Zero Coupon Bond Default
Formula Reference
Default probability in terms of price
\begin{eqnarray}\label{eqA10}
p & = & \frac{1 - \alpha}{1 - R}\\
\alpha & = & \frac{P}{P_0}\nonumber\\
P & = & \mbox{price of the bond}\nonumber\\
P_0 & = & \mbox{price of the risk free bond}\nonumber\\
R & = & \mbox{recovery rate, between 0 and 1}\nonumber
\end{eqnarray}
Default probability in terms of discount rate
\begin{eqnarray}\label{eqA20}
p & = & \frac{1 - \alpha}{1 - R}\\
\alpha & = & \frac{1 - r_1 \frac{T}{360}}{1 - r_0 \frac{T}{360}}\nonumber\\
r_1 & = & \mbox{discount rate of the bond}\nonumber\\
r_0 & = & \mbox{discount rate of the risk free bond}\nonumber\\
T & = & \mbox{number of days to maturity}\nonumber\\
R & = & \mbox{recovery rate, between 0 and 1}\nonumber
\end{eqnarray}
Approximate default probability in terms of discount rate
\begin{eqnarray}\label{eqA30}
p & = & \frac{(r_1 - r_0)\frac{T}{360}}{1 - R}\\
r_1 & = & \mbox{discount rate of the bond}\nonumber\\
r_0 & = & \mbox{discount rate of the risk free bond}\nonumber\\
T & = & \mbox{number of days to maturity}\nonumber\\
R & = & \mbox{recovery rate, between 0 and 1}\nonumber
\end{eqnarray}
Default probability in terms of yield
\begin{eqnarray}\label{eqA40}
p & = & \frac{1 - \alpha}{1 - R}\\
\alpha & = & \frac{1 + y_0 \frac{T}{360}}{1 + y_1 \frac{T}{360}}\nonumber\\
y_1 & = & \mbox{yield of the bond}\nonumber\\
y_0 & = & \mbox{yield of the risk free bond}\nonumber\\
T & = & \mbox{number of days to maturity}\nonumber\\
R & = & \mbox{recovery rate, between 0 and 1}\nonumber
\end{eqnarray}
Approximate default probability in terms of yield
\begin{eqnarray}\label{eqA50}
p & = & \frac{(y_1 - y_0)\frac{T}{360}}{1 - R}\\
y_1 & = & \mbox{yield of the bond}\nonumber\\
y_0 & = & \mbox{yield of the risk free bond}\nonumber\\
T & = & \mbox{number of days to maturity}\nonumber\\
R & = & \mbox{recovery rate, between 0 and 1}\nonumber
\end{eqnarray}
Converting discount rate to yield
\begin{eqnarray}\label{eqA60}
y & = & \frac{r}{1 - r\frac{T}{360}}\\
r & = & \mbox{discount rate}\nonumber\\
T & = & \mbox{number of days to maturity}\nonumber\\
\end{eqnarray}
Converting yield to discount rate
\begin{eqnarray}\label{eqA70}
r & = & \frac{y}{1 + y\frac{T}{360}}\\
y & = & \mbox{yield}\nonumber\\
T & = & \mbox{number of days to maturity}\nonumber\\
\end{eqnarray}