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CFA Level II Exam · Pricing and Valuation of Forward Commitments

Pricing and Valuation of Interest Rate Swaps

Updated 7 October 2026 · Fact-checked

A plain vanilla swap is priced by setting its fixed rate so the swap has zero value at start. The fixed rate equals (1 − final discount factor) ÷ sum of discount factors. Afterwards, value it as fixed-rate bond minus floating-rate bond, or as a series of FRAs, using current discount factors.

Understand Pricing and Valuation of Interest Rate Swaps

A plain vanilla interest rate swap exchanges fixed payments for floating payments on the same notional. Nothing is exchanged at the start, so a fair swap has zero value at initiation. The swap fixed rate is the rate that makes this true.

Think of the swap as two bonds. The fixed-rate payer is short a fixed-rate bond and long a floating-rate bond. A floating-rate bond is worth par on any reset date. So at the start, the fixed leg must also be worth par. That means the fixed coupons plus the notional, discounted, equal 1 per unit of notional. Solve for the coupon and you get the swap rate formula.

The second view is a series of FRAs. Each floating payment is set by a forward rate. The swap fixed rate is a weighted average of those forward rates, weighted by discount factors. Early on, the PV of the FRAs to the payer nets to zero. Some FRAs are positive and some are negative.

After initiation, rates move and the discount factors change. The old fixed rate no longer equals the new par swap rate. The swap now has value. If market swap rates have fallen below your contract rate, the fixed receiver gains and the fixed payer loses. Reverse that if rates have risen.

In an item set, you are usually given discount factors (or spot rates you must convert) and a notional. Your job is to pick out the right ones, apply the formula, and watch the sign.

Key formulas to remember

Discount factor from a spot rate
Z_t = 1 ÷ (1 + S_t)^t
Use the spot rate for the same maturity. For a period of less than a year, use the periodic rate.
Par swap fixed rate (periodic)
Fixed rate per period = (1 − Z_N) ÷ Σ Z_i
Sum runs over all payment dates 1 to N. For an annualized rate, multiply by the number of periods per year (e.g. ×4 for quarterly).
Par swap rate with day count
S = (1 − Z_N) ÷ Σ (days_i ÷ 360 × Z_i)
Use when the question gives actual day counts. Annual payments with equal periods reduce to the simple form.
Implied forward rate
f_t = Z_(t−1) ÷ Z_t − 1
This is the one-period forward rate for period t that sets each floating payment in the FRA view.
Value to fixed receiver after initiation
V = [ (S_old − S_new) × Σ Z_i(new) ] × Notional
S_new is the current par swap rate for the remaining term. Sum is over the remaining payment dates.
Bond approach to value
V(receiver) = PV(fixed bond at S_old) − PV(floating bond)
Right after a reset, the floating bond is worth 1 per unit of notional. Between resets, it equals (1 + next floating rate × period) × Z to the next payment date.
Payer value
V(payer) = − V(receiver)
Swap value is zero sum between the two parties.

How to solve Pricing and Valuation of Interest Rate Swaps questions

Use this order for any swap pricing or valuation question in an item set.

  1. 1Read the vignette and note the notional, payment frequency, remaining term, fixed rate, and whether you are payer or receiver.
  2. 2Find the discount factors for each remaining payment date. If given spot rates, convert with Z = 1 ÷ (1 + S)^t. If given forward rates, build Z by chaining 1 ÷ (1 + f).
  3. 3For a new swap, compute the sum of the discount factors and apply (1 − Z_N) ÷ Σ Z. Annualize if the payments are not annual.
  4. 4For a swap already in place, get the current par swap rate using the new discount factors and the remaining term.
  5. 5Compute the value per unit of notional: (S_old − S_new) × Σ Z_new for the receiver. Check by the bond method: PV of fixed coupons plus PV of notional minus 1.
  6. 6Multiply by the notional. Flip the sign if you are the payer.
  7. 7Sense-check. If market rates are below your fixed rate, the receiver should show a gain. If not, recheck.

Quickest way: Receiver value in one line

When to use it: Use when you are given current discount factors and the old fixed rate, and the valuation date is a reset date.

  1. Add the new discount factors. Call this sum A.
  2. Compute the fixed leg PV per 1 of notional: S_old × A + Z_N.
  3. Subtract 1 for the floating leg (it is at par on a reset date).
  4. Multiply by notional. That is the receiver value. The payer value is the negative.
  5. If the date is between resets, replace the 1 with (1 + next floating rate × period) × Z of the next payment date.

Common mistakes in Pricing and Valuation of Interest Rate Swaps

  • Using only the final discount factor in the denominator

    Students mix up the swap rate formula with a zero-coupon yield.

    Fix: The denominator is the sum of all discount factors on payment dates. Only the numerator uses the last one.

  • Forgetting to annualize a quarterly or semiannual rate

    The formula gives a per-period rate, and the answer looks plausible.

