Skip to content

FRM Exam Part I · Swaps

Plain Vanilla Interest Rate Swap Valuation Explained

Updated 11 October 2026 · Fact-checked

A plain vanilla swap exchanges fixed payments for floating payments on the same notional. Value it as a fixed-rate bond minus a floating-rate bond, or as a set of forward rate agreements. Discount all cash flows with current discount factors. The par swap rate is the fixed rate that makes the swap worth zero at the start.

Understand Plain Vanilla Interest Rate Swap Valuation

A plain vanilla interest rate swap has one party paying a fixed rate and the other paying a floating rate. Both rates apply to the same notional, and only the net interest is exchanged. The notional itself never changes hands.

The bond view is the easiest way to see the value. The fixed-rate payer behaves like someone who has issued a fixed-rate bond and bought a floating-rate bond with the proceeds. So the value to the fixed-rate payer = floating bond value − fixed bond value. The fixed-rate receiver has the opposite position.

The floating bond is simple. Just after a reset date, a floating-rate bond is worth its par value, because it pays the current market rate. Between reset dates, you know the next floating payment. The bond is then worth (notional + next floating payment) discounted from the next payment date only.

The FRA view gives the same answer. Each floating payment is replaced by its forward rate. Each period is then a forward rate agreement: the fixed payer receives (forward rate − fixed rate) × accrual × notional, discounted to today. Add up all periods.

The par swap rate is the fixed rate at which the fixed bond is worth par and the swap is worth zero at inception. It depends only on today's discount factors. After inception, rates move and the swap gains value for one side and loses it for the other.

Key formulas to remember

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.

How to solve Plain Vanilla Interest Rate Swap Valuation questions

Use this method for any swap valuation question. Decide first whether you are on a reset date or between resets.

  1. 1Identify your side: are you paying or receiving fixed? Note notional L, fixed rate k and payment frequency.
  2. 2List the remaining payment dates and the accrual fraction τ (for example 0.5 for semiannual).
  3. 3Get the discount factors d(t) for each date. If you are given zero rates, convert: d = 1 ÷ (1 + z/m)^(m×t), or e^(−zt) if continuous.
  4. 4Value the fixed bond: fixed coupons (k × τ × L) discounted, plus L × d(T).
  5. 5Value the floating bond: L if you are on a reset date. Otherwise (L + known next floating payment) × d(t₁).
  6. 6Take the difference. Pay fixed = B_float − B_fix. Receive fixed = B_fix − B_float.
  7. 7Cross-check with the FRA method or the par-rate shortcut if time allows, and check the sign makes sense.

Quickest way: Par-rate shortcut

When to use it: Use it when the swap has just reset or started and you can compute the par rate from discount factors.

  1. Compute the annuity A = Σ τ × d(tᵢ).
  2. Compute the par rate s = (1 − d(T)) ÷ A.
  3. Value of pay-fixed swap = (s − k) × A × L.
  4. If s > k, the pay-fixed side gains. If s < k, the receive-fixed side gains.
  5. Between resets, fall back on the bond method, because the next floating payment is already fixed.

Common mistakes in Plain Vanilla Interest Rate Swap Valuation

  • Using par value for the floating bond when the swap is between reset dates.

    Students remember 'floating bond = par' as a universal rule.

    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.

    The two sides are mirror images and it is easy to swap them.

    Fix: Fixed payer = floating bond − fixed bond. Write the formula first. Then check: if rates rose, the fixed payer should gain.

  • Forgetting to include the notional in the fixed bond's final payment.

    No notional is exchanged in a swap, so students leave it out.

    Fix: The bond approach adds L to both bonds at the end. It cancels out in the economics, but you must include it in the fixed bond's value so that it matches the floating bond.

  • Ignoring the accrual fraction for semiannual or quarterly swaps.

    Students use the annual rate directly as the coupon.

    Fix: Coupon = k × τ × L. For semiannual, divide the annual rate by 2.

