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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?

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that may be too long for a parametric method to estimate pnl. can't you reprice your cds with today's interest rate curve and cds spreads?

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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 $

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