CDS Spreads & Hazard Rates

You already know what a credit default swap is: a contract where the protection buyer pays a steady premium — the spread — and the seller makes them whole if a named borrower defaults. So far, so much insurance. But here’s the thing nobody tells you on day one: that spread is not just a price. It is an opinion about the odds of default, encoded in basis points, and with a little arithmetic you can pull those odds right back out. A CDS quoted at 200bp is the market whispering “this name defaults roughly 3% a year.” This lesson is about doing that decode — turning a spread into a probability of default, a survival curve, and ultimately a number you can trade against. The machinery is the hazard rate, the credit triangle, and the risky annuity, and by the end you’ll mark a CDS to market the way a desk does.

A spread is the price of default risk

Start with the intuition, because the formula falls right out of it. If you sell protection on Acme, every year you collect the spread, and in exchange you’re on the hook for a payout if Acme defaults. For that to be a fair trade, the premium you pocket each year should roughly equal the loss you expect to pay out each year. And your expected annual payout is just two things multiplied together: how likely default is, and how much you lose when it happens.

That “how much you lose” piece has a name. When a borrower defaults, bondholders don’t usually lose everything — they recover some fraction of face value through the bankruptcy process, salvaged assets, restructuring. Call that fraction the recovery rate $R$. What you actually lose is the rest: the loss given default, $\text{LGD} = 1 - R$. If recovery is 40 cents on the dollar, your loss given default is 60%.

So, in words:

$\text{annual spread} \;\approx\; (\text{annual default probability}) \times (1 - R)$

Rearrange and you’ve got an implied default probability straight from the quote:

$\text{annual default probability} \;\approx\; \frac{\text{spread}}{1 - R}$

Let’s run real numbers. You sell protection on $10M of Acme at a spread of 200bp (2.00%), with the market-standard recovery assumption of 40%, so $1 - R = 0.60$.

Read that bottom row out loud: the $200,000 a year you collect is the fair price for insuring against a 60%-of-notional loss that arrives with roughly 3.33% probability each year. The spread was a default-odds quote all along; we just divided by the loss given default to reveal it.

Hazard rates and the survival curve

The “3.33% per year” we just computed is a fine back-of-envelope number, but it’s secretly sloppy: it treats default probability as if it ticked up in equal slabs year after year. Real survival doesn’t work that way, and to price multi-year CDS properly we need a cleaner object — the hazard rate.

Analogy. Think of a lightbulb that can burn out at any instant. The hazard rate $\lambda$ is the bulb’s instantaneous failure intensity: given that it’s still glowing right now, $\lambda$ is the rate at which it’s about to die in the next sliver of time. For a credit, $\lambda$ is the instantaneous conditional default intensity — given the name has survived this far, the rate at which it defaults in the next instant. It’s a “per year” rate, but applied continuously, like continuously compounded interest applied to dying.

Precise definition. If the hazard rate is a constant $\lambda$, the probability of surviving (no default) all the way to time $t$ is the survival probability:

$Q(t) = e^{-\lambda t}$

and therefore the cumulative probability of having defaulted by horizon $t$ is

$\text{PD}(t) = 1 - e^{-\lambda t}.$

That $e^{-\lambda t}$ is the survival curve: it starts at 100% (everyone’s alive at $t=0$) and decays exponentially. It’s the exact same shape as radioactive decay or a discount factor — survival compounds, it doesn’t subtract in straight lines.

Let’s tabulate it for $\lambda = 3.33\%$ (the hazard implied by our 200bp / 40%-recovery name):

When to use it

Reach for the hazard-rate / survival-curve machinery whenever the horizon matters — pricing a 5-year vs a 10-year CDS, computing the present value of premium payments that only happen if the name survives, or comparing default odds across maturities. The one-line “spread / (1−R)” shortcut is great for a quick implied-PD sanity check; the survival curve is what you actually integrate when money changes hands over time.

The credit triangle

Now we tie the two ideas together into the single most useful relationship in credit. We said the fair spread compensates for the expected annual loss, $\lambda \times (1-R)$. That’s the credit triangle:

$\boxed{\;\text{spread} \;\approx\; \lambda \,(1 - R)\;}$

It’s called a triangle because three quantities — spread, hazard rate $\lambda$, and recovery $R$ — are locked together, and knowing any two pins down the third. (The ”≈” is because the clean equality holds exactly in the idealized continuous, flat-hazard limit; for ordinary spreads it’s an excellent approximation, and it’s exact under continuous flat-hazard pricing.) Rearrange it three ways depending on what you’re solving for:

• Solve for hazard (the usual move — you see a spread, you want default odds): $\quad\lambda \approx \dfrac{\text{spread}}{1 - R}$
• Solve for implied recovery (you have spread and a hazard view): $\quad R \approx 1 - \dfrac{\text{spread}}{\lambda}$
• Solve for spread (you have a default-intensity model, you want the quote): $\quad\text{spread} \approx \lambda\,(1 - R)$

Three worked cases, all in basis points and percent, with the standard $R$ unless noted:

Look at rows one and three: same 200bp spread, different recovery, different implied hazard. Lower recovery means each default costs more, so a given spread can be “paid for” by fewer defaults — the implied hazard drops from 3.33% to 2.50%. That’s the whole subtlety in one comparison.

The interactive below is the credit triangle, alive. Drag the CDS spread slider (25–800bp) and the recovery slider; it computes $\lambda = (\text{spread}/10000)/(1-R)$ and draws the survival curve $e^{-\lambda t}$ across ten years, shading the cumulative-default region and marking the 5-year point. Try this: push the spread up or drag recovery down, and watch the survival curve bend downward faster — both mean “more default,” so the curve falls away from 100% more steeply and the shaded default region swells.

