Definition

Consider an asset’s log-return series $r_t=\mu +{\epsilon }_t$, where $\mu$ is the expected return and ${\epsilon }_t$ is a zero-mean white noise. The total log-return between date $t$ and date $t+k$ is then naturally defined as:$$r_{t,t+k}=\sum _{i=1}^k{r}_{t+k}$$The standard definition of the $k$-day ahead $\mathrm{VaR}$ of a position in this asset is the 1% or 5% quantile of the return distribution for $r_{t,t+k}$. V-Lab’s long run risk measures use simulation based methods to calculate the $\mathrm{VaR}$ at horizons of $k=30$ and $k=365$.

Estimation

One way to calculate $\mathrm{VaR}$ is to simulate future realizations of the return process and use the resulting simulations to calculate $\mathrm{VaR}$. For both of the models prescribed in the models section of the documentation, a volatility model is fit to historical data on each day. The resulting model is then simulated ahead 10,000 times, for a horizon of 1-year in advance. All simulations are bootstrapped - that is historical shocks to the return process are drawn at random to simulate each path. $\mathrm{VaR}$ is then calculated using both the 1% and 5% quantile of the 10,000 simulated return paths. Finally, logarithmic returns are converted back to arithmetic returns.

For a given day $t$, the following $\mathrm{GJR-GARCH}$ model is fit to historical data:

For more details on estimation refer to the GJR-GARCH section of the volatility analysis in V-Lab. Estimation delivers the parameters $\alpha$, $\gamma$, $\beta$ as well as a historical variance series $g_t$. Subsequently, a historical series of residuals, or shocks, is easily computed by ${\epsilon }_t=r_t/\sqrt{g_t}$. Using this historical residual series and the estimated parameters, 10,000 future return paths rt,t+k are simulated to deliver a measure of $\mathrm{VaR}$.

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