Approximating CDS PnL with Coupon-Adjusted Risky Duration
Summary
The document outlines a first-order way to estimate the mark-to-market PnL of a CDS as its maturity shortens and its market spread changes. From the protection buyer’s perspective, it expresses value using the difference between the market spread and the contractual coupon, multiplied by CS01. Comparing the current contract value with the original value gives an approximate trade PnL, with the example accounting for a roll from a five-year to a four-year tenor.
It motivates the approximation by defining CS01 as the discounted, survival-weighted risky duration of the contract. Under constant rates and hazard, that integral can be simplified, and for sufficiently small combined rates it is approximated by time to maturity. The worked estimate assumes this first-order treatment and supplies a numerical result. It is a quick approximation, not a full valuation: clean versus dirty CS01, accrued coupon treatment, changing curves, and more exact survival and discount assumptions can affect realized PnL.
Key ideas
- CDS mark-to-market can be approximated using the market spread less the contractual coupon, multiplied by CS01.
- CS01 represents discounted risky duration and can be expressed as an integral over survival-weighted discount factors.
- With constant rates and hazard, CS01 has a closed-form expression.
- For small combined rates, time to maturity provides a first-order CS01 approximation.
- A PnL estimate compares coupon-adjusted values at the original and current maturities.
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Full text
# CDS Credit Default Swap PnL
# CDS Credit Default Swap PnL
I estimate daily pnl on a CDS position using the spread change times the CS01.
However I would like to estimate the PnL for a longer trade that has gone from a 5Y CDS to a 4Y with associated coupon payments.
Lets consider:
- Trade date 2018-08-01: Sell Protection Nominal 1,000,000 at 455 Spread on 5Y CDS maturity Jun 23
- Coupon 500 Bps
- Current 5Y Spread 415
- Current 4Y Spread 336
How can I calculate the current PnL for this trade?
## Answer by reheno (score 1)
https://quant.stackexchange.com/a/47592
You could estimate a CDS MtM from the protection buyer's perspective by `MtM = (s-c)CS01`. This would be a clean or dirty MtM depending on whether the CS01 is clean or dirty. For reasonable levels of spreads and interest rates, we can approximate the CS01 with the time to maturity. This should allow you to calculate a quick approximation of the PnL using the data you have.
CS01 is essentially a risky duration. It's the dollar value of one spread/coupon unit. To get to the approximation above, let's consider continuously compound interest and hazard rate. $DF_t$ and $Q_t$ respectively denote discount factor and probability of survival.
$$ \begin{eqnarray*} CS01 &=& \int_0^T DF_t . Q_t dt\, \\ \end{eqnarray*} $$
Let's also consider constant interest rate r and constant hazard rate $\lambda$ over the life of the contract.
$$ \begin{eqnarray*} CS01 &=& \int_0^T e^{-(r+\lambda)t}dt\, \\ &=& \frac{1}{\lambda + r} [ 1 - e^{-(\lambda + r)T}]\, \\ \end{eqnarray*} $$
Now let's assume $\lambda + r$ is small enough
$$ \begin{eqnarray*} CS01 &\approx& T - \frac{\lambda + r}{2} T^2 .... \end{eqnarray*} $$
Getting back to the original question, and sticking to a first order approximation of the CS01. From the perspective of the protection buyer :
$PnL \approx 10^6 * [ (336 - 500) * 4 - (455 - 500) * 5 ] * 10^{-4} \approx -144k $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.