2. A New Method for Estimating Value-at-Risk of Brady Bond Portfolios Ron D'Vari & Juan C. Sosa State Street Research & Management CIFEr, New York March 30th, 1999

3. VaR of Brady Bonds - Ron D'Vari, et al. Objectives Estimate 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

4. VaR of Brady Bonds - Ron D'Vari, et al. Methodology Requirements Accuracy Robustness Feasible automation and maintenance

5. VaR of Brady Bonds - Ron D'Vari, et al. Modeling Alternatives Rolling Variance-Covariance (Multivariate) GARCH We suggest a hybrid approach –Univariate GARCH with Persistent Jumps –Rolling white noise correlation matrix –Exogenized jump frequencies

6. VaR of Brady Bonds - Ron D'Vari, et al. Data Set JP 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

7. VaR of Brady Bonds - Ron D'Vari, et al. Approximating Returns Brady Bond portfolio returns can be decomposed into –US Term Structure Movements –Country Risk Changes –Bond Issue Specifics We are concerned only about the second

8. VaR of Brady Bonds - Ron D'Vari, et al. Spread Returns For a N-country portfolio, our return formula is given by r  = w 1 r 1 + w 2 r 2 +…+ w N r N  - w 1 d 1  s 1 - w 2 d 2  s 2 -…- w N d N  s N d i and  s i are the duration and spread change for country i bonds over the return horizon w i is the weight of country i bonds in the portfolio

9. VaR of Brady Bonds - Ron D'Vari, et al. Rolling Var-Covar Var t (r  ) = (w 1 d 1... w N d N )   (w 1 d 1... w N d N )` where   is the sample var-covar matrix of the spread change vector over the past 3-months

10. VaR of Brady Bonds - Ron D'Vari, et al. 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

11. VaR of Brady Bonds - Ron D'Vari, et al. Rolling GARCH-PJ (univariate) We consider a variation of GARCH(1,1) that features Bernoulli-style jumps  s t = a 0 + e t, where e t = sqrt(h t )u t + j t, with u t ~ N(0,1) i.i.d. h t = g 0 + g 1 e 2 t-1 + g 2 h t-1 j t ~ N(  j,  j 2 ) with probability p 0 with probability 1-p

12. VaR of Brady Bonds - Ron D'Vari, et al. Rolling 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

13. VaR of Brady Bonds - Ron D'Vari, et al. 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

14. VaR of Brady Bonds - Ron D'Vari, et al. 0255075100125150175200225250 -3 -2 0 1 2 3 Brazil: Daily Series of 1-day spread changes, Jan/01/98-Jan/22/99 and 90%&99% Var-Covar VaR estimates

15. VaR of Brady Bonds - Ron D'Vari, et al. 0255075100125150175200225250 -3 -2 0 1 2 3 4 Brazil: Daily Series of 1-day spread changes, Jan/01/98-Jan/22/99 and 90%&99% GARCH(1,1) VaR estimates

16. VaR of Brady Bonds - Ron D'Vari, et al. 0255075100125150175200225250 -3 -2 0 1 2 3 4 Brazil: Daily Series of 1-day spread changes, Jan/01/98-Jan/22/99 and 90%&99% GARCH-PJ(1,1) VaR estimates

17. VaR of Brady Bonds - Ron D'Vari, et al. 0255075100125150175200225250 -3 -2 0 1 2 3 4 Brazil: Daily Series of 1-day spread changes, Jan/01/98-Jan/22/99 and 90%&99% GARCH-PJ(1,1) w/ Exogenized Jumps VaR estimates

18. VaR of Brady Bonds - Ron D'Vari, et al. Model Choice The 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 excluded

19. VaR of Brady Bonds - Ron D'Vari, et al. 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   

20. VaR of Brady Bonds - Ron D'Vari, et al. Model Choice (cont’d) Pearson Goodness-of-Fit SeriesEMBIArgent.BulgariaBrazilEcuador Num Obs2050146510691798923 Var-Covar197.56138.3281.477136.6826.933 0.00%0.00%0.00%0.00%0.27% GARCH85.16544.58736.8260.47516.441 0.00%0.00%0.01%0.00%8.77% GARCH-PJ21.0988.85746.573224.8744.6923 2.04%54.57%76.50%0.56%91.08% GARCH-PJ 17.54717.5569.625917.1849.9252 (exogenized)6.31%6.29%47.39%7.04%44.71%

21. VaR of Brady Bonds - Ron D'Vari, et al. Model Choice (cont’d) Pearson Goodness-of-Fit SeriesMexicoPanamaPeruPolandVenezuela Num Obs179850744410691798 Var-Covar140.0746.2944.18643.94193.301 0.00%0.00%0.00%0.00%0.00% GARCH83.03513.67934.80637.31470.317 0.00%18.81%0.01%0.00%0.00% GARCH-PJ14.0166.1581.445427.2129.7889 17.23%80.18%99.91%0.24%45.92% GARCH-PJ 18.074 5.20918.652712.179.2327 (exogenized)5.37%87.68%56.54%27.38%51.02%

22. VaR of Brady Bonds - Ron D'Vari, et al. Hit Rates (1-day 90%,95%, 97.5% and 99% VaR) ArgentinaBulgariaBrazilMexicoPolandVenezuela 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%

23. VaR of Brady Bonds - Ron D'Vari, et al. Hit Rates (1-week 90%,95%, 97.5% and 99% VaR) ArgentinaBulgariaBrazilMexicoPolandVenezuela 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%

24. VaR of Brady Bonds - Ron D'Vari, et al. Hit Rates (1-month 90%,95%, 97.5% and 99% VaR) ArgentinaBulgariaBrazilMexicoPolandVenezuela 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%

25. VaR of Brady Bonds - Ron D'Vari, et al. Multivariate ARCH Issues Multivariate ARCH models suffer from estimation problems, deriving from the inclusion of correlation parameters Our ad-hoc approach: a 3-month sample correlation matrix estimated from (non- jump) standardized innovations

26. VaR of Brady Bonds - Ron D'Vari, et al. Portfolio VaR We 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

27. VaR of Brady Bonds - Ron D'Vari, et al. Portfolio VaR Hit Rates Rolling Var-Covar 90%95%97.50%99% LatAm1-day90.60%93.70%94.90%96.50% 1-week87.80%91.70%93.50%95.20% 1-month80.30%85.90%88.00%90.50% Global 1-day91.10%94.00%95.70%96.50% (East Europe)1-week87.70%91.50%93.50%95.00% 1-month81.80%86.50%90.00%91.80% Global 1-day91.30%94.50%95.90%96.80% (LatAm)1-week88.40%91.90%93.80%95.90% 1-month82.20%87.70%90.40%92.70%

28. VaR of Brady Bonds - Ron D'Vari, et al. Portfolio VaR Hit Rates GARCH 90%95%97.50%99% LatAm1-day91.50%94.70%95.90%97.40% 1-week87.70%92.10%95.00%96.60% 1-month85.20%91.00%93.80%94.50% Global1-day91.60%94.80%96.50%98.00% (East Europe)1-week88.70%92.80%94.70%96.20% 1-month86.50%92.30%94.10%95.10% Global 1-day91.80%95.10%96.30%97.30% (LatAm)1-week89.90%93.80%95.30%96.60% 1-month90.10%93.80%94.90%96.00%

29. VaR of Brady Bonds - Ron D'Vari, et al. Portfolio VaR Hit Rates GARCH-PJ (exogenized jumps) 90%95%97.50%99% LatAm1-day89.30%94.00%96.80%99.00% 1-week90.00%95.00%96.80%97.90% 1-month91.90%94.70%96.10%97.60% Global 1-day90.40%94.30%97.20%98.80% (East Europe)1-week90.60%94.70%96.60%98.20% 1-month93.50%95.30%96.50%97.30% Global 1-day90.20%94.10%96.40%98.70% (LatAm)1-week90.60%94.70%96.50%97.80% 1-month95.00%95.90%96.40%97.50%

30. VaR of Brady Bonds - Ron D'Vari, et al. Conclusions and Comments GARCH-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

31. VaR of Brady Bonds - Ron D'Vari, et al. Fin