1. Value-at-risk
This course on GARCH models in R started with the Wall Street wisdom that there are old traders and bold traders, but almost no old bold traders. Traders need to quantify their risk of losing money. The GARCH volatility is one such risk measure. Also the 5% value-at-risk is popular. It measures how much you expect to lose in the best of the 5% worst cases.
2. Value-at-risk
This 5% value-at-risk measure coincides with the 5% quantile of the return distribution. As you can see, it is the best return in the 5% worst scenarios. Since risk is time-varying, also this 5% value-at-risk changes through time.
3. Time-varying quantiles
You can see here that for the daily S&P 500 returns, the 5% quantile computed on windows of one year varies between -0.5% and -4.39%.
4. Forward looking approach is needed
These value-at-risk estimates are computed at the end of the year. They tell you after the facts what the 5% quantile has been over the past year. Risk managers need to be forward looking and predict the 5% quantile of the future return. For this the GARCH model is useful as it gives you directly a predicted distribution and thus also the predicted 5% quantile. To do so, you can use the method quantile and apply it to a `ugarchroll` object.
5. Workflow to obtain predicted 5% quantiles from ugarchroll
More specifically, you need to first specify the GARCH model to use. Then you need to indicate which type of rolling window estimation you wish. Finally you obtain the predicted quantile at 5% using the quantile method and setting the probs argument at 5%. Alternatively you can take another loss probability such as 1% or 2-point-5%.
6. Value-at-risk plot for loss probability 5%
Through the VaRplot function you can then compare the actual returns in gray with the predicted value at risk values in black. Note the time-variation in the predicted value-at-risk. In the middle of the plot you can see the high levels of downside risk around the 2008 financial crisis. The red crosses correspond to the losses that are larger than predicted by the value at risk.
7. Exceedance and VaR coverage
Those extreme losses are called exceedances. The frequency of observing exceedances is called the coverage of the test. In the example where we use a probability of 5%, we find that the coverage is close to 5%. This is also what you expect in case of a valid value at risk prediction model.
8. VaR coverage and model validation
The VaR model is accurate as long as the coverage is close the loss probability alpha used. If the coverage is much higher, then the predicted quantile was not negative enough and you have underestimated the actual risk. The reverse is true in case the coverage is much lower than your loss probability alpha: you were too conservative and overestimated the risk leading to a too low number of exceedances.
9. Factors that deteriorate the performance
Other modeling choices lead to a worse performance. If we assume a symmetric distribution and take a student t distribution instead of a skewed student t distribution, then the percentage of exceedances becomes 5.8%.
10. Further deterioration
If in addition we choose to use a standard GARCH model instead of a GJRgarch model, then the percentage violation is 6% instead of the desired level of 5%.
11. Even further deterioration
Finally, if we re-estimate the model every one thousand observations instead of every one hundred observations, than the percentage number of exceedances is 6-point-2%.
12. Downside risk means thinking about predicted quantiles.
The take away message is that analyzing the risk of losing money involves analyzing the quantiles predicted by the GARCH model.