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Sosa State Street Research & Management CIFEr, New York March 30th, 1999_>!,# ObjectivesEstimate short-term spread-driven VaR statistics for Brady Bond portfolios Model accurately the dynamics of country spread time series: time-varying volatility and persistent shock-events Allow for exogenous factors: contagion, sentiment indicators, macroeconomic variables$ CMethodology Requirements7Accuracy Robustness Feasible automation and maintenance Modeling AlternativesRolling Variance-Covariance (Multivariate) GARCH We suggest a hybrid approach Univariate GARCH with Persistent Jumps Rolling white noise correlation matrix Exogenized jump frequencies6OjO; $,N C  Data SetJP Morgan s EMBI database of country-representative Brady Bond indices Current countries: Argentina, Bulgaria, Brazil, Ecuador, Mexico, Panama, Peru, Poland, and Venezuela Longest daily data sets start in 1992  Approximating ReturnsBrady Bond portfolio returns can be decomposed into US Term Structure Movements Country Risk Changes Bond Issue Specifics We are concerned only about the secondH4G'4G Spread ReturnsFor a N-country portfolio, our return formula is given by rp = w1r1+ w2r2+& + wNrN - w1d1Ds1 - w2d2Ds2 -& - wNdNDsN di and Dsi are the duration and spread change for country i bonds over the return horizon wi is the weight of country i bonds in the portfolio: <   Q 3b9 P3 Rolling Var-Covar$ Vart(rp) = (w1d1 ... wNdN)S (w1d1 ... wNdN)` where S is the sample var-covar matrix of the spread change vector over the past 3-months  $( $(Jl  = Rolling GARCH (univariate) We consider the popular GARCH(1,1) version of the model Model parameters are reestimated daily using all previously available spread change data VaR estimates are produced via simulation M c Rolling GARCH-PJ (univariate) We consider a variation of GARCH(1,1) that features Bernoulli-style jumps Dst = a0 + et, where et = sqrt(ht)ut + jt, with ut ~ N(0,1) i.i.d. ht = g0 + g1 e2t-1 + g2ht-1 jt ~ N(mj,sj2) with probability p 0 with probability 1-pKK  /I-3 HRolling GARCH-PJ (univariate) cont d$ Jump occurrences in this model will induce a volatility spike in subsequent days Bernoulli, rather than Poisson jumps, simplify and speed up the parameter estimation procedure VaR estimates are also produced via simulation(Rolling Exogenized GARCH-PJ (univariate),  Jump frequencies are also allowed to depend on exogenous