Probability of default

We can model the probability of default based on the expected payoff. The expected value of the risky bond investment, should be equal to the value of the risk-free treasury bond.

Thus, in the initial case, where the bond is paying 3.5%, let the company have a default probability of $p$ . We then have

$p \times 0 + ( 1-p) \times ( 1.035 ) = 1.02$

Solving this gives us $p = 0.0145$ .

Subsequently, when the interest rate increased to 5.5%, let the company have a default probability of $q$ . We then have

$q \times 0 + ( 1 - q) \times ( 1.055 ) = 1.02$

Solving this gives us $q = 0.0332$ .

Hence, the implied increase in probability is $3.32 \% - 1.45 \% = 1.87 \%$