Interest Rates, Benchmarks and Yield Curves

Introduction

Interest rates are the basic prices of time, liquidity and credit in financial markets. In foreign exchange and money markets, they determine the value of deposits, loans, swaps, forward exchange contracts and short-term securities. A modern market participant must therefore understand not only how rates are quoted, but also how reference rates are produced, how curves are constructed, and how instruments are priced from those curves.

This subject has changed substantially since the global financial crisis. Unsecured interbank offered rates were found to contain bank-credit and liquidity components, while transaction volumes in some tenors declined. Regulators and administrators consequently developed more robust risk-free rates (RFRs), usually based on overnight transactions. The transition from IBORs to RFRs has affected loan documentation, derivatives, valuation systems, accounting and risk management.

A yield curve is not merely a line on a chart. It is a compact representation of discount factors, forward expectations, liquidity conditions, term premia and, in some markets, credit risk. Correct interpretation requires attention to conventions, collateral, compounding, day-count rules and the distinction between a quoted rate and an economically equivalent price.

Learning objectives

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

• distinguish term IBORs, transaction-based RFRs and credit-sensitive reference rates;
• explain benchmark governance, fallback provisions and the principal consequences of benchmark reform;
• construct discount, OIS and forward curves from market instruments;
• interpret normal, inverted and humped yield curves;
• price forward rate agreements, interest-rate futures and short-term swaps;
• identify basis, convexity, liquidity, model and operational risks; and
• apply these concepts to treasury, trading and investment-banking decisions.

1. Rates, quotations and conventions

A simple annualised money-market rate is commonly quoted as:

where P is the present price or principal, M is the maturity value, d is the number of calendar days in the accrual period, and 360 is the assumed day-count denominator. The result is a simple annualised rate. A 360-day convention must not be substituted for a 365-day convention without adjustment.

For a simple-compounded deposit, the maturity value is:

where r is the annualised rate and α is the year fraction. If the convention is Actual/360, α = actual days ÷ 360; if Actual/365, α = actual days ÷ 365. The formula assumes one accrual period, no interim cash flows and simple interest.

For periodic compounding at m periods per year:

where R is the nominal annual rate, T is the term in years and mT is the number of compounding periods. The equivalent continuously compounded rate is:

Rates with different compounding and day-count conventions are not directly comparable until converted to a common basis.

Common market conventions

A quoted rate is meaningful only with its currency, tenor, fixing time, settlement convention, day-count basis, compounding method and payment dates.

2. Benchmarks and reference-rate reform

A benchmark is a published reference value used to determine payments, valuations or performance. An interest-rate benchmark can be classified by its underlying market and methodology.

IBOR-type benchmarks

Traditional interbank offered rates attempted to represent the rate at which banks could borrow unsecured funds for specified terms, such as one, three or six months. They were useful because they provided a forward-looking term rate. However, their weaknesses included reliance on expert judgement, limited transactions in some tenors, susceptibility to manipulation and an embedded bank-credit component.

Risk-free rates

RFRs are generally overnight rates derived from substantial transaction markets, often secured or nearly risk-free. Examples include:

• SOFR for US dollar secured overnight financing;
• SONIA for sterling unsecured overnight transactions;
• €STR for euro unsecured overnight transactions;
• SARON for Swiss franc secured overnight transactions; and
• TONAR for Japanese yen overnight transactions.

An RFR is not necessarily literally risk-free. It may contain small liquidity, market-structure or collateral effects. It is better understood as a low-credit-risk reference rate appropriate to its market design.

Because overnight RFRs are backward-looking, a compounded term rate is normally calculated over the interest period. For daily rates rᵢ and daily accrual fractions αᵢ, the compounded return is:

where the product covers all business-day accrual intervals and α_total = ∑αᵢ. This is an annualised rate for the whole observation period. It assumes rates are applied sequentially and that the published rates and calendar conventions are correct.

A term RFR may instead be produced from derivatives, such as futures or OIS transactions. A term rate and a compounded-in-arrears rate are economically related but not identical: one is known at the beginning of the period, while the other is determined substantially during or at the end of the period.

