Tail Risk in Pakistani Financial Markets — VaR and ES in Python

Value at Risk (VaR) and Expected Shortfall (ES) estimation for the KSE-100 equity index and COMEX Gold futures using three methods — Historical Simulation, Parametric Normal, and GARCH(1,1) Filtered Historical Simulation — with formal backtesting via the Kupiec POF and Christoffersen Independence tests.

Ahmer | GitHub: ahmer-econ | 2026

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Assets and Sample

Asset	Ticker	Source	Sample
KSE-100 Index	—	Kaggle historical dataset	Jan 2015 – Aug 2024
COMEX Gold Futures	GC=F	Yahoo Finance (yfinance)	Jan 2015 – Aug 2024

2,288 daily log return observations after inner join on common trading dates.

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Methods

Historical Simulation (HS) VaR estimated as the empirical percentile of the return distribution. No distributional assumption. ES is the average of all returns below the VaR threshold.

Parametric (Normal) VaR estimated as mean minus z-score times standard deviation, assuming normally distributed returns. ES derived analytically from the normal density.

GARCH(1,1) Filtered Historical Simulation (GARCH-FHS) GARCH(1,1) fitted to extract conditional volatility. Standardised residuals computed. Historical Simulation applied to standardised residuals and scaled by the one-step-ahead volatility forecast. Adapts to current volatility conditions while retaining the empirical tail distribution.

Confidence levels: 95% and 99% Rolling backtest window: 250 trading days

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Key Results

GARCH(1,1) Parameter Estimates

Parameter	KSE-100	COMEX Gold
ω (omega)	0.000713	0.000139
α (alpha)	0.1352	0.0340
β (beta)	0.8088	0.9509
Persistence (α+β)	0.9440	0.9850

VaR and ES Estimates (full sample)

Asset	Level	Method	VaR	ES
KSE-100	99%	Historical Simulation	−3.24%	−4.34%
KSE-100	99%	Parametric	−2.53%	−2.90%
KSE-100	99%	GARCH-FHS	−2.71%	−3.23%
Gold	99%	Historical Simulation	−2.53%	−3.41%
Gold	99%	Parametric	−2.20%	−2.53%
Gold	99%	GARCH-FHS	−2.65%	−3.77%

Backtesting Summary (2,038 out-of-sample observations)

Asset	Method	Level	Violations	Expected	Kupiec p	Christ. p	Pass?
KSE-100	HS	95%	119	102	0.090	0.000	❌
KSE-100	Parametric	95%	106	102	0.679	0.001	❌
KSE-100	GARCH-FHS	95%	119	102	0.090	0.000	❌
KSE-100	HS	99%	34	20	0.006	0.019	❌
KSE-100	Parametric	99%	37	20	0.001	0.179	❌
KSE-100	GARCH-FHS	99%	34	20	0.006	0.019	❌
Gold	HS	95%	109	102	0.475	0.710	✅
Gold	Parametric	95%	107	102	0.607	0.867	✅
Gold	GARCH-FHS	95%	109	102	0.475	0.710	✅
Gold	HS	99%	28	20	0.108	0.400	✅
Gold	Parametric	99%	37	20	0.001	0.703	❌
Gold	GARCH-FHS	99%	28	20	0.108	0.400	✅

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Main Findings

• For COMEX Gold, Historical Simulation and GARCH-FHS pass all Kupiec and Christoffersen tests at both confidence levels. Violations arrive at the correct frequency and independently over time.
• For KSE-100, all methods fail backtesting. Violations cluster during crisis periods (2017, 2020) rather than arriving randomly — a structural feature of Pakistan's frontier equity market, not a failure of the methods.
• The Parametric Normal method is the worst performer at 99% for both assets, generating the most violations and failing the Kupiec test. The normality assumption cannot capture fat-tailed return distributions (excess kurtosis: KSE-100 = 4.25, Gold = 10.48).
• GARCH-FHS matches HS in violation counts while incorporating a time-varying volatility forecast, making it more suitable for real-time risk management.

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Requirements

pandas numpy scipy matplotlib seaborn yfinance arch statsmodels

Install with:

pip install yfinance arch statsmodels seaborn

pandas, numpy, scipy, and matplotlib are included in the Anaconda base distribution.

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How to Reproduce

1. Clone the repository
2. Install dependencies (see above)
3. Run scripts in order: 01_data.py through 06_violations_plot.py
4. Each script sets its working directory at the top — update the path if your folder location differs
5. All outputs are saved automatically to outputs/figures/ and outputs/tables/

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Related Work

This project replicates the methodology of an earlier R-based implementation of the same analysis. The Python version demonstrates equivalent results using pandas, NumPy, SciPy, and the arch library in place of R's rugarch and PerformanceAnalytics packages.

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References

• Christoffersen, P. (1998). Evaluating interval forecasts. International Economic Review, 39(4), 841–862.
• Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50(4), 987–1007.
• Kupiec, P. (1995). Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives, 3(2), 73–84.
• McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management (Revised ed.). Princeton University Press.
• Sheppard, K. (2024). arch: ARCH and other tools for financial econometrics. https://github.com/bashtage/arch

Repository Structure