Swaps and Interest-Rate Derivatives

Introduction and Context

A swap is a contract in which two parties exchange cash flows according to predetermined rules. Unlike an exchange-traded futures contract, a swap is usually negotiated over the counter, although standardized derivatives may be centrally cleared. Swaps allow institutions to transform the economic characteristics of assets, liabilities, revenues, or costs without refinancing the underlying position.

Interest-rate swaps are central to modern financial management. A borrower with floating-rate debt may exchange floating payments for fixed payments; an investment manager may obtain floating-rate exposure from fixed-rate securities; and a bank may hedge the duration and repricing risk of its balance sheet. Related instruments—including forward-rate agreements (FRAs), overnight-indexed swaps (OIS), caps, floors, collars, and swaptions—provide increasingly flexible methods of managing interest-rate uncertainty.

Contemporary valuation is based on collateralized discounting and multiple yield curves. The former single-curve approach, in which one curve generated both discount factors and forward rates, became inadequate when basis spreads and collateral conventions became economically significant. A robust valuation framework therefore distinguishes the curve used to forecast each reference rate from the curve used to discount collateralized cash flows.

Learning Objectives

By the end of this chapter, the reader should be able to:

• explain the economic structure and uses of interest-rate, currency, equity, and commodity swaps;
• construct discount factors and forward curves from market instruments;
• value fixed-for-floating swaps, FRAs, OIS, caps, floors, and swaptions;
• distinguish single-curve from multi-curve valuation;
• interpret basis spreads, collateralization, credit risk, and settlement conventions;
• analyze the risks, governance requirements, and practical limitations of swap portfolios.

1. Swap Structure and Terminology

A plain-vanilla interest-rate swap exchanges a fixed-rate leg for a floating-rate leg on a notional principal. The notional is generally not exchanged. If the notional is N, the fixed rate is K, the floating rate observed for period i is Lᵢ, and the accrual fraction is αᵢ, then the payments for a receiver-fixed swap are:

The net payment is normally the difference between the two amounts. The notional is a calculation reference rather than a payment of principal. Payment dates, reset dates, day-count conventions, business-day adjustments, calendars, compounding rules, and payment lags are contractual terms, not minor administrative details.

A party receiving fixed and paying floating is called receiver-fixed or payer-floating. A party paying fixed and receiving floating is payer-fixed or receiver-floating. The swap rate is the fixed rate that makes the initial value of the contract zero, assuming no credit or funding adjustment at inception.

Economic Uses

Swaps are used to:

1. hedge interest-rate exposure;
2. alter the duration of a balance sheet;
3. obtain synthetic fixed or floating borrowing;
4. manage currency and basis risk;
5. separate funding decisions from investment decisions;
6. express views on rates, inflation, credit, commodities, or equities.

A swap does not eliminate risk. It replaces one set of risks with another, including basis risk, curve risk, counterparty risk, collateral and liquidity risk, model risk, and operational risk.

2. Discount Factors, Zero Rates, and Forward Rates

Discount Factors

The discount factor P(0,T) is the present value at time 0 of one unit paid at time T. Under annual compounding, a continuously compounded zero rate z(0,T) satisfies:

where T is measured in years. Under annual compounding with frequency m, the equivalent relationship is:

The discount factor is dimensionless, while z is an annualized rate and T is measured in years. These expressions assume the quoted zero rate and compounding convention are consistently defined.

For a cash flow C paid at T, present value is:

For multiple deterministic cash flows, values are additive:

Forward Rates

The continuously compounded forward rate between times Tᵢ₋₁ and Tᵢ implied by discount factors is:

For simple annualized forward rates, the corresponding relationship is:

Here αᵢ is the year fraction under the relevant day-count convention. Forward rates are not forecasts in a pure statistical sense; they are no-arbitrage-implied rates under the pricing measure and market conventions. They may differ from expected future spot rates because of risk premia and convexity effects.

Curve Construction

A curve is commonly bootstrapped from deposits or overnight instruments, futures or FRAs, and fixed-for-floating swaps. The basic procedure is:

1. collect clean, synchronized market quotes;
2. define calendars, settlement rules, day-count conventions, and interpolation;
3. solve the shortest maturity discount factors first;
4. use each subsequent instrument to solve the next unknown discount factor or zero rate;
5. validate by repricing all input instruments;
6. generate forward rates and sensitivity measures.

Interpolation is economically important. Interpolating zero rates, discount factors, log-discount factors, or instantaneous forward rates produces different results. A professional implementation documents the choice and tests whether it creates implausible forward-rate spikes or arbitrage-like behavior.

