CFA Level I Exam · Pricing and Valuation of Interest Rate and Other Swaps

Swap Pricing and Par Swap Fixed Rate Calculation

Updated 7 October 2026 · Fact-checked

Swap pricing means finding the fixed rate that makes a new swap worth zero. Convert the spot or forward curve into discount factors. Then set the PV of fixed payments equal to the PV of floating payments: fixed rate = (1 − final discount factor) ÷ sum of discount factors, adjusted for day-count fractions.

Understand Swap Pricing and Swap Fixed Rate Calculation

A plain vanilla interest rate swap exchanges fixed payments for floating payments on a notional amount. At initiation, neither side pays the other anything up front. So the swap must have a value of zero. Pricing a swap means finding the one fixed rate that makes this true. This is the par swap rate, also called the swap fixed rate.

The key idea is that a floating-rate leg that resets to the market rate is worth par at each reset date. Think of it as a floating-rate bond. Right after a reset, it is worth its notional. So the PV of the floating leg, per 1 of notional, equals 1 − Z_N, where Z_N is the discount factor for the final payment date. You do not need to forecast every floating rate to get this result.

A discount factor is today's price of 1 received at a future date. It comes from the spot curve: Z_t = 1 ÷ (1 + S_t)^t for annual periods. It can also come from the forward curve: each period's discount factor is the previous one divided by (1 + that period's forward rate). Both routes give the same discount factors if the curves are consistent.

The fixed leg is an annuity. Each payment is the fixed rate times the period's day-count fraction, and each is discounted with its own discount factor. Set PV(fixed) = PV(floating) and solve for the rate. When all periods are equal in length, the result is a weighted average of the forward rates, with the discount factors as weights. With unequal periods, each weight must also include the period's day-count fraction. The swap rate therefore lies between the lowest and highest forward rates. With equal periods, it sits closer to the forward rates of earlier periods because they carry larger discount factors.

Key formulas to remember

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.

How to solve Swap Pricing and Swap Fixed Rate Calculation questions

Use this order for any question that asks for the swap fixed rate or the swap price from a curve.

1Identify what the curve gives you: spot rates, forward rates, or discount factors directly. Note the payment frequency and the day-count convention.
2Convert the curve to discount factors. For spot rates use Z_t = 1 ÷ (1 + S_t)^t. For forward rates chain them: Z_n = Z_(n−1) ÷ (1 + F).
3Keep at least six decimal places for the discount factors. Rounding early moves the answer in the third decimal of the rate.
4Compute the numerator: 1 − Z_N, using the discount factor of the last payment date.
5Compute the denominator: the sum of the discount factors, each multiplied by its period's day-count fraction if the periods are not annual.
6Divide the numerator by the denominator. This is the periodic fixed rate if you used the period fractions as 1, or the annualised rate if you used year fractions.
7Check that the answer lies between the smallest and largest forward rates and is close to the spot rate for the swap's maturity.
8If the question asks for a currency amount, multiply the rate by the notional to get the fixed payment per period.

Quickest way: Numerator-over-annuity shortcut

When to use it: Use it whenever you can get discount factors quickly and the question asks for the par fixed rate or the fixed payment.

Write the discount factors in a column and add them up. Do not build the two legs separately.
Take 1 − Z_N as the floating leg PV. Skip the forward rates for the floating side.
Divide. Then compare with the options: for a normally shaped, upward-sloping curve, the swap rate is typically a bit below the longest spot rate.
Use the calculator: for 1.03^3 on the BA II Plus press 1.03 [y^x] 3 [=] then [1/x]. On the HP 12C press 1.03 [ENTER] 3 [y^x] [1/x].
Store each discount factor in a memory register to save time. On the HP 12C, use STO+ to accumulate the sum in a register, then RCL it. On the BA II Plus, use STO and RCL, or add each factor to the running total in the display.

