A New Method for Estimating Value-at-Risk of Brady Bond Portfolios Ron D'Vari & Juan C. Sosa State...

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

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


Methodology Requirements

• Accuracy

• Robustness

• Feasible automation and maintenance


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


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


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


Spread Returns• For a N-country portfolio, our return formula is

given by

r = w1r1+ w2r2+…+ wNrN

- w1d1s1 - w2d2s2 -…- wNdNsN

di and si 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


Rolling Var-Covar

Vart(r) = (w1d1 ... wNdN)(w1d1 ... wNdN)`

where is the sample var-covar matrix of the

spread change vector over the past 3-months


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


Rolling GARCH-PJ (univariate)

• We consider a variation of GARCH(1,1) that features Bernoulli-style jumps

st = 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(j,j2) with probability p

0 with probability 1-p


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


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


0 25 50 75 100 125 150 175 200 225 250

-3

-2

-1

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


0 25 50 75 100 125 150 175 200 225 250

-3

-2

-1

0

1

2

3

4 Brazil: Daily Series of 1-day spread changes, Jan/01/98-Jan/22/99and 90%&99% GARCH(1,1) VaR estimates


0 25 50 75 100 125 150 175 200 225 250

-3

-2

-1

0

1

2

3

4

Brazil: Daily Series of 1-day spread changes, Jan/01/98-Jan/22/99and 90%&99% GARCH-PJ(1,1) VaR estimates


0 25 50 75 100 125 150 175 200 225 250

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-2

-1

0

1

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Brazil: Daily Series of 1-day spread changes, Jan/01/98-Jan/22/99and 90%&99% GARCH-PJ(1,1) w/ Exogenized Jumps VaR estimates


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


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


Model 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%


Model 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%


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%


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


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%


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


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%


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


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


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%


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%


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%


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


Fin

A New Method for Estimating Value-at-Risk of Brady Bond Portfolios Ron D'Vari & Juan C. Sosa State...

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