A corporate bond has a cumulative probability of default equal to 20% in the first...

Note that marginal probabilities of default are the probabilities for default for a given period, conditional on survival till the end of the previous period. Cumulative probabilities of default are probabilities of default by a point in time, regardless of when the default occurs. If the marginal probabilities of default for periods 1, 2... n are p1, p2...pn, then cumulative probability of default can be calculated as Cn = 1 - (1 - p1)(1-p2)...(1-pn).

For this question, we can calculate the marginal probability of default for year 2 by solving the equation [1 - (1 - 20%)(1 - P2) = 45%] for P2. Solving, we get the marginal probability of default during year 2 as 31.25%. Since this is the annual marginal probability of default, we will need to convert it to a monthly number, which we can do by solving the following equation where M1 is the monthly marginal probability of default.

1 - 31.25% = (1 - M1)^12, implying M1 = 3.07%