Explain Interest-Rate Swap P&L with Curve Factors and Gamma
Summary
The document asks how daily profit and loss on an interest rate swap is normally attributed, proposing accrual carry, curve roll-down, and residual change in net present value as possible components. The response focuses on explaining curve-driven P&L for a linear product. A first-order approach multiplies sensitivities to hedging or curve-fitting instruments by their market moves; key-rate duration is offered as one way to express those sensitivities. Another view decomposes yield-curve moves into historical principal components and attributes results to the leading components and the unexplained remainder.
The answer adds that second-order rate risk can improve the explanation, especially for long-dated swaps, and notes that rate gamma can be represented across tenor pairs or summarized with delta weighting. Cross effects between rates and time may help further, while using futures for the short end can warrant sensitivities in both futures and swap-rate terms. Calendar or model changes can also affect P&L. The response does not prescribe a complete daily reporting template or quantify these effects.
Key ideas
- First-order P&L can be estimated from instrument sensitivities multiplied by market moves.
- Historical principal components can explain broad yield-curve movements and associated P&L.
- Rate gamma may matter for long-maturity swaps and reduce unexplained P&L.
- Futures-based short-end curves may call for attribution in both futures and swap-rate terms.
- Calendar and model changes can also affect reported P&L.
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# Interest rate swap Profit and loss attribution # Interest rate swap Profit and loss attribution I am wondering how the IRS daily PnL will normally be attributed from trader's/market risk perspective. Will it be broken down into three parts daily carry (Daily interest accrual), daily roll-down (Assuming static yield curve rolling down one day) and the daily change in yield curve (Any remaining change in the NPV) or there are other kinds of attribution? Thank you. ## Answer by Dimitri Vulis (score 2) https://quant.stackexchange.com/a/77903 Regarding the P&L impact of > daily change in yield curve For a linear product, you want to view this in multiple ways: 1- for each hedging/fitting instrument, you had this much delta to this instrument (or key rate duration if you prefer), and the instrument moved that much, and we just multiply the delta sensitivities by the changes for 1st order Taylor analysis 2- for more insight, explain the movements of the yield curve in terms of its historical Principal Components (PC; typically, the first 3), and attribute the P&L to the PCs and to any curve movement not explained by the first 3 PCs. Second order risks may help reduce the unexplained P&L (UPL) even for swaps that we normally think of as linear products. If your swaps have 20+ years to maturity, then you should consider the interest rate gamma. It would be more accurate to view the gamma as a tenors$\times$tenors matrix, but you can use a single number, risk-weighted by the deltas. Cross-gammas between interest rates and the time are not very material, but may help further minimize the UPL if you wish. If you use interest rate futures, rather than swap rates, to build the short end of your curve, then you may want to see the factor sensitivities and the P&L explain using both the interest rate futures and the swap rates. Although this is seldom needed, it is good to be able to quantify the P&L impact of environment/model changes, such as change in calendars, e.g. a Queen's funeral or a new King's coronation in London or a day of mourning for a former president in the U.S.
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