# Calculating the cumulative probability of default from recovery rate, yield and coupon rate

I have the following details: A 10-year U.S.Treasury strip has a yield of 6% and a 10-year zero issued by XYZ Inc, rated A by S&P and Moody's, has 7% (semi-annual compounding). Assuming a recovery rate of 45% What is the cumulative probability of XYZ Inc, defaulting during next 10 years?

How do I calculate the cumulative probability?

## 1 Answer

Let us denote with $$r_f$$ the yield of the 10 U.S.Treasury strip and $$r_{A}$$ the yield of the risky bond issued by XYZ Inc. We denote with $$p$$ the cumulative default probability, with $$P$$ the bond face value, with $$R$$ the recovery rate and with $$T$$ the bond maturity. In the absence of arbitrage, we have

$$\dfrac{(1-p)\times FV+p \times R \times FV}{ \left(1+\frac{r_f}{2} \right)^{2 \times T}}=\dfrac{FV}{ \left(1+\frac{r_A}{2} \right)^{2 \times T}}$$

Solving the above equation for $$p$$, we get the implied default probability. In your specific example, we get:

• Thank you so much for this explanation. I noticed you mentioned 'in the absence of arbitrage', how would this effect the value?
– May
Dec 13, 2020 at 12:52
• Note that the cumulative default probability in your case is a risk-neutral one. We discount with the risk-free rate the contingent bond payoff at maturity. This is not the physical cumulative default probability, because in the valuation of bonds other factors are priced as well (for instance, liquidity risk, the taxation, demand/supply pressures) Dec 14, 2020 at 13:45