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?

• Commonly, you may assume a certain market-price-of-risk for credit default risk (and other risks) priced into the traded credit spread of an instrument. I think you could take eq. 3.3 on p 70 of this here mediatum.ub.tum.de/download/736705/736705.pdf as a reference / starting point. Aug 2, 2021 at 12:26
• Thanks, Kermittfrog. However, I was looking after a back-of-envelope calculation given the above transitions and default rates. For instance, the price change of a bond would be expressed as -duration*(spread changes given rating change)*probability of rating change Aug 2, 2021 at 12:37
• Could you please provide your own ansatz to your question so that we could discuss the goal you're after? The way it is stated, I do not fully understand your question, sorry. Aug 3, 2021 at 12:54
• Given the 1-year prob. of rating migrations and 1-year default rates I would like to compare the credit spread of BB, B and CCC indices. Outright or market spread may be 300, 450 and 700 respectively but if I should incorporate the transition matrix and defaults so I can compare the spreads across ratings like-for-like/adjusted for defaults. So adjusted for defaults the spread may be more similar, for instance 260, 330 and 300 respectively. I tried matrix multiplication between the two vectors (transition mtx and 1-year default) and multiply by the outright spread, but would that correct? Aug 3, 2021 at 15:34

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?

• I appreciate your help, Kermittfrog. Could you explain how I could use the last formula you have presented to calculate the adjusted spread for B bonds given the numbers in the question? Aug 2, 2021 at 14:34