Modeling Swap Curve Risk in Monte Carlo VaR with Key Rate Factors

# Answer by Dimitri Vulis (score 1)


# Adding vanilla IR swaps to an existing multi-asset Monte Carlo VaR (Excel): simulate par swap rate shocks (Δbps) and revalue via DV01?












I already have an Excel-based Monte Carlo VaR engine working for FX and commodities:

- build historical factor series

- estimate volatilities + correlation matrix

- generate correlated shocks via Cholesky convert shocks to scenario PnL and compute 1-day VaR from simulated PnL distribution.

Now I want to integrate vanilla interest rate swaps (e.g., EUR swaps on Euribor 3M/6M, USD swaps on SOFR OIS). I do not want to bootstrap/rebuild full curves inside Excel.

My plan is a sensitivities-based Monte Carlo VaR:

- For each swap, I choose a market rate risk factor consistent with the swap’s curve and maturity, e.g.: 2YR Swap RCV x% EUR Pay 6M EURIBOR EUR use the par swap rate 2Y EUR006M Index.

- From the historical par swap rate series, compute daily absolute changes in bps

- I estimate vol/corr on these Δbps series (and include cross-asset correlations with FX/commodities), then simulate correlated 1-day shocks Δbps via the same Cholesky engine.

- I revalue swaps approximately using DV01 obtained from Bloomberg SWPM

Questions:

- Is this approach considered sound for 1-day VaR on vanilla swaps, as an alternative to full curve bootstrapping and full repricing?

- If the portfolio contains swaps across many maturities, should I move from “one factor per swap maturity” to a small set of key-rate tenors (e.g. 1Y/2Y/5Y/10Y) and use key-rate DV01, even in a simplified Excel setup?

## Answer by Dimitri Vulis (score 1)

https://quant.stackexchange.com/a/85410

If all your factor sensitivities are linear, then you might get away with calculating VaR by matrix multiplication without a need for any Monte Carlo.

But whether you use matrix multiplication or Monte Carlo, there are problems with using just 1 market factor for the entire curve. The most immediate one is that the historical volatility of the rates <1Y can be less than 1/3 of the volatility of the 5Y rate. Depending on what you want to do with the VaR, it may be OK to conservatively use the higher volatility for all tenors, overstating the VaR.

Also the correlation between the rates at different tenors can be much less than 1/2, and you're effectively assuming that all correlations are 1. If all your dv01s have the same sign, this assumption is conservative. But suppose that your dv01 is +\$100 at 3 month, and -\$100 at 5 years. Your spreadsheet thinks these two sensitivities perfectly offset each other - you're flat, while in reality the historical correlation may be around 0.4. The slope of the curve may change, these two rates will move by different amounts, and you will have P&L.

Modeling each key rate as a market factor in MC, havings its own volatility and correlations, is better, and may be acceptable to you / your model validators / regulators :). Some notes:

- if you run MC on the historical volatilities and pairwise correlations, MC will generate some unrealistic scenarios and P&Ls, which may overstate the VaR for some shapes of sensitivities, again erring on the conservative side. Some people control this by simulating historical principal components of the curve, rather than individual tenors. This takes some work - may be worth exploring if your book has a lot of positive and negative sensitivities at different tenors. At least, examine the scenarios in the tail and see if they are too unrealistic for your taste.

- if, e.g., you have a swap maturing in 2 years and 4 months... it's probably "close enough" to allocate 2/3 of the parallel-shift DV01 from SWPM to the 2Y bucket and the remaining 1/3 to the 3Y bucket. But someone like model validation may insist that you calculate the sensitivity to each tenor bucket.

- if you have swaps maturing in more than approximately 20 years, then convexity (IR gamma) may be material-ish. You may want to estimate how much P&L that would contribute, and decide whether to include it, which is not hard.

- if you're working with multiple currencies: there will be materially different correlations between fx rates and different tenors of interest rate curves and precious metals. Not sure about your other commodities.

- if you include cross-currency basis in valuing your positions, then you should include it, with term structure, in your VaR calculation.

> small set of key-rate tenors (e.g. 1Y/2Y/5Y/10Y)

this set may be too small, or good enough, depending on your factor sensitivities. Look at your current / anticipated sensitivities and decide what choice of tenor buckets makes you comfortable. As above, if your dv01 is +\$100 at 6 months, and -\$100 at 1 year, and you put them both in the same 1Y bucket, then you assume the same volatility and perfect correlation inside the bucket. But in reality the two rates may move very differently, creating P&L.