Interest Rate Swaps - Lecture Notes
Background and Context
The lecturer begins by noting a recent monetary policy move: the Fed funds rate target was lowered by 45 basis points earlier in the afternoon. He acknowledges a busy session ahead and expresses optimism about getting through the material efficiently, while also inviting students to slow him down if needed. He mentions that he will cover interest rate swaps, with background reading on chapter 22, and previews that the class will later discuss SOFR-based swaps (so-called “SOFR” swaps) which use daily-rate compounding between swap payment dates. He explains that he will first present a simple, intuitive valuation method based on rates that match the actual payment periods, and that there are more complex bells-and-whistles for the newer SOFR-based framework which he will not teach in detail today. He points to a pay paper of his for the more complete treatment, and situates swaps in a historical context, noting that forwards and futures are much older securities, while swaps are relatively new innovations (currency swaps began in 1981, interest rate swaps in 1982, credit default swaps in 1995).
What is a Swap?
A plain vanilla (simple) interest rate swap is a bilateral financial contract where two counterparties agree to exchange cash flows: one leg pays a floating rate on a notional amount, while the other pays a fixed rate on the same notional. The contract has a stated maturity, with periodic intermediate payment dates. The notional itself is not exchanged at the start or end; only the net interest payments are exchanged. In practice, the market is netted: the net amount owed from one side to the other is exchanged.
The two legs are often called the floating leg and the fixed leg. The floating leg’s payments are based on a reference rate (the rate for the period between payment dates) times the notional, while the fixed leg’s payments are a fixed rate times the notional. In real markets, collateral is typically posted to guarantee performance, and there are credit and liquidity considerations, including limits on how much notional one can transact.
In the class example, two illustrative counterparties are introduced: Floating Cruisers (which has a floating-rate loan) and Fixed Towers (which has a fixed-rate loan). They decide to swap to align with their preferences: Cruisers wants floating, Towers wants fixed. Intermediaries (like Goldman Sachs) facilitate the match, acting as the counterparty on either side or hedging the exposure with other instruments. The lecturer emphasizes that a dealer can hedge the swap by creating a synthetic position from other securities; this netting removes the dealer’s own rate risk and earns a service fee for creating the hedge.
Why Swaps Are Used
Swaps let counterparties avoid issuing new debt to change their interest-rate exposure. For example, a company with a floating-rate loan can swap its floating payments for fixed payments, avoiding the costs and hassles of retiring the old loan and issuing new debt. Conversely, a company with a fixed-rate loan can swap into floating exposure without taking on the transaction costs of refinancing. The example illustrates how both parties ultimately benefit by swapping only the payment streams rather than the loan itself.
Valuing a Simple Swap (No Model, Intuition-Based)
The lecturer presents two key ideas (sometimes called tricks) used to value straight, plain-vanilla swaps in a simple framework:
Trick 1: pretend notional is exchanged. In reality it is not, but conceptually separating the two legs as if notional were exchanged helps intuition; the netting will still yield the correct cash flows.
Trick 2: add the notional back into both legs. With the notional added back, each leg resembles a standard loan: one leg becomes a fixed-rate loan, the other a floating-rate loan. Thus, valuing a swap reduces to valuing two standard loans and taking their difference.
The method relies on a market framework that is frictionless, competitive, and arbitrage-free. The lecturer notes two major model-building steps that will come later (in HDM/HJM-type models), but for this simple valuation those detailed dynamics are not required.
Floating Leg Valuation (Par on Payment Dates)
In the simplified, discrete-time framework, a floating-rate loan is always valued at par on the payment dates. The intuition is straightforward: if you borrow N today, you owe N immediately (the loan’s par). After the next coupon payment, you owe the same par value N again, so the present value of known, upcoming cash flows equals the par amount. Consequently, the floating leg’s value is N (the notional) on each payment date, and the sequence reverts to par between payments.
