# 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?

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: