Estimating VaR with Time-Varying Volatility Models
Summary
The document asks why historical return quantiles can be used for Value-at-Risk when returns are dependent, as in ARCH-type models. The response clarifies that dependence among returns does not prevent modeling their marginal distribution: volatility models describe changing conditional variance, while their innovations may be independent. Sorting observations to calculate quantiles does not itself eliminate the time dependence represented by the model.
For simple volatility specifications, VaR may have a closed-form calculation; more complex models can be fitted to data and used to simulate many future paths. VaR is then computed for each path, with the resulting estimates summarized by an average or, preferably, a confidence interval. The same broad approach extends to multivariate settings, where models such as dynamic conditional correlation GARCH represent changing relationships among assets. The explanation is conceptual and does not give a worked calibration, backtest, or guidance on model selection and tail-risk validation.
Key ideas
- Dependent returns can be modeled using time-varying conditional variance and independent innovations.
- A univariate return distribution does not imply that observations are independent over time.
- Simulation from a fitted volatility model can estimate VaR when closed-form methods are unavailable.
- Confidence intervals can communicate uncertainty across simulated VaR estimates.
- Multivariate volatility models can represent changing correlations between assets.
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# VaR estimation when returns are not independent, e.g. ARCH
# VaR estimation when returns are not independent, e.g. ARCH
Time series of returns, $r_t$, in finance are often modeled with some type of conditional heteroskedasticity model, e.g. ARCH(1):
$$r_t = \sigma_t z_t$$ $$\sigma_t^2 = a_0 +a_1 r_{t-1}^2$$
where, say, $z_t \tilde{} N(0,1)$, which implies that
$$r_t = z_t \sqrt{a_0 +a_1 r_{t-1}^2}$$
and hence the returns are not independent. However, Value-at-risk (VaR) seems to be estimated by constructing a single empirical distribution of the observed returns and taking some quantile. But since the returns are not independent, they cannot be represented by one univariate distribution, so why this procedure to calculate VaR is considered valid? It seems to me that, at the very least, VaR estimated in this way would be biased, but I am not sure I have seen corrections being applied / discussed for this.
Add 1 It seems to me that given a time series with a persistent volatility (e.g. one with a high order of the ARCH term above, say 100), any finite period of observations, say 250, is likely to have explored a smaller part of the total space compared to IID returns, so I would have thought that one has to correct somehow for this when estimating VaR from historical observations in this case.
## Answer by user22108 (score 0)
https://quant.stackexchange.com/a/35460
The fact that you have a univariate distribution does not imply that the returns are independent. If fact it is plenty of univariate models that take into account the dependence of the time series. In volatility models the innovations (${{Z}_{t}}$ in your question) are independent, not the returns.
The procedure to caluclate VaR with volatiltiy models is consistent. When you compute quantiles, you may think that the time dependence is broken apart by ordering the returns, but this is not the case, because the returns are modeled using time dependence variance.
I try to explain with an example. There are closed formulas to complute VaR using simple volatility models. Most of the time, for advanced models you have to use simulation. For example you estimate a GARCH model with your data. Then you simulate N (N>1000) paths from this model and for each path you compute VaR. Finally you can compute the mean of tha VaRs or better you compute a confidence interval.
In multivariate models the procedure is the same. You have just more parameters to model the time-varying correlation between assets (see GARCH-DCC model)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.