Interest Rate Swaps: Trading Fixed for Floating

Swaps are one of the biggest markets in finance. The total notional principal is comfortably in the hundreds of trillions of dollars. Chapter 14 of Wilmott’s book explains how they work, why they exist, and how they connect to the bond pricing we covered in the previous post.

The basic idea is almost too simple: two parties agree to exchange different types of interest payments. That is it. But this simple concept has become so fundamental that the swap market essentially defines the interest rate curve.

The Vanilla Interest Rate Swap

In a standard (vanilla) interest rate swap, one party pays a fixed interest rate and the other pays a floating rate, both calculated on the same notional principal. The principal itself is never exchanged. Only the interest payments change hands.

Let me give you Wilmott’s example. You enter a five-year swap on August 4, 2006. Semi-annual payments. You pay a fixed rate of 6% on a notional principal of $100 million. The counterparty pays you six-month LIBOR.

Every six months, two things happen:

• You pay 3% of $100 million = $3 million (the fixed leg, always the same)
• You receive whatever six-month LIBOR was, applied to $100 million (the floating leg, different each time)

One important detail that Wilmott highlights: the LIBOR rate is set six months before it is paid. So on the first payment date, you already know both sides of the exchange because the LIBOR rate was set when the contract started. This “minor” detail actually makes swap pricing much cleaner, as we will see.

Why set it six months early? Because LIBOR is the rate on a six-month fixed-term deposit. Each floating leg of the swap is like a six-month deposit made at the previous payment date. The interest rate is known at the start of the period and paid at the end.

There is also a variation called the LIBOR in arrears swap where the floating rate paid on a given date is the rate set on that same date, not six months earlier. This creates extra complexity.

Why Swaps Exist: Comparative Advantage

Swaps were originally created to exploit comparative advantage in borrowing. Here is how it works.

Company A and Company B both want to borrow $50 million for two years. The market quotes them different rates:

Company B is charged more for both fixed and floating because it has higher credit risk. But the gap between the two companies is different for fixed versus floating. The fixed rate difference is 1.2%, but the floating rate difference is only 0.7%. This mismatch creates an opportunity.

If A borrows at fixed (7%) and B borrows at floating (LIBOR + 100 bps), their combined interest cost is LIBOR + 8%. If instead A had borrowed floating and B fixed (what they actually want), the combined cost would be LIBOR + 8.5%. There is a 0.5% saving available.

By setting up a swap, they can capture this saving. A borrows fixed at 7% but enters a swap to receive fixed and pay floating. B borrows floating at LIBOR + 1% but enters a swap to pay fixed and receive floating. After netting everything out, both companies end up paying 25 basis points less than they would have without the swap.

In practice, an intermediary (usually a bank) sits between the two parties and takes a cut.

Wilmott notes that comparative advantage was the original reason for swaps, but it is no longer the main driver. Swaps have become so liquid and standardized that they are essentially a commodity product. They exist in many maturities and are more liquid than most bonds.

The Swap Curve

When a swap is first created, the fixed rate is chosen so that the contract has zero value to both parties. The present value of the fixed payments equals the present value of the expected floating payments.

The fixed rates at various maturities form the swap curve. This curve is quoted by the market and is one of the most watched indicators in fixed income. Because swaps are so liquid across many maturities, the swap curve is often a better representation of interest rates than government bond yields.

Swaps and Zero-Coupon Bonds

Here is the elegant part. A swap can be decomposed exactly into a portfolio of zero-coupon bonds. No models needed. No assumptions about interest rate dynamics.

The fixed leg is straightforward. If the fixed rate is r_s and payments are spaced by time tau, the present value of all fixed payments is:

r_s * tau * sum of Z(t; T_i) for all payment dates T_i

where Z(t; T) is the price of a zero-coupon bond maturing at T.

The floating leg is the clever part. Wilmott walks through it with diagrams. A single floating payment at time T_i equals the LIBOR rate set at T_i minus tau, multiplied by the notional. By adding and subtracting $1 at each payment date, you can show that each floating leg is equivalent to being long a zero-coupon bond maturing at T_i minus tau and short a zero-coupon bond maturing at T_i.

When you add up all the floating legs, most of the intermediate bonds cancel out. You are left with:

Value of floating side = Z(t; T_0) - Z(t; T_N)

where T_0 is the first payment date and T_N is the last. In other words, the entire floating side of a swap is worth the difference between two zero-coupon bonds. This is a beautiful result.

Combining both sides, the value of the swap (to the fixed-rate receiver) is:

Value = r_s * tau * sum of Z(t; T_i) + Z(t; T_N) - Z(t; T_0)

This is completely model independent. No assumptions about volatility, no stochastic differential equations, no Monte Carlo simulation. Just bond prices.

For a par swap (zero initial value), the swap rate r_s is:

r_s = [Z(t; T_0) - Z(t; T_N)] / [tau * sum of Z(t; T_i)]

Bootstrapping from Swaps

Because swaps are more liquid than bonds and available for many maturities, practitioners use swap prices to build the yield curve through bootstrapping, rather than the other way around.

The process is the same as we saw for bonds in the previous post. Start with the shortest-maturity swap, solve for the zero-coupon bond price. Use that to solve for the next one. Keep going.

For the first point: Z(t; T_1) = 1 / (1 + r_s(T_1) * tau)

For subsequent points: Z(t; T_{j+1}) = [1 - r_s * tau * sum of Z(t; T_i) for i=1 to j] / (1 + r_s * tau)

This gives you the entire discount factor curve from swap quotes.

Exotic Swaps

The vanilla swap is just the beginning. Here are some variations:

Callable and puttable swaps: One party can terminate the swap early. This is essentially an American-style option embedded in the swap. It makes the pricing model-dependent because you need to figure out when it is optimal to exercise.

Extendible swaps: The holder can extend the swap beyond its original maturity at the same fixed rate. Another option feature.

Index amortizing rate swaps: The notional principal decreases over time, and the rate of decrease depends on some index (like LIBOR). If rates drop, the principal amortizes faster. These are complex and definitely need a model.

Basis rate swaps: Both legs are floating, but based on different rates. For example, the prime rate versus LIBOR. Banks use these to manage basis risk, the risk that two floating rates they are exposed to diverge.

Equity swaps: One leg is an interest rate, the other is the total return on an equity index. The principal is not exchanged. An equity basis swap exchanges returns on two different equity indices.

Currency swaps: Exchange interest payments in different currencies. Unlike interest rate swaps, the principals are exchanged at the beginning and end of the contract (in different currencies). To value them, compute the present value of cashflows in each currency and convert using the spot exchange rate.

Wilmott’s Warning

The chapter ends with an important warning that captures Wilmott’s practical philosophy. Swaps are so liquid that you should not price them using a theoretical model. They are almost like an underlying asset themselves. The market price is the price.

If a set of cashflows can be perfectly, statically, and model-independently hedged by other cashflows, you must use the market-implied prices. Any mispricing from a model, no matter how small, could expose you to large and risk-free losses.

This is a recurring theme in the book: use models only when you have to, and never when simple no-arbitrage arguments will do.

Key Takeaways

Swaps are beautiful in their simplicity. Two parties exchange interest payments, and through the magic of LIBOR’s structure, the whole thing decomposes exactly into zero-coupon bonds. No models required for vanilla swaps. The swap market is so large and liquid that it essentially defines the interest rate curve for the rest of the financial system.

The practical lessons: the swap curve is more important than the government bond yield curve for most purposes, bootstrapping from swaps gives you discount factors, and you should never use a fancy model to price something that can be replicated exactly with traded instruments. Save the models for when you actually need them.

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