or past data We consider a contagion variable: the average implicit jump probability across all countries in the sample over the past month$U h         Model ChoiceThe skewness and kurtosis of the standardized innovations support GARCH-PJ Brazil 1992-1999: Skewness Kurtosis Rolling Var-Covar 5.94 99.67 GARCH 2.96 47.20 GARCH-PJ * 0.16 3.50 GARCH-PJ Exo* 0.12 3.42 *jump days excluded0Lb Q b.*Model Choice (cont d)Pearson goodness-of-fit statistics concentrated at the 90% tails also support (Exogenized) GARCH-PJ In this example, the Pearson goodness-of-fit statistics are distributed c2(10)2 O [dModel Choice (cont d) Pearson Goodness-of-Fit Series EMBI Argent. Bulgaria Brazil Ecuador Num Obs 2050 1465 1069 1798 923 Var-Covar 197.56 138.32 81.477 136.68 26.933 0.00% 0.00% 0.00% 0.00% 0.27% GARCH 85.165 44.587 36.82 60.475 16.441 0.00% 0.00% 0.01% 0.00% 8.77% GARCH-PJ 21.098 8.8574 6.5732 24.874 4.6923 2.04% 54.57% 76.50% 0.56% 91.08% GARCH-PJ 17.547 17.556 9.6259 17.184 9.9252 (exogenized) 6.31% 6.29% 47.39% 7.04% 44.71% 15 AP-  $dModel Choice (cont d) Pearson Goodness-of-Fit Series Mexico Panama Peru Poland Venezuela Num Obs 1798 507 444 1069 1798 Var-Covar 140.07 46.29 44.186 43.941 93.301 0.00% 0.00% 0.00% 0.00% 0.00% GARCH 83.035 13.679 34.806 37.314 70.317 0.00% 18.81% 0.01% 0.00% 0.00% GARCH-PJ 14.016 6.158 1.4454 27.212 9.7889 17.23% 80.18% 99.91% 0.24% 45.92% GARCH-PJ 18.074 5.2091 8.6527 12.17 9.2327 (exogenized) 5.37% 87.68% 56.54% 27.38% 51.02% t/2!A$P,  ',Hit Rates (1-day 90%,95%, 97.5% and 99% VaR) Argentina Bulgaria Brazil Mexico Poland Venezuela Rolling Var-Covar 90.0% 91.5% 91.3% 90.6% 91.1% 91.4% 91.0% 95.0% 93.2% 93.9% 92.9% 94.2% 94.9% 94.3% 97.5% 94.8% 95.3% 94.9% 95.7% 96.6% 96.2% 99.0% 96.3% 97.0% 96.7% 96.6% 97.6% 97.3% GARCH 90.0% 90.9% 90.8% 91.2% 91.2% 93.7% 90.5% 95.0% 94.3% 94.6% 94.6% 94.3% 96.0% 94.3% 97.5% 96.2% 96.0% 96.1% 96.2% 97.0% 95.9% 99.0% 97.4% 97.6% 97.6% 96.9% 98.5% 97.4% GARCH-PJ (exogenized jump) 90.0% 91.0% 90.3% 89.8% 89.9% 90.2% 90.2% 95.0% 94.9% 94.4% 94.0% 94.4% 94.5% 93.9% 97.5% 97.0% 96.7% 96.6% 97.1% 96.3% 97.2% 99.0% 99.0% 99.0% 98.5% 98.9% 99.0% 98.9% 5    (  (  (  &    (  (  (  &    (  (  (  ' >=z .Hit Rates (1-week 90%,95%, 97.5% and 99% VaR) /- Argentina Bulgaria Brazil Mexico Poland Venezuela Rolling Var-Covar 90.0% 88.4% 88.3% 87.0% 89.0% 90.4% 87.3% 95.0% 91.6% 91.9% 90.7% 92.8% 94.2% 91.4% 97.5% 93.4% 93.3% 93.0% 94.8% 95.7% 