Historical Simulation VaR for a Plain Vanilla Interest Rate Swap
Summary
The document explains how to estimate 10-day historical-simulation VaR for a fixed-for-floating interest rate swap when only a history of swap rates is available. Under a single-curve assumption, bootstrap the swap rates into historical zero curves; the notional is unnecessary for the calculation if the swap is valued per unit notional.
Construct scenarios from changes in the curves, apply each change to the current curve, and revalue the swap. The resulting scenario NPVs relative to today’s value form the P&L distribution, whose lower-tail quantile gives VaR at the chosen confidence level. The answer recommends non-overlapping 10-day changes, noting that overlapping windows may understate VaR. It also cautions that scaling one-day VaR by the square root of time relies on independent returns, an assumption that may be inaccurate for mean-reverting rates. The method still depends on suitable curve construction and swap valuation details, which the document does not spell out.
Key ideas
- Bootstrap swap rates into a history of zero curves under the monocurve assumption.
- A unit notional is sufficient when calculating VaR per unit notional.
- Apply historical curve changes to today’s curve and revalue the swap to generate scenario P&Ls.
- Estimate VaR from the selected lower-tail quantile of those P&Ls.
- Square-root-of-time scaling assumes independent returns and may be unsuitable when rates mean-revert.
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# Value at Risk for a plain vanilla interest rate swap
# 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
## Answer by byouness (score 2)
https://quant.stackexchange.com/a/39666
To value your swap, you need the zero rates. Assuming a monocurve setup, you could compute your value at risk as follows:
- 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.
- 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.
- 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).
- Get the relevant quantile of this P&L vector, this is your value at risk.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.