# 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

## 1 Answer

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.