Swaps: formula sheet

Key formulas

Fixed payment per period

Fixed payment = Notional × fixed rate × (days in period ÷ day-count basis)

For a simple annual or semiannual swap, use rate ÷ payments per year. Use the stated day count if given.

Floating payment per period

Floating payment = Notional × floating rate set at the start of period × accrual fraction

The rate is set at the start and paid at the end of the period (set in advance, paid in arrears).

Net settlement

Net payment = Fixed payment − Floating payment

The fixed payer pays this amount if positive. If negative, the fixed payer receives the absolute value.

Total comparative advantage gain

Total gain = |Spread difference in fixed market − Spread difference in floating market|

Gain is split by negotiation, and any dealer fee reduces what the two firms share.

Swap as bond portfolio

Value to fixed payer = B_float − B_fixed

Fixed payer is short a fixed-rate bond and long a floating-rate bond. Floating bond is worth par just after a reset.

Swap value, bond approach

V(pay fixed) = B_float − B_fix; V(receive fixed) = B_fix − B_float

Both bonds are valued with the same current discount factors and the same notional.

Fixed bond value

B_fix = Σ [k × τ × L × d(tᵢ)] + L × d(T)

k is the fixed rate, τ the accrual fraction (0.5 for semiannual), L the notional, d(t) the discount factor for time t.

Floating bond value

B_float = (L + floating payment due next) × d(t₁)

On a reset date this equals L. The next payment is already fixed from the last reset rate.

Forward rate from discount factors

F = [d(t₁) ÷ d(t₂) − 1] ÷ τ

This is the forward rate for the period from t₁ to t₂ with simple accrual τ.

Swap value, FRA approach

V(pay fixed) = Σ [(Fᵢ − k) × τ × L × d(tᵢ)]

The first period uses the known floating rate if the swap is between resets.

Par swap rate

s = [1 − d(T)] ÷ Σ [τ × d(tᵢ)]

Valid for a swap starting today. The sum runs over all fixed payment dates.

Value from rate difference

V(pay fixed) ≈ (s_new − k) × Σ [τ × d(tᵢ)] × L

Useful when the new par rate is given. Sign: pay fixed gains if market par rate is above your fixed rate.

Value to the party receiving foreign currency (bond method)

V = S0 × B_F − B_D

S0 is the spot rate in domestic currency per 1 unit of foreign currency. B_F is the foreign bond value in foreign currency. B_D is the domestic bond value in domestic currency. The party receives foreign and pays domestic.

Value to the party receiving domestic currency

V = B_D − S0 × B_F

The negative of the other party's value.

Bond value for each leg

B = Σ [c × e^(−r_t × t)] + L × e^(−r_T × T)

c is the fixed payment, L is the principal, and r_t is the zero rate for that currency. With discrete compounding, use the matching discount factors.

Forward exchange rate (continuous compounding)

F_t = S0 × e^((r_D − r_F) × t)

Domestic currency per 1 foreign. r_D is the domestic rate and r_F is the foreign rate, both for maturity t. If r_D > r_F, then F_t > S0: the foreign currency trades at a forward premium and the domestic currency at a forward discount.

Forward method value

V = Σ (foreign cash flow × F_t − domestic cash flow) × e^(−r_D × t)

Use net cash flow at each date, signed for the party you value. Discount at the domestic rate.

Value of a forward contract to receive foreign currency

PV = (foreign cash flow × F_t − domestic cash flow) × discount factor

Applies to each date, including the final principal exchange.

Par swap rate (from discount factors)

s = (1 − d_n) ÷ Σ(τ_i × d_i)

d_i is the discount factor for payment date i, τ_i the accrual fraction. For annual payments τ = 1, so s = (1 − d_n) ÷ Σ d_i.

Bootstrapping the last discount factor (annual pay)

d_n = (1 − s_n × Σ_{i<n} d_i) ÷ (1 + s_n)

Use earlier discount factors already found. Start with d_1 = 1 ÷ (1 + s_1).

Zero rate from discount factor (annual compounding)

z_n = (1 ÷ d_n)^(1/n) − 1

With continuous compounding use z_n = −ln(d_n) ÷ n instead. Check which one the question wants.

Swap spread

Swap spread = swap rate − Treasury par yield (same maturity)

Quoted in basis points. 1 bp = 0.01%.

