Motivation

Prior to the financial crisis of 2007 and 2008, standard risk management focused primarily on short-run risks such as 1-day value at risk ($\mathrm{VaR}$) or 10-day $\mathrm{VaR}$. Engle (2009) points out that most assets in portfolios are held longer than these horizons, and as such longer term risk measures are important from a risk management perspective. For example, in periods of low volatility excess leverage likely carries little short-run risk. However, volatility is mean reverting and will rise in the long run; hence, any short-term risk measure will understate the risk of a leveraged portfolio. The Long Run $\mathrm{VaR}$ section of V-Lab is designed to account for mean-reversion of volatility and build $\mathrm{VaR}$ models that account for “the risk that risk will change”.

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.