# Fixed Income VaR: Yield Vol vs Cash Flow Mapping

I have come across two ways of measuring VaR for Fixed Income instruments thus far:

• Express the volatility in of basis points and the position in terms of sensitivity to a 1 basis point movement in yields and then multiply it by the desired largest possible movement (95% or 99%); This method is described on page 17 of this document.

• Map the cash flows of an instrument (a coupon bond) into buckets, get the zero rates (interpolate if needed), find PV01, volatility of these zero rates, get the correlation matrix for the zero rates, find the total variance and calculate the VaR.

The first one seems relatively simple. However, what happens if there are many bonds in the portfolio? Is it OK to find individual VaRs for each bond and then simply sum them up?

On the other hand, the second approach does deal with covariance of rates in different time horizons. But there might be a small error due to bucket specification. Theoretically, we could get an infinite number of buckets. But this is obviously very complicated.

Any thoughts?

• Any ideas/thoughts?
– AK88
Apr 15, 2017 at 7:05
• I'm not a 100% sure what your actual question is. The first method can be used for a portfolio of many bonds by summing up the BPVs of all bonds first and then calculate the VaR. Method 1 is a special case of Method 2 only with a single bucket, if I understood you correctly. Apr 18, 2017 at 22:44
• In the first methodology (or in general), you would certainly not sum up the VaRs. Suppose you're long some bond (the VaR comes from scenarions where yields go up), long another bond (VaR from scenarios where yields go down), the VaR of the portfolio is the sum only if the correlation of the yields is -1, which is not realistic. Assuming correlation 1 is better. Assuming some correlation <1 is even better. Aug 8, 2019 at 14:49
• I took a look at Ken Abbott's MIT lecture notes that you cited. I think it's wrong to suggest that if you know the sensitivity (first order, delta) to a 1 basis point in yield, you can just linearly extrapolate the impact of a couple of standard deviations, which might be hundreds of basis points. You should somehow consider second order (convexity, gamma). Aug 8, 2019 at 17:01

According to the RiskMetrics cash flow mapping method for the bond portfolio, the correlation matrix(not covariance matrix) includes the key time(1m, 3m, 6m, 1y ...50y etc.) zero coupon rates(or spot rates). Since all the actual cash flows can be mapped to 2 cash flows on the nearest 2 key time points, you only need the key time correlation matrix.

The first approach like you said is simplistic, and the second is just a generalisation to multiple factors/assets. Note sensitivity and PV01 represent same concept. If you use 1 for multiple assets/factors, and then add their VaR, you are essentially assuming that all rates are perfectly correlated (called undiversified VaR as you are not accounting for diversification.

As you said, the second approach accounts for this diversification by bringing in the covariance, so it is just a generalisation of 1.

You can have as many factors as you like but normally you are limited by the availability of liquid market data, and computational resources.