Spread Duration in Credit: Survival-Weighted PV and CDS Sensitivity

# Definition of spread duration


# Definition of spread duration












is there anyone who can explain the concept of spread duration, both from a mathematical point of view and an intuitive one? Providing a practical example would be highly appreciated. Thanks in advance.

## Answer by Mehness (score 3)

https://quant.stackexchange.com/a/31160

will try and describe a simplified mathematical description, and hopefully some of it will be intuitive :)

I am going to use a definition of spread duration used throughout credit markets at least - this may not be what you are getting at so do let me know if this is the case.

Spread duration is a risky duration, that is, the survival probability adjusted discount factor - weighted present value of a one basis point annuity / coupon paid on the bond. (This is almost the same things as price sensitivity, at least to a first order approximation as we'll see).

Spread duration or risky present value of a basis point is given by: $PV_{BP_{risky}} = \displaystyle{\int_0^Tc_t\,\mathbb{P}(\tau\geq t)}D_{0,t}\mathrm{dt} =\int_0^T{c_te^{\int_0^t{\lambda_sds}}e^{\int_0^t{r_sds}}\mathrm{dt}}$

where $D_{0,t}$ is the credit riskless discount, $c_t$ is the coupon stream (assumed continuous for simplicity) pf 1bp and $\mathbb{P}(\tau\geq t)$ is the $t$ survival probability. Please note - to make things simple I am assuming here that credit and rates are uncorrelated - in many applications this is NOT suitable. Am also ignoring accrued for clarity of exposition.

Let's now make things even simpler and assume a flat hazard rate and flat interest rate 'curve'. We then have simply that:

$\displaystyle{PV_{BP_{risky}} =\int_0^T{e^{(\lambda+r)t}\mathrm{dt}} }=\left\{\frac{1-e^{(\lambda+r)T}}{\lambda+r}\right\}$

This is a nice 'quick and dirty' formula for the risky duration which is surprisingly useful for calculating CDS MTMs or CR01 (which latter is the PV change. Now, let's have a look at a CDS MTM:

$PV_{CDS}=PV_{loss\,leg}-PV_{premium\,leg} =(1-R)\int_0^T{\lambda e^{(\lambda+r)t}\mathrm{dt}} -\kappa\int_0^T{e^{(\lambda+r)t}\mathrm{dt}}$

The loss leg integrand houses the survival density, $\kappa$ is the CDS premium or strike as it's often called, and the whole thing reduces to the familiar:

$PV_{CDS}=\{\lambda(1-R)-\kappa\}\cdot\left\{\frac{1-e^{(\lambda+r)T}}{\lambda+r}\right\}$

This is just $(par\,spread - strike)*risky\,duration$

Given this we can see that for CDS, a 1bp move in the spread does indeed deflect the MTM by one unit of the spread duration, but for the fact that the duration is negatively convex for a bought protection position. As the hazard rate widens, the risky duration shortens. A name moving from say 100bp to 500bp can easily lose 15% of its delta on say a 5y CDS struck at 1%, which is material.

Finally then just to write down a simple formula in a similar vein for a risky bond, we have:

$\displaystyle{PV_{risky\,bond} =\kappa\int_0^T{e^{(\lambda+r)t}\mathrm{dt}} +e^{-(\lambda+r)T}+R\int_0^T{\lambda e^{(\lambda+r)t}\mathrm{dt}}}$

This is risky coupon + principal on survival + recovery on default (integrated against density again). The quick and dirty formula here is:

$PV_{risky\,bond}=\{\kappa+R\lambda\}\cdot\left\{\frac{1-e^{(\lambda+r)T}}{\lambda+r}\right\}+e^{-(\lambda+r)T}$

Once again you can see the risky duration term, enclosed in $\{\cdot\}$ making an appearance, however this time we have a different default payoff of $R$ and not $1-R$, together with a term survival principal.

Hope that's a helpful summary of some simplified credit pricing showing where spread duration crops up. The reduced form formulae are handy for playing around with in a spreadsheet. Probably ridden with typos I'm afraid so caveat lector, cheers!