Benchmark reform mechanisms

Benchmark reform commonly involves:

1. selecting a robust replacement rate;
2. defining a spread adjustment for the economic difference between the old and new benchmark;
3. inserting contractual fallback language;
4. changing system, valuation and reporting infrastructure; and
5. managing the legal, tax, accounting and conduct consequences.

A fallback may specify a replacement rate plus a fixed spread adjustment. The spread is intended to preserve, as far as practicable, the historical median difference between the discontinued benchmark and the replacement rate over a defined lookback period. It does not remove all basis risk.

Governance is essential. Administrators require transparent methodologies, control frameworks, audit trails, benchmark statements and procedures for errors or cessation. Users must avoid manipulating submissions or trading around publication windows. Benchmark conduct is both a regulatory and an ethical responsibility.

3. The mathematics of discounting and curves

A discount factor is the present value of one unit of currency received at a future date. Under simple compounding:

Under continuous compounding:

The appropriate formula depends on the quoted convention. A discount factor is dimensionless; a rate is normally expressed as a decimal per year.

The zero rate for maturity T is the single rate that prices a zero-coupon cash flow at T. A forward rate is the rate implied between two future dates T₁ and T₂. With discount factors D₁ and D₂ and accrual fraction α:

This relation follows from no-arbitrage: investing to T₂ directly must be equivalent to investing to T₁ and then from T₁ to T₂. It assumes the same currency, collateral and compounding framework for both discount factors.

The instantaneous continuously compounded forward rate is related to the discount function by:

This expression is useful for analysing smooth curves, but in practice curves are built from discrete market instruments and interpolation choices.

Multiple curves

Modern collateralised derivatives generally use at least two curves:

• a discount curve, often derived from overnight indexed swaps and used to present-value collateralised cash flows; and
• one or more projection curves, used to forecast term benchmark cash flows.

Before the crisis, a single curve was often used for both discounting and projection. That simplification became unreliable once credit and liquidity bases became material. In a multi-curve framework, the discount curve and projection curve need not imply the same forward rates.

4. Constructing money-market and OIS curves

Curve construction is a bootstrapping process. The objective is to infer discount factors or zero rates from instruments whose prices are observed in the market.

Typical instrument sequence

The exact set depends on currency, liquidity and market practice.

Bootstrapping logic

Suppose a par fixed-for-floating swap has payment dates T₁,…,Tₙ, fixed rate K, accrual fractions αᵢ and discount factors Dᵢ. If the floating leg is represented consistently by the discount curve, the par condition is:

Thus, if D₁ through Dₙ₋₁ are known, the final discount factor is:

The equation assumes a par OIS-style swap with a unit notional, no exchange of principal, settlement at the payment dates and a floating leg whose value is consistent with the overnight discounting framework. For a conventional term swap in a multi-curve system, projected floating cash flows must be calculated from the relevant projection curve, so the equation cannot be used indiscriminately.

Worked example: two-period curve extraction

Assume semiannual payments, Actual/360 approximated here by α = 0.5, and an already known six-month discount factor D₁ = 0.9850. A one-year par OIS rate is K = 4.00%. The two-period par condition gives:

0.04 × (0.5D₁ + 0.5D₂) = 1 − D₂

Substituting D₁:

0.04 × (0.4925 + 0.5D₂) = 1 − D₂

0.0197 + 0.02D₂ = 1 − D₂

Therefore:

1.02D₂ = 0.9803

D₂ ≈ 0.96108

The one-year continuously compounded zero rate is:

r_c = −ln(0.96108) ÷ 1 ≈ 3.97%

The result is approximately 3.97% because the par swap rate uses semiannual accrual while the reported zero rate here is continuously compounded. Rounding and the simplified calendar assumptions affect the last decimals.

5. Reading the yield curve

A normal or upward-sloping curve generally indicates that longer maturities have higher yields than shorter maturities. Possible explanations include expected future increases in short rates, positive term premia, inflation uncertainty, stronger future growth or greater duration risk.

An inverted curve occurs when short rates exceed long rates. It may reflect expectations of future monetary easing, recession risk or restrictive current policy. Inversion is not a mechanical recession forecast: term premia, technical demand and central-bank balance-sheet policies can also affect the shape.

A flat curve indicates similar rates across maturities. A humped curve has a local maximum, often reflecting a period of expected policy tightening followed by easing. The curve should be analysed jointly with inflation expectations, central-bank communication, liquidity, credit spreads and market positioning.

The slope between maturities T₁ and T₂ may be measured as:

where y denotes a yield under a common compounding basis. The curvature measure can compare a medium maturity with a linear interpolation of two surrounding maturities. Such measures are descriptive, not structural explanations.

6. Pricing short-term instruments

Forward rate agreements

An FRA locks an interest rate for a future borrowing or lending period. Let N be the notional, K the contract rate, L the observed reference rate, α the accrual fraction and δ the settlement discount fraction. For a borrower receiving compensation when rates rise, the payment at the start of the underlying period is commonly:

If L > K, the borrower receives a positive amount. If L < K, the borrower pays. The denominator discounts the end-of-period interest difference back to the settlement date using the realised reference rate. If the contract is settled at period end instead, the denominator is omitted, subject to documentation.

Example: N = $10,000,000, K = 5.00%, L = 5.60%, α = 90/360 = 0.25. Then:

Settlement = 10,000,000 × 0.006 × 0.25 ÷ (1 + 0.056 × 0.25)

Settlement ≈ $14,791

The borrower receives approximately $14,791 at the start of the period.

Interest-rate futures

A money-market interest-rate futures contract often quotes:

Thus a quoted price of 95.25 implies a rate of 4.75%, subject to the contract’s exact specification. A one-basis-point change is worth approximately:

For N = $1,000,000 and α = 0.25, BPV is $25 per basis point. Futures are marked to market daily, unlike a typical FRA settled once at the beginning of the underlying period. This daily settlement creates convexity differences between futures-implied rates and forward rates, especially when rates are volatile.

Short-term interest-rate swaps

In a plain-vanilla pay-fixed, receive-floating swap, the value is:

where K is the contractual fixed rate and N is notional. Under a standard par swap at inception, V = 0. In a simple single-curve setting, the floating-leg present value is approximately N × [1 − D(0,Tₙ)]. In a multi-curve setting, each projected floating cash flow is forecast using the projection curve and discounted using the discount curve.

The fixed rate of a par swap is therefore:

The formula assumes deterministic payment dates and a defined collateral and discounting framework. Actual valuation must include settlement lags, calendars, payment conventions, notional changes, credit support terms and any embedded optionality.

7. Basis, risk and governance

A basis exists when two economically related rates differ. Important examples include:

• term benchmark versus overnight RFR;
• unsecured versus secured funding;
• futures-implied versus OIS-implied rates;
• different currencies in cross-currency swaps; and
• two projection curves used for different tenors.

Basis risk cannot be eliminated by assuming that all short-term rates move together. It should be measured through historical scenarios, stress tests and sensitivity analysis.

Key risks include:

• Market risk: changes in zero rates, forward rates and curve shape;
• Basis risk: divergence between reference rates or curves;
• Convexity risk: nonlinear effects caused by daily margining or rate volatility;
• Liquidity risk: unreliable prices and wider bid–offer spreads;
• Model risk: interpolation, extrapolation, compounding or bootstrapping errors;
• Legal risk: inadequate fallback language or disagreement about replacement rates;
• Operational risk: incorrect calendars, fixings, data feeds or system mappings; and
• Conduct risk: manipulation, misleading disclosures or unsuitable benchmark selection.

A robust control framework includes independent price verification, version-controlled curve construction, data-quality checks, reconciliation of fixings, documented assumptions, model validation and clear escalation procedures. Artificial intelligence can assist with anomaly detection and data validation, but it should not replace governance, explainability, human review or accountability for benchmark and valuation decisions.

8. Illustrative case: benchmark transition in a corporate treasury

Background

A fictional multinational corporate, Meridian Components, has a USD loan of $250 million referencing three-month USD LIBOR, with quarterly interest payments and a maturity three years after the proposed transition date. Its treasury department also holds an interest-rate swap that historically hedged the loan.

