Interest Rate Swap Valuation with AI: Fixed vs Floating, SOFR Discounting, and DV01

Interest Rate Swap Valuation: The Core Framework

An interest rate swap is one of the most liquid derivatives in the world — the global notional outstanding in IRS exceeds $350 trillion. The basic structure is straightforward: two counterparties exchange fixed and floating interest rate cash flows on a notional principal. No principal is exchanged. The valuation — the MTM of the position — depends on where rates are now relative to where they were when the swap was struck.

Post-LIBOR transition (2023), virtually all new USD swaps reference SOFR, and the discounting curve for collateralized trades is the SOFR OIS curve. EUR swaps use ESTR (Euro Short-Term Rate). The mechanics of valuation didn't change with the IBOR transition — but traders and analysts need to be precise about which curve is used for projection (estimating the floating cash flows) and which for discounting (computing present values).

ClaudeFinanceLab's Quantitative Finance and Compliance & Risk templates include interest rate derivative valuation tools, DV01 calculation, SA-CCR exposure analysis, and XVA overlay calculators for swaps books.

Fixed Leg Valuation

The fixed leg is straightforward: it's a series of fixed coupon payments on the notional. Each payment is known in advance (fixed rate × notional × day count fraction), and the PV is each payment discounted at the appropriate OIS rate for that payment date. The day count convention matters — USD swaps typically use 30/360 for the fixed leg and Actual/360 for the floating leg.

• "Fixed leg PV calculation for a 5-year USD IRS: Notional $100M. Fixed rate 4.75%. Semi-annual payments (6-month payment intervals). Day count: 30/360 for fixed leg. SOFR OIS discount rates (zero rates): 6M 5.12%, 1Y 4.98%, 1.5Y 4.89%, 2Y 4.84%, 2.5Y 4.81%, 3Y 4.79%, 3.5Y 4.78%, 4Y 4.77%, 4.5Y 4.76%, 5Y 4.75%. Each semi-annual payment = 4.75% × $100M / 2 = $2,375,000. Calculate: (1) discount factor for each payment date using continuous compounding from the zero rates, (2) PV of each fixed cash flow, (3) total PV of fixed leg, (4) how does the fixed leg PV change if the discount curve shifts up 50bps in parallel?"
• "Floating leg PV — SOFR-based: For the same 5Y $100M swap: the floating leg pays daily-compounded SOFR in arrears, resetting every 6 months. Under standard SOFR OIS pricing: (1) at inception, the floating leg PV = notional (par value) for a fully collateralized swap under SOFR discounting — confirm this by showing the relationship between forward SOFR rates and par, (2) 18 months into the swap, rates have risen 75bps across the curve. The first 3 coupon periods have been paid. The remaining floating leg covers 3.5 years. Calculate the current PV of the remaining floating leg: the resets will reflect the new higher SOFR curve, so the floating leg is now worth more than 18 months ago. Estimate the floating leg PV using the shifted discount curve."
• "MTM of a seasoned pay-fixed swap: 2 years ago, we entered a 10-year pay-fixed IRS at a fixed rate of 3.80% on $75M notional. Today's 8-year USD SOFR par swap rate is 4.95%. We have 8 years (16 semi-annual payments) remaining. Fixed coupon: $75M × 3.80% / 2 = $1,425,000 per period. Today's 8-year par swap rate 4.95% implies our fixed rate is below market — our position should have negative MTM (we're paying below-market fixed, so the counterparty would pay to exit). Calculate: (1) PV of remaining fixed leg at 4.95% discount level, (2) PV of floating leg (use approximation: floating leg ≈ par for a collateralized swap + adjustment for received SOFR vs. 4.95% OIS), (3) net MTM and whether it is an asset or liability on our books."

Par Swap Rate Calculation

The par swap rate is the fixed rate that makes a new swap have zero NPV at inception. It is derived entirely from the forward rate curve — no market judgment is required. For a standard fixed-for-SOFR swap, the par rate equals the sum of discounted forward SOFR rates divided by the annuity factor (sum of discount factors). This is the rate you'd get if you called your dealer for a quote on a new at-the-money swap.

• "Par rate calculation for a 3-year semi-annual USD swap: SOFR OIS zero rates: 6M 5.10%, 1Y 4.95%, 1.5Y 4.88%, 2Y 4.82%, 2.5Y 4.78%, 3Y 4.75%. Steps: (1) calculate discount factors for each semi-annual period using continuous compounding, (2) calculate forward rates for each 6-month period from the zero curve (f(t1, t2) = (r2×t2 - r1×t1)/(t2-t1) for continuous rates, adjusted for semi-annual compounding), (3) calculate the expected floating cash flow for each period = forward rate × $100M × 0.5, (4) PV of floating cash flows = sum of (expected cash flow × discount factor), (5) annuity factor = sum of discount factors, (6) par fixed rate = PV of floating cash flows / annuity factor. What is the 3-year par swap rate from this curve?"