  • Computing the par swap rate as the average of the spot rates.

    It looks plausible and is quick.

    Fix: Use s = (1 − d(T)) ÷ Σ τ d(tᵢ). It is a discount-factor-weighted measure, not a simple average.

  • Using the swap's fixed rate to discount cash flows.

    Students confuse the contract rate with market rates.

    Fix: Always discount with the current market discount curve, not the swap's own fixed rate.

Worked examples

Example 1

A 2-year annual-pay swap has just reset. Notional is $10 million. You pay fixed at 4.50% and receive floating. Discount factors: d(1) = 0.9615 and d(2) = 0.9246. What is the value to you, and what is the par swap rate? (Options for value: −$94,745; +$94,745; −$9,474; +$1,009,475.)

Show the solution
  1. Annuity A = 0.9615 + 0.9246 = 1.8861.
  2. Par rate s = (1 − 0.9246) ÷ 1.8861 = 0.0754 ÷ 1.8861 = 3.998%.
  3. Fixed bond per $1: 0.045 × 1.8861 + 0.9246 = 0.0848745 + 0.9246 = 1.0094745. For $10 million: $10,094,745.
  4. Floating bond is at par on the reset date: $10,000,000.
  5. Pay-fixed value = 10,000,000 − 10,094,745 = −$94,745.
  6. Check: (3.998% − 4.50%) × 1.8861 × 10,000,000 ≈ −$94,700. The small gap is rounding of the par rate.

Answer: The swap is worth about −$94,745 to you (the fixed payer). The par swap rate is about 3.998%, and you pay 4.50%, so you are overpaying.

Example 2

A semiannual swap with $50 million notional has 9 months left, with payments in 3 months and 9 months. You receive fixed at 4.8% and pay floating. The floating rate set at the last reset was 5.0% (annual rate). Discount factors: d(0.25) = 0.9900 and d(0.75) = 0.9650. Find the swap value to you.

Show the solution
  1. Fixed coupon = 0.048 × 0.5 × 50,000,000 = $1,200,000.
  2. Fixed bond = 1.2 × 0.9900 + (50 + 1.2) × 0.9650 in $ millions = 1.188 + 49.408 = $50.596 million.
  3. Next floating payment = 0.05 × 0.5 × 50,000,000 = $1,250,000.
  4. Floating bond = (50 + 1.25) × 0.9900 = 50.7375, so $50.7375 million.
  5. Receive-fixed value = B_fix − B_float = 50.596 − 50.7375 = −$0.1415 million.

Answer: The swap is worth about −$141,500 to you as the fixed receiver, which is +$141,500 to the fixed payer.

Exam tips

  • 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.
  • Do a sign check at the end. If market rates rose above your fixed rate, the fixed payer should have a positive value.
  • On a financial calculator, store the discount factors in memory and compute the annuity once. Reuse it for both the par rate and the fixed bond.

Practice questions from Swaps

Plain Vanilla Interest Rate Swap Valuation in other exams

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

Plain Vanilla Interest Rate Swap Valuation: frequently asked questions

How do I value an interest rate swap using bonds?

Treat the fixed leg as a fixed-rate bond and the floating leg as a floating-rate bond. Discount both with current discount factors. The fixed payer's value is the floating bond minus the fixed bond. The receiver's value is the reverse.

How is valuing a swap with FRAs different?

You replace each floating payment with the forward rate for its period. Each period then becomes an FRA. You discount the net payment (forward rate minus fixed rate, times accrual and notional) and add them up. The result equals the bond method.

How do I calculate the par swap rate?

Take one minus the final discount factor and divide by the sum of accrual-weighted discount factors across all payment dates. This is the fixed rate that makes the swap's value zero at the start.

What is the value of a swap after a reset date?

Just after a reset, the floating bond is worth par. The swap value is then the notional minus the fixed bond value for the fixed payer. Between resets, the floating bond is worth the notional plus the next known floating payment, discounted from the next payment date.