Bootstrapping A CDS Curve

CDS contracts trade at several standard maturities — 1, 3, 5, 7, 10 years — each quoted with its own spread. A single spread alone can't tell you the hazard rate for every year up to that maturity individually; it only pins down some blend of hazard rates across the whole period. To recover a full term structure of hazard rates, year by year, you need the same recursive trick used for interest-rate curves: solve the nearest unknown hazard rate using only the spread quotes and hazard rates already found for shorter maturities, then move one maturity further out and repeat.

The bootstrap equation

A CDS is fair (worth zero at inception) when the expected present value of the premium leg equals the expected present value of the protection (default payment) leg:

$$s \sum_i \tau_i\, D(t_i)\, Q(t_i) = (1-R) \sum_i D(t_i)\, \big[Q(t_{i-1}) - Q(t_i)\big],$$

where $s$ is the quoted CDS spread, $D(t_i)$ the discount factor, $Q(t_i)$ the survival probability to time $t_i$, and $R$ the assumed recovery rate on default. In plain English: the left side is what the protection buyer pays (spread times survival-weighted time, since payments stop once default happens), and the right side is what the protection seller expects to pay out (the loss given default, times the probability of defaulting in each period). Everything is known except the hazard rates buried inside the newest $Q(t_i)$ term, which is solved for using the previously-bootstrapped $Q$'s at shorter maturities.

Worked example 1 — the first segment

A 1-year CDS quotes $s_1 = 100$bp with recovery $R = 40\%$, and assume flat discounting at $D(1) \approx 0.97$ for simplicity, with the hazard rate assumed constant over $[0,1]$. A commonly used simplified (mid-point) approximation for the first, single-period segment is $s_1 \approx (1-R) \times h_1$, giving $h_1 = s_1 / (1-R) = 0.01 / 0.60 = 0.0167$, about 1.67% per year. Survival to 1 year: $Q(1) = e^{-0.0167} = 0.9834$.

Worked example 2 — the second segment builds on the first

A 2-year CDS quotes $s_2 = 140$bp, same recovery. The new unknown is $h_2$, the hazard rate applying only to the second year (years 1 to 2), with $h_1 = 1.67\%$ and $Q(1) = 0.9834$ already fixed from the previous step. Using the same simplified per-period logic extended to two periods, the second spread has to compensate for both years' expected loss blended together: approximately $s_2 \times [\tau_1 Q(1) + \tau_2 Q(2)] \approx (1-R)[Q(0)-Q(1) + Q(1) - Q(2)]$. Solving this for $Q(2)$ with $s_2 = 140$bp typically yields something like $Q(2) \approx 0.9629$, implying an average 2-year hazard higher than the first year's, i.e. $h_2 \approx 2.7\%$ for year two alone — steeper than $h_1$, consistent with the market pricing in rising default risk further out, which is exactly what an upward-sloping CDS curve ($s_1 < s_2$) should imply once bootstrapped.

The same staircase logic from a discount curve applies here — each maturity's hazard rate is solved using only the survival probabilities already pinned down by shorter CDS spreads, moving strictly left to right.

What this means in practice

A bootstrapped hazard curve feeds directly into pricing bespoke credit products — CDS with off-market maturities, first-to-default baskets, or the tranches of a CDO — anything that needs default probabilities at dates the market doesn't quote directly. The recovery rate assumption $R$ is not observed from CDS spreads alone; it has to be assumed (commonly 40% for senior unsecured debt) or taken from recovery swaps, and changing it noticeably shifts the bootstrapped hazard rates even holding spreads fixed.