Credit Derivatives: formula sheet

Key formulas

Annual premium

Premium = CDS spread × notional

Quarterly payment ≈ spread × notional × day-count fraction, for example 0.25 for a quarter.

Protection payout

Payout = notional × (1 − recovery rate)

Same for cash settlement (auction recovery) and physical settlement (par minus market value of delivered bond).

Upfront payment (approximation)

Upfront ≈ (market spread − fixed coupon) × risky annuity × notional

Positive means the buyer pays the seller. Risky annuity is the PV of 1 per year paid while the entity survives.

Credit triangle (approximation)

Spread ≈ PD (hazard rate) × (1 − recovery rate)

A rule of thumb with constant hazard rate. Gives annual risk-neutral default intensity from a spread.

Accrued premium on default

Accrual = spread × notional × (days since last payment ÷ day-count basis)

Paid by the buyer at the credit event.

Credit triangle

s ≈ λ × (1 − R)

s is the annual CDS spread, λ the constant hazard rate, R the recovery rate. An approximation.

Hazard rate from spread

λ ≈ s ÷ (1 − R)

Gives a risk-neutral intensity. Convert s from basis points to decimals first.

Survival probability (constant hazard rate)

Q(t) = e^(−λt)

Cumulative default probability is 1 − e^(−λt).

Premium leg PV

PV(premium) = s × Σ [Δt × Q(tᵢ) × D(tᵢ)] (plus accrual-on-default term)

Δt is the accrual period, D the discount factor. The sum is often called the risky annuity or RPV01.

Protection leg PV

PV(protection) = (1 − R) × Σ [(Q(tᵢ₋₁) − Q(tᵢ)) × D(tᵢ)]

Q(tᵢ₋₁) − Q(tᵢ) is the probability of default in period i. Multiply by notional for money amounts.

Par spread

s* = PV(protection) ÷ risky annuity

The spread that sets the CDS value to zero at inception.

Value of an existing CDS (to protection buyer)

V ≈ (s_market − s_contract) × risky annuity × notional

Positive if spreads have widened since the contract was struck.

CDS-bond basis

Basis = CDS spread − bond spread

Negative basis: CDS spread below bond spread.

Credit triangle

spread ≈ λ × (1 − R)

λ is the annual hazard rate and R the recovery rate. Use it as an approximation for flat curves.

Index upfront payment

Upfront ≈ (market spread − fixed coupon) × risky duration × notional

Positive means the protection buyer pays the seller. Risky duration is also called RPV01.

Loss per name in an equal-weighted index

Payout = (Notional ÷ N) × (1 − R)

N is the number of names in the original index. After the default, notional falls by Notional ÷ N.

Index basis

Basis = index spread − weighted average of constituent spreads

Typically small. It is driven by liquidity and supply and demand.

FTD spread bounds

max(si) ≤ FTD spread ≤ Σ si

The lower bound applies at perfect correlation. The upper bound applies at zero correlation: with independence and flat hazard rates, the credit-triangle FTD spread equals the sum of the spreads. In practice it is at or slightly below the sum once discounting and payment timing are considered.

Correlation effect on nth-to-default

FTD spread falls as correlation rises; high-n spread rises as correlation rises

The effect for middle n depends on the portfolio. Do not assume a single direction.

Tranche width

Width = Detachment point − Attachment point

Expressed as a percentage of total pool notional.

Tranche loss

Tranche loss = min(max(L − A, 0), D − A)

L is pool loss, A attachment, D detachment, all in the same units. Divide by (D − A) for percentage of tranche lost.

Pool loss from defaults

L = number of defaults × exposure × (1 − recovery rate)

Assumes equal exposures. Divide by pool notional for a percentage.

Expected pool loss

E[L] = PD × LGD × notional (equal exposures)

Does not depend on correlation. The sum of tranche expected losses equals this.

Correlation effect (rule of thumb)

Higher correlation: equity value ↑, senior value ↓

Mezzanine is ambiguous and depends on position and spread. Do not say it always falls.

TRS receiver net payment

Net to receiver = (Coupons + Price change) − (Floating rate × Notional)

Price change is negative if the asset falls, so the receiver then pays that amount as well.

CLN investor loss on default

Loss = Face value × (1 − Recovery rate)

The investor's principal is cut to the recovery value; the coupon premium compensates for this risk.

Credit spread option payoff (call on spread)

Payoff ≈ Notional × Duration × max(Spread at expiry − Strike, 0)

An approximation only, reasonable for small spread moves and a European-style option. Duration converts a spread move into a price change; the actual payoff depends on the contract terms. Use the reverse for a put on price.

