Chapter 7: Swaps

Mechanics of Interest Rate Swaps

An interest rate swap (plain vanilla) is an agreement to exchange fixed-rate interest payments for floating-rate interest payments on a specified principal at specified future dates.

• Fixed rate payer: Pays fixed, receives floating (like a long bond position + short floating-rate note)
• Floating rate payer: Pays floating, receives fixed (the opposite)

No principal is exchanged — only net interest payments change hands.

Typical terms:

• Notional principal: e.g., $100 million
• Maturity: 2-30 years
• Fixed rate: determined at initiation
• Floating rate: reset at each payment date based on a reference rate (historically LIBOR, now SOFR/SONIA etc.)

Example: A $3$-year swap, notional $100million.CompanyApays5$100M × (2.5% - 2.525%) = net receipt of $12,500 for that period.

The Comparative-Advantage Argument

Why do swaps exist? The classic explanation involves comparative advantage. Company A can borrow fixed at 5% and floating at SOFR + 0.3%. Company B can borrow fixed at 6.4% and floating at SOFR + 1.0%.

• A has an absolute advantage in both markets
• B has a comparative disadvantage of only 0.7% in floating (vs. 1.4% in fixed)
• A borrows fixed, B borrows floating, and they swap — both benefit

CAUTION: This argument ignores the fact that lenders price credit risk. If B's credit spread widens over time, the floating-rate lenders will charge more, eroding the theoretical gain. The argument is no longer widely used in practice — swaps are more about managing interest rate exposure.

Valuation of Interest Rate Swaps

An interest rate swap can be valued as:

1. The difference between two bonds: Value of fixed-rate bond minus value of floating-rate bond

$V_{\text{swap}}=B_{\text{fixed}}-B_{\text{floating}}$

2. A portfolio of Forward Rate Agreements (FRAs): Each exchange in the swap is an FRA

At initiation, the swap rate is set so that $V_{\text{swap}}=0$:

$R_{\text{swap}}=\frac{1-P(0,T_n)}{{\sum }_{i=1}^{n}P(0,T_i)}$

where $P(0,T_i)$ is the discount factor (price of a zero-coupon bond paying $1at$T_i$).

During the life of the swap: The floating-rate bond is always worth close to par on a payment date. Between payment dates, it equals par plus the next floating payment, discounted.

Example: A swap with $k$ payments remaining where you pay fixed at rate $R$ and receive floating. Value:

$V={\sum }_{i=1}^k\text{floating cash flow received}\cdot P(0,T_i)-{\sum }_{i=1}^{k}R\cdot \text{principal}\cdot {\tau }_i\cdot P(0,T_i)$

Currency Swaps

A fixed-for-fixed currency swap involves exchanging principal and fixed interest payments in one currency for principal and fixed interest payments in another currency.

At initiation, principal amounts in the two currencies are approximately equal in value. During the swap, fixed interest payments are exchanged. At maturity, the principal amounts are exchanged back.

Valuation: Value the cash flows in each currency separately and convert to the base currency:

$V_{\text{swap}}=S_0\cdot B_{\text{foreign}}-B_{\text{domestic}}$

where $S_0$ is the spot exchange rate and $B_{\text{foreign/domestic}}$ are the values of the bonds denominated in each currency.

Credit Default Swaps (Introduction)

A CDS provides insurance against the default of a reference entity. The buyer pays regular premiums (spread) and receives a payoff if the reference entity defaults. The payoff is typically the face value minus the recovery value of the defaulted bond (cash settlement) or face value of bonds in exchange for delivering the defaulted bonds (physical settlement).

The CDS spread can be approximated from bond yields:

$\text{CDS spread}\approx \text{Bond yield}-\text{Risk-free rate}$

Other Types of Swaps