The risky annuity (RPV01) and marking a CDS to market

So far we’ve priced the protection side. But a CDS has two legs, and to value a real position you need both — and the survival curve is what makes the premium leg honest.

The premium leg is the stream of spread payments the buyer makes. Here’s the catch: those payments stop the instant the name defaults. So when you present-value them, you can’t just discount each coupon by the risk-free factor — you must also multiply by the probability the name is still alive to pay it. Discount by the risk-free rate and by survival.

That survival-weighted present value of a 1-unit-per-year premium stream has a name: the risky annuity, or RPV01 (risky PV of a basis point). Formally, it’s the PV of receiving 1 unit of premium per year for the life of the swap, where each coupon is discounted by both the risk-free discount factor and the survival probability $Q(t)$ — plus a small accrual-on-default term for the fraction of a coupon owed if default lands mid-period.

$\text{Premium-leg PV} = \text{spread} \times \text{RPV01}, \qquad \text{Protection-leg PV} \approx (1-R) \times (\text{PV of default})$

The fair spread is the one that makes those two legs equal — the spread at which the swap is worth zero at inception. That’s just the credit triangle again, now written with proper discounting.

Why RPV01 is the key to mark-to-market. Since the Big Bang, single-name CDS trade with standardized fixed coupons (e.g. 100bp or 500bp) and an upfront payment to settle the difference between that fixed coupon and the current fair spread. The mark-to-market of a position is, to an excellent approximation:

$\text{Upfront} \;\approx\; (\text{spread} - \text{coupon}) \times \text{RPV01}$

and the gain on a position when spreads move is the change in spread times RPV01 times notional. Buy protection at a tight spread, watch it widen, and you profit — because you’re paying a coupon that’s now cheap relative to the protection’s current value.

Worked mark-to-market. You bought protection on a name with a standardized 100bp coupon. Months later the name has deteriorated and the market fair spread is now 250bp. The 5-year RPV01 is about 4.3, and your notional is $10M. Your position has gained:

You bought cheap protection (paying only 100bp) on a name that now deserves 250bp of protection — and the RPV01 of 4.3 tells you each basis point of that 150bp edge is worth about 4.3 years of survival-discounted annuity. Multiply through and the position is up roughly $645,000. Mark a CDS to market and that’s the calculation, every time.

Bootstrapping the hazard curve

One last upgrade. A single CDS quote and a single recovery give you one hazard number — a flat curve. But real names trade a strip of CDS at 1, 3, 5, 7 and 10 years, and those spreads usually aren’t equal. To honor every quote at once you build a term structure of hazard rates, exactly the way you bootstrapped a zero-coupon yield curve from a strip of bonds.

The procedure, step by step:

1. Take the 1-year CDS quote. Solve for the constant hazard $\lambda_1$ that re-prices it. That fixes survival out to year 1.
2. Take the 3-year quote. Holding $\lambda_1$ fixed for the first year, solve for the forward hazard $\lambda_{1\to3}$ over years 1–3 that makes the 3-year CDS fair. That extends the survival curve to year 3.
3. Take the 5-year quote, hold everything earlier fixed, peel off the next forward hazard $\lambda_{3\to5}$. Repeat for 7y, 10y.

The result is a piecewise-constant hazard curve: a different (constant) forward default intensity in each maturity bucket, stitched together so the survival curve simultaneously re-prices every CDS in the strip. It’s the credit twin of zero-curve bootstrapping — instead of peeling off forward interest rates, you peel off forward default intensities.

The intuition for the slope. An upward-sloping CDS curve (longer maturities quoted wider) implies rising forward hazards — the market thinks near-term default is unlikely but the danger grows with time (typical of a currently-healthy name with long-run uncertainty). An inverted (downward-sloping) curve implies the opposite: high near-term forward hazard that’s expected to fall if the name survives the immediate scare — the classic signature of a name in acute distress, where the market is pricing “blow up soon, or be fine.”

Putting it together

A CDS spread is an opinion about default odds wearing a basis-point costume. Strip away the costume with the credit triangle, $\text{spread} \approx \lambda\,(1-R)$, and you recover the hazard rate $\lambda$ — the instantaneous conditional default intensity — via $\lambda \approx \text{spread}/(1-R)$, where $1-R$ is the loss given default. Compound that hazard and you get the survival curve $Q(t) = e^{-\lambda t}$, with cumulative default $1 - e^{-\lambda t}$ — a curve that decays geometrically, not in straight-line slabs, so the naïve ”$\lambda$ per year” guess overstates long-horizon default. Pricing the premium leg honestly means discounting each coupon by both the risk-free factor and survival, giving the risky annuity RPV01; the fair spread equates premium and protection legs. From RPV01 falls the desk’s everyday move — mark-to-market $\approx (\text{spread} - \text{coupon}) \times \text{RPV01} \times \text{notional}$ — so 150bp of widening on a 4.3-RPV01, $10M position is worth about $645,000. And to honor a whole strip of maturities you bootstrap a piecewise-constant forward hazard curve, the credit twin of zero-curve bootstrapping. Get all that, and a wall of CDS quotes becomes a clean, tradeable picture of who defaults, when, and what it’s worth.

Next up — CDS indices: we’ll take the single-name machinery you just built and bundle dozens of names into CDX and iTraxx, read the index basis that opens up between the index and its constituents, and see why the whole can trade at a different spread than the sum of its parts.