DV01, Duration, and Rate Sensitivity

DV01 is the primary risk metric for interest rate swaps — it measures P&L sensitivity to a 1 basis point move in interest rates. For a pay-fixed swap, a rise in rates is beneficial (floating receipts increase, fixed payments unchanged), so DV01 is positive. For a receive-fixed swap, DV01 is negative (rising rates hurt the position). Risk managers use DV01 to aggregate interest rate risk across entire swap books and to size hedges.

• "DV01 calculation for a swap portfolio: We have 4 swaps outstanding. Swap 1: pay-fixed 5Y $50M at 4.50%, inception 6M ago. Swap 2: receive-fixed 7Y $80M at 4.85%, inception 1Y ago. Swap 3: pay-fixed 2Y $30M at 5.10%, inception 3M ago. Swap 4: receive-fixed 10Y $20M at 4.20%, inception 3Y ago. Approximate modified duration: 5Y pay-fixed ≈ 4.5Y, 7Y receive-fixed ≈ 6.0Y, 2Y pay-fixed ≈ 1.8Y, 10Y receive-fixed ≈ 7.8Y. Calculate: (1) DV01 for each swap (direction and magnitude), (2) net portfolio DV01, (3) is the portfolio net long or short duration?, (4) what notional of a 5Y swap would hedge the net DV01 to within ±$5,000?"
• "Bucket DV01 and yield curve sensitivity: Instead of a parallel shift, we want to know our sensitivity to individual tenor points on the SOFR curve. Swap: pay-fixed 10Y $100M, entered ATM. Approximate DV01 by tenor bucket: 1Y bucket (sensitivity to 1Y rate move) ≈ $1,200, 2Y ≈ $2,800, 3Y ≈ $4,100, 5Y ≈ $7,800, 7Y ≈ $9,500, 10Y ≈ $12,400, >10Y ≈ $1,800. Total DV01 ≈ $39,600. Now: the yield curve steepens — 2-year rate falls 15bps, 10-year rate rises 10bps, everything else unchanged. Calculate the P&L impact on the swap position using the bucket DV01. Is a steepening good or bad for a pay-fixed 10Y swap?"
• "DV01 hedge with Treasury futures: We have a pay-fixed 10Y $200M swap with DV01 $80,000. We want to hedge with 10-year Treasury note futures. 10Y Treasury future: 1 contract = $100,000 notional, modified duration of CTD bond approximately 8.2 years, DV01 per contract = $100,000 × 8.2 × 0.0001 = $82. Calculate: (1) number of TY futures contracts needed to hedge the swap DV01, (2) should we buy or sell the futures (remember: pay-fixed swap has positive DV01 — benefits from rising rates — so to hedge we need a position that loses when rates rise, i.e., short futures), (3) what basis risk exists between the swap DV01 and the futures hedge (swap uses SOFR OIS, futures track Treasury yield)?"

Swap Pricing: Bid-Offer and Mid-Market Rates

In practice, when a client calls a bank to enter a swap, they don't transact at the mid-market par rate — they transact at either the bid (the fixed rate the bank will pay if the client is receive-fixed) or the offer (the fixed rate the bank will receive if the client is pay-fixed). The bid-offer spread on vanilla USD IRS is typically 0.5-2bps depending on tenor and liquidity. Understanding the bid-offer economics helps when negotiating swap pricing for a corporate hedging program.

• "Swap pricing for a corporate interest rate hedge: A €500M revenue industrial company wants to hedge its floating-rate debt. Outstanding debt: €180M term loan at EURIBOR 3M + 210bps, 4-year remaining maturity. The treasurer wants to convert to fixed to remove rate uncertainty. Current 4-year EUR swap (ESTR-based par rate): 3.42% mid. Bank offers: 3.44% (client pays fixed, bank receives fixed — the 'offer' side). Alternative: 4-year €180M pay-fixed IRS at 3.44%, effectively locking in total borrowing cost of 3.44% + 210bps = 5.54%. Compare to: 5-year fixed-rate bond refunding (indicative rate 5.72%). Show: (1) all-in cost of the swap-hedged floating loan vs. fixed refunding, (2) break-even analysis — how much must EURIBOR rise for the unhedged position to exceed 5.54%?, (3) what is the mark-to-market if EURIBOR falls 100bps in year 1 — does this create P&L volatility under IAS 39/IFRS 9 if hedge accounting is not applied?"
• "SOFR compounding vs. term SOFR: Our company's new revolving credit facility references term SOFR (CME 3M Term SOFR) rather than daily-compounded SOFR in arrears. We want to enter a matching interest rate swap. The swap desk quotes: (1) pay-fixed vs. daily-compounded SOFR at 4.82%, (2) pay-fixed vs. 3M Term SOFR at 4.78%. The 4bp spread reflects the fact that daily-compounded SOFR is the risk-free benchmark while Term SOFR includes a small term risk premium. If we enter the Term SOFR swap to match our loan, we need a receive-Term-SOFR swap to create the hedge. Explain: (1) why Term SOFR typically trades 3-5bps above SOFR OIS, (2) does this basis risk create P&L exposure?, (3) does ISDA documentation distinguish between SOFR swap types, (4) hedge accounting qualification: does the swap-to-debt documentation need to specify Term SOFR basis?"

