# How do you calculate value at risk on a portfolio of fixed income instruments

I'm curious about this question both for a parametric "Delta" style approach and a Monte Carlo full revaluation approach and I will lead one question into the next.

Taking the "Delta" approach first. Your calculation looks something like this (?):

$$VaR = conf * \sqrt{w * cov*w}$$ where conf is your confidence factor, so 2.33 for 99% and 1.65 for 95% VaR and cov is the covariance matrix. The weights should be easy to calculate here by just taking the duration of your instruments, you could also extend this with convexity easily. My question is how do you handle the covariance matrix, are you allowed to do the same thing you do with equities? I.e. just calculate the variance and covariance as the sample variance and covariance from the time series of the underlying interest rates for each time to maturity?

If the above is how you do it for the "Delta" style approach then how is Monte Carlo handled? Do you simulate it just using a driftless geometric brownian motion with the volatility calculated from the time series of the underlying interest rate? I know that interest rates are typically modeled as mean-reverting so this seems like an oversimplification, but on the other hand fitting a mean reverting process like the Cox Ingersoll Ross process for each of your underlying interest rates seems far too computationally intensive. What is the correct approach?

• CreditMetrics approach. May 9 '20 at 8:13

how do you handle the covariance matrix, are you allowed to do the same thing you do with equities? I.e. just calculate the variance and covariance as the sample variance and covariance from the time series of the underlying interest rates for each time to maturity

Many people do exactly what you wrote: they collect a historical time series for each instrument used to fit the interest rate curve, and they construct a covariance matrix from the curve fitting instruments.

It's not that bad, but: if you run a Monte Carlo simulation using this covariance matrix (which you may or may not need here) and print out the market scenarios that led to the 99%+ losses, you will immediately observe that most of these scenarios are unrealistic - the IR curve can't move like that! But for the purposes of VaR or expected shortfall, this is not a fatal flaw. You're just being conservative by considering the P&L under unrealistic adverse scenarios. :) However this may overstate your VaR so if you backtest, you will find that your actual P&L never gets close to your VaR, so someone may question why your VaR is so conservative.

A better approach, which is also common, is to calculate the historical pricipal components of all your interest rate curves. (Note that 3 principal components will not suffice; 6 or more) and have a ovariance matrix of curve PCs rather than of curve fitting instruments. You can then run MC on the PCs and get scenarios that are explained by the PCs. You will not get scenarios that cannot be explained by the PCs.

Another approach (I have seen this done; I don't think it works will) is run MC on fitting instruments, but then use PCs to effectively replace the scenarios that cannot be explained by PCs by those that can.

just taking the duration of your instruments

What you need to do here depends on the products in your book. Are they all zero-coupon bonds? That only happens in homework problems. :) Or are there coupons and amortizations before maturity? Then using duration as one number, which basically tells you the sensitivity to the parallel shift of the interest rate curve, loses too much information about the instrument's sensitivity to the shape of the curve. You can look instead at "key rate durations" instead - the sensitivity to each of the instruments that you use to build your IR curve.

Further, for VaR purposes, it is more convenient to use not durations, but the P&L change from perturbing your market data.

You can perturb separately each instrument used to fit the IR curve (e.g. 1 basis point) and reprice your portfolio.

Or you can perturb each PC - if your book is non-linear as discussed below, you perturb each PC up and down 1, 2, 3 its standard deviations and reprice your portfolio.

You can make it more complicated by calculating the sensitivities to one set of instruments (e.g. ED futures) and then transform them to sensitivities to another set of instruments (e.g. 1Y, 2Y, 3Y IR swaps) using inverse Jacobian. This is fine if you just want to see the sensitivity to these instruments, but it is better not to use such transofmed sensitivies as inputs into your VaR calculation.

you could also extend this with convexity

Again, it depends on the products. If the products are nearly linear, like vanilla IR swaps, then for the purposes of 99% VaR the convexity is immaterial - don't bother with it. The linear sensitivity to the fitting instruments will suffice. Further, you don't need any Monte-Carlo simulation. A matrix multiplication that you wrote, using the dollar sensitivities to the curve fitting instruments, and the covariance of the curve fitting instruments will do just fine.

But if the products are non-linear, e.g. any callables, caps/floors, swaptions... then using 2 terms of Taylor expansion to estimate the P&L impact of a 99%+ move will be extremely inaccurate. (This is true not only for fixed income, but other options.) But you can run MC on PCs as described above. Ideally you would fully reprice your book under each MC scenario. But this would probably require too much computation. If you have precomputed your book's sensitivies to 1, 2, 3 standard deviations of the PCs, you can quickly interpolate the P&L impact of the PC scenarios in your MC.

• Thank you that was very informative. About the approach I mentioned producing higher VaR than what can be observed, why is that? Since I'm essentially fitting a normal distribution to the change in interest rates does that mean that interest rate changes have less heavy tails than the normal distribution (as opposed to equities having more heavy)? Also, so I'm understanding you right, just duration in a "delta VaR" approach is fine for nearly linear options like IRS, but using that approach even with both duration and convexity is still not enough for Swaptions, caps etc. usually? Jun 7 '20 at 14:48
• 1 consider for example a hypothetical book whose interest rate sensitivities are positive to 1bp increase in 1y, 3y, 5y swap rates; and negative to 1bp increase in 2y, 4y, 6y swap rates. (All these rates should have historical correlation above .8.) If your Monte Carlo simulates each rate independently then the scenarios, adverse to your book, where 1y, 3y, 5y go down and 2y, 4y, 6y go up look more frequent in the simulation, having the same as historical volatilties and correlations, then they have been historically or in a Monte Carlo that simulates principal components. Jun 7 '20 at 15:07
• 2 yes not just IR swaption - if the underlying is FX, stock, commodity, whatever - estimating the P&L of non-linear product when the underlying moves a lot using only the first 2 terms of the Taylor explansion is very likely to give P&L estimate very different from full revaluation. Just play with some numeric examples, you will see it right away. Jun 7 '20 at 15:10

I've also had to deal with something similar in one of my fixed income projects. I suggest you to, like you said, simulate interest rate curves (the simplest one would be vasicek), price/calculate NPV according to those generated curves, and calculate the portfolio P&L over each scenario. The VaR would be the nth% percentile according to your confidence interval. Using the "delta" approach, I advise the BPV method. Market Risk Analysis from Alexander (2008) may give you a good understanding over this particular topic. You may be able to use it if your portfolio is composed by vanilla bonds.