CDS Par Spread Derivation Under Constant Hazard Rate
Consider a par Credit Default Swap (CDS) with maturity $T$. The premium leg pays continuously at rate $s$ until default or maturity. The protection leg pays $1 - R$ at the time of default. Assume a constant hazard rate $h$, constant recovery rate $R$, and constant risk-free rate $r$, all under the risk-neutral measure.
(a) Derive the par spread $s$ in closed form as a function of $r$, $h$, $R$, and $T$.
(b) Show that as $T \to \infty$, the par spread satisfies $s \to (1 - R)h$. State any assumptions you use.
(c) Given $R = 40\%$ and an observed par spread $s = 180$ bps for a long-maturity CDS, estimate the hazard rate $h$.
Hints
- Write the PV of each leg as an integral against the discount factor $e^{-rt}$ and the survival probability $Q(t) = e^{-ht}$. Both integrals have the same structure.
- Both legs share the same annuity factor $\frac{1 - e^{-(r+h)T}}{r+h}$, which cancels when you set them equal. This is why the par spread has such a clean form.
- For part (c), just invert the asymptotic formula: $h = s / (1-R)$. Convert 180 bps to a decimal first.
Worked Solution
How to Think About It: A CDS is just a swap where one leg compensates for credit losses and the other pays a running premium for that protection. At par, the present value of premiums paid equals the present value of the protection received. The math simplifies enormously under a constant hazard rate because the survival probability is just $e^{-ht}$ -- an exponential decay. So both legs reduce to integrals of exponentials, which are elementary. Before diving in, note the limiting intuition: if you extend the maturity to infinity, boundary effects vanish, and the spread must just reflect the instantaneous expected loss rate, which is $(1-R)h$.
Quick Sanity Checks: The par spread should increase with $h$ (more default risk means higher premium), increase with $1-R$ (lower recovery means bigger losses to insure), and be independent of notional. For long maturities, the spread should flatten out -- it should not blow up or go to zero. The answer $s = (1-R)h$ in the limit is dimensionally correct (both sides are rates in units of per-year).
Derivation:
(a) Par Spread in Closed Form
The survival probability to time $t$ is $Q(t) = e^{-ht}$, so the default density is $f(t) = h e^{-ht}$.
Premium leg PV: The protection buyer pays $s$ continuously, but only while alive. Discounting at rate $r$:
$$PV_{\text{prem}} = s \int_0^T e^{-rt} Q(t) \, dt = s \int_0^T e^{-(r+h)t} \, dt = s \cdot \frac{1 - e^{-(r+h)T}}{r + h}$$
Protection leg PV: The protection seller pays $1 - R$ at default (if default occurs before $T$). The default density at time $t$ is $h e^{-ht}$, so:
$$PV_{\text{prot}} = (1-R) \int_0^T e^{-rt} h e^{-ht} \, dt = (1-R) h \int_0^T e^{-(r+h)t} \, dt = (1-R) h \cdot \frac{1 - e^{-(r+h)T}}{r+h}$$
Setting $PV_{\text{prem}} = PV_{\text{prot}}$ for the par condition:
$$s \cdot \frac{1 - e^{-(r+h)T}}{r+h} = (1-R) h \cdot \frac{1 - e^{-(r+h)T}}{r+h}$$
The annuity factors $\frac{1 - e^{-(r+h)T}}{r+h}$ cancel (they are strictly positive for $T > 0$), giving:
$$\boxed{s = (1-R)h}$$
Notice something striking: under continuous premium payment and constant parameters, the par spread does not depend on $r$ or $T$ at all. The discounting hits both legs identically, so it cancels.
(b) Asymptotic Behavior as $T \to \infty$
From part (a), we already have $s = (1-R)h$ exactly for all $T > 0$, so the limit is immediate:
$$\lim_{T \to \infty} s = (1-R)h$$
Assumptions used: - Constant hazard rate $h$ (time-homogeneous Poisson default process) - Constant risk-free rate $r > 0$ - Continuous premium payments (no accrual adjustment at default) - No counterparty risk - Recovery $R$ is a fixed fraction of notional, paid at default time
If any of these assumptions are relaxed (e.g., term structure of hazard rates, discrete premium payments with accrued interest), then $s$ would depend on $T$ and $r$, and the relationship $s = (1-R)h$ would only hold approximately for large $T$.
(c) Estimating $h$ from Market Data
For a long-maturity CDS, we use the asymptotic relationship:
$$s \approx (1-R)h \implies h \approx \frac{s}{1-R}$$
Plugging in $s = 180$ bps $= 0.0180$ and $R = 0.40$:
$$h \approx \frac{0.0180}{1 - 0.40} = \frac{0.0180}{0.60} = 0.03$$
So the implied hazard rate is $h \approx 3\%$ per year, or equivalently about 300 bps. This means the risk-neutral default intensity is 3% annually.
Practical Interpretation: The formula $s = (1-R)h$ is the single most important relationship in credit derivatives. Traders use it constantly to convert between spreads and implied default probabilities. If you see a 5-year CDS at 180 bps with standard 40% recovery, you immediately know the market is pricing about 3% annual default probability. The one-year survival probability is $e^{-0.03} \approx 97\%$, and the 5-year survival probability is $e^{-0.15} \approx 86\%$.
Answer: (a) $s = (1-R)h$ -- under constant hazard rate with continuous premium payments, the par spread is independent of both $r$ and $T$. (b) The limit $s \to (1-R)h$ as $T \to \infty$ is immediate from (a); with non-constant parameters, it requires $r > 0$ and bounded hazard rates so boundary terms vanish. (c) $h \approx 0.03 = 3\%$ per year.
Intuition
The formula $s = (1-R)h$ is the credit markets' equivalent of "price equals expected loss." Over any tiny interval $dt$, the probability of default is $h \, dt$ and the loss given default is $1-R$, so the expected loss rate is $(1-R)h \, dt$ -- and that is exactly what the premium leg must compensate. Under constant parameters with continuous payments, this local balance holds at every instant, so it also holds globally regardless of maturity or interest rates. The discount rate drops out because it affects both legs symmetrically.
This is why traders treat $s/(1-R)$ as a quick-and-dirty hazard rate estimate -- it works surprisingly well even when the constant-hazard assumption is only approximate. The main ways this breaks down in practice are discrete premium payments (quarterly with accrual), term structure effects in the hazard rate, and recovery uncertainty. But as a first-pass mental model for converting between spreads and default probabilities, $s \approx (1-R)h$ is the single most useful formula in credit.