93.9% 99.0% 94.3% 95.7% 94.8% 95.8% 96.7% 95.4% GARCH 90.0% 86.8% 89.2% 89.8% 88.6% 93.5% 86.2% 95.0% 92.2% 93.1% 93.3% 92.7% 96.2% 90.9% 97.5% 94.6% 94.9% 95.1% 95.9% 97.3% 93.9% 99.0% 96.8% 96.6% 96.4% 97.6% 98.3% 96.1% GARCH-PJ (exogenized jumps) 90.0% 89.6% 91.5% 89.9% 91.1% 92.7% 88.7% 95.0% 94.5% 94.7% 94.7% 95.9% 96.2% 94.7% 97.5% 96.6% 96.3% 97.5% 97.7% 97.8% 96.9% 99.0% 98.3% 98.0% 98.6% 98.7% 99.1% 98.6%  1     (  (  (  &    (  (  (  &    (  (  (  & >=z .Hit Rates (1-month 90%,95%, 97.5% and 99% VaR) Argentina Bulgaria Brazil Mexico Poland Venezuela Rolling Var-Covar 90.0% 80.6% 80.4% 79.4% 83.4% 82.7% 80.8% 95.0% 85.8% 85.7% 84.3% 88.2% 88.2% 85.3% 97.5% 88.2% 88.6% 86.8% 91.0% 91.8% 88.7% 99.0% 91.3% 91.8% 89.4% 93.5% 94.4% 91.5% GARCH 90.0% 83.5% 87.1% 84.4% 87.6% 94.2% 80.9% 95.0% 88.6% 90.8% 89.1% 91.5% 94.9% 88.1% 97.5% 92.0% 92.5% 91.8% 93.9% 95.9% 92.4% 99.0% 94.5% 95.2% 93.9% 95.9% 96.8% 94.5% GARCH-PJ (exogenized jumps) 90.0% 89.0% 90.5% 89.9% 92.4% 93.7% 91.1% 95.0% 93.0% 93.2% 94.6% 95.0% 96.7% 94.4% 97.5% 95.7% 95.9% 97.2% 96.7% 98.1% 96.8% 99.0% 97.6% 97.8% 99.0% 97.5% 99.1% 98.0% 7    (  (  (  &    (  (  (  &    (  (  (  & >>{ "Multivariate ARCH Issues  Portfolio VaRWe consider 3 equally-weighted sample portfolios LatAm: Argentina, Brazil, Mexico, Venezuela Global (EastEurope): Bulgaria, Mexico, Poland Global (LatAm): Argentina, Brazil, Bulgaria Current spread durations were used 61$1$>0/ $D%>Portfolio VaR Hit Rates Rolling Var-Covar$4 90% 95% 97.50% 99% LatAm 1-day 90.60% 93.70% 94.90% 96.50% 1-week 87.80% 91.70% 93.50% 95.20% 1-month 80.30% 85.90% 88.00% 90.50% Global 1-day 91.10% 94.00% 95.70% 96.50% (East Europe) 1-week 87.70% 91.50% 93.50% 95.00% 1-month 81.80% 86.50% 90.00% 91.80% Global 1-day 91.30% 94.50% 95.90% 96.80% (LatAm) 1-week 88.40% 91.90% 93.80% 95.90% 1-month 82.20% 87.70% 90.40% 92.70% ""& "&",<R $/Portfolio VaR Hit Rates GARCH 90% 95% 97.50% 99% LatAm 1-day 91.50% 94.70% 95.90% 97.40% 1-week 87.70% 92.10% 95.00% 96.60% 1-month 85.20% 91.00% 93.80% 94.50% Global 1-day 91.60% 94.80% 96.50% 98.00% (East Europe) 1-week 88.70% 92.80% 94.70% 96.20% 1-month 86.50% 92.30% 94.10% 95.10% Global 1-day 91.80% 95.10% 96.30% 97.30% (LatAm) 1-week 89.90% 93.80% 95.30% 96.60% 1-month 90.10% 93.80% 94.90% 96.00% ""& !&" ,;R!!DPortfolio VaR Hit Rates GARCH-PJ (exogenized jumps)3  90% 95% 97.50% 99% LatAm 1-day 89.30% 94.00% 96.80% 99.00% 1-week 90.00% 95.00% 96.80% 97.90% 1-month 91.90% 94.70% 96.10% 97.60% Global 1-day 90.40% 94.30% 97.20% 98.80% (East Europe) 1-week 90.60% 94.70% 96.60% 98.20% 1-month 93.50% 95.30% 96.50% 97.30% Global 1-day 90.20% 94.10% 96.40% 98.70% (LatAm) 1-week 90.60% 94.70% 96.50% 97.80% 1-month 95.00% 95.90% 96.40% 97.50% "!& "'",=Q")Conclusions and CommentsDGARCH-PJ s fit to Emerging Market spread data is superior to that of GARCH and Var-Covar approaches Hybrid univariate GARCH fit/empirical correlation matrix VaR approach is flexible, accurate, fast, robust and easily automated Application of methodology in other contexts is straightforward$#6@PD #*   Fin$6/H$%&'()*+,- . / 0 1 23456789:;<=>?@AP  ` ___f3f` ___3f` 999MMM>?" dd@,|?" dd@   " @ `"  n?" dd@   @@``PR    @ ` ` p>>0 S K   (  b8  r0b r 0b2  C BUCENG#H8ZIPQf3? 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X | 0?YTjB | B1?7gjB | B1?OgjB | B1?OvvgjB | B1?OgjB | B1?OFFgjB  | B1?7gjB  | B1?OgjB  | B1?O}}gjB  | B1?OgjB  | B1?OKKgjB | B1?7gjB | B1?OgjB | B1?OgjB | B1?OgjB | B1?ORRgjB | B1?7gjB | B1?O!!gjB | B1?OgjB | B1?OgjB | B1?OXXgjB | B1?7gjB | B1?O' ' gjB | B1?O gjB | B1?O gjB | B1?O^ ^ gjB | B1?7 gjB | B1?O- - gjB | B1?O gjB  | B1?O gjB !| B1?Od d gjB "| B1?7 gjB #| B1?O3 3 gjB $| B1?O gjB %| B1?OgjB &| B1?OjjgjB '| B1?7gjB (| B1?O99gjB )| B1?OgjB *| B1?O  gjB +| B1?OppgjB ,| B1?7gjB -| B1?O@@gjB .| B1?OgjB /| B1?OgjB 0| B1?OwwgjB 1| B1?7gjB 2| B1?OEEgjB 3| B1?OgjB 4| B1?OgjB 5| B1?O||gjB 6| B1?7gjB 7| B1?OLLgjB 8| B1?OgjB 9| B1?OgjB :| B1?OgjB ;| B1?hi <| Tz ?g 50 =| Ttz ?sg 625 >| Tz ?yg 650 ?| T4{ ?!g 675 @| T{ ?hD g 7100 A| T{ ?o K g 7125 B| TT| ?u Q g 7150 C| T| ?{Wg 7175 D| T} ?]g 7200 E| Tt} ?cg 7225 F| T} ?ig 7250pB G| H1?  jB H| B1?WXjB I| B1?jB J| B1? jB K| B1?WXjB L| B1? jB M| B1? jB N| B1?G H jB O| B1? jB P| B1? jB Q| B1?6 7 jB R| B1? jB S| B1? jB T| B1?& ' jB U| B1? jB V| B1? jB W| B1?p q jB X| B1?  jB Y| B1? jB Z| B1?_ ` jB [| B1?  jB \| B1?  jB ]| B1?jB ^| B1?OPjB _| B1?jB `| B1?jB a| B1?>?jB b| B1?jB c| B1?jB d| B1?./jB e| B1?jB f| B1?xyjB g| B1?xyjB h| B1?jB i| B1?jB j| B1?ghjB k| B1? jB l| B1?jB m| B1?WXjB n| B1?jB o| B1?jB p| B1?FGjB q| B1?jB r| B1?jB s| B1?h t| T4~ ?X 6-3 u| T~ ? ,  6-2 v| T~ ? f  6-1 w| TT ?   50 x| T ?? 51 y| T ?y 52 z| Tt ?M 53 {| TԀ ? 54  ||  B`CDE$F,1?  k}*j>aSahj|hpinuiw o"o7L`~u0DYn(=Rf{ !6J_t.CXl'<Pez  4I^s^llvso-pB~Vvkdvn ~& r: _O Yd `y h l z  u u k z t3 H |] yq   v m { |   , @ U j         % 9 @N c w       M q 2 G [ p / + [ l ~ *?