Fixed-Income VaR: Yield Volatility and Cash-Flow Mapping
Summary
The document compares two ways to estimate value at risk for a fixed-income portfolio. A simple method combines yield volatility, measured in basis points, with position sensitivity such as PV01. For multiple bonds, summing individual VaRs assumes their rate exposures move in perfect correlation, so it does not reflect diversification.
A more general cash-flow mapping method assigns bond cash flows to key maturities, estimates sensitivities to zero rates, and combines rate volatilities using their correlation structure. Mapping cash flows to nearby maturity points makes the calculation manageable, though bucket choices introduce approximation error. The discussion emphasizes that the number of risk factors is practically constrained by available liquid market data and computing capacity. It offers a conceptual comparison rather than portfolio examples, calibration guidance, or empirical VaR results.
Key ideas
- Summing individual bond VaRs assumes perfect correlation across their rate exposures.
- PV01 and sensitivity describe exposure to a small yield change.
- Cash-flow mapping represents a bond portfolio through sensitivities to zero rates at selected maturities.
- A correlation matrix captures diversification across rate horizons.
- Maturity buckets simplify calculations but introduce mapping approximation.
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# Fixed Income VaR: Yield Vol vs Cash Flow Mapping # 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? ## Answer by Aria (score 1) https://quant.stackexchange.com/a/42152 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. ## Answer by Magic is in the chain (score 1) https://quant.stackexchange.com/a/42163 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.
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