    Fix: Multiply the per-period result by the number of periods per year. Check that the answer is close to spot rates in size.

  • Getting the sign wrong for payer versus receiver

    Students remember a rate difference but not who benefits.

    Fix: Receiver gains when the market swap rate is below the contract rate. Payer gains when it is above. Always sanity-check the sign.

  • Using the original discount factors to value the swap later

    The initial table is still in the vignette.

    Fix: Use discount factors for the current date and the remaining payment dates only. Drop payments already made.

  • Treating the floating leg as worth zero or as the PV of fixed payments

    Confusion between swap value and bond value.

    Fix: On a reset date the floating bond equals par, so the swap value equals the fixed bond minus 1 per unit of notional. Between resets, include the known next floating payment.

  • Mixing up the new swap rate with the old fixed rate in the formula

    Both rates appear in the vignette and look alike.

    Fix: Label them S_old (contract) and S_new (market par rate for the remaining term) before you calculate.

Worked examples

Example 1

A vignette gives these discount factors for annual dates: Z1 = 0.9709, Z2 = 0.9426, Z3 = 0.9151. An analyst prices a new 3-year annual-pay plain vanilla swap. Q1: What is the swap fixed rate? Q2: What is the implied forward rate for year 3?

Show the solution
  1. Q1: Sum the discount factors: 0.9709 + 0.9426 + 0.9151 = 2.8286.
  2. Numerator: 1 − Z3 = 1 − 0.9151 = 0.0849.
  3. Fixed rate = 0.0849 ÷ 2.8286 = 0.0300, or about 3.00%.
  4. Q2: f3 = Z2 ÷ Z3 − 1 = 0.9426 ÷ 0.9151 − 1 = 0.0301, or about 3.0%.
  5. The curve is flat at about 3%, so the forward rate is close to the swap rate.

Answer: Q1: about 3.00%. Q2: about 3.0% for year 3.

Example 2

Two years remain on a swap with annual payments and notional ₹5,00,00,000. You pay fixed at 3.50% and receive floating. The next floating payment is set at 2.00% (1 ÷ Z1 − 1). Current discount factors: Z1 = 0.9804, Z2 = 0.9426. Q1: What is the implied year-2 forward rate? Q2: What is the value of the swap to you, the fixed payer?

Show the solution
  1. Q1: f2 = Z1 ÷ Z2 − 1 = 0.9804 ÷ 0.9426 − 1 = 0.0401, or about 4.01%.
  2. Q2, FRA view. Year 1 net per 1 of notional: (0.019992 − 0.035) × 0.9804 = −0.014714.
  3. Year 2 net: (0.040102 − 0.035) × 0.9426 = +0.004809.
  4. Sum = −0.009905 per 1 of notional.
  5. Check with bonds: floating PV = 1 − Z2 = 0.0574. Fixed PV = 0.035 × (0.9804 + 0.9426) = 0.067305. Payer = 0.0574 − 0.067305 = −0.009905. It matches.
  6. Multiply by notional: −0.009905 × ₹5,00,00,000 = −₹4,95,250.

Answer: Q1: about 4.01%. Q2: the swap is worth about −₹4,95,250 to the fixed payer (a liability). The fixed receiver would hold +₹4,95,250.

Exam tips

  • Write the sum of discount factors first. Almost every swap calculation in an item set uses it.
  • Decide payer or receiver before you calculate, and write the expected sign of the answer. Market rates below your fixed rate favour the receiver.
  • Check whether the valuation date is a reset date. If so, the floating leg is at par and the calculation is short.
  • When the vignette gives spot rates instead of discount factors, convert them first. Keep four decimal places to avoid rounding drift, since the answer options can be close together.
  • Expect some questions to ask for a forward rate or the swap rate rather than the value. Read the last line of each question before you start.

Pricing and Valuation of Interest Rate Swaps in other exams

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

Pricing and Valuation of Interest Rate Swaps: frequently asked questions

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

Take one minus the final discount factor and divide by the sum of all discount factors on payment dates. This gives a per-period rate. Multiply by the number of periods per year to annualize it.

Why is a swap's value zero at initiation?

The swap fixed rate is set so the fixed bond and the floating bond have the same present value. The floating bond is worth par on the reset date, so the fixed bond is also set to par. The net value is then zero for both sides.

How is a swap a series of FRAs?

Each floating payment date works like an FRA where one side pays a fixed rate and the other receives the floating rate set by the forward rate. The swap value is the sum of the PVs of these FRAs. The swap fixed rate is the single rate that makes the total zero at the start.

How do I find the value of a fixed receiver swap after initiation?

Find the current par swap rate for the remaining term and multiply the difference (S_old − S_new) by the sum of current discount factors and the notional. You can also compute the fixed bond PV minus the floating bond PV. A positive result means the receiver has a gain.