Fixed Leg Valuation
By contrast, a fixed-rate loan has a known sequence of cash flows: the fixed coupon payments on each payment date. The value of the fixed leg is the present value of these fixed cash flows, which requires discounting each fixed cash flow by the appropriate zero-coupon bond price, and then summing them up. In the simplified approach that includes the notional for intuition, the fixed leg’s value is the discounted present value of the fixed coupons plus the notional, reflecting the decomposition into a fixed-rate loan plus the notional exchange idea.
Worked Numerical Example (Illustrative Numbers)
To illustrate, suppose the notional is and the fixed rate on the swap is . The fixed leg pays in each period. In the example with three remaining payment dates, the present value of the fixed leg is shown as approximately (including the notional, for intuition). The floating leg, in this simple model, is valued at par, i.e. The net value of the swap, if you are long the floating leg and short the fixed leg, is therefore
Thus, in this example, the holder of the fixed leg (short fixed) is in a liability position of . The lecturer notes that a counterparty can exit the swap by paying the current value to the other party, or may transfer the obligation by finding a third party to assume the position (or by entering into an offsetting swap).
Symbolic Valuation and Swap Rate
Symbolically, the value of the swap can be written as the present value of the floating leg minus the present value of the fixed leg, with both notionals included as part of the decomposition into two loans:
where the floating leg is par in this simple model, so The fixed leg has known fixed cash flows, discounted using zero-coupon bond prices and periodic year fractions :
where the sum runs over all payment dates in the swap’s life.
The swap rate is defined as the fixed rate that sets the swap value to zero at inception (i.e., the par swap rate). In the simple setting, the par swap rate satisfies
where $Tm$ is the swap’s final maturity date and $P(0,t)$ are the current zero-coupon prices. The numerator $1 - P(0,T_m)$ reflects the present value of receiving the notional at maturity in the floating leg, while the denominator is the present value of a basis-fixed annuity (the sum of discount factors times accrual periods). In the simplified three-period example, the same logic applies, and the swap rate aligns with the market so that the value is zero at inception.
SOFR-Based and Classical-Rate Distinctions
The lecturer notes that the simple valuation method aligns with the old practice of LIBOR-based (or, historically, three-month) rate references, where the floating payment period matched the reference rate’s maturity. With SOFR-based swaps (SOFR = Secured Overnight Financing Rate, often compounded or with replacement rates), the rate used for floating payments is not exactly the same as the rate that would produce a perfectly smooth cash-flow match. The practitioner’s adjustment is typically discussed in the cited paper, which covers how to adapt the simple valuation to the SOFR framework. In practice, this means recognizing that the “three-month rate” used in a classic valuation becomes a slightly different effective rate when daily compounding and overnight-tenor conventions are used in the floating leg.
Real-World Rates and Market References
The lecturer mentions real-world reference curves and provides a node for practical reference, including current treasury rates and term swap rates for short- and longer-term maturities, noting that the quotes are often based on daily compounding rather than a fixed three-month rate. The over-the-counter nature of plain vanilla swaps persists, but embedded options may be present in more complex structures; the lecturer introduces the concept of embedded options as caps (call options on rates) and floors (put options on rates), which add another layer of valuation complexity.
Embedded Options in Swaps
An embedded option within a swap is a feature that gives one party the right to alter the flow of payments if rates move unfavorably. The simplest embedded option is a cap swap, where the floating payments are capped above a certain rate; a cap is a portfolio of caplets. A caplet is an individual European call option on the forward rate for a period. Conversely, a floor is a portfolio of floorlets, i.e., European put options on forward rates, with the underlying being the reference rate for a given period. The terminology maps as follows: caps correspond to portfolios of caplets (European calls on forward rates), and floors correspond to portfolios of floorlets (European puts on forward rates). The lecturer humorously notes the humorous terminology and etymology surrounding the terms American and European with respect to options, including anecdotes about the origin of the terms.
Two Key Concepts: Forwards, Futures, and Their Role in Swaps
The course notes mention four fundamental derivatives: forwards, futures, calls, and puts. Understanding these four basics allows one to decompose many other derivatives. Forwards and FRAs (forward rate agreements) trade as instruments that lock in a rate for a future period. Futures are exchange-traded and typically more liquid, with daily settlement, whereas FRAs are over-the-counter and have different liquidity and settlement characteristics. The discussion then extends to the construction of a swap from FRAs and zero-coupon bonds, highlighting the equivalence of a swap and a carefully chosen portfolio of FRAs and zero-coupon bonds under no-arbitrage conditions.
Synthetic Swaps and Hedging (Goldman Sachs Example)
A central theme is that a traded swap (the actual bilateral contract) can be replicated by a synthetic swap, built from other securities that produce identical cash flows. If Goldman Sachs is the dealer that writes the swap with a customer, it can hedge its own exposure by creating a synthetic position using zero-coupon bonds, FRAs, and possibly futures. This synthetic portfolio would replicate the cash flows of the traded swap such that the net exposure on Goldman’s books is removed. The dealer earns a service or hedging fee for providing this replication. The method relies on constructing a portfolio whose cash flows at each date match the swap’s cash flows, given the same notional and payment structure.
Synthetic Swap Construction: An Illustrative Outline
Suppose we want to replicate a $100 million, 3-year swap in which the fixed leg pays 6% (as in the example) and the floating leg pays the 1-year (or next-period) floating rate. The replication uses a combination of zero-coupon bonds maturing at the swap’s payment dates and forward rate agreements (FRAs) that mature at those dates. The basic idea is to choose the number of zero-coupon bonds maturing at times 1, 2, and 3 such that their cash flows, together with the FRAs, reproduce the exact fixed-rate cash flows and the floating-rate cash flows of the swap. At time zero, FRAs have zero initial value, so we only need to determine the appropriate private positions so that the sum of their terminal cash flows equals the swap’s cash flows. The portfolio is formed such that the net cash flows at each payment date replicate the swap’s long floating short fixed cash flows. Under the no-arbitrage and frictionless market assumptions, the cost today of the synthetic portfolio equals the swap’s value, demonstrating the equivalence.
Schedule of Real-World Implementation and Practicalities
The lecturer discusses practicalities, including the costs of dealing with notional exchange and the potential use of futures versus FRAs for constructing synthetic exposures. Futures offer lower transaction costs and greater liquidity, but introduce basis risk relative to FRAs due to cash-settlement mechanics and differing term structures. The choice between using futures or FRAs depends on liquidity, cost, and how closely the hedge must track the exact cash flows of the traded swap. The pedagogical aim is to ensure students understand the economic logic of hedging and replacement strategies, even if the precise operational details vary by institution and market environment.
Closing Remarks and Next Steps
The lecturer transitions to other derivative types (call and put options, and the broader derivatives landscape) and previews the next topics, including building the HTM model (likely HJM in common terminology) to price a wide array of interest-rate derivatives and to study risk management. He also mentions forthcoming discussions of other basic derivatives and the notion that a deep understanding of forwards, futures, calls, and puts provides a strong foundation for understanding more complex derivatives. He ends by acknowledging missed points from the previous class, clarifying the connection between FRAs and zero-coupon bonds in synthetic swaps, and noting that futures can substitute for FRAs with some basis risk, depending on practical considerations in real markets.
Recap of Key Formulas and Concepts
Basic swap structure: floating leg vs fixed leg with notional N and payment dates ti with year fractions Δi.
Floating leg discounting (par): on payment dates in the simple model.
Fixed leg present value (with notional included for intuition):
Swap value (long floating, short fixed):
Par swap rate (zero value at inception):
Embedded options terminology: caps (caplets) and floors (floorlets) are European-style options on rates; caps are portfolios of caplets (call options on forward rates), floors are portfolios of floorlets (put options on forward rates).
Basis risk: the hedging difference when using futures instead of FRAs or when other instrument substitutes are used.
Synthetic swaps: replication of a traded swap using zero-coupon bonds and FRAs; the value of the swap equals the value of the replicating portfolio under no-arbitrage; dealers hedge to neutralize risk and earn a fee.
If you want, I can convert any section into a more compact cheat-sheet layout or expand any single formula with a numerical example using your preferred numbers.