Coupon Bond Default
Formula Reference
Default probability
Solve equation numerically for \(z\). Default probability is \(1-z/d\).
\begin{eqnarray}\label{eqA80}
P & = & \left(Iz + R(1-z/d)\right)\frac{1-z^N}{1-z} + z^N\\
P & = & \mbox{price per face value of the bond}\nonumber\\
I & = & \frac{C}{F}\nonumber\\
C & = & \mbox{coupon payment}\nonumber\\
F & = & \mbox{face value of the bond}\nonumber\\
R & = & \mbox{recovery rate, between 0 and 1}\nonumber\\
d & = & (1 + f)^{-1} = \mbox{coupon period discount factor}\nonumber\\
f & = & \mbox{risk free interest rate}\nonumber\\
z & = & p d\nonumber\\
p & = & \mbox{survival probability}\nonumber\\
1 - p & = & \mbox{default probability}\nonumber\\
N & = & \mbox{number of remaining coupon payments}\nonumber
\end{eqnarray}
Initial guess for z
\begin{eqnarray}\label{eqA90}
z & = & \frac{P - R}{P - R/d + I}\\
P & = & \mbox{price per face value of the bond}\nonumber\\
I & = & \mbox{coupon interest rate} = \frac{C}{F}\nonumber\\
R & = & \mbox{recovery rate, between 0 and 1}\nonumber\\
d & = & (1 + f)^{-1} = \mbox{coupon period discount factor}\nonumber\\
f & = & \mbox{risk free interest rate}\nonumber
\end{eqnarray}