalized exreme value (GEV) describes he limiing behavior of normalized maxima of iid disribued random variables Block maxima: This approach consiss of dividing he series ino non-overlapping blocks of he same lengh and choosing he maximum from each block The corresponding block maxima mehod is he radiional mehod applied in fields like reliabiliy, reinsurance, hydrology and environmenal science GPD: The generalized Pareo disribuion (GPD) describes he ail of a disribuion above a given high hreshold. Thus, his second approach focuses on he realizaions which exceed some specified hreshold, he so-called peak over hreshold (POT) mehod The POT mehods use daa more efficienly and seem o become he choice mehod in recen applicaions

3 3 4 Two Principal Theorems Generalized Pareo Disribuion Fisher-Tippe Theorem: essenially says ha he GEV is he only possible limiing disribuion for (normalized) block maxima Pickland, Balkema-de Haan Theorem: for sufficienly high hresholds, he disribuion funcion of he excesses, may be well approximaed by he GPD G (x) where x X e x, if x /, if < if if 5 6 EVT Dynamic.6%.4%.2% Hisorical () Use GARCH (2) Apply mehod on he sandardized residuals Probabiliy.%.8%.6%.4% EVT.2% Normal.% -35.% -3.% -25.% -2.% -5.% -.% -5.%.% P/L r μ μ Z is he whie noise σ Z σ Z Quanile Quanile

4 References Reurn Dynamic EVT Saic EVT McNeil, A., 2(a), Exreme Value Theory for Risk Managers, Exremes and Inegraed Risk Managemen, Risk Books, Chaper, pp.3-8 McNeil, A., 2(b), Reading he Riskomeer, Exremes and Inegraed Risk Managemen, Risk Books, Chaper 9, pp.7-4 McNeil, A. and R. Frey, 2, Esimaion of Tail-Relaed Risk Measures for Heeroscedasic Financial Series: An Exreme Value Approach, Journal of Empirical Finance, Auumn, pp.27-3 Tsay, R., 22, Analysis of Financial Series, John Wiley & Sons Backesing Backesing Basel rule Kupiec es Chrisoffersen approach Crnkovic-Drachman procedure Berkowiz approach Lopez approach Hendricks measures 22 Basel Rule Wih a confidence level of 99%, here are inherenly very few excepions, so a large sample is required. This leads o backesing models using daily daa, even if is inended o be used wih a longer horizon The Basel Rule: daily excepions of 99% over las year Expeced excepions = 2.5 Green:~4; Yellow:5~9; Red:+ 23 Kupiec Tes () The simples backesing mehod: based on failure raes Le x = number of excepion observed T = oal sample size x/t = failure rae = hreshold for The number of excepions x follows a binomial probabiliy disribuion 24

5 Kupiec Tes (2) The probabiliy of observing x excepions in a sample of size T is T x Tx prob(x) ( ) x Le * = x/t, H: * = Kupiec (995) develops he appropriae likelihood raio saisic wih an asympoic 2 () disribuion Accep H if p-value >.5 25 Condiional Coverage Models Chrisoffersen (998): condiional coverage models Excepions could bunch closely in ime, which invalidae he model The combine es saisics for condiional coverage is: LR cc = LR uc + LR ind ~ 2 (2) LR uc : frequency-of-ail-losses es LR ind : independen es 26 Disribuion Forecas Models Crnkovic & Drachman (996) and Berkowiz (999): disribuion forecas models Backesing could exploi he fac ha an esimae of he enire probabiliy disribuion underlies he esimae raher han focus on he quanile of he disribuion Le X be he realized P/L F = he disribuion funcion of X If he esimae F is acually he disribuion funcion of X, hen u = F(X ) should be disribued uniformly on [,], and he u s from differen daes should be independen 27 Lopez Loss Funcions Moivaed by he evaluaion mehods used o rank he forecass of macroeconomic models Does no give us any formal saisical indicaion of model adequacy Does no suffer from he low power of sandard ess such as he Kupiec es 28 Frequency-of-ail-losses C Size-adjused frequency approach 29 Hendricks Risk Measures Compare risk measures o he model-average measures Le T = ime periods, M = number of models MRB T T i i i M 3 M i i C (R ) 2 RMSRB T T i i i 2

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