There are two ratings in a very simple world: non-default (ND)and default (D).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 default (D).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

The price of a default-risk-bearing three-year unit face value zero-coupon bond can be calculated by considering the probability of the bond not defaulting over the three years and discounting the face value by the risk-free rate. The probability of not defaulting over three years is the cube of the probability of staying in the non-default state for one year, which is $0.90^3 = 0.729$. The present value of receiving 1 unit in three years, discounted at a continuously compounded rate of 4%, is $e^{-0.04*3} = e^{-0.12}$. Therefore, the price of the bond is $0.729 * e^{-0.12} \approx 0.65$.

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 $Q = \left[ \begin{array} { c c } 0.90 & 0.10 \\0 & 1\end{array} \right]$