A payment due next month and a payment due in ten years should not normally be discounted using the same single rate. A yield curve describes the relationship between time and the rates or discount factors used to value future payments. Building a curve connects observable market quotes to the prices and risks of loans, bonds, swaps and other instruments.

Bootstrapping works through suitable market instruments in sequence. Once earlier discount factors are known, the next instrument can help determine an additional point on the curve. The module also distinguishes spot rates from forward rates and shows how interpolation fills the gaps between quoted maturities. Different interpolation choices can change valuations and sensitivities even when the original quotes are matched.

Start with discount factors and then follow the quote-to-curve calculations. Change a quote, rebuild the curve and examine the resulting valuation change. The later examples explain overnight-index discounting and the use of separate curves for projecting and discounting cash flows. The simplified instruments and compounding conventions are stated explicitly so that the arithmetic can be followed.

Continue your learning: Risk sensitivities · Interest-rate risk in the banking book · Treasury risk

Look up a term as you work through the examples. Definitions explain the terminology; the source references and assumptions beside each lesson explain the scope of its calculation.

44 definitions

Accrual

Accrual allocates interest or other income and expense to the period in which it is earned or incurred. Accrued interest can make a settlement amount differ from a quoted clean bond price.

Accrued interest

Accrued interest is interest earned or owed since the last payment date but not yet paid. Its calculation depends on the contract’s day-count and payment conventions.

Basis risk

Basis risk arises when two related prices or rates do not move together as assumed. A hedge tied to one benchmark may therefore leave exposure to a different benchmark, location or contract specification.

Bid–ask spread

The bid is the price at which a dealer offers to buy, and the ask is the price at which the dealer offers to sell. Their difference is a trading cost and can widen when markets become less liquid.

Bond

A bond is a debt security under which an issuer promises specified payments to investors. Its value depends on the timing of those payments, discount rates and the risk of non-payment.

Bootstrapping

Bootstrapping builds a curve sequentially by using market instruments to determine additional discount factors or rates once earlier points are known. The result depends on instrument selection and pricing conventions.

Calibration

Calibration selects model parameters to match chosen data, market prices or target risk properties. A calibrated model can still be inappropriate if its structure or the selected evidence does not suit the task.

Cash flow

A cash flow is a payment or receipt at a specified time. Its amount, date, currency and legal owner affect both valuation and liquidity.

Collateral

Collateral is cash or another eligible asset provided to secure an obligation. Its protection depends on value, legal enforceability, availability and the time required to realise it.

Compounding

Compounding describes how interest accumulates on principal and previously earned interest. Under continuous compounding, accumulation is represented by an exponential function rather than a sequence of periodic additions.

Concentration

Concentration means that a large share of exposure or funding depends on a small number of sources or common drivers. The Herfindahl–Hirschman Index, or HHI, sums squared shares to measure size concentration but does not by itself measure behavioural dependence.

Coupon

A coupon is an interest payment on a bond or loan. The coupon rate determines the contractual payment; it is not necessarily the yield available to an investor at the current market price.

Credit spread

A credit spread is the additional yield or premium associated with credit exposure relative to a reference rate. It can reflect default risk, liquidity and other market effects, so it is not a direct default probability.

Day-count convention

A day-count convention converts dates into year fractions for interest and valuation. For example, actual/365 uses the actual number of days divided by 365; other conventions produce different accrual fractions.

Deposit beta and lag

Deposit beta measures the proportion of a market-rate change passed into the customer deposit rate. Lag measures the delay before that repricing occurs; neither is a deposit withdrawal rate.

Discount factor

A discount factor is the multiplier that converts a future cash flow into present value. Its value depends on maturity and the discounting convention.

Discount factor extrapolation

Extrapolation estimates a curve beyond its observed maturities. Unlike interpolation, it extends outside the available data and can make long-dated values particularly sensitive to the chosen assumptions.

Discount rate

A discount rate is a rate used to value future cash flows in present terms. The appropriate rate depends on the valuation framework and the characteristics of the cash flow.

Finite difference

A finite difference estimates a sensitivity by repricing after a small change in an input. The bump size, direction and units determine how the observed price change is converted into a sensitivity.

Floating rate

A floating rate resets according to a benchmark or contractual rule. Repricing occurs when the rate used to calculate a payment is updated, which need not coincide with repayment of principal.

Forward contract

A forward contract fixes today the terms of a transaction to occur at a future date. The value of an existing forward changes as current market prices and financing conditions change.

Forward rate

A forward rate is a rate implied for a future interval by the selected curve and its conventions. It is consistent with that curve’s pricing relationships but is not necessarily a forecast of the future realised rate.

Gamma

Gamma measures how delta changes as the underlying changes. Convexity describes curvature more generally; bond convexity usually refers to the second-order relationship between price and yield.

Implied volatility

Implied volatility is the volatility parameter that makes a selected pricing model match an observed option price. A volatility smile or surface describes how implied volatility varies across strikes and expiries.

Index decomposition

Index decomposition represents a position through exposures to its constituents when the relevant framework permits or requires it. Delta, vega and curvature can have different index treatