I'm familiar with the expression for a (continuously compounded) credit spread of the form $$ c(t,T) = -\frac{1}{T-t} \ln \frac{v(t,T)}{p(t,T)},$$ where $p(t,T)$ denotes the time $t$ price of a $T$-maturity default-free bond, and $v(t,T)$ denotes the time $t$ price of a $T$-maturity defaultable bond. Using some standard assumptions from credit risk models (that the default time $\tau$ is independent of the short rate $r_t$ process so the $v(t,T)=p(t,T)\left( \delta +(1\color{red}{-}\delta) \mathbb{Q}(\tau > T)\right)$), the spread can be written as $$c(t,T) = - \frac{1}{T-t} \ln \left( \delta +(1\color{red}{-}\delta) \mathbb{Q}(\tau > T)\right), $$ where $\delta$ is the recovery rate and $\mathbb{Q}(\tau > T)$ the risk-neutral probability of the defaultable bond's issuer survival.

I came across the following expression for the "semi-annually compounded" spread, given by $$c(t,T)=2\left[ (\delta +(1-\delta))\mathbb{Q}(\tau > T)) ^{\color{red}{-}\frac{1}{2T}}-1 \right].$$

I don't understand how is this expression derived, could somebody explain it to me? The way I see it, the semi-annually compounded spread should be equal to the difference $y_v-y_p$, where yields $y_v$ and $y_p$ are given implicitly through: $$p(t,T)\left( 1+\frac{y_p}{2} \right)^{2T}=1, $$ and $$v(t,T)\left( 1+\frac{y_v}{2} \right)^{2T}=1. $$ In that case, shouldn't spread be equal to: $$ c(t,T)= 2 \left[ p(t,T)^{-\frac{1}{2T}} \left( 1- \left( \delta +(1+\delta) \mathbb{Q}(\tau > T)\right)^{-\frac{1}{2T}} \right) \right] ? $$

I appreciate any insights on this, many thanks.


1 Answer 1


This is all based on how you define the spread. In continuous compounding case, you can define the spread $c(t, T)$ by the formula \begin{align*} v(t, T) = e^{-c(t, T) (T-t)} p(t, T). \end{align*} While, in the semi-annual compounding case, by the formula \begin{align*} v(t, T) = \left(1+\frac{c(t, T)}{2}\right)^{-2 (T-t)} p(t, T). \end{align*}


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.