Credit Default Swaps

Car insurance has a simple structure: you pay a premium every month, and if you crash, the insurer pays out. A credit default swap, or CDS, is that exact structure applied to a bond or a borrower instead of a car. The protection buyer pays a regular fee — quoted as a spread, in basis points per year on some notional amount — to the protection seller. If the reference borrower defaults during the contract's life, the seller pays the buyer the loss, and the contract ends. If the borrower never defaults, the buyer has paid premiums for nothing, exactly like a driver who never crashes. The entire pricing question is the same one an insurance actuary asks: what premium exactly balances what you expect to collect against what you expect to pay out?

The two legs, and the balance between them

A CDS has a premium leg (what the buyer pays, as long as the reference entity survives) and a protection leg (what the seller pays, only if it defaults). At the moment the contract is struck, a fair spread makes these two legs worth exactly the same today.

$$s \times \sum_{t} Q(t)\, D(t) \;=\; (1-R) \sum_{t} \big[Q(t-1) - Q(t)\big]\, D(t) .$$

In plain English: $s$ is the CDS spread, the annual premium rate being solved for. $Q(t)$ is the survival probability to time $t$ — the chance the borrower hasn't defaulted by then — and $D(t)$ is the discount factor for time $t$. The left side is the premium leg: the spread times a "risky annuity," the present value of $1 received each period, weighted down by the chance the borrower has already defaulted and stopped paying. The right side is the protection leg: $(1-R)$, one minus the recovery rate (the fraction of notional a bondholder typically recovers after a default), times the probability of defaulting in each specific period, $Q(t-1) - Q(t)$, discounted back to today. Setting the two sides equal and solving for $s$ is exactly setting an insurance premium so the insurer breaks even in expectation.

Worked example 1 — the quick formula and the careful calculation

The shortcut. If the hazard rate (the constant annual probability of default, conditional on having survived so far) is $\lambda = 2\%$ and the recovery rate is $R = 40\%$, a widely used approximation says the fair spread is simply $s \approx \lambda (1 - R) = 0.02 \times 0.60 = 0.012$, or 120 basis points a year. The logic: each year there's roughly a 2% chance of losing 60% of notional, so the expected annual loss rate is $2\% \times 60\% = 1.2\%$, and a fair premium just charges exactly that.

The careful version. Build it out year by year for a 3-year CDS, notional $10,000,000, flat discount rate 3%. Survival probabilities under constant hazard: $Q(1) = e^{-0.02} = 0.9802$, $Q(2) = e^{-0.04} = 0.9608$, $Q(3) = e^{-0.06} = 0.9418$. Annual default probabilities: year 1, $1 - 0.9802 = 0.0198$; year 2, $0.9802 - 0.9608 = 0.0194$; year 3, $0.9608 - 0.9418 = 0.0190$. Discount factors at 3%: $D(1) = 0.9709$, $D(2) = 0.9426$, $D(3) = 0.9151$.

Protection leg, per $1 of notional: $0.60 \times \big[0.0198 \times 0.9709 + 0.0194 \times 0.9426 + 0.0190 \times 0.9151\big] = 0.60 \times \big[0.01922 + 0.01829 + 0.01739\big] = 0.60 \times 0.05490 = 0.03294$.

Risky annuity: $Q(1)D(1) + Q(2)D(2) + Q(3)D(3) = 0.9802 \times 0.9709 + 0.9608 \times 0.9426 + 0.9418 \times 0.9151 = 0.9517 + 0.9057 + 0.8620 = 2.7194$.

Solving $s \times 2.7194 = 0.03294$: $s = 0.03294 / 2.7194 = 0.01211$, or 121.1 basis points. The quick formula said 120bps; the full calculation, accounting for exactly when defaults and discounting happen, said 121.1bps — close enough that the shortcut is genuinely useful for a quick sanity check, and precise enough that a trading desk uses the full calculation for anything real.

Worked example 2 — marking a position to market

An investor bought $10,000,000 of protection at a spread of 120bps, using the approximate fair value from example 1. Six months later, the reference borrower's credit has deteriorated and the market CDS spread has widened to 200bps — the market now demands a much higher premium for the same protection, meaning the market thinks default is more likely. The buyer's existing contract, still paying only 120bps, is now worth more than a freshly struck one, because they locked in the cheap premium before the credit worsened. A standard approximation for the mark-to-market gain: the spread change times the risky annuity times the notional. Using the risky annuity of 2.7194 from example 1: $(0.0200 - 0.0120) \times 2.7194 \times 10{,}000{,}000 = 0.0080 \times 2.7194 \times 10{,}000{,}000 = 217{,}552$, or $217,552. The protection buyer is sitting on roughly a $217,550 gain, without the borrower having defaulted at all — exactly like an insurance policy becoming more valuable to hold the moment the insured party starts looking riskier, well before anything actually goes wrong.

What this means in practice

CDS spreads are watched as a real-time market read on credit risk, often reacting faster than the underlying bond's own price. A widening spread — as in worked example 2 — signals the market pricing in more default risk before rating agencies or headlines catch up. CDS were also central to the 2008 financial crisis: AIG sold enormous amounts of protection without holding capital against the possibility of actually having to pay the protection leg, and when defaults spiked, the payout obligations nearly brought the firm down.

Practice

1. Using the shortcut formula, what fair CDS spread would you quote for a borrower with hazard rate $\lambda = 3\%$ and recovery rate $R = 25\%$?
2. A protection seller wrote $5,000,000 of protection at a spread of 150bps. The market spread later tightens to 100bps (credit improved). Using the mark-to-market approximation from worked example 2 and a risky annuity of 3.0, is the seller sitting on a gain or a loss, and how large is it?