How do you value an interest rate swap after initiation? I can't figure out which leg to discount at which rate.

Swap valuation trips up a lot of candidates, but it's really just bond math in disguise. Let me walk through it. ## The Key Insight A plain vanilla interest rate swap is equivalent to a long position in one bond and a short position in another: - **Fixed-rate payer** = Long floating-rate bond + Short fixed-rate bond - **Fixed-rate receiver** = Long fixed-rate bond + Short floating-rate bond To value the swap for the fixed-rate payer: **V_swap = Value of floating-rate bond - Value of fixed-rate bond** ## Worked Example: Trident Pharmaceuticals **Original swap terms (initiated 1 year ago):** - Notional: $50 million - Trident pays fixed 4.80% annually - Trident receives 1-year SOFR (floating), reset annually - Remaining term: 2 years (payments at Year 1 and Year 2) - Payment frequency: Annual **Current market data (1 year after initiation):** - Current 1-year spot rate: 5.20% - Current 2-year spot rate: 5.45% - SOFR rate just reset (for next payment): 5.20% ## Step 1: Value the Fixed-Rate Bond The fixed leg pays 4.80% x $50M = **$2.40M** at Year 1 and **$2.40M + $50M = $52.40M** at Year 2. PV of fixed bond = $2.40M / (1.052) + $52.40M / (1.0545)^2 = $2.2814M + $47.1456M = **$49.4270M** ## Step 2: Value the Floating-Rate Bond A floating-rate bond resets to par on each coupon date. Since the next floating payment is based on the just-reset SOFR of 5.20%: Next floating payment = 5.20% x $50M = **$2.60M** After this payment, the floating bond resets to par ($50M) PV of floating bond = ($2.60M + $50M) / (1.052) = $52.60M / 1.052 = **$50.0000M** (The floating-rate bond is worth exactly par right after a rate reset — this is the key insight that simplifies the calculation.) ## Step 3: Value the Swap From Trident's perspective (fixed-rate payer): V_swap = Floating bond value - Fixed bond value = $50.0000M - $49.4270M = **+$0.5730M** (or +$573,000) Trident is **in the money** by $573,000 because rates rose after the swap was initiated. Trident locked in paying 4.80% but the market now demands 5.20-5.45%, so Trident's fixed payments are below-market — that's valuable. ## Sanity Check When rates rise, the fixed-rate payer benefits (they're paying below-market fixed and receiving above-market floating). When rates fall, the fixed-rate receiver benefits. This is consistent with our result. ## Forward Rate Agreement Connection A single-period swap is equivalent to an FRA. The CFA exam sometimes asks you to value a swap as a series of FRAs, where each FRA's value is the present value of the difference between the fixed rate and the implied forward rate for that period. Using our spot rates: - Implied 1-year forward rate one year from now: f(1,1) = [(1.0545)^2 / (1.052)] - 1 = **5.70%** - Year 2 FRA value (per $50M): (5.70% - 4.80%) x $50M / (1.0545)^2 = $0.45M / 1.1120 = **$0.4046M** - Year 1 "FRA" value: (5.20% - 4.80%) x $50M / 1.052 = $0.20M / 1.052 = **$0.1901M** - Total: $0.4046 + $0.1901 = **$0.5947M** (Small difference from $0.573M due to rounding in intermediate steps.) For more swap and FRA valuation practice, explore AcadiFi's CFA Level II Derivatives section.