(T#i}Sel#8Mav+ki1EZo)>Sg| "7K `"u2 / D Yn%(+=Q)f{ ~ !=5Jo_|@  }|  B`CDE$F,1?  AO*<>Sh2|@J?9;? I"M7mLm`Suqqzo0DYn(=Rf{ !6J_t.CXl'<Pez  4I^s-?Lb=Q-@B[VSk2MilepH ^& 7: 2O !d !y ; 8 G e W @ = ; \3 [H Y] :q O U _ E M X u e , @ U tj  t   k   ~ r% 9 N c w  F N      2 +G J[ rp   4   1 *s?T%i} sz#'8[MKa%vmZ1EZo=XCgh0)>Sg| .$F v"y7K`u]-]t/tDYn/651T:(=Qf{ !5J_@  ~|  B`C DE$F,1?  e*>iSh|e r"i7L`%uJWe@\0wD!Y@nEwN3iJeee(a=SREf{3raw| .!6J7_t3rw737.CXNl|<nJ*@<W<'e<%Perzw\3N E .4 IJ^as9wJ-BVk%n\%a & : O d ry  n       *3 H ] q N     w |  *, 7@ U j  n  S \    @% 9 N %c kw   q   H p  2 $G [ Np   ' e  S Q*?WT i}B$#8M1avQD1EqZo]Y;e)>;Slg\|g'3 Q"7KW`u>+W/9DYnEr (=Q f1{Eh   !5WJ_@d | T4 ?w N uABrazil: Daily Series of 1-day spread changes, Jan/01/98-Jan/22/99BB | T ?>}  X$and 90%&99% GARCH(1,1) VaR estimates%%H | 0޽h ? ___f3f?[  ZZ@gZ( ]Dt     fT gֳgֳ ?   X  0?YTjB  B1??ojB  B1?Y  pjB  B1?YuupjB  B1?YpjB  B1?YDDpjB  B1??ojB  B1?YpjB  B1?Y{{pjB  B1?YpjB  B1?YJJpjB  B1??ojB  B1?YpjB  B1?YpjB  B1?YpjB  B1?YQQpjB  B1??ojB  B1?Y  pjB  B1?YpjB  B1?YpjB  B1?YWWpjB  B1??ojB  B1?Y' ' pjB  B1?Y pjB  B1?Y pjB  B1?Y^ ^ pjB  B1?? ojB  B1?Y- - pjB  B1?Y pjB  B1?Y pjB ! B1?Yd d pjB " B1?? ojB # B1?Y4 4 pjB $ B1?Y pjB % B1?YpjB & B1?YkkpjB ' B1??ojB ( B1?Y::pjB ) B1?YpjB * B1?Y  pjB + B1?YqqpjB , B1??ojB - B1?YAApjB . B1?YpjB / B1?YpjB 0 B1?YxxpjB 1 B1??ojB 2 B1?YGGpjB 3 B1?YpjB 4 B1?YpjB 5 B1?Y~~pjB 6 B1??ojB 7 B1?YMMpjB 8 B1?YpjB 9 B1?YpjB : B1?YpjB ; B1?pq < T ?p 50 = T ?rp 625 > Tt ?xp 650 ? Tԃ ?!p 675 @ T4 ?hD p 7100 A T ?n J p 7125 B T ?u Q p 7150 C TT ?{Wp 7175 D T$m ?^p 7200 E Tm ?dp 7225 F Tm ?kp 7250pB G H1?4 5 jB H B1?fgjB I B1?jB J B1?jB K B1?fgjB L B1? jB M B1? jB N B1?\ ] jB O B1?  jB P B1? jB Q B1?R S jB R B1? jB S B1? jB T B1?H I jB U B1? jB V B1? jB W B1? jB X B1?> ? jB Y B1? jB Z B1? jB [ B1?4 5 jB \ B1?4 5 jB ] B1?jB ^ B1?jB _ B1?*+jB ` B1?jB a B1?yzjB b B1? !jB c B1?jB d B1?opjB e B1?jB f B1?jB g B1?jB h B1?dejB i B1?  jB j B1?jB k B1?Z[jB l B1?jB m B1?jB n B1?PQjB o B1?jB p B1?jB q B1?FGjB r B1?jB s B1?jB t B1?p u TDn ?g 6-3 v Tn ? F  6-2 w To ?   6-1 x Tdo ?5   50 y To ?z 51 z T$p ?X 52 { Tp ? 53 | Tp ?G 54  }  BdC?DE$F,1?  *>Sh|#7Lau"0$E$Z!n&+,110777)8>:S9g>|.'(*,+' '"'7K`$u"&%'#/DYn "$(+/(/=3R*g{   !!6%K_t&$/CXm ' < Q f {         5 J _ s         . C W l         ' < lP Ae Yz n  9   .   5 I ^ s D D     S-ABnWk-k0y&;PduyN.