A New Method for Estimating Value-at-Risk of Brady Bond Portfolios
Ron D’Vari, CFA[1] and Juan C. Sosa[2]
State Street Research & Management
December 11, 1998
Abstract
Value-at-Risk (VAR) statistics are often calculated via a variance-covariance matrix methodology. While simple to implement, such an approach ignores the well documented fact that high-frequency financial data tend to substantially deviate from the Gaussian distribution. This feature is particularly pronounced in country spread data (over treasuries) for emerging markets. This study addresses the problem of estimating VAR statistics for Brady bond portfolios by using a modified GARCH(1,1) model with a superimposed ‘jump’ innovation which affects not only instantaneous spreads but subsequent volatilities. This approach incorporates the stochastic volatility feature of ARCH models, while allowing for occasional large shocks to country spreads that persist over time. This new methodology is evaluated by estimating one-day, one-week and one-month VAR measures on a daily basis for several risk tolerance levels for spread-driven returns of sample portfolios. The introduction of a persistent ‘jump’ component improves upon the risk ‘confidence intervals’ estimated by both standard GARCH(1,1) and rolling variance-covariance methods. However, this result is not homogeneous across countries. Our methodology deals with multiple-country portfolios without the computationally problematic large number of parameters of standard Multivariate ARCH models. Instead of parametrically estimating cross-country correlations, we extract them from each country’s individual “non-jump” standardized innovations on a rolling basis. Furthermore, our results also show that by allowing the jump frequencies to depend on variables such as contagion effects, the accuracy of VAR estimates may be improved. All model parameters are re-estimated daily using prior historical data. Therefore our testing is performed out-of-sample.
Extended Abstract
VAR statistics are often calculated via a variance-covariance matrix methodology. While simple to implement, such an approach ignores the well documented fact that high-frequency financial data tend to substantially deviate from the Gaussian distribution. We introduce a new methodology for the modeling of country spread movements, aimed at the estimation of VAR measurements for emerging market fixed income portfolios, that addresses such deviations. We propose an extension of the standard GARCH(1,1) model, Bollerslev(86), that includes a superimposed ‘jump’ component. Previous literature, such as Vlaar & Palm (93), Nieuwland, Vershoor & Wolff (94) and Kim & Moe(94) provide examples of this approach. In our study, we introduce two significant innovations. First, we model emerging market daily spread change data with a GARCH specification that features volatility-persistent jumps. Such large-shock persistence is particularly conspicuous in emerging market spread movements, and the accuracy of our VaR estimates supports our model choice. Second, our methodology deals with multiple-country portfolios without the computationally problematic large number of parameters of standard Multivariate ARCH models. Instead of parametrically estimating cross-country correlations, we extract them from each country’s individual “non-jump” standardized innovations on a rolling basis. VAR estimates are then produced via Monte Carlo simulations. This procedure is updated daily using only prior historical data. Hence our testing is performed out-of-sample.
This study considers three alternatives for the modeling of a country’s daily spread changes: Variance-Covariance, GARCH(1,1) and GARCH(1,1) with Jumps (GARCH-PJ). The specification for the latter is given by:
yt = a0 + et, where
et = ut + jt, with ut ~ N(0,1), i.i.d.,
jt ~ N(mj,sj2) with probability pt, 0 with probability 1-pt, and
ht = g0 + g1 e2t-1 + g2ht-1
In our context
St = country spread on day t in %, and
yt = St – St-1, daily spread change
These three frameworks are compared by estimating, on a daily basis, confidence intervals for daily/weekly/monthly spread-driven returns on representative Brady bond portfolios. A duration based approach was selected to approximate the spread-related returns of these portfolios in terms of the spread changes of each individual country. Confidence intervals are generated via tabulated cutoff values in the Variance-Covariance case and Monte-Carlo simulations in the GARCH and GARCH-PJ cases. We consider tolerance levels of 90%, 95%, 97.5% and 99% and 1-day, 1-week and 1-month return horizons. For example, in the 1-week 90% risk tolerance case, a cutoff value a1-week,90%(T) is generated on each day T in such way that the 1-week portfolio return starting at time T be above this value with a 90% probability (all other cases are treated analogously). Now, to deal with a multivariate portfolio, we must estimate rolling cross-country correlations (as in the Var-Covar model). For this, we use the 2-month correlation matrix of standardized innovations, in the GARCH(1,1) case, or the non-jump standardized innovations in the GARCH(1,1) with Jumps model. Once this matrix is estimated, we use the individual parameters from each country to simulate the returns of the portfolio and estimate confidence intervals. This method allows us to avoid the estimation of a very large number of parameters, a characteristic of Multivariate GARCH models.
