Estimating Credit Spreads from Default Risk under a Merton Model

# Credit spreads adjusted for rating migration and default


# Credit spreads adjusted for rating migration and default












Given the below 1-year rating transition matrix and cumulative default rates, I am interested in calculating credit spread adjusted for defaults so I can compare this with the outright credit spread.

So for instance, if the Euro credit spreads were BB 250, B 450 and CCC 700, how would the outright spread of 450 for B rated credits be compared to the credit spread adjusted for rating transition and default?

## Answer by Kermittfrog (score 1)

https://quant.stackexchange.com/a/66269

According to the setup in Bluhm/Overbeck/Wagner (2003) An Introduction to Credit Risk Modelling, you could follow KMV's Merton style credit default risk approach (eq. 6.16 ff. in that version)

Under the Merton model, the (physical) cumulative default probability from time zero up to time $t$, $\mathrm{PD}_t^{real}$ is given by the probability that the asset process $A_t$ will fall short of the liability level $C$ after some time $t$:

$$ \mathrm{PD}_t^{real}=N\left(\frac{\log(A_0/C)+(\mu-\sigma^2/2)t}{\sigma\sqrt{t}}\right) $$

and in the risk neutral world, this would result in the risk-neutral default probability $\mathrm{PD}_t^{rn}$,

$$ \mathrm{PD}_t^{rn}=N\left(\frac{\log(A_0/C)+(r-\sigma^2/2)t}{\sigma\sqrt{t}}\right) $$ where the expected asset return $\mu$ has been replaced by the risk free rate $r$. If you substitute the two equations into each other and re-aarange, you come up with equation 6.18 of that book:

$$ \mathrm{PD}_t^{rn}=N\left(N^{-1}\left(\mathrm{PD}_t^{real}\right)+\frac{\mu-r}{\sigma}\sqrt{t}\right) $$ I.e.: The risk neutral default probability equals the empirical PD plus a risk-adjustment (under the model).

The book then explains how you could come up with an estimate of the premium.

For now, we simply let $\pi \equiv \frac{\mu-r}{\sigma}\sqrt{t}$ and thus $$ \mathrm{PD}_t^{rn}=N\left(N^{-1}\left(\mathrm{PD}_t^{real}\right)+\pi\right) $$

We know that the credit-risky present value of a zero coupon bond equals

$$ \begin{align} e^{-(r_t+s_t)t}&=e^{-rt}\left[(1-\mathrm{PD}_t^{rn})+(1-LGD)\mathrm{PD}_t^{rn}\right]\\ &=e^{-rt}\left[1-LGD\times\mathrm{PD}_t^{rn}\right] \end{align} $$

Thus, our spread equals $$ s_t=-\frac{1}{t}\ln\left(1-LGD\times N\left(N^{-1}\left(\mathrm{PD}_t^{real}\right)+\pi\right)\right) $$

You could use this expression to derive at some approximation for the change in spread due to a change in cumulative default probability. To a very first approximation and under the strong assumption that $\pi=0$, you'd get

$$ \mathrm{d}s_t=\frac{e^{s_tt}}{t}LGD \times \mathrm{d}\mathrm{PD}_t^{real}\approx \frac{LGD}{t}\times \mathrm{d}\mathrm{PD}_t^{real} $$

You could then, of course, also assess the change in PD due to a change in year-on-year PD or the like.

NB To be clear, the empirical default data alone is not sufficient to come up with an average impact of rating transitions on spread levels, even when we assume that it only credit events that drove spreads in the first place. You'd need to (at least) set up and identify/calibrate some model that connects physical and risk neutral PDs or spreads.

HTH?