> $`4;IN]/rUlpl2-BAV=k?FHW`]~&f;OdyXc,2 cs3Hd]@q6Ag>`x,AVj  %:FOcx@   ~  BdCQDE$F,1?  *Y>lS[hP|meqv{H#7nLau50$EZnX(4!vAE+)>=ShgP|%(#UH "7K#`u<!/DYny#$'(=R'g{   !(6K_t &*QX7/CdX?mtYn/ 8' n< eQ Lf c{ v   ]  o  n 5 J |_ hs    L i    . C W l  O  .     ' N< zP le z  F   1 +   u5 |I n^ s      f 8-B|W kak[v^&k;Pdy # ;4Im]Br-0BoV7k~gv&;OdFyt1a z3H]q#AiW,(AVj(%F:O~c@    BdCg DE$F,1?  D*>ISh|cDR#I7_Laullh*7Du!;0VEZ!n%V.I*DDD)@>2S%g|hR@V[_ "~7_K`uuRV/pDlY.n[M* !7(D=RgR{Vl;. %!6K*_@tc _lVo*/CXmM;@ ' < Q f R{  M    l ~ y 5 J _ _s .   y  V [ c . C W l  M ~ 2 ;    !' < P e Pz   \   5 g  5 I ^ .s    D  2 P-B7Wz kc&;Pdy=h l4IJ]rCf,D-BVEk;L+&;O7dy$n 73 H]qic%R,AVj %Swjf %j:7Oc@m  TDq ?>m ) uABrazil: Daily Series of 1-day spread changes, Jan/01/98-Jan/22/99BB  Tr ?280  ['and 90%&99% GARCH-PJ(1,1) VaR estimates((H  0޽h ? ___f3fu[   [[PZ(     fdr gֳgֳ ?   X  0?YTjB  B1??ojB  B1?Y  pjB  B1?YuupjB  B1?YpjB  B1?YDDpjB  B1??ojB  B1?YpjB  B1?Y{{pjB  B1?YpjB  B1?YJJpjB  B1??ojB  B1?YpjB  B1?YpjB  B1?YpjB  B1?YQQpjB  B1??ojB  B1?Y  pjB  B1?YpjB  B1?YpjB  B1?YWWpjB  B1??ojB  B1?Y' ' pjB  B1?Y pjB  B1?Y pjB  B1?Y^ ^ pjB  B1?? ojB  B1?Y- - pjB  B1?Y pjB  B1?Y pjB ! B1?Yd d pjB " B1?? ojB # B1?Y4 4 pjB $ B1?Y pjB % B1?YpjB & B1?YkkpjB ' B1??ojB ( B1?Y::pjB ) B1?YpjB * B1?Y  pjB + B1?YqqpjB , B1??ojB - B1?YAApjB . B1?YpjB / B1?YpjB 0 B1?YxxpjB 1 B1??ojB 2 B1?YGGpjB 3 B1?YpjB 4 B1?YpjB 5 B1?Y~~pjB 6 B1??ojB 7 B1?YMMpjB 8 B1?YpjB 9 B1?YpjB : B1?YpjB ; B1?pq < Tr ?p 50 = T$s ?rp 625 > Ts ?xp 650 ? Ts ?!p 675 @ TDt ?hD p 7100 A Tt ?n J p 7125 B Tu ?u Q p 7150 C Tdu ?{Wp 7175 D Tu ?^p 7200 E T$v ?dp 7225 F Tv ?kp 7250pB G H1?4 5 jB H B1?fgjB I B1?jB J B1?jB K B1?fgjB L B1? jB M B1? jB N B1?\ ] jB O B1?  jB P B1? jB Q B1?R S jB R B1? jB S B1? jB T B1?H I jB U B1? jB V B1? jB W B1? jB X B1?> ? jB Y B1? jB Z B1? jB [ B1?4 5 jB \ B1?4 5 jB ] B1?jB ^ B1?jB _ B1?*+jB ` B1?jB a B1?yzjB b B1? !jB c B1?jB d B1?opjB e B1?jB f B1?jB g B1?jB h B1?dejB i B1?  jB j B1?jB k B1?Z[jB l B1?jB m B1?jB n B1?PQjB o B1?jB p B1?jB q B1?FGjB r B1?jB s B1?jB t B1?p u Tv ?g 6-3 v TDw ? F  6-2 w Tw ?   6-1 x Tx ?5   50 y Tdx ?z 51 z Tx ?X 52 { T ? 53 | T ?G 54  }  BdC0DE$F,1?  *>Sh|#7Lau  0EZn"#")())*>,S+g/|  "7 K `u /DYn  !( =$Rg{   !6K _t/CXm ' < Q f {         5 J _ s         . C W l         ' < XP e Gz V  &      5 I ^ ss  D     N-8BeWk$n,W&;Pd[y8&  W42IE]&rLfhb*- B:V4k8>>MVSsx&\;OdywNY#& Xh3HY]8q-5[4Vn,AV{j%:@Oci@   ~  BdC7DE$F,1?  *B>WSEh:|WN[_r<#7Lau0EZn '-+) >,S{g6