Out-of-sample volatility forecasting and Value-at-Risk backtesting for four NIFTY 50 stocks (TCS, Infosys, Asian Paints, Bajaj Finance) using GARCH(1,1) with Student-t errors, benchmarked against EWMA (RiskMetrics) and historical volatility.
Out-of-sample period: roughly January 2021 to December 2024 (979 trading days per stock), with the model refitted daily on a rolling 500-day window.
1. VaR backtesting (99%, one day). GARCH-t passed both the Kupiec and Christoffersen tests for all 4 stocks. EWMA with normal errors passed for only 1 of 4, breaching almost twice as often as a 99% VaR should.
| Stock | GARCH-t breaches (expected 9.8) | GARCH-t breach rate | EWMA-normal breaches | EWMA-normal breach rate | GARCH-t passes | EWMA passes |
|---|---|---|---|---|---|---|
| TCS | 10 | 1.02% | 18 | 1.84% | Yes | No |
| Infosys | 10 | 1.02% | 14 | 1.43% | Yes | Yes |
| Asian Paints | 15 | 1.53% | 25 | 2.55% | Yes | No |
| Bajaj Finance | 15 | 1.53% | 20 | 2.04% | Yes | No |
| Average | 1.28% | 1.97% | 4 / 4 | 1 / 4 |
2. Forecast accuracy (QLIKE loss, lower is better). GARCH reduced forecast loss by 4.9% on average versus 21-day historical volatility, significant at 5% for 3 of 4 stocks (Diebold-Mariano). Against EWMA the average gain was 1.2%, which was not statistically significant for any stock.
| Stock | QLIKE gain vs 21-day hist | DM p-value | QLIKE gain vs EWMA | DM p-value |
|---|---|---|---|---|
| TCS | 5.4% | 0.004 | 2.2% | 0.051 |
| Infosys | 1.9% | 0.319 | 0.6% | 0.670 |
| Asian Paints | 6.1% | 0.038 | 0.8% | 0.687 |
| Bajaj Finance | 6.2% | 0.014 | 1.3% | 0.515 |
Takeaway: EWMA tracks day-to-day volatility almost as well as GARCH, which is expected because EWMA is effectively a GARCH model with persistence fixed near 1. The real advantage of GARCH-t is its fat-tailed error distribution: the normal distribution used by RiskMetrics understates tail losses, which is exactly what the 99% VaR backtest exposes.
- Data. Daily prices from Yahoo Finance (January 2019 to December 2024), adjusted for splits and bonus issues. Log returns are scaled by 100 to help the optimiser converge. Any one-day move above 20% is flagged so that unadjusted corporate actions can be caught before modelling.
- Full-sample model. GARCH(1,1) with a constant mean and Student-t errors, reporting persistence (alpha + beta), volatility half-life and long-run volatility.
- Rolling forecasts. Each day's volatility is forecast using only the previous 500 trading days (about two years), with parameters refitted daily. No future data is used.
- Benchmarks.
- EWMA (RiskMetrics): sigma²(t+1) = 0.94 · sigma²(t) + 0.06 · r²(t)
- Historical volatility: rolling 21-day standard deviation
- Forecast evaluation. The next-day squared return is used as the realised variance proxy. Models are scored with MSE and QLIKE (Patton, 2011), and GARCH is compared with each benchmark using a Diebold-Mariano test with Newey-West standard errors.
- VaR backtesting. One-day VaR at 99% and 95%. GARCH VaR uses the forecast mean, forecast volatility and the fitted Student-t quantile; EWMA VaR uses the normal quantile. Each model is tested with:
- Kupiec POF test: is the breach rate consistent with the VaR level?
- Christoffersen test: are breaches independent, or do they cluster?
- Volatility summary and forecast. Average volatility over the last 3, 6 and 9 months, plus a 14-trading-day forecast path converging towards the long-run level.
| Stock | alpha | beta | alpha + beta | Half-life (days) | nu (t d.o.f.) |
|---|---|---|---|---|---|
| TCS | 0.044 | 0.919 | 0.962 | 18.1 | 4.76 |
| Infosys | 0.075 | 0.828 | 0.903 | 6.8 | 4.23 |
| Asian Paints | 0.322 | 0.184 | 0.506 | 1.0 | 3.60 |
| Bajaj Finance | 0.068 | 0.925 | 0.994 | 105.7 | 3.41 |
All four models converged. Every stock has nu below 5, confirming heavy tails and supporting the choice of Student-t over normal errors.
- Asian Paints shows weak volatility clustering. Persistence is only 0.51 and beta is not significant (p = 0.07), so a volatility shock fades in about one day. Its volatility in this period was driven by isolated large moves rather than sustained turbulent regimes. The VaR plot shows this clearly: VaR widens sharply after each large move and returns to normal the next day.
- Bajaj Finance's long-run volatility estimate is unreliable. With persistence at 0.994, the implied long-run volatility (57% annualised) is highly sensitive to small parameter changes and far above the realised full-sample volatility (37.7%).
- The flagged Bajaj Finance move is genuine. The −26.4% return on 23 March 2020 was the COVID-19 market crash, not an unadjusted corporate action (its price ratio of 0.77 does not match any split or bonus ratio).
- 95% VaR is weaker than 99%. At 95%, GARCH-t breach rates are close to 5% for every stock (Kupiec passes for all four), but breaches cluster for Infosys and Asian Paints (Christoffersen p < 0.05). EWMA-normal passes at 95% for 3 of 4 stocks. Fat tails matter most at the extreme 99% level.
- MSE and QLIKE disagree for two stocks. On MSE, EWMA beats GARCH for Asian Paints and Bajaj Finance. MSE is dominated by a few extreme days, which is why QLIKE is the preferred loss for comparing volatility models.
- Scope. Four large-cap stocks over one period. Results may differ for other stocks, periods or asymmetric models such as GJR-GARCH or EGARCH.
pip install numpy pandas matplotlib scipy arch yfinance
jupyter notebook GARCH_Volatility_VaR.ipynbRun all cells. With daily refitting, the rolling forecasts take a few minutes; set REFIT_EVERY = 5 in the configuration cell for a faster run.
To use local CSV files instead of Yahoo Finance, set USE_YFINANCE = False and place files named tcs.csv, infosys.csv, asianpaints.csv and bajaj.csv in a data/ folder, with a Date column and an Adj Close (or Close) column.
