13.4 Swaps

Swaps

FMP–20 closes the Financial Markets and Products sequence with the workhorse OTC derivative: the interest-rate swap, plus currency swaps and credit-exposure intuition. Treat swaps as packages of bonds or FRAs—you already have the pricing tools from earlier readings.

Plain Vanilla IRS: Mechanics and Cash Flows

A plain vanilla interest-rate swap is an agreement to exchange fixed-rate payments for floating-rate payments on a notional principal for a tenor. In a standard fixed-for-floating IRS:

• Notional is used only to compute interest; it is not exchanged
• Fixed leg pays a fixed rate K on an agreed day-count (often 30/360 semiannual in USD classical convention; market conventions evolved with SOFR)
• Floating leg pays a reference rate (historically 3M LIBOR; now typically SOFR-based) reset in advance or in arrears per confirmation
• Net settlement is common: only the difference exchanges each period

Pay-fixed swap: pay fixed, receive floating (duration like a short fixed bond / long floater). Receive-fixed: opposite.

Worked net cash flow

Notional $10 million; semiannual fixed 3.00% (30/360); floating SOFR sets at 2.40% for the period (ACT/360 approximate for illustration—follow the problem’s day-count). Fixed payment ≈ 10m × 0.03 × 0.5 = $150,000. Floating ≈ 10m × 0.024 × (180/360) = $120,000. Pay-fixed party pays net $30,000 that period.

At initiation, a par swap has fixed rate (swap rate) set so NPV = 0. Later, if rates fall, a receive-fixed swap gains value.

Asset and Liability Transformation

Swaps are balance-sheet transformers:

A bank with floating-rate loans and fixed-rate deposits may use swaps to manage duration gap. A corporate with floating debt that wants fixed payments enters pay-fixed swaps—often cheaper operationally than refinancing the entire loan.

Intermediaries and Confirmations

Historically, dealers warehoused swaps and hedged residual risk in the interdealer market. Today many IRS are centrally cleared (LCH, CME) with variation margin; uncleared swaps face uncleared margin rules. Legal docs typically include an ISDA Master Agreement plus a confirmation specifying notional, fixed rate, floating index, reset dates, day-counts, calendars, and business-day conventions. Confirmations reduce operational risk—mismatched notionals or day-counts are expensive.

Comparative Advantage Critique

Textbook stories claim firm A has absolute advantage in fixed markets and firm B in floating, so they swap to share surplus. The critique:

1. Apparent advantage often reflects credit spreads priced differently across markets—not free lunch
2. Once credit risk, bid–ask, and funding are included, the “arbitrage” shrinks
3. Modern markets quote swaps efficiently; corporates use swaps for risk transformation and convenience, not magical comparative advantage profits

Exam takeaway: understand the classic numerical example if presented, but prefer valuation and credit explanations over naive comparative-advantage claims.

Classic numbers (then the critique)

Suppose A pays fixed 4% or floating SOFR+0.30%; B pays fixed 5.20% or floating SOFR+1.00%. A’s advantage is 1.20% in fixed vs 0.70% in floating—a 0.50% comparative gap. After a swap and dealer spread, textbooks split the 0.50%. Critique: B’s higher spreads embed higher credit risk; you cannot pocket the gap without bearing that risk somehow.

Discount Rates for Swap Valuation

Post-GFC practice: forecast floating legs with the appropriate forward curve for the index, and discount expected cash flows with an OIS / SOFR discount curve reflecting funding and collateral. Multi-curve framework: projection curve ≠ discount curve when the floating index is not the discount rate. For FRM Part I, many problems still use a single set of LIBOR-style zero rates for both; read the vignette.

Valuing an IRS as Bonds

Receive-fixed swap value:

V_receive-fixed = B_fixed − B_floating

Pay-fixed value is the negative. Immediately after a floating payment (for a LIBOR-style set-in-advance floater), B_floating ≈ notional (par floater identity, ignoring spreads). Between resets, the floater differs from par by the next known coupon’s PV effects.

B_fixed = PV of fixed coupons + PV of notional at maturity (even though notional is not exchanged—this is a valuation device).

Worked bond-method valuation

Notional 100. Fixed swap remaining: two annual payments of 5, then 105 conceptually at year 2 for the fixed bond. Zeros: r1 = 4%, r2 = 4.5% (annual compounding).

B_fixed = 5/1.04 + 105/1.045^2 = 4.808 + 96.152 = 100.960.

If just after a floating reset B_floating ≈ 100, then V_receive-fixed ≈ 100.960 − 100 = 0.960 per 100 notional.

Valuing an IRS as a Sequence of FRAs

Each future period’s net payment is like an FRA settlement on the floating rate versus fixed. Value = sum of PVs of expected net payments using forward rates for expectation (under the right measure) and discount factors.

Expected floating rate for period ≈ forward rate F from the projection curve. Net expected payment ≈ notional × (K − F) × τ for a pay-fixed view (sign depends on side), then discount.

Worked two-period FRA view (receive-fixed)

Notional 1; annual periods; K = 5%; forwards F0,1 = 4.0%, F1,2 = 5.5%; discount factors DF1 = 0.9615, DF2 = 0.9150.

PV ≈ DF1 × (0.05 − 0.04) + DF2 × (0.05 − 0.055) = 0.9615 × 0.01 + 0.9150 × (−0.005) = 0.009615 − 0.004575 = 0.00504 per unit notional.

Same economics as the bond method when curves are consistent.

Currency Swaps Valuation

A fixed-for-fixed currency swap exchanges principal at start and re-exchanges at end (or uses off-market principals), plus interest in two currencies. Valuation:

V = B_domestic − S0 × B_foreign

(for the party that receives domestic fixed and pays foreign fixed; S0 = spot FX as domestic per foreign). Each B is the bond in that currency discounted on that currency’s curve. Floating–floating and fixed–floating cross-currency swaps add floating legs; cross-currency basis spreads appear in market pricing.

Worked currency-swap sketch

Receive USD bond worth $10.20m; pay EUR bond worth €9.00m; spot 1.10 USD per EUR. V_USD receiver ≈ 10.20 − 1.10 × 9.00 = 10.20 − 9.90 = $0.30m.

Other Swap Types

Know the qualitative payoff; deep pricing of variance swaps is more VRM/Part II territory.

Swap Credit Exposure

Swap credit exposure is the cost to replace a defaulted swap when mark-to-market is positive to you (you are owed value). Key points:

• Exposure is asymmetric: only positive MTM creates current replacement risk
• Expected exposure evolves with time: often hump-shaped for IRS as uncertainty grows then remaining maturity shrinks
• Netting (ISDA) and collateral (CSA / VM/IM) sharply reduce exposure
• Clearing replaces bilateral counterparty risk with CCP risk and margin
• Current exposure vs potential future exposure (PFE) appear in credit-limit frameworks

Worked exposure intuition

You receive fixed on a 10-year swap; rates fall; MTM = +$2m. If the counterparty defaults and spreads have moved, you may replace the swap at a cost near $2m (plus bid–ask)—that is the credit loss if uncollateralized. If $2m variation margin is already posted to you, current exposure ≈ 0 (ignoring gap risk).

Exam Synthesis

Map every swap question to a side (pay/receive fixed), then value with bonds or FRAs. For currency swaps, include FX conversion of the foreign bond. Treat comparative advantage stories skeptically. For risk, remember exposure equals positive replacement cost under netting and collateral rules.