What is the difference between the premium leg and the protection leg of a CDS, and how do they determine the CDS spread?

A credit default swap has two legs, and understanding both is essential for FRM Part I credit risk questions.

The Two Legs

Premium Leg (paid by the protection buyer):

The buyer makes periodic payments (usually quarterly) of a fixed spread s on the notional amount, conditional on the reference entity NOT having defaulted yet. The PV of the premium leg is:

PV_premium = s x N x SUM[ delta_i x DF_i x Q_i ]

Where:

• s = CDS spread (annualized)
• N = notional
• delta_i = accrual fraction for period i
• DF_i = risk-free discount factor to time i
• Q_i = survival probability to time i

Protection Leg (paid by the protection seller):

If the reference entity defaults, the seller pays (1 - Recovery) x Notional. The PV is computed over all possible default times:

PV_protection = (1 - R) x N x SUM[ DF_j x (Q_{j-1} - Q_j) ]

Where (Q_{j-1} - Q_j) represents the marginal default probability in period j.

Fair Spread Derivation

At initiation, the CDS has zero value, meaning:

PV_premium = PV_protection

Solving for s:

s = [(1 - R) x SUM(DF_j x (Q_{j-1} - Q_j))] / [SUM(delta_i x DF_i x Q_i)]

Numerical Example

Suppose Northfield Industries is the reference entity. Notional = $10M, recovery = 40%, and using a simplified 2-year framework with annual periods:

PV_protection = 0.60 x [0.9615 x 0.03 + 0.9246 x 0.04] = 0.60 x [0.02885 + 0.03698] = 0.60 x 0.06583 = 0.03950

Risky annuity = 1.0 x 0.9615 x 0.97 + 1.0 x 0.9246 x 0.93 = 0.9327 + 0.8599 = 1.7926

Fair spread = 0.03950 / 1.7926 = 2.20% or 220 bps

Check our FRM question bank for more CDS pricing problems.