LIBOR-OIS spread

LIBOR-OIS spread = LIBOR (term) − OIS rate (same term)

A wider spread signals higher perceived bank credit and liquidity stress.

Compounded overnight rate over a period

Rate = [Π(1 + r_i × n_i ÷ 360) − 1] × 360 ÷ D

r_i is the overnight rate on day i, n_i the number of days it applies, D total days. 360 for USD SOFR (use 365 for SONIA).

Credit exposure of a swap

Exposure = max(V, 0)

V is the swap's value to you. Only a positive value is lost if the counterparty defaults.

Netted exposure

Netted exposure = max(V1 + V2 + ... + Vn, 0)

Applies only if a legally enforceable netting agreement covers all the trades.

Exposure after collateral

Net exposure = max(V − Collateral held, 0)

Ignores thresholds, minimum transfer amounts and the margin period of risk.

Expected credit loss

Expected loss = PD × LGD × Exposure

Use expected exposure at default. Loss given default (LGD) = 1 − recovery rate.

Amortizing swap payment

Payment in period t = Rate × Notional outstanding at the start of period t

Notional falls on the schedule, so payments fall too.

Quick revision

• A swap exchanges cash flows on a notional; in an interest rate swap the notional is normally not exchanged.
• Pay-fixed swap = long floating-rate bond + short fixed-rate bond.
• Value to fixed payer = PV of floating leg − PV of fixed leg.
• Just after a reset date, a floating-rate bond is worth its par value.
• The swap rate makes the swap worth zero at start.
• Par swap rate = (1 − final discount factor) ÷ Σ (accrual × discount factors).
• FRA approach: value each period as (forward rate − fixed rate) × notional × accrual × discount factor.
• Currency swaps usually exchange notional at start and end; swap value is the difference of two bond values in one currency.
• Modern practice discounts collateralised swaps on the OIS curve, not on LIBOR.
• Only a party with positive swap value faces credit loss; exposure is the positive replacement value.
• Netting and collateral reduce credit exposure; central clearing moves it to a clearing house.
• Interest rate swap exposure tends to be lower than a loan of the same size, because only net flows and value changes are at risk.

Common mistakes

• Treating the notional principal as an amount that is exchanged in an interest rate swap. Fix: In a single-currency interest rate swap the notional is only used to calculate interest and is not exchanged.
• Using the floating rate at the payment date instead of the reset date. Fix: The floating rate is set at the start of the period and paid at the end.
• Using par value for the floating bond when the swap is between reset dates. Fix: Par only holds on a reset date. Otherwise use (L + next floating payment) × d(t₁).
• Getting the sign wrong for the payer and receiver. Fix: Fixed payer = floating bond − fixed bond. Write the formula first. Then check: if rates rose, the fixed payer should gain.
• Treating principal as notional only, as in an interest rate swap. Fix: In a currency swap, principal is exchanged at the end, and usually at the start. Include the final principal in both bonds.
• Discounting both currencies at the same rate. Fix: Discount each currency's cash flows at that currency's own zero rates.
• Treating LIBOR as a risk-free rate Fix: LIBOR was unsecured interbank borrowing and carried bank credit and liquidity premiums. OIS and SOFR are the near risk-free rates.
• Saying SOFR is an unsecured rate or a term rate by nature Fix: SOFR is an overnight rate secured by Treasuries in repo. Term SOFR is a separate, forward-looking measure derived from derivatives.
• Treating the notional as the amount at risk. Fix: In an interest rate swap the notional is not exchanged. Only the net value of the swap is at risk.
• Counting a negative value as a loss to the counterparty's default. Fix: Use max(V, 0). If V is negative, your credit exposure is zero.

Exam tips

• Read the direction carefully: pay-fixed versus receive-fixed decides every sign.
• Write the legs on separate lines before netting. This avoids sign errors.
• Know the contrast between bilateral OTC (netting, collateral, credit risk) and cleared (novation, margin, CCP).
• For comparative advantage, compute spread differences first. Do not compare absolute rates.
• Remember that interest rate swap notionals are not exchanged, while currency swap principals usually are.
• Look for the words 'just reset' or 'immediately after a payment'. They tell you the floating bond equals notional.
• If the question gives zero rates, convert them to discount factors first. Check the compounding convention stated.
• If a question gives a new par rate for the remaining maturity, the rate-difference shortcut saves time.