VaR: Impact of Confidence Level and Horizon (FRM Part 1)
1. Context
Here, we look at what impact the choice of two important parameters: confidence level and horizon have the magnitude of VaR. The details of the reading in which this topic appears are given below:
| Area | Valuation and Risk Management |
| Reading | Measures of Financial Risk |
| Reference | Dowd, Kevin, Ch2. Measures of Financial Risk. In Measuring Market Risk, 2nd Edition, West Sussex, England: John Wiley & Sons, 2005. |
2. Symbols
We use the following symbols in the sections that follow:
| $T$ | Horizon or holding period |
| $c$ | Confidence Level |
| $V_0$ | Current value of a portfolio |
| $V_T$ | Future value of portfolio (at $T$) |
| $VaR$ | Value-at-Risk |
| $P/L$ | Profit-or-Loss |
| $\mu$ | Annualized mean of changes in portfolio value |
| $\sigma$ | Annualized volatility of changes in portfolio value |
3. Defining VaR
To recap, we define VaR as the maximum loss over a target horizon ($T$), such that there is a low, prespecified volatility ($p$) that the actual loss will be higher. This is equivalent to saying that: $$\Pr(P/L > -VaR)=c \Leftrightarrow \Pr(P/L < -VaR)=p$$ With this definition in mind, we now dig into what impact choice of $c(=1-p)$ and $T$ have on $VaR$.
4. A Simple Model for P/L
To calculate $VaR$, we put down a simple model for evolution of the future value of our portfolio (valued at $V_0$ today). Over a small time instant $dt$, our portfolio value changes by: $$dV_t = \mu dt + \sigma dz_t$$ where, $dz_t = \varepsilon \sqrt(dt); \varepsilon \sim N(0,1)$. Owing to constant coefficients, the above process yields: $$V_T – V_0 = P/L = \mu T + \sigma \sqrt{T} \varepsilon$$ We can now compute the VaR using the above guiding equation for the $P/L$: $$\begin{align} \Pr(P/L < -VaR)&=p \\ \Pr(\mu T + \sigma \sqrt{T} \varepsilon < -VaR) &=p\\ \Pr \left(\varepsilon < \frac{-VaR - \mu T}{\sigma \sqrt{T}} \right) &=p \\ N\left( \frac{-VaR - \mu T}{\sigma \sqrt{T}} \right) &=p \\ \end{align}$$ $$VaR = -\left [\mu T + \sigma \sqrt{T} N^{-1}(p) \right]$$
5. Impact of Confidence Level
As the confidence level $c$ increases, probability $p(=1-c)$ decreases. The second term in the square bracket $\left(\sigma \sqrt{T} N^{-1}(p)\right)$ of $VaR$ expression becomes more negative and VaR increases. The increase in VaR is not linear with $c$, as shown in figure below:
6. Impact of Horizon
In general, the impact of horizon ($T$) on $VaR$ is positive, but depends on the sign and magnitude of mean $\mu$. One can easily check from the above $VaR$ equation that for $\mu=0$, $VaR$ should increase with horizon, and with a $\sqrt{T}$ dependence. When $\mu>0$, the VaR is lower (since sign of first term is positive) and it doesn’t increase as markedly as in the $\mu=0$ case because the first term ($\mu T$) and second term $\left(\sigma \sqrt{T} N^{-1}(p)\right)$ tend to offset each other. The behavior is shown below:
