Defining Credit Spreads Under Semiannual Compounding
Summary
The document asks how to express a defaultable bond’s credit spread when yields are quoted with semiannual compounding. It begins with the continuously compounded spread as the rate that relates the defaultable bond price to the default-free bond price over the time to maturity. Under stated assumptions about recovery, survival, and independence from the short rate, the question also relates the bond price ratio to risk-neutral survival probability.
The response emphasizes that a spread depends on its compounding convention. For continuous compounding, the spread is defined through an exponential discount factor applied to the default-free bond price. For semiannual compounding, the corresponding relationship uses a periodic factor raised to twice the remaining maturity. This provides the defining equation from which a semiannually compounded spread can be solved. The exchange does not work through the algebra or resolve apparent errors in the question’s proposed formulas, so care is needed with signs, parentheses, recovery conventions, and the use of time to maturity rather than calendar maturity.
Key ideas
- A credit spread is defined relative to a default-free bond price and a chosen compounding convention.
- Continuous compounding expresses the price ratio with an exponential discount factor.
- Semiannual compounding expresses the ratio using a periodic factor applied twice per year.
- Use the time remaining to maturity consistently when setting the exponent.
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# Right expression for the semi-annually compounded credit spreads?
# Right expression for the semi-annually compounded credit spreads?
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.
## Answer by Gordon (score 0, accepted)
https://quant.stackexchange.com/a/29497
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*}Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.