Problem

The loan and swap contain different fallback language. The loan converts to a compounded RFR plus a spread adjustment, while the swap documentation references a separate industry fallback. The two instruments may therefore produce different cash flows after transition.

Analysis

Treasury maps every contract, identifying currency, benchmark, tenor, observation shift, payment lag, day-count basis, spread adjustment and fallback trigger. It then constructs projected cash flows under several scenarios: unchanged rates, a steepening curve, a widening term-RFR basis and a delayed payment convention. The hedge ratio is recalculated using the new discount and projection curves.

Approach and outcome

Meridian negotiates aligned fallback language, confirms the applicable spread adjustment, updates its treasury-management system and performs parallel valuation for three reporting periods. It also establishes a daily control comparing published fixings with independent data. The hedge remains effective, but its sensitivity changes because the loan references a compounded overnight rate while the hedge contains a term-rate component. A small basis hedge is added.

Learning points

Benchmark transition is not a simple rate substitution. It is a legal, quantitative, operational and governance project. Contract inventories, common conventions, curve consistency and post-transition basis monitoring are as important as the replacement rate itself.

Contemporary developments and best practices

Current market practice increasingly emphasises transaction-based overnight benchmarks, resilient fallback language, collateral-consistent discounting and transparent compounded-rate conventions. Market participants should:

1. identify every benchmark exposure, including loans, bonds, derivatives and internal transfer-pricing arrangements;
2. document day-count, observation and payment conventions precisely;
3. use separate discount and projection curves where required;
4. validate curves against independent market prices;
5. stress both parallel shifts and non-parallel curve movements;
6. monitor basis and liquidity, not only outright rates; and
7. communicate changes clearly to clients, counterparties and senior management.

Chapter Summary

Interest-rate markets combine quoted conventions, reference benchmarks, discount factors and forward expectations. Traditional IBORs offered forward-looking term rates but contained credit and liquidity components and became less robust when transaction activity declined. Overnight RFRs are generally more transaction-based and are commonly compounded over an accrual period.

Yield-curve construction uses deposits, futures, FRAs, OIS and swaps to infer discount factors and forward rates. The fundamental no-arbitrage relationship is that forward rates are implied by ratios of discount factors. Modern derivatives valuation often requires separate discount and projection curves. FRAs compensate for the difference between a contracted and realised rate; futures are daily marked to market and can differ from forwards because of convexity; and swaps exchange fixed and floating cash flows whose value depends on both curves and conventions.

Professional practice requires careful treatment of day counts, calendars, collateral, basis, fallback language, data quality and governance. Benchmark reform is therefore simultaneously a market, legal, technological and risk-management challenge.

Key Terms

Benchmark: A published reference value used to determine payments, valuations or performance.

IBOR: An interbank offered rate intended to represent unsecured bank borrowing for a specified tenor.

Risk-free rate: A low-credit-risk reference rate, usually based on overnight transactions.

SOFR: A US dollar secured overnight financing reference rate.

SONIA: A sterling overnight index average rate.

€STR: A euro short-term unsecured overnight reference rate.

OIS: An overnight indexed swap exchanging a fixed rate for compounded overnight interest.

Discount factor: The present value of one unit payable at a future date.

Zero rate: The single rate applicable to a cash flow at a particular maturity.

Forward rate: A rate implied for a future accrual period.

Bootstrapping: Sequentially deriving discount factors or rates from market instruments.

Projection curve: A curve used to forecast future floating benchmark cash flows.

Discount curve: A curve used to present-value future cash flows.

FRA: A contract that fixes an interest rate for a future borrowing or lending period.

Convexity: The nonlinear relationship between prices and interest rates.

Basis risk: Risk that related interest rates or curves move differently.

Fallback: A contractual method for replacing a benchmark after cessation or unavailability.

Term premium: Compensation investors require for holding longer-maturity instruments.

Day-count convention: A rule for converting calendar dates into an accrual fraction.

Par swap rate: The fixed rate that gives a swap zero value at inception.