CDS vs TRS risk transfer

CDS: credit event risk only. TRS: credit risk + market (spread and rate) risk

A conceptual rule to remember when comparing them.

CDS-bond basis

Basis = CDS spread − bond credit spread

Positive basis means CDS is more expensive than the bond spread implies. Negative basis means the reverse. Use the same maturity and the same spread benchmark.

Counterparty exposure at default

Exposure = max(V, 0)

V is the CDS market value to you. You lose only when the contract is in the money to you. With a netting agreement, use the net value across the netting set.

Expected counterparty loss (simple form)

Expected loss ≈ PD × LGD × EAD

Wrong-way risk means EAD and PD are positively linked, so using independent averages understates the loss.

Protection leg payout on default

Payout = Notional × (1 − Recovery rate)

Typical for cash or auction settlement. This is what the buyer hopes to receive from the seller.

Quick revision

• A CDS buyer pays a periodic spread and receives a payment if a credit event occurs.
• The protection buyer is short credit risk on the reference entity (gains if credit quality worsens); the seller is long credit risk (earns the spread, bears the loss).
• Payout on default = notional × (1 − recovery rate), under standard settlement.
• Approximate spread = annual default probability × (1 − recovery rate); this is a simplification.
• A higher recovery rate lowers the implied default probability for a given spread.
• A credit index is a standardized basket of single-name CDS.
• In a CDO, losses hit the equity tranche first, then mezzanine, then senior.
• Higher default correlation raises the chance of very large portfolio losses, which hurts senior tranches.
• Equity tranches usually gain value as default correlation rises, because the chance of few or no defaults increases; senior tranches lose value.
• A credit-linked note embeds a CDS in a bond and substantially reduces counterparty risk for the protection buyer, since the seller funds the note upfront and the cash collateralizes the obligation. The risk is not fully removed.
• Wrong-way risk arises when exposure to a counterparty rises as its credit quality falls.
• Basis risk is the gap between a CDS spread and a comparable bond spread, or between the hedge and the exposure.

Common mistakes

• Using the recovery rate as the payout instead of 1 − recovery. Fix: Payout is the loss: notional × (1 − recovery). Recovery is what the buyer keeps through the bond's value.
• Getting the upfront direction wrong. Fix: Compare market spread with the fixed coupon. Higher market spread means the buyer must compensate the seller upfront.
• Using spread as the hazard rate, ignoring recovery Fix: Always divide by (1 − R). With a 40% recovery, λ is s ÷ 0.60, which is larger than s.
• Mixing basis points and decimals Fix: Convert first. Write 120 bp = 0.0120 before any calculation.
• Saying higher correlation raises every basket spread. Fix: For FTD, higher correlation lowers the spread because defaults cluster. For high-n nth-to-default, it raises the spread.
• Treating the index as a single entity that ends after one default. Fix: An index continues after a default. Only that name's weight is removed and settled. An FTD ends after the first default.
• Saying higher correlation increases the expected pool loss. Fix: Expected pool loss depends on PD and LGD only. Correlation changes the shape of the loss distribution and shifts risk between tranches.
• Saying equity always loses value when correlation rises. Fix: Equity is long a call-like payoff on defaults being few. Higher correlation makes zero or few defaults more likely, so equity value rises.
• Saying a TRS only transfers credit risk. Fix: Remember a TRS transfers total economic return, including market and interest rate effects, as well as credit risk.
• Treating the CLN as unfunded. Fix: The CLN investor pays cash up front. The protection buyer receives that cash, so its counterparty risk is small. The investor bears the reference entity risk and also the issuer's risk (or the SPV collateral risk).

Exam tips

• Always settle direction first: buyer pays premium and upfront (if market spread above coupon), seller pays on default.
• Expect questions linking spread to default probability through the credit triangle. Use it only as an approximation and say so.
• Know the restructuring clause names and what each restricts. Questions often ask which clause favours the buyer or seller.
• For settlement questions, remember the auction sets cash recovery and the buyer picks cheapest-to-deliver in physical settlement.
• Watch units: bp, annual versus quarterly, and notional currency.
• Check whether the question wants risk-neutral or real-world probabilities. CDS-implied means risk-neutral.
• Read the answer options for the sign of the CDS-bond basis before calculating, since many items are interpretation only.
• Do the credit triangle first. Only build the legs if discount factors and payment dates are given.