# Value at Risk for a plain vanilla interest rate swap

Hello I have question regarding the computations of the Value at Risk for a plain vanilla interest rate swap (i.e. same currency and fixed-for-floating).

I have a data set consisting of the Swap Rates from 2017-12-31 to date in the relevant currency, and I would like to use historical simulation (rather than variance-covariance or monte carlo method) to compute the 10 day VaR at level p=0.01.

How would I go on and do this? Are the Swap rates sufficient to compute? I don't have any data regarding principals etc, just the swap rates. I think we can assume a monocurve setup.

Thanks in advance

## 1 Answer

To value your swap, you need the zero rates. Assuming a monocurve setup, you could compute your value at risk as follows:

1. Get zero coupon rates from the swap rates series by bootstrap, to get a zero curves history. For this you don't need the notional, simply assume that the notional is equal to 1 for example, for all your swaps.

2. Deduce your value at risk scenarios from this serie (evaluation of zero rates at each time step), if what you want is a 10d VaR, you can consider 10d variations:

• I would go for non overlapping returns, as some authors argue that using overlapping returns leads to a underestimation of the VaR, e.g: https://www.risk.net/risk-management/1500264/error-var-overlapping-intervals)
• Computing a 1d VaR and rescaling by $\sqrt{10}$ works only when the realizations are independent. Here, it might not be accurate because of mean reversion of the interest rates.
3. For each VaR scenario, apply the scenario to today's zero curve, and value your swap using the resulting zero curve. This will give you a vector of NPVs and hence a vector of P&Ls (wrt today's NPV).

4. Get the relevant quantile of this P&L vector, this is your value at risk.

• note that data availability from end-dec-2017 to 6-may-18 is a problem here, particularly if used with non-overlapping 10day returns since that restricts roughly 90-odd trading days to only 9 periods. If you consider daily market movements to be independent then you could use a statistical bootstrap procedure to collect more samples. This is technically not Monte Carlo since you are resampling the data non-parametrically. – Attack68 May 9 '18 at 20:34