# How to derive the expected loss from the credit risk of a bond?

I am trying to work out a formula to derive the expected loss from the credit risk of a bond.

My idea is to tie the credit risk to credit valuation adjustment and derive the expected loss from there, but then how would I relate credit risk and credit valuation adjustment?

Help would be greatly appreciated.

Thanks in advance!

## 1 Answer

One way to estimate the credit risk of a bond is to look at the price of insuring the bond using a Credit Default Swap, which will cost roughly the spread between the bond's yield and the yield of a risk free bond with the same maturity (typically we use a government bond here).

Using simple assumptions, this leads us to the Credit Triangle, $$K = (1-R)\lambda$$ where $$K$$ is the credit spread, $$R$$ is the recovery-given-default (ie. what you expect to get back out of 100 if there's a credit event), and $$\lambda$$ is the probability of default per unit time. Given any two of these by the market, we can calculate the third.

Note that if we can only see $$K$$, we need to make some assumptions about either $$R$$ or $$\lambda$$ - the market will charge the same for a bond with high probability of default but low loss-given-default as it will for a bond with low probability of default but high loss-given-default.

If you want to calculate CVA of a contract expiring at time $$T_f$$, that is typically done using either the credit spreads from either CDS contracts or calculated spreads between the company's bonds and the risk free bonds to calculate values for $$R$$ and $$\lambda$$, and then solving $$CVA = (1-R) \int_0^{T_f} EPE(t) \cdot P(0,t) \cdot S'(t) dt$$ where $$S'(t)$$ is the derivative of the probability of survival until $$t$$ (depends on modelling assumptions, but this often assumed to be $$-\lambda e^{-\lambda t}$$), $$P(0,t)$$ is the discount factor to time $$t$$, and $$EPE(t)$$ is the expected positive exposute at time $$t$$. This complicates things a bit (especially of $$S$$ and $$EPE$$ are correlated...) and turns CVA calculations even for simple instruments into a derivtives pricing problem - the procedure is discussed in more depth in this article