Exam 31: Reduced-Form Models of Default Risk

There are two ratings in a very simple world: non-default (ND) and defaultd. The risk-neutral rating transition matrix per year is given by: $Q = \left[ \begin{array} { c c } 0.90 & 0.10 \\0 & 1\end{array} \right]$ i.e., the probability of defaulting when the current state is non-default is 0.10, and a defaulted bond never leaves that state and has zero recovery. The three-year zero-coupon risk-free rate is 4% (continuously-compounded). The price of a default-risk-bearing three-year unit face value zero-coupon bond is:

The probability of a firm defaulting each year, given that it has not defaulted in prior years is 10%. What is the probability that it will have defaulted at some time in the first 10 years? Approximately:

If the rate of defaults per year in a set of companies is given by $\lambda = 5$ , what is the probability of four or more defaults in half a year?