CFA Level I · CFA Level I Exam

Pricing and Valuation of Interest Rate and Other Swaps: formula sheet

Full chapter guide

Key formulas

Fixed leg payment: Fixed payment = Notional × Swap fixed rate × (days in period ÷ days in year); For annual payments, days ÷ year is 1. For semiannual, use 0.5, quarterly 0.25, unless the day count is given.
Floating leg payment: Floating payment = Notional × (Reference rate set at start of period + spread) × (days ÷ year); Use the rate set at the start of the period, not the rate at payment date. Payment is made at period end.
Net payment: Net payment = Notional × (Floating rate − Fixed rate) × period fraction; Positive means the fixed-rate payer receives. Negative means the fixed-rate payer pays.
Initial value: Value at initiation = 0 (PV of fixed leg = PV of floating leg); The swap fixed rate is set so neither party pays upfront.
Discount factor from spot rate: Z_t = 1 ÷ (1 + S_t)^t; S_t is the t-year spot rate with annual compounding. For a period of a fraction of a year, use the periodic rate and the number of periods.
Discount factor from forward rates: Z_n = Z_(n−1) ÷ (1 + F_(n−1,1)), with Z_0 = 1; F_(n−1,1) is the one-period forward rate starting at time n−1. Divide by (1 + F × day-count fraction) if periods are not annual.
Par swap fixed rate (annual payments): Fixed rate = (1 − Z_N) ÷ (Z_1 + Z_2 + ... + Z_N); Z_N is the discount factor at the swap's final payment date. The denominator is the annuity factor.
Par swap fixed rate with day counts: Fixed rate = (1 − Z_N) ÷ Σ(Δ_i × Z_i); Δ_i is the day-count fraction of period i, for example 0.25 for quarterly payments on a simple basis. This gives the rate per year.
Swap rate as weighted average of forwards: Fixed rate = Σ(F_i × Z_i) ÷ Σ(Z_i); Holds only when all periods are equal in length. With unequal periods, use Σ(F_i × Δ_i × Z_i) ÷ Σ(Δ_i × Z_i). It is a useful check on the answer.
Swap value at initiation: PV(fixed leg) = PV(floating leg) = 1 − Z_N per 1 of notional; Multiply by the notional for a currency amount.
Discount factor from a spot rate: Z = 1 ÷ (1 + r × days/360); Use the same day-count basis as the swap. For a multi-period case, Z_n is the discount factor to payment date n.
Forward rate from discount factors: FR(i-1, i) = (Z_(i-1) ÷ Z_i − 1) × (360 ÷ days in period); This is the expected floating rate for the period, from the current curve.
Fixed-leg PV per unit of notional: PV(fixed) = [Fixed rate × (days/360) × Σ Z_i] + Z_n; Z_n is the discounted principal, so the fixed leg is valued as a bond. On a reset date, compare this with 1, the value of the floating bond at par. The difference is the receive-fixed value per unit of notional.
Swap fixed rate at the current market: Current swap rate = (1 − Z_n) ÷ [Σ (days/360) × Z_i]; This is the rate that gives a new swap a zero value today. For annual periods the period fraction is 1, so it reduces to (1 − Z_n) ÷ ΣZ_i. Use it for the quick method.
Value to the receive-fixed party (on a reset date): V(receive-fixed) = Notional × Σ [(Old fixed rate − Current swap rate) × period fraction × Z_i]; Valid on a reset date, when the floating leg is worth par. Between reset dates the floating leg is not at par, so you must adjust for the next floating coupon already set. Pay-fixed value is the negative of this.
Value to the pay-fixed party: V(pay-fixed) = PV(floating) − PV(fixed) = −V(receive-fixed); The two sides of the swap always sum to zero, ignoring credit risk.
Notional in the second currency: Notional (USD) = Notional (EUR) × S0, where S0 is USD per 1 EUR; Use the spot rate at initiation. Check the quote direction: multiply when the rate is price of the notional currency in the other currency.
Periodic fixed rate for each currency: Fixed rate per period = (1 − Z_N) ÷ (Z_1 + Z_2 + … + Z_N); Z are discount factors for that currency. Annual rate = periodic rate × periods per year. Do it once per currency.
Value of a fixed leg (own currency): PV = C × (Z_1 + … + Z_N) + Notional × Z_N; C is the fixed coupon per period in currency units. Use current discount factors.
Value of a floating leg at a reset date: PV = Notional (today, just reset); Between resets: PV = (Notional + next floating payment) × Z for the time to the next payment.
Value of the swap to the receiver of currency A: V (in currency B) = PV(leg A, in A) × S_t (B per A) − PV(leg B, in B); Convert only the leg stated in currency A, so the result is in currency B. Leg B is already in B and is not converted. Use the spot rate at time t, not at initiation.
Fixed swap rate per period: Fixed rate = (1 − Zₙ) ÷ (Z₁ + Z₂ + … + Zₙ); Zᵢ are discount factors for each settlement date. Multiply by the number of periods per year to annualize. This makes the swap value zero at initiation.
Fixed payment each period: Fixed payment = Notional × periodic fixed rate; Use the periodic rate (annual rate ÷ periods per year, or days ÷ 360 if the convention is given).
Equity payment each period: Equity payment = Notional × (Sₜ ÷ Sₜ₋₁ − 1); Add dividends if the swap is a total return swap. A negative result means the equity payer receives money.
Equity leg value between resets: Equity leg value = Notional × (Sₜ ÷ S last reset); This is the total value of the leg, including the payment due, not just the change. Just after a reset the ratio is 1, so it equals the notional.
Fixed leg value: Fixed leg = Σ (Notional × fixed rate × Zᵢ) + Notional × Zₙ; Value it like a fixed-rate bond using current discount factors for the remaining payments.
Swap value to equity receiver (fixed payer): V = Equity leg value − Fixed leg value; Reverse the sign for the equity payer. At initiation V = 0.
Payer swaption payoff (per unit of notional, per period): max(0, market swap rate − exercise rate); Like a call on the swap rate. The payoff is received as an annuity over the swap's life, not as one lump sum. Present value uses the annuity of the swap's payments.
Receiver swaption payoff (per unit of notional, per period): max(0, exercise rate − market swap rate); Like a put on the swap rate. In the money when the market swap rate is below the strike.
Annual payoff on notional: Payoff = notional × max(0, rate difference) × (days ÷ 360 or year fraction); Use the year fraction for the settlement period. Check the stem's day-count basis.
Swap versus swaption value at start: Swap value at initiation ≈ 0; swaption value = premium > 0; A swaption buyer pays a premium. A swap normally has no upfront payment.
Maximum loss for swaption buyer: Premium paid; The writer's loss can be large. Gain for the buyer is potentially large if the swap rate moves far.

