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

Question

AcadiFi · Accepted Answer

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.

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

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