CVA Overlay on Swap Valuation

The MTM of a swap with a creditworthy counterparty is reduced by CVA (Credit Valuation Adjustment) — the market value of the risk that the counterparty defaults before the swap matures. For a collateralized swap under a standard ISDA CSA, CVA is small because daily margining limits exposure. For an uncollateralized corporate swap (e.g., a corporate treasury hedging its debt with a one-way CSA or no CSA), CVA can be material.

• "CVA for an uncollateralized corporate swap: Our bank has a 5Y pay-fixed $50M IRS with an uncollateralized BBB-rated corporate counterparty (CDS spread 95bps, LGD 60%). The swap is currently in-the-money to us by $2.8M (counterparty owes us). Using simplified CVA: CVA ≈ LGD × (sum of EPE(t) × PD_marginal(t)) over quarterly intervals. Assume EPE declines linearly from current exposure $2.8M to zero at maturity. Marginal default probability from 95bp flat CDS: approximately 1.9% per year using standard conversion. Calculate: (1) simplified CVA in dollars, (2) CVA as percentage of current MTM, (3) CVA spread in bps on the notional, (4) if the counterparty's CDS widens to 180bps, what is the new CVA?"

Where to Start

For derivatives traders and risk analysts working with interest rate swaps, the Quantitative Finance templates include Swap Pricer (fixed/floating PV, par rate, DV01), Curve Risk Analyzer (bucket DV01, carry-and-roll), and CVA Overlay for uncollateralized counterparties. For corporate treasury teams using swaps to hedge debt, the Compliance & Risk category has IFRS 9 Hedge Documentation Assistant and Fair Value Disclosure Drafter. See also Claude AI for XVA and CVA Trading Desks and XVA Explained.

Frequently Asked Questions

What is the difference between a par swap and an off-market swap?

A par swap is struck at the current market par rate — no upfront payment, zero NPV at inception. An off-market swap has a fixed rate that differs from the current par rate, creating a non-zero NPV at inception. Off-market swaps are used in several contexts: (1) amending an existing swap that is in- or out-of-the-money without breaking and rebooking (the "blended rate" approach), (2) buy-side clients who want to embed a margin into a trade, (3) structured products where the swap economics are packaged with other features. Off-market swaps require an upfront payment to compensate the counterparty for the below/above-market fixed rate, or alternatively the counterparty accepts a spread on other terms.

How does hedge accounting affect swap valuation under IFRS 9?

Under IFRS 9 (and IAS 39), swaps designated as cash flow hedges are carried at fair value on the balance sheet, but the effective portion of the gain or loss goes to OCI (Other Comprehensive Income) rather than P&L. Only the ineffective portion hits income. When the hedged item (the floating rate debt) affects P&L, the OCI balance is recycled to P&L at the same time. Without hedge accounting designation, every period's MTM change on the swap hits P&L directly, creating volatility that doesn't match the economics of the hedging relationship. IFRS 9 hedge accounting qualification requires prospective and retrospective effectiveness testing — the swap terms should closely match the hedged item (same notional, maturity, and reference rate) to pass effectiveness testing.

What is the convexity adjustment in swap valuation?

Convexity arises because the relationship between swap price and yield is not perfectly linear — it's curved. For vanilla fixed-for-floating IRS, convexity is generally small relative to other risks and often ignored in first-order risk management. However, convexity becomes important for: (1) very long-tenor swaps (30-50 year) where the duration-weighted DV01 approximation breaks down for large rate moves, (2) swaptions (options on swaps), where convexity significantly affects delta and the hedging requirement, (3) CMS (Constant Maturity Swap) products, where the floating leg references a swap rate rather than SOFR — CMS swaps require a convexity correction because the expected value of a swap rate in the future is not the forward swap rate. Most XVA desks model convexity explicitly for positions sensitive to it.