Quick revision

A swap has zero value at initiation, which sets the fixed rate.
Plain vanilla swap: one party pays fixed and receives floating; the other does the opposite.
Swap fixed rate = (1 − final discount factor) ÷ Σ discount factors, with period fractions included in the sum when relevant.
Value to the fixed payer = PV of floating leg − PV of remaining fixed payments.
Value to the fixed receiver is the opposite sign of the fixed payer's value.
If rates rise after initiation, the fixed payer gains because the fixed rate it pays is below the new market swap rate; the fixed receiver loses.
Just after a reset, a floating-rate leg is worth its notional, ignoring credit effects.
Currency swap: value each leg in its own currency, then convert at current spot.
Currency swaps usually exchange notional at start and end; interest rate swaps do not.
Equity swap: value the equity leg at the current equity value (notional adjusted for the return since the last reset), value the fixed or floating leg by discounting, and take the difference.
A payer swaption gives the right to pay fixed; a receiver swaption gives the right to receive fixed.
Credit exposure sits with the party for whom the swap has positive value.

Common mistakes

Believing the notional principal is exchanged at the end. Fix: In a plain vanilla interest rate swap, only interest is exchanged, and only the net amount. The notional is a reference value.
Using the floating rate at the payment date instead of the start of the period. Fix: Floating is set in advance and paid in arrears. Use the rate observed at the start of the period.
Using the final spot rate as the swap rate Fix: The swap rate is an average over the whole curve. It differs from the final spot rate unless the curve is flat. Always compute (1 − Z_N) ÷ ΣZ.
Using the wrong discount factor in the numerator Fix: The numerator is always 1 − Z_N, where N is the swap's final payment. Z_N is the smallest discount factor in your list.
Getting the sign wrong for the party being valued Fix: Pay-fixed gains when rates rise. Receive-fixed gains when rates fall. Check the sign against this before you choose an answer.
Using the original swap rate to discount the remaining cash flows Fix: Always discount with current discount factors from today's curve. The original fixed rate only sets the size of the fixed payments.
Using the spot rate in the fixed rate formula. Fix: Price each currency's fixed rate only from that currency's discount factors. The spot rate only converts notionals and values.
Using the initiation spot rate when valuing the swap later. Fix: Use the notionals as fixed, but convert the present values at the current spot rate.
Treating the equity leg like a known floating rate. Fix: Remember the equity payment can be negative. If the index falls, the equity payer receives money.
Valuing the equity leg as the full index level instead of the ratio. Fix: Use notional × (current level ÷ level at last reset).

Exam tips

Circle which side pays fixed before any arithmetic. Most wrong answers come from direction errors.
Options are listed smallest to largest, so check for options that equal half or double the correct size, which signals a period-fraction trap.
If a stem says the notional is exchanged, treat it as a distractor for a plain vanilla swap.
When the stem gives a spread, add it to the floating reference rate, not to the fixed rate.
When only the net is needed, subtract the rates first, then multiply by the notional and the period fraction, working in decimals. For the first worked example: 4.60% − 4.00% = 0.60% = 0.006, so net = 20,000,000 × 0.006 × 0.5 = 60,000.
Questions give either spot rates or forward rates. Decide the conversion to discount factors in the first ten seconds, then follow the same formula.
Eliminate options by logic. The par swap rate lies between the lowest and highest forward rates, so any option outside that range is wrong.
For a normally shaped curve, the swap rate is typically below the spot rate of the same maturity when the curve slopes upward, and above it when the curve slopes downward. This is a guide, not a rule for every curve shape, so use it to eliminate an option only when the choices are clearly apart.