FINM6041 · Applied Derivatives
Interest-Rate Contracts & Swaps
Interest-Rate Contracts & Swaps (Lecture 3) applies no-arbitrage thinking to the rates market. It starts with the term structure: zero rates, and the forward rate implied between two of them, r_{x,y} = (R_y·T_y − R_x·T_x)/(T_y − T_x) in continuous compounding, together with the forward-rate agreement (FRA) that locks such a rate on a notional. The centrepiece is the interest-rate swap and the five-step comparative-advantage recipe: identify who is relatively cheaper in each market, compute the total gain as the difference of the two spread differences, split it (equally unless told otherwise), design the exchange under the 'floating paid all the way across' convention, and confirm each party ends better off than borrowing alone. Finally, an existing swap is valued as the difference of two bonds, V = B_fix − B_fl, where the crucial trick is that the floating leg is worth par plus the next known coupon, discounted only to the next reset — never the present value of all future floating coupons.
What this chapter covers
- 01Rate types: Treasury, LIBOR as the floating proxy, repo, zero rates and forward rates
- 02The continuous forward rate r_{x,y} = (R_y·T_y − R_x·T_x)/(T_y − T_x)
- 03Forward-rate agreements (FRAs): fixing a rate on a notional that is never exchanged
- 04The five-step comparative-advantage swap recipe
- 05Gain from trade = difference of differences (fixed spread − floating spread); default equal split
- 06The 'floating paid all the way across' convention that pins down a unique answer
- 07Swap valuation as two bonds: V = B_fix − B_fl (sign flips with the position)
- 08The floating leg = par + next coupon, discounted only to the next reset date
Interest-rate swap by comparative advantage (equal split)
- 2 marksCompute the two spread differences. Fixed spread = 7.0% − 5.0% = 2.0%; floating spread = (L + 0.9%) − (L + 0.3%) = 0.6%. Total gain from trade = the difference of the differences = 2.0% − 0.6% = 1.4%.
- 1 markIdentify comparative advantage. P is cheaper in both markets (absolute advantage), but its edge is largest in the fixed market, so P has comparative advantage in fixed and Q in floating. Each borrows where it is comparatively strong: P borrows fixed 5.0%, Q borrows floating L + 0.9%.
- 1 markSplit the 1.4% gain equally: 0.7% each. P's own best floating alternative is L + 0.3%, so its target is L − 0.4%; Q's own best fixed alternative is 7.0%, so its target is 6.3%.
- 2 marksDesign the exchange (floating paid all the way across): P pays Q LIBOR, and Q pays P a fixed 5.4%. P net = −5.0% (external) − L (to Q) + 5.4% (from Q) = L − 0.4%, i.e. a floating cost, as wanted.
- 1 markConfirm Q: Q net = −(L + 0.9%) (external) − 5.4% (to P) + L (from P) = −6.3%, a fixed cost of 6.3%, as wanted. Each party is 0.7% better off, and the combined saving is 1.4%. ✓
Key terms
- Forward rate
- The interest rate for a future period implied by today's zero rates. In continuous compounding between times T_x and T_y it is r_{x,y} = (R_y·T_y − R_x·T_x)/(T_y − T_x), where R_x and R_y are the zero rates to those horizons.
- Forward-rate agreement (FRA)
- An over-the-counter contract that fixes an interest rate on a notional principal for a future period. The notional is never exchanged; at settlement one party compensates the other for the difference between the agreed rate and the realised market rate.
- Comparative advantage
- The basis for an interest-rate swap: even if one firm can borrow more cheaply in both markets, each has a relatively larger edge in one of them. Each borrows where it is comparatively strong and swaps, sharing the difference-of-differences gain.
- Interest-rate swap
- An agreement to exchange fixed-rate for floating-rate interest payments on a notional principal over a set life. The notional is not exchanged; only the net interest difference changes hands on each payment date.
- Swap valuation (two-bond method)
- An existing swap is valued as the difference of two bonds: for a receive-fixed party, V = B_fix − B_fl. The value is zero at inception and drifts thereafter; the sign reverses for the pay-fixed counterparty.
- Floating-rate leg
- In swap valuation, the floating side is worth par immediately after its next reset, so it is priced as par plus the next known coupon, discounted only to the next payment date — not as the present value of all future floating coupons.
Interest-Rate Contracts & Swaps FAQ
What is comparative advantage in an interest-rate swap?
It is the idea that two firms can both borrow more cheaply by borrowing where each has its relatively larger edge and then swapping. Even if one firm is cheaper in both the fixed and floating markets (absolute advantage), the gap between them differs across the two markets — and the total gain available to share is that difference of differences. Each firm borrows in its comparatively strong market and the swap redistributes the cash flows so both end up better off.
How do you value an interest-rate swap in FINM6041?
As the difference of two bonds. For a party that receives fixed and pays floating, V = B_fix − B_fl: price the fixed leg as the present value of its coupons and principal at the continuously compounded zero rates, and price the floating leg as par plus its next known coupon discounted only to the next reset. The value is zero at inception and non-zero afterwards, and the sign flips for the pay-fixed side.
Why is the floating leg of a swap worth par at the reset date?
Because a floating-rate bond always resets its coupon to the current market rate, so immediately after each reset it is priced to yield the market rate and is therefore worth its face value (par). To value it partway between resets, you take that par value plus the next already-fixed coupon and discount the sum back to the next payment date. Treating the floating leg as the present value of all future floating coupons is a flagged trap.
Is LIBOR still used in FINM6041?
Yes — the course keeps LIBOR as the floating-rate proxy throughout the swap and FRA material, matching the lecture content. For exam purposes, treat LIBOR as the reference floating rate in comparative-advantage and valuation questions. The mechanics you learn (resetting to the market rate, par at reset) carry over to any successor floating benchmark.
What is an FRA and how does it relate to swaps?
A forward-rate agreement fixes an interest rate on a notional principal for a single future period; the notional is never exchanged, and at settlement one side pays the other the difference between the locked rate and the realised market rate. It is effectively a one-period building block: an interest-rate swap can be seen as a portfolio of FRAs, one for each payment date over the swap's life.
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Exam move
Swaps reward a fixed procedure, so memorise the five-step comparative-advantage recipe and run it the same way every time: (1) fixed spread minus floating spread gives the total gain; (2) name who has comparative advantage in each market; (3) split the gain (equally unless told otherwise); (4) design the exchange under 'floating paid all the way across'; (5) re-derive each party's net rate and confirm it beats their standalone cost. For valuation, drill the two-bond method V = B_fix − B_fl and burn in the one trap that separates strong answers: the floating leg is par plus the next coupon discounted to the next reset, never the PV of every future floating coupon. Both a swap-design and a swap-valuation task are heavy, repeat-every-year exam items.