Our results have shown how our GARCH-PJ methodology represents a clear improvement upon the standard GARCH and Var-Covar models for Value-at-Risk estimation when Latin-American individual spread data are considered. The situation is reversed for the Philippines and Nigeria, and for those portfolios that include them. A key to explain this conflict is the fact that GARCH-PJ models appear to overestimate the risk of the two non-Latin countries, while the Var-Covar and GARCH approaches underestimate the risk of the Latin countries. The country spread time series data indicate that the observed simultaneity in jumps across Latin-America does not extend to the Philippines and Nigeria. These two countries, in fact, exhibit jump patterns that seem to blend with the overall high volatility levels of these countries’ daily spread changes. That is, while Latin-American countries exhibit sudden dramatic volatility inducing events, such as the ’94 Crash of the Mexican peso, the other two countries exhibit “smoother” volatility patterns. As a result, the models that incorporate jump effects overestimate the risk levels of the spread data. In contrast, Latin-American countries appear to experience distinct shock events that occur simultaneously across countries, consistently with the GARCH-PJ assumptions. Consequently, Var-Covar and GARCH confidence intervals are too narrow, as risk is underestimated.
When we concentrate on the Latin countries and portfolios, we find that, although it is the most accurate approach, GARCH-PJ tends to underestimate risk for the longer horizons. Our simulation procedures assume a number of conditions. Of these, possibly the most debatable are those regarding the joint distribution of jump events. We have assumed that jumps occur virtually simultaneously throughout Latin-America, and that they are of similar magnitude (and direction). This decision should, if anything, result in risk overestimation as our simultaneity assumptions reduce the effect of diversification in our portfolio. Moreover, the longer horizon risk underestimation problem also occurs at the individual level. The distribution of our non-jump innovations appears to be normal, consistently with the GARCH-PJ specification. Therefore, our model may be incorrectly assuming the magnitude of our jump events to be normally distributed also. Future research could consider alternative, leptokurtotic distributions for the jump magnitudes and their effect on confidence interval estimation.
Finally, the comparison between the constant and time-varying jump-probability specifications of GARCH-PJ yields mixed results. For the three forecast horizons, especially the two longer ones, the risk of a representative portfolio consisting of Brazil, Mexico, and Venezuela is better approximated by the constant jump probability GARCH-PJ. Yet, the risk of a second portfolio that adds Argentina is better approximated by the time-varying version of the model. Therefore, further testing is necessary to gauge the gains in risk-predictive power from using variables other than our ‘contagion’ indicator.
References
Bollerslev, T., 1986, “Generalized Autoregressive Conditional Heteroskedasticity”, Journal of Econometrics, Vol. 31, pages 307-327
Bollerslev T., Engle R.F., and Nelson, D.B., 1994, ARCH models, Handbook of Econometrics IV, pp. 2959-3038 (Engle and McFadden, editors), Elsevier Science B.V.
Davidson, R., and MacKinnon, J.G., 1993, Estimation and Inference in Econometrics, Oxford University Press
Drost, F.C., Nijman, T.E., and Werker, B.J., 1998, Estimation and Testing in Models Containing Both Jump and Conditional Heteroscedasticity, Journal of Business and Economic Statistics, Vol. 16, N. 2, April 1998, pages 237-243
Engle, R.F., 1995, ARCH Selected Readings, R. Engle editor, Oxford University Press, 1995
Engle, R.F., and Kroner, K.F., 1993, Multivariate Simultaneous Generalized ARCH, Working Paper, UCSD, June 1993
Gay, R.S., and Ross, C.W., 1997, Risk Indicators for Avoiding Losers, Emerging Markets: Economic & Strategy, BT Alex. Brown Research, September 1997
Hamilton, J.D., 1994, Time Series Analysis, Princeton
University Press
Hamilton, J.D., and Susmel, R., 1994, Autoregressive Conditional Heteroskedasticity and Changes in Regime, Journal of Econometrics, Vol. 64, 1997, pages 307-333
Kim, H.Y., and Moe, T.H., 1994, Forecasting Volatility in the Hong Kong Equity Market, Equity Derivatives Research/Asia-Pacific Quantitative Analysis, Salomon Brothers, February 1994
Nieuwland, F.G., Verschoor, W.F., and Wolff, C.C., 1994, Stochastic Trends and Jumps in EMS Exchange Rates, Journal of International Money and Finance, Vol. 13, N. 6, December 1994, pages 699-727
Persaud, A.D., 1998, Event Risk Indicator Handbook, Global Foreign Exchange Research: Technical Series, JP Morgan January, 1998
Rosenberg, M.R., 1998, Currency Crises in Emerging Markets, Fixed Income Research, Merril Lynch, July 1998
Susmel, R., Switching Volatility in Latin American Emerging Equity Markets, Emerging Markets Quarterly, Spring 98, pages 44-56
Vlaar, P.J., and Palm, F.C., The Message in Weekly Exchange Rates in the European Monetary System: Mean Reversion, Conditional Heteroscedasticity, and Jumps, Journal of Business & Economic Statistics, July 1993, Vol. 11, No. 3
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[1] V.P., portfolio manager and head of quantitative fixed-income research
[2] Fixed Income Analyst, and Ph.D. Candidate, Boston College, MA