Common mistakes in Swap Pricing and Swap Fixed Rate Calculation

Using the final spot rate as the swap rate
Both look like a rate for the same maturity, so students assume they are equal.
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
Students use Z_1 or the average discount factor instead of the discount factor of the last payment date.
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.
Forgetting the day-count fraction on non-annual swaps
The annual formula is memorised and applied to quarterly or semi-annual swaps.
Fix: Multiply each discount factor by its period fraction in the denominator. The result is then a per-year rate. Without it you get a per-period rate.
Chaining forward rates with the wrong base
Students divide Z_1 by the forward rate instead of by (1 + forward rate).
Fix: Use Z_n = Z_(n−1) ÷ (1 + F). Start from Z_0 = 1 and check that each Z is smaller than the one before it when rates are positive.
Rounding discount factors too early
Four-decimal rounding looks tidy, but the error grows when you divide a small numerator by a sum.
Fix: Keep six decimals, or use calculator memory. Round only the final rate.
Confusing swap pricing with swap valuation
Both use discount factors, so the two tasks blur together.
Fix: Pricing sets the fixed rate so that the value is zero at initiation. Valuation after initiation uses a fixed rate that is already agreed and finds a non-zero value.

Worked examples

Example 1

The annual spot rates are 2.00% for 1 year, 2.50% for 2 years and 3.00% for 3 years. A 3-year annual-pay plain vanilla swap is initiated today. Which is the closest par swap fixed rate? A) 2.50% B) 2.98% C) 3.00%

Show the solution

Z_1 = 1 ÷ 1.02 = 0.980392.
Z_2 = 1 ÷ 1.025² = 1 ÷ 1.050625 = 0.951814.
Z_3 = 1 ÷ 1.03³ = 1 ÷ 1.092727 = 0.915142.
Sum of discount factors = 0.980392 + 0.951814 + 0.915142 = 2.847348.
Numerator = 1 − 0.915142 = 0.084858.
Fixed rate = 0.084858 ÷ 2.847348 = 0.029803, which is about 2.98%.
Compare with the options: 2.98% matches B. The swap rate is just below the 3-year spot rate because the curve slopes upward.

Answer: B) 2.98%

Example 2

A 3-year annual-pay swap is priced from forward rates: the 1-year rate today is 1.50%, the 1-year rate one year forward is 2.50%, and the 1-year rate two years forward is 3.50%. Find the par swap fixed rate.

Show the solution

Z_1 = 1 ÷ 1.015 = 0.985222.
Z_2 = 0.985222 ÷ 1.025 = 0.961192.
Z_3 = 0.961192 ÷ 1.035 = 0.928687.
Sum of discount factors = 0.985222 + 0.961192 + 0.928687 = 2.875101.
Numerator = 1 − 0.928687 = 0.071313.
Fixed rate = 0.071313 ÷ 2.875101 = 0.024803, or 2.48%.
Check with the weighted-average method: (0.015 × 0.985222 + 0.025 × 0.961192 + 0.035 × 0.928687) ÷ 2.875101 = (0.014778 + 0.024030 + 0.032504) ÷ 2.875101 = 0.071312 ÷ 2.875101 = 2.48%. The rate lies between 1.50% and 3.50%, as expected.

Answer: The par swap fixed rate is about 2.48%.

Exam tips

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.
Read the payment frequency carefully. Quarterly and semi-annual swaps need the day-count fraction in the denominator.
Options are listed from smallest to largest and are often close. Carry six decimals through the discount factors, and round only at the end.

Practice questions from Pricing and Valuation of Interest Rate and Other Swaps

Swap Pricing and Swap Fixed Rate Calculation in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Swap Pricing and Swap Fixed Rate Calculation: frequently asked questions

What is the swap fixed rate formula in CFA Level I?

The par swap fixed rate is (1 − Z_N) ÷ Σ(Δ_i × Z_i). Z_N is the final discount factor and Δ_i is the day-count fraction of each period. For annual payments, Δ_i equals 1 and the denominator is just the sum of the discount factors.

How do I calculate the par swap rate from discount factors?

Subtract the last discount factor from 1. Divide that by the sum of all the discount factors, each adjusted for its day-count fraction. The result is the fixed rate that makes the swap's value zero at initiation.

Why is the floating leg worth 1 − Z_N?

A plain vanilla swap does not exchange notional. The notional is only a device for valuing the floating leg. A floating-rate bond that resets to market rates and repays the notional at the end is worth 1 per unit of notional at initiation. Subtract the PV of that final notional repayment, which is Z_N, and you are left with the PV of the floating payments: 1 − Z_N. You therefore do not need to forecast each floating payment.

How is the swap rate different from the spot rate?

The spot rate is the yield on a single cash flow at one maturity. The swap rate is one fixed rate that applies to every payment date, so it is a weighted average of the forward rates. It matches the spot rate only when the curve is flat.