# How to predict VaR changes on a DoD basis?

I am trying to predict change in VaR on a DoD basis. So let's say at t=0, I have my VaR based on full valuation. On t=1, I will have another VaR based on full valuation. I am trying to predict this VaR on t=1 without full valuations. I also have Composite VaR and Incremental VaR at t=0. At t=1, I will also know how my risk factors have changed and if there is any new or dropped trades in my portfolio as well.

What will be the best way to proceed? Links to any reference material will also be highly appreciated.

• When you talk about full valuation I assume you mean historic simulation, right? – Ami44 Aug 23 '16 at 21:36
• Yes by full valuation I mean historic simulation. So without doing the full valuation I want to approximate VaR for the next day. Something like a Taylor Series Expantion, where we can predict change in security with the help of Delta and Gamma – Deb Aug 24 '16 at 15:41

## 2 Answers

Lets say you want the 99% daily VaR using a historic simulation over 250 days. In this case you have a risk vector with 250 values $v_{1}, ..., v_{250}$ with $v_{n}$ being the change in the portfolio value, if the market would exhibit the same daily movements from $t=0$ to $t=1$ as had happened from $n-1$ days ago to $n$ days ago. Lets call these market movements $d_{i}(n, n-1)$ where $i$ indexes the relevant risk factors. The VaR at $t=0$ is the third lowest value of this risk vector. Also given are the sensitivities $\delta_{i}$ to each risk factor at $t=0$, the gammas $\gamma_{i,i}$ and cross gammas $\gamma_{i,j}$ and a forecast for the change in each riskfactor $\hat{d}_{i}$ from $t=0$ to $t=1$. I also assume, that you know the new and vanishing positions from $t=0$ to $t=1$ along with their sensitivities.

Estimating the VaR at $t=1$:

1. Update your risk vector according to the new and vanishing positions. Multiply the sensitivities of that positions with market movements $d_{i}(n, n-1)$ and add or subtract that value from the risk vector to get an updated risk vector ${v}'_{1}, ..., {v}'_{250}$. Also update your sensitivities and gammas of the portfolio accordingly to ${\delta}'_{i}$ and ${\gamma}'_{i,j}$.

2. If you want to stay in linear approximation, things are easy. Calculate ${v}'_{0}=\sum_{i} {\delta}'_{i} * \hat{d}_{i}$ and substitute ${v}'_{250}$ with ${v}'_{0}$ to get the new risk vector.

3. If you insist on factor in gamma effects, it gets more complicated. First you have to update your deltas to $t=1$ by $\delta^{\ast}_{i} = {\delta}'_{i} + \sum_{j} \hat{d}_{i} * {\gamma}'_{i, j}$. Than update your risk vector by $$v^{\ast}_{n} = {v}'_{n} + \sum_{i} ( \delta^{\ast}_{i} - {\delta}'_{i} ) * d_{i}(n, n-1)$$ Than calculate $$v^{\ast}_{0}= \sum_{i} \delta^{\ast}_{i} * \hat{d}_{i} + 0.5 * \sum_{i, j} {\gamma}'_{i, j} * \hat{d}_{i} * \hat{d}_{j}$$ Substitute $v^{\ast}_{250}$ with $v^{\ast}_{0}$

4. Determine the third lowest value of your new risk vector to get your VaR estimate.

Remarks:

To 1: Feel free to add gamma effect to calculate updated risk vector. If you happen to have the risk vector on positions level you can of course also update the risk vector directly by adding/substracting the change from new/vanishing positions from full valuation.

if you need to correct for theta effects (unlikely for daily VaR), than add the difference in the theta effect from $t=0$ and $t=1$ to every element of your risk vector.

Monte Carlo simulation comes to mind: Run a number of simulations. In each simulation, for each of your variables (risk factors, new/dropped trades etc.), pick an outcome according to the (assumed or known) probability distribution. For this set of outcomes, calculate VaR (as best you can - Monte Carlo can be used here as well, you might have faster ways depending on your exact scenario).

From this set of simulated VaRs, you can calculate not only the most probably VaR value, but also the confidence interval (the more simulations you run, the smaller it will get).

• MC will be again close to Full Valuation only. I just want to approximate – Deb Aug 24 '16 at 15:41