A New Method for Estimating ValueatRisk of Brady Bond Portfolios

1
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

2
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

3
Methodology Requirements

• Accuracy
• Robustness
• Feasible automation and maintenance

4
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

5
Data Set

• JP Morgans EMBI database of country-representativ
  e Brady Bond indices
• Current countries Argentina, Bulgaria, Brazil,
  Ecuador, Mexico, Panama, Peru, Poland, and
  Venezuela
• Longest daily data sets start in 1992

6
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

7
Spread Returns

• For 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

8
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

9
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

10
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-p

11
Rolling GARCH-PJ (univariate) contd

• 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

12
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

13
Brazil Daily Series of 1-day spread changes,
Jan/01/98-Jan/22/99
3
and 9099 Var-Covar VaR estimates
2
1
0
-1
-2
-3
0
25
50
75
100
125
150
175
200
225
250
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Brazil Daily Series of 1-day spread changes,
Jan/01/98-Jan/22/99
4
and 9099 GARCH(1,1) VaR estimates
3
2
1
0
-1
-2
-3
0
25
50
75
100
125
150
175
200
225
250
15
Brazil Daily Series of 1-day spread changes,
Jan/01/98-Jan/22/99
and 9099 GARCH-PJ(1,1) VaR estimates
4
3
2
1
0
-1
-2
-3
0
25
50
75
100
125
150
175
200
225
250
16
Brazil Daily Series of 1-day spread changes,
Jan/01/98-Jan/22/99
and 9099 GARCH-PJ(1,1) w/ Exogenized Jumps VaR
estimates
4
3
2
1
0
-1
-2
-3
0
25
50
75
100
125
150
175
200
225
250
17
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

18
Model Choice (contd)

• 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)

19
Model Choice (contd) 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

20
Model Choice (contd) 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

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Hit Rates (1-day 90,95, 97.5 and 99 VaR)

• Argentina Bulgaria Brazil Mexico Poland Venezue
  la
• 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

22
Hit Rates (1-week 90,95, 97.5 and 99 VaR)

• Argentina Bulgaria Brazil Mexico Poland Venezue
  la
• 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

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Hit Rates (1-month 90,95, 97.5 and 99 VaR)

• Argentina Bulgaria Brazil Mexico Poland Venezue
  la
• 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

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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

25
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

26
Portfolio VaR Hit Rates
Rolling Var-Covar

• 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

27
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

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Portfolio VaR Hit Rates GARCH-PJ
(exogenized jumps)

• 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

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Conclusions and Comments

• GARCH-PJs 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

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• Fin