Module 11 · Advanced
Credit default swaps & hazard rates
A credit default swap (CDS) is insurance on a borrower. The protection buyer pays a spreadevery quarter; if the borrower defaults, the seller pays out the loss — (1 − recovery) of the notional. CDS are how the market prices, and trades, the chance of default.
To value one we need the odds of default over time. The standard trick is a hazard rate — a constant chance of defaulting in any given year. From it, the survival probability(still alive at time t) decays exponentially. Balance the premium you'd pay against the expected payout and you get the fair spread, which works out to roughly the default probability × loss given default. Lock in a spread below fair and your protection is worth money.
🎛 CDS pricer
Fair (par) spread
181 bp
≈ default prob × (1 − recovery)
5y survival probability
86%
chance it doesn't default
Your protection MtM
+$126,565
you locked a cheap spread
Survival probability over time
A CDSis insurance on a bond: you pay a spread each quarter, and if the borrower defaults you're paid (1 − recovery). We model default with a constant hazard rate — a fixed chance of defaulting each year — so the survival probability decays exponentially. The fair spreadis roughly the default probability times the loss given default. Buy protection below the fair spread and it's worth money to you. Educational tool — not investment advice.
Where the default probabilities come from
A CDS spread and a default probability are two sides of one coin, linked by the credit triangle:
hazard rate λ ≈ spread ÷ (1 − recovery)
The hazard rate λ is the chance of defaulting in any given year. From it, the survival probability— still alive at time t — decays exponentially, and its mirror is the cumulative chance of default:
survival Q(t) = e^(−λ·t) cumulative default = 1 − Q(t)
Worked example. Set the calculator to hazard 3.5%/yr, recovery 40%, 5-year, $10M — a name whose fair (par) 5-year spread comes out to 211 bp. Its default odds build up like this:
1y: survival 96.6% cumulative default 3.4% 3y: survival 90.0% cumulative default 10.0% 5y: survival 83.9% cumulative default 16.1%
Recovery does the rest: if it defaults you don't lose everything, only the loss given default = 1 − recovery = 60%. A higher recovery means less to lose, so the same default odds justify a tighter spread.
How a CDS is marked to market
A CDS has two legs, and its value is the difference between them:
- Premium leg — the spread you pay each quarter, but only while the name is still alive: spread × Σ (Δt · DF · Q(t)). That sum, Σ Δt · DF · Q, is the risky annuity or RPV01 — the present value of paying 1 per year until default.
- Protection leg — what you receive on default: (1 − recovery) × Σ DF · (Q(t−1) − Q(t)), summing the loss over each period's default chance.
At inception the contract spread is set so the two legs cancel (value ≈ 0). Later, as the name's spread moves, the position gains or loses. The shortcut every credit trader uses:
MtM (to protection buyer) ≈ (today's spread − contract spread) × RPV01 × notional
Worked example. You bought protection on that name at 100 bp; its spread has since widened to the 211 bp par level (credit deteriorated). With a risky annuity RPV01 ≈ 4.075on $10 M:
MtM ≈ (211 − 100) bp × 4.075 × $10M ≈ +$452,000
Your protection is now worth about $452k— you locked in cheap insurance and the risk got worse. (The full two-leg calculation gives the same number.) The calculator's “protection MtM” tile runs exactly this: set the contract spread below the fair spread and it's in your favour.
Things to try
- • Double the hazard rate — the fair spread roughly doubles and the survival curve drops faster.
- • Raise the recovery toward 80% — the fair spread shrinks (less to lose on default).
- • Set your contract spread below the fair spread — your protection shows a positive mark-to-market.