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:
A) 0.55
B) 0.60
C) 0.65
D) 0.70

Question

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:
A) 0.55
B) 0.60
C) 0.65
D) 0.70

Quizplus · Accepted Answer

To find the price of a default-risk-bearing three-year unit face value zero-coupon bond, we first calculate the probability of the bond not defaulting over the three years. Using the given transition matrix $Q$, the probability of staying non-default after one year is 0.90. Since the matrix shows that once defaulted, the bond stays in default (with a transition probability of 1 to stay in default), we can calculate the probability of not defaulting over three years as $0.90^3 = 0.729$.Given the risk-free rate is 4% (continuously compounded), the risk-free price of a three-year zero-coupon bond with a face value of 1 is calculated using the formula $P = e^{-rt}$, where $r$ is the risk-free rate and $t$ is the time in years. Substituting $r = 0.04$ and $t = 3$, we get $P = e^{-0.04*3} = e^{-0.12} \approx 0.8869$.The price of the default-risk-bearing bond is then the product of the risk-free price and the probability of not defaulting, which is $0.8869 * 0.729 \approx 0.646$, which rounds to 0.65, making option C the correct answer.

There Are Two Ratings in a Very Simple World: Non-Default Q=[0.900.1001]Q = \left[ \begin{array} { c c } 0.90 & 0.10 \\0 & 1\end{array} \right]Q=[0.900​0.101​]

There Are Two Ratings in a Very Simple World: Non-Default $Q = \left[ \begin{array} { c c } 0.90 & 0.10 \\0 & 1\end{array} \right]$