3. Valuation of a Plain-Vanilla Interest-Rate Swap

Consider a fixed-for-floating swap with payment dates T₁,…,Tₙ. Under a single-curve framework, let P(0,Tᵢ) be discount factors and Fᵢ the forward floating rate for the relevant accrual period. The present value of the fixed leg is:

The present value of the floating leg is:

For a receiver-fixed position:

For a payer-fixed position, the value is the negative of this amount. The par swap rate K* is the fixed rate producing zero value:

In the special single-curve case, where the forward rates are generated from the same discount curve and the floating leg begins at par, the floating-leg value can simplify to N[P(0,T₀) − P(0,Tₙ)]. Thus:

This telescoping identity should not be applied automatically in a multi-curve framework.

Worked Example: Single-Curve Swap

Assume a two-year annual-pay swap with notional N = $10,000,000. Discount factors are P(0,1) = 0.9700 and P(0,2) = 0.9350. The one-year forward rates are F₁ = 3.09% and F₂ = 3.74%, and α₁ = α₂ = 1. The fixed rate is K = 3.50%.

The fixed-leg value is:

The floating-leg value is:

Using the stated rounded inputs: 0.0309×0.9700 = 0.029973 and 0.0374×0.9350 = 0.034969; their sum is 0.064942, producing $649,420. Therefore:

Rounding in the quoted forward rates explains why the result may not exactly equal a value obtained from the telescoping identity. In production valuation, internally consistent curve data rather than independently rounded rates must be used.

4. Multi-Curve Valuation

Following the financial crisis, the rate used to forecast a term index such as three-month reference interest need not equal the rate used to discount collateralized cash flows. Let Pᵈ(0,Tᵢ) denote discount factors from the collateral or overnight discount curve, and let Fˡᵢ denote the forward rate for index L. Then:

The fixed leg is:

The multi-curve par swap rate is therefore:

The forecasting curve for L is bootstrapped from instruments referencing L, while the discount curve is generally built from overnight-indexed swaps consistent with collateral remuneration. The difference between term-index swap rates and overnight swap rates is often described through basis spreads, though the basis may reflect several economic and technical factors.

Multi-curve valuation requires consistency between index definitions, observation lags, payment lags, compounding, fixing calendars, and collateral terms. It also requires careful treatment of instruments used in curve construction because a quoted basis swap may itself depend on several curves.

5. Forward-Rate Agreements

An FRA locks an interest rate for a future borrowing or investment period. At the fixing date T₀, the reference rate L is compared with the contract rate K. Settlement usually occurs at T₀ rather than at the end T₁ of the interest period. For a notional N and accrual factor α, the borrower’s payoff when the market rate exceeds K is commonly:

The denominator discounts the end-period interest difference back to the settlement date using the realized reference rate, subject to market convention. An FRA is economically similar to a single-period interest-rate swap, but its settlement timing and payoff convention must be respected.

FRAs are useful for hedging anticipated borrowings, but they expose the user to fixing risk, basis risk, and counterparty risk. A hedge based on an FRA referencing one index may not perfectly hedge funding linked to another index.

6. Overnight-Indexed Swaps

An OIS exchanges a fixed rate for a compounded overnight rate over each accrual period. If daily overnight rates are r₁,…,r_m and the daily accrual fractions are δ₁,…,δ_m, the compounded floating accumulation factor is:

The floating cash flow on notional N is N(A − 1). In arrears compounding, the exact realized daily rates are needed before the payment amount is known. Observation shifts, lookbacks, lockouts, and payment lags alter the operational formula.

An OIS par rate is the fixed rate K that equates the discounted fixed and floating legs:

OIS curves are widely used for discounting collateralized derivatives because collateral interest is often linked to an overnight rate. This is a convention grounded in the collateral agreement and market practice, not a universal law independent of contract terms.

7. Caps, Floors, and Collars

A cap is a portfolio of caplets. A caplet pays when a reference rate exceeds a strike K. For a period beginning at Tᵢ₋₁ and ending at Tᵢ, its payoff at Tᵢ is:

A floor is a portfolio of floorlets with payoff:

A cap protects a floating-rate borrower from rising rates while preserving the benefit of falling rates, in exchange for an upfront premium. A floor protects a floating-rate lender or supports a minimum return. A collar combines a purchased cap with a sold floor, or vice versa, reducing premium cost while limiting favorable movements.

Under a simplified Black-style model, a caplet’s value is often written as:

where:

F is the forward rate, σ is the quoted volatility, T₀ is the option expiry in years, Φ is the standard normal cumulative distribution function, and Pᵈ is the discount factor to payment. This model assumes lognormal forward-rate dynamics, constant volatility over the option life, and suitable treatment of the payoff timing. When rates can be near or below zero, normal or shifted-lognormal models may be more appropriate. The model is a pricing approximation, not an observable fact about rate behavior.

8. Swaptions

A swaption is an option to enter an interest-rate swap at a future date. A payer swaption gives the right to pay fixed and receive floating; it benefits from rising swap rates. A receiver swaption gives the right to receive fixed and pay floating; it benefits from falling swap rates.

For a payer swaption, the exercise payoff at expiry T₀ is conceptually:

where S_T₀ is the prevailing forward swap rate and A_swap is the present-value annuity of the underlying swap’s fixed leg at expiry. A receiver swaption replaces the positive part with max(K − S_T₀, 0). Swaptions are valued using models such as Black’s swaption model, normal models, short-rate models, or volatility-surface approaches. The selected model must match the quoted volatility convention, rate regime, exercise style, and collateral framework.

A swaption’s risks include delta to the swap rate, vega to volatility, gamma, theta, correlation and curve-shape exposure. A portfolio can be approximately delta-neutral yet materially exposed to volatility or changes in the shape of the yield curve.

9. Currency, Equity, and Commodity Swaps

Currency Swaps

A cross-currency swap exchanges interest and often principal cash flows in two currencies. Principal exchanges may occur at inception and maturity, commonly using a contractual exchange rate. The instrument combines interest-rate exposure, foreign-exchange exposure, cross-currency basis, and counterparty risk. Valuation requires discount curves for both currencies, projected curves for each floating index, and an FX spot or forward relationship consistent with market conventions.

A currency swap can transform a euro liability into a dollar liability, but it does not remove the economic cost of funding. Cross-currency basis spreads may arise from funding demand, balance-sheet constraints, liquidity, collateral, and market segmentation.

Equity Swaps

An equity swap exchanges the return on an equity, index, or basket for a fixed or floating financing rate. A total-return receiver obtains price appreciation, dividends, and other specified distributions, while paying financing. Valuation requires expected dividends, financing spreads, corporate-action treatment, collateral, and the precise definition of total return. Equity swaps create leverage and synthetic exposure, so disclosure, suitability, margin, and concentration controls are essential.

Commodity Swaps

A commodity swap exchanges a fixed commodity price for a floating reference price, often based on an index or settlement quotation. The notional may be defined in physical units, such as barrels or metric tonnes. The contract can hedge input costs or revenues but introduces basis risk if the firm’s actual grade, location, timing, or quality differs from the reference commodity. Storage, transportation, convenience yield, seasonality, and delivery rules affect valuation and hedging effectiveness.

10. Risk Measurement and Governance

Key sensitivities include:

• DV01 or PV01: change in value for a one-basis-point parallel rate shift;
• key-rate duration: sensitivity to movements at selected maturities;
• basis sensitivity: exposure to changes between reference-index curves;
• vega: sensitivity to implied volatility;
• gamma: change in delta as the underlying rate changes;
• cross-gamma and correlation risk: interaction between risk factors;
• CS01 and CVA exposure: sensitivity to counterparty credit spreads and valuation adjustments.

A first-order approximation for a small yield change is:

where ∆y_bp is the yield change measured in basis points and DV01 is defined consistently as value change per basis point. The approximation becomes unreliable for large movements, nonlinear instruments, or substantial curve reshaping.

Governance should include independent price verification, model validation, collateral monitoring, limit systems, legal confirmation, margin-call controls, fallback language, and stress testing. Scenario analysis should include parallel and nonparallel curve shifts, volatility shocks, basis widening, counterparty default, collateral disputes, and liquidity deterioration.

Regulatory reforms in many jurisdictions have promoted central clearing for standardized derivatives, margin requirements for uncleared transactions, trade reporting, and documentation standards. The precise obligations depend on jurisdiction, entity classification, product, and clearing status. Legal agreements such as the ISDA Master Agreement and Credit Support Annex define close-out, netting, collateral eligibility, haircuts, thresholds, and dispute procedures.

11. Illustrative Case: Hedging a Floating-Rate Corporate Loan

Background

A manufacturing company has a three-year loan with a principal balance of $50 million. Interest is paid quarterly at a term reference rate plus a 1.80% credit spread. Management expects rates to rise but wants to retain some benefit if rates fall.

Problem

A fixed-rate swap would provide strong protection but would eliminate all benefit from lower rates and could create mark-to-market losses if rates decline. The treasury team must compare a swap with a cap and a collar.

Analysis

The company’s primary risk is the reference-rate component, not the 1.80% contractual spread. A payer-fixed, receiver-floating swap would stabilize the reference-rate cost, but the company would still pay its credit spread. A cap would limit the reference rate above the strike while preserving downside participation, but requires an upfront premium. A collar could finance part of the cap premium by selling a floor, thereby agreeing to pay a minimum reference rate if rates fall sufficiently.

The hedge ratio should reflect the loan’s actual reset dates, amortization, index, and payment conventions. If the loan references a term index while the hedge uses an overnight index, the residual basis exposure must be measured rather than assumed to be negligible.

Approach and Outcome

Suppose management values cash-flow certainty most highly because debt-service coverage is tight. It selects a payer swap on 80% of projected principal and a cap on the remaining 20%. The partial hedge reduces over-hedging if the loan is prepaid or amortizes faster than expected. Treasury establishes limits for hedge ineffectiveness, collateral calls, counterparty concentration, and basis movements.

The outcome is not simply “lower interest cost.” The swap improves certainty but may generate collateral requirements and an accounting or economic mark-to-market loss when rates fall. The cap costs a premium but preserves favorable downside movements. The mixed strategy aligns hedge design with risk appetite, liquidity capacity, and forecast uncertainty.

Learning Points

1. Hedge the exposure actually present in the loan contract.
2. Separate reference-rate risk from the borrower’s credit spread.
3. Evaluate premium, collateral, liquidity, basis, and termination risk together.
4. Match maturity and amortization assumptions to realistic cash flows.
5. Treat hedge designation, documentation, and monitoring as part of the transaction—not as afterthoughts.

12. Technology, Data, and Contemporary Practice

Modern derivatives platforms integrate market data, curve construction, pricing, risk, collateral, and accounting. Automated calibration can improve speed but cannot correct poor instrument selection, bad dates, stale quotes, or inappropriate model assumptions. Artificial intelligence may assist with anomaly detection, trade classification, document extraction, or scenario generation; however, it should not replace independent validation, explainability, controls, or human accountability.

Data lineage is particularly important. A valuation should be reproducible from the market snapshots, curve versions, conventions, model parameters, and trade records used at the valuation time. Model risk committees should test sensitivity to interpolation, volatility extrapolation, rate floors, collateral assumptions, and fallback methodologies.

Challenges and Limitations

Swap valuation is exposed to several limitations:

• market quotes may be sparse or inconsistent;
• curves depend on interpolation and bootstrapping choices;
• option values depend strongly on volatility and distribution assumptions;
• collateral and funding terms may change over time;
• counterparty default and close-out conventions complicate theoretical values;
• liquidity and bid–offer spreads can dominate small theoretical mispricings;
• hedge effectiveness may be impaired by basis, timing, volume, or index mismatch.

Consequently, a model price is an estimate conditional on conventions and assumptions. Professional judgment is required when interpreting results, especially during stressed markets.

Chapter Summary

Swaps transform financial exposures by exchanging contractual cash flows without necessarily exchanging principal. Plain-vanilla interest-rate swaps are valued by discounting fixed and floating legs, with the par rate equating their present values. Discount factors and forward rates are constructed from market instruments under explicit conventions. In modern practice, overnight discount curves and separate index-forward curves support multi-curve valuation.

FRAs hedge a single future interest period; OIS exchange fixed payments for compounded overnight rates; caps and floors provide option-based protection; and swaptions give rights to enter future swaps. Currency, equity, and commodity swaps extend the framework but add foreign-exchange, dividend, financing, basis, physical-market, and index risks. Effective management requires sensitivity analysis, stress testing, collateral and counterparty controls, legal documentation, model governance, and reliable data. The central professional principle is that valuation and hedging are inseparable from contractual detail and risk governance.

Key Terms

Swap: Contract exchanging cash flows according to predetermined terms.

Notional principal: Reference amount used to calculate payments.

Fixed leg: Swap cash flows based on a predetermined rate.

Floating leg: Swap cash flows based on a reference rate.

Discount factor: Present value of one unit paid at a future date.

Zero rate: Rate associated with a single maturity under a specified compounding convention.

Forward rate: Rate implied for a future accrual period by current market prices.

Par swap rate: Fixed rate that gives a swap zero initial value.

Single-curve valuation: Framework using one curve for discounting and forecasting.

Multi-curve valuation: Framework separating discounting from index forecasting.

FRA: Agreement fixing an interest rate for a future borrowing or investment period.

OIS: Swap exchanging fixed payments for compounded overnight-index payments.

Cap: Portfolio of options limiting a floating rate above a strike.

Floor: Portfolio of options establishing a minimum floating rate.

Collar: Combination of bought and sold rate options.

Swaption: Option to enter an interest-rate swap in the future.

Basis risk: Risk that related rates or prices do not move together.

DV01: Change in value associated with a one-basis-point rate movement.

Collateral: Assets or cash posted to secure derivatives obligations.

CVA: Adjustment reflecting expected counterparty credit loss.

Review Questions

Short Answer Questions

1. Why is the notional principal of a plain-vanilla interest-rate swap usually not exchanged?
2. Define a discount factor and explain its role in valuation.
3. What is the economic difference between a payer-fixed and receiver-fixed swap?
4. Why did multi-curve valuation become important in modern derivatives markets?
5. What risk does a cap protect against, and what does the buyer pay for that protection?

Long Answer Questions

1. Derive the par fixed rate of a plain-vanilla interest-rate swap and explain the assumptions behind the single-curve telescoping result.
2. Describe a practical bootstrapping process for constructing discount and forward curves, including the role of interpolation.
3. Compare FRAs, OIS, caps, floors, collars, and swaptions as interest-rate risk-management instruments.
4. Explain how collateral, counterparty credit, netting, and margin affect the economic value of an over-the-counter swap.
5. Discuss the principal risks and governance requirements associated with currency, equity, and commodity swaps.

Analytical / Application Questions

1. A firm has floating-rate debt linked to a term index but can hedge only with an OIS. Identify the residual risks and propose a monitoring framework.
2. A treasury department must choose between a payer swap, a cap, and a collar. Analyze the choice under rising, falling, and highly volatile interest-rate scenarios.
3. A swap portfolio is delta-neutral but experiences a large loss after a volatility and curve-basis shock. Explain how this can occur and recommend additional risk measures.

Multiple Choice Questions with Answers

Question 1

Which party benefits directly when market floating rates rise in a payer-fixed interest-rate swap?

A. The payer-fixed party, because it receives floating and pays fixed

B. The receiver-fixed party, because it receives fixed and pays floating

C. Both parties equally

D. Neither party, because the notional is not exchanged

Correct Answer: A. The payer-fixed party, because it receives floating and pays fixed

Question 2

Using annual accrual periods, P(0,1)=0.97 and P(0,2)=0.94, what is the simple forward rate for year two?

A. 2.00%

B. 3.19%

C. 3.50%

D. 6.00%

Correct Answer: B. 3.19%

Question 3

In a standard multi-curve framework, which curve generally discounts collateralized cash flows?

A. The equity dividend curve

B. The overnight or collateral curve

C. The commodity futures curve

D. The borrower’s credit-spread curve alone

Correct Answer: B. The overnight or collateral curve

Question 4

What is the payoff of a caplet at the payment date, ignoring discounting?

A. Nα(K−L)

B. Nα max(L−K,0)

C. Nα max(K−L,0)

D. N(K+L)

Correct Answer: B. Nα max(L−K,0)

Question 5

An OIS floating leg is typically based on:

A. A compounded overnight rate

B. A fixed coupon only

C. An equity dividend yield

D. A commodity spot price

Correct Answer: A. A compounded overnight rate

Question 6

A receiver swaption benefits most directly from:

A. Higher future swap rates

B. Lower future swap rates

C. Higher commodity storage costs

D. A fall in the notional amount

Correct Answer: B. Lower future swap rates

Question 7

Why may a term-index swap and an OIS with similar maturity have different par rates?

A. Because all swaps exchange principal

B. Because term-index forecasting and overnight discounting can differ, creating basis

C. Because discount factors are never used for swaps

D. Because OIS contracts have no cash flows

Correct Answer: B. Because term-index forecasting and overnight discounting can differ, creating basis

Question 8

An FRA commonly settles:

A. At the beginning of the interest period using a discounted interest difference

B. Only at maturity with principal exchange

C. By delivering physical commodities

D. Through an equity dividend

Correct Answer: A. At the beginning of the interest period using a discounted interest difference

Question 9

Which measure approximates value sensitivity to a one-basis-point rate change?

A. Vega

B. DV01

C. Gamma

D. CVA

Correct Answer: B. DV01

Question 10

A commodity swap most directly creates basis risk when:

A. The hedged commodity differs in location, quality, or timing from the reference index

B. The notional is denominated in currency

C. The contract has a fixed price

D. The swap is documented legally

Correct Answer: A. The hedged commodity differs in location, quality, or timing from the reference index