Attributing P&L in a Duration-Neutral Swap Spread Trade
Summary
The document explains how to estimate P&L for a position long a government bond and paying fixed on an interest rate swap. It starts with each leg’s DV01 and its yield change: multiplying them gives the bond and swap P&Ls, which sum to the trade’s total. A numerical example shows that the two legs can have slightly different rate sensitivities and moves, leaving both spread exposure and residual outright-rate exposure.
The same total P&L can be expressed using different risk coordinates. One convention assigns the bond’s risk to the swap spread and leaves residual swap risk; another assigns swap risk to the spread and leaves residual bond risk. The resulting component figures differ, though the total agrees. The document says spread positions are generally designed to be duration neutral, while noting that the right hedge depends on the risk measure and modeling choices. Its examples are approximate and do not cover more elaborate curves or portfolios.
Key ideas
- Estimate each leg’s P&L from its DV01 and the relevant yield move.
- A bond and swap position can retain residual outright rate exposure when their DV01s differ.
- Different risk-coordinate conventions allocate P&L components differently while preserving total P&L.
- Swap spread trades are generally structured to reduce duration exposure, but the hedge depends on risk measures.
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# Calculating PnL on Swap Spread Trade
# Calculating PnL on Swap Spread Trade
Say I am a long a 10Y US govt. bond and short (i.e., pay fixed) a 10Y US IRS. The swap spread moves, for example, from -0.25 to -0.3. What do I need to calculate P&L (approximately)?
Are such swap spread trades typically made duration neutral by weighting appropriately according to the duration of the two legs?
I'm new to relative-value/swap spreads so would appreciate any relevant information and links.
Thanks, V
## Answer by Attack68 (score 3, accepted)
https://quant.stackexchange.com/a/40874
The fundamental underlying PnL you have is PnL on a bond and PnL on a swap, but you can choose to arbitrarily allocate this in different perspectives.
Say you have the following DV01s: Bond +102, Swap -99, and say the market movements are: Bond +2bp, Swap: +1.7bp. The corresponding PnLs are: Bond +204, Swap -168, Total: +36.
Your question thus becomes how do you allocate this PnL to a "swapspread" instrument. Let me give you a mathematical representation...
## Basic Outright Instruments
Risk: $\begin{bmatrix} Bond \\ Swap \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \begin{bmatrix} +102 \\ -99 \end{bmatrix}$, Change: $\begin{bmatrix} Bond \\ Swap \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \begin{bmatrix} +2.0 \\ +1.7 \end{bmatrix}$ Total PnL = $Risk \cdot Change = \begin{bmatrix} +102 \\ -99 \end{bmatrix} \cdot \begin{bmatrix} +2.0 \\ +1.7 \end{bmatrix} = \sum \begin{bmatrix} +204 \\ -168.3 \end{bmatrix}= +35.7$
## Swapspread and Swap Instrument
This model attributes all bond risk to swapspread and any residual is allocated to swap delta. Risk: $\begin{bmatrix} Swapspd \\ Swap \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 1 & 1 \\ \end{bmatrix} \begin{bmatrix} +102 \\ -99 \end{bmatrix}$, Change: $\begin{bmatrix} Swapspd \\ Swap \end{bmatrix} = \begin{bmatrix} 1 & -1 \\ 0 & 1 \\ \end{bmatrix} \begin{bmatrix} +2.0 \\ +1.7 \end{bmatrix}$ Total PnL = $Risk \cdot Change = \begin{bmatrix} +102 \\ +3 \end{bmatrix} \cdot \begin{bmatrix} +0.3 \\ +1.7 \end{bmatrix} = \sum \begin{bmatrix} +30.6 \\ +5.1 \end{bmatrix} = +35.7$
## Swapspread and Bond Instrument
This model attributes all swap risk to swapspread and any residual is allocated to bond delta. Risk: $\begin{bmatrix} Bond \\ Swapspd \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 0 & -1 \\ \end{bmatrix} \begin{bmatrix} +102 \\ -99 \end{bmatrix}$, Change: $\begin{bmatrix} Swapspd \\ Swap \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 1 & -1 \\ \end{bmatrix} \begin{bmatrix} +2.0 \\ +1.7 \end{bmatrix}$ Total PnL = $Risk \cdot Change = \begin{bmatrix} +3 \\ +99 \end{bmatrix} \cdot \begin{bmatrix} +2 \\ +0.3 \end{bmatrix} = \sum \begin{bmatrix} +6.0 \\ +29.7 \end{bmatrix} = +35.7$
Therefore depending upon your perspective (and each one is equally valid) you attain different numbers. As a swap trader I employed the second one, but I sat next to bond traders who preferred the third option, primarily since the outright delta is expressed in the native liquid hedge of the product traded. As for your question specifically about swapspreads - yes they are attempted to be duration neutral but due to different types of risk measure this can be quite nuanced how to actually do this right.
Note calculating bond DV01 and including more bonds and swaps in your model makes it more complicated but completely do-able. A good reference for this kind of model building is Darbyshire: Pricing and Trading Interest Rate Derivatives.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.