## Calculating Value at Risk for Swaps (Rates & Currency) using Historical Simulation

*in*Computational Finance

# Value at Risk for Interest Rate and Cross Currency Swaps using Historical Simulation

Valuing and marking to market over the counter interest rate (IRS) and cross currency swaps (CCS) has always been a painful topic. Model assumptions, liquidity and open interest in any given tenor varies from one valuation interval to the next and there is generally a big issue around model transparency and dependability on the executing bank for rates. God forbid you need to unwind early or seek termination before the scheduled end date.

To this context add Value at Risk (VaR) estimates for the same securities. How can you calculate VaR for a security where you are not sure about the final liquid exit price? Once upon a time in a different life we wrote a model for valuing a portfolio of IRS and CCS positions using Monte Carlo simulation. It worked perfectly in an ideal world. It was a disaster in the real world. Let’s just say that we were naive and simple and hopelessly confused and the model was much worse.

The primary problem with using Monte Carlo simulation for IRS and CCS Value at Risk is that rates (FX & Interest rates) are not normally distributed. The same is also true for all other price risk. However for vanilla instruments we are lucky to have trade prices which can be used to calibrate the pricing and VaR model. That is unfortunately not always true for IRS and CCS exposures. Which means that the model is already out of alignment and drifts even further from reality as we pile on assumptions after assumptions.

The solution is a simple tweak in the valuation model itself. Here is the model that we finally evolved after a number of high profile drubbings by the trading desk and the sales team. Not that this model is any better but it is an improvement as it uses the actual distribution of results rather than an assumed normal or log normal distribution.

**Figure 1 The Historical simulation VaR model for IRS & CCS VaR calculations **

a) Assume that the swap will retain the same life (constant maturity).

b) Collect historical interest rates and currency rates for the last 2 – 4 years.

c) Use historical rates to mark to market the constant maturity swap and generate a historical (simulated) price series for the IRS/CCS.

d) Use the simulated price series to calculate daily returns over the last two years.

e) Generate a Histogram from the daily return series.

f) Use the Histogram to calculate Value at Risk using the historical simulation approach.

One note of caution though. This is a full valuation approach which for a single IRS and CCS turns out to be a handful. If you have a large portfolio your safest (possibly only bet) is a platform or a software that does all the valuation, bootstrapping, curve building work for you. Doing all of that in Excel will quickly become an exercise.

See the MTM & Valuation of a simple Interest Rate Swap post for the valuation model and the Bootstrapping Zero and Forward Curves from treasury term structure data post for building the underling interest rate model.

If you are interest in an Excel Model that illustrates the principal of applying historical simulation to a single Interest Rate Swap, check out the** what’s new section of our store** later tonight. We will be uploading a brand new Excel sheet that shows how to build the Value at Risk using Historical Simulation model in Excel for an IRS. The same model can be extended easily to at Cross Currency Swap.

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