Review Questions

Short Answer Questions

1. Why are transaction-based overnight RFRs generally considered more robust than many traditional IBOR settings?
2. What is the difference between a discount curve and a projection curve?
3. Define a forward rate in terms of discount factors.
4. Why can a futures-implied rate differ from an FRA-implied forward rate?
5. What information is specified by a day-count convention?

Long Answer Questions

1. Explain the economic and governance reasons for the transition from IBORs to RFRs.
2. Describe the bootstrapping process for constructing an OIS discount curve.
3. Compare normal, inverted, flat and humped yield curves and discuss possible interpretations.
4. Explain the pricing and settlement mechanics of an FRA, including the settlement discounting adjustment.
5. Discuss the risks and control requirements associated with multi-curve valuation.

Analytical / Application Questions

1. A borrower enters a $20 million 90-day FRA at 4.80%. At fixing, the reference rate is 5.20%. Using Actual/360 and settlement at the start of the period, calculate the borrower’s receipt.
2. A treasury desk observes that a loan has transitioned to compounded overnight RFR while its hedge references a term benchmark. Identify the principal risks and design a monitoring framework.
3. Given discount factors for two maturities, derive the implied forward rate and explain what assumptions are required for the result to be used in pricing.

Multiple Choice Questions with Answers

Question 1

Which characteristic most directly distinguishes an overnight RFR from a traditional term IBOR?

A. It is always quoted with a six-month tenor.

B. It is generally based on overnight transactions rather than bank judgement for a forward term.

C. It contains a guaranteed bank-credit spread.

D. It is never compounded.

Correct Answer: B. It is generally based on overnight transactions rather than bank judgement for a forward term.

Question 2

If D(0,T₁) = 0.98, D(0,T₂) = 0.95 and α = 0.25, what is the implied simple forward rate?

A. 12.63%

B. 3.16%

C. 0.31%

D. 5.00%

Correct Answer: A. 12.63%

Question 3

A quoted money-market rate of 5% applies for 90 days under Actual/360. What is the simple interest on a principal of $1,000,000?

A. $5,000

B. $12,500

C. $25,000

D. $4,500

Correct Answer: B. $12,500

Question 4

In a standard FRA settled at the beginning of the underlying period, why is the rate difference discounted?

A. Because the notional is exchanged twice.

B. Because the compensation represents interest payable at the end of the underlying period.

C. Because the reference rate is always negative.

D. Because futures are marked to market daily.

Correct Answer: B. Because the compensation represents interest payable at the end of the underlying period.

Question 5

A money-market futures price of 95.25 conventionally implies an annualised rate of:

A. 95.25%

B. 5.25%

C. 4.75%

D. 0.25%

Correct Answer: C. 4.75%

Question 6

Which instrument is most directly associated with constructing an overnight discount curve beyond the very short maturities?

A. OIS

B. Equity forward

C. Commodity option

D. Spot FX transaction

Correct Answer: A. OIS

Question 7

What is the principal purpose of a fallback provision?

A. To increase the loan notional after maturity.

B. To specify how a benchmark is replaced if it becomes unavailable or ceases.

C. To eliminate all market risk.

D. To convert every rate into a fixed rate permanently.

Correct Answer: B. To specify how a benchmark is replaced if it becomes unavailable or ceases.

Question 8

A pay-fixed, receive-floating swap benefits, all else equal, when the relevant projected floating rates:

A. Fall substantially below the fixed rate.

B. Rise above the fixed rate.

C. Become exactly zero only.

D. Stop being published.

Correct Answer: B. Rise above the fixed rate.

Question 9

Which statement best describes basis risk?

A. The risk that related reference rates move differently.

B. The risk that a payment date is a weekend.

C. The risk that a notional is denominated in one currency.

D. The risk eliminated by using any two curves.

Correct Answer: A. The risk that related reference rates move differently.

Question 10

Using N = $10,000,000, K = 5.00%, L = 5.60% and α = 0.25, the beginning-of-period FRA settlement is approximately:

A. $15,000

B. $14,791

C. $60,000

D. $5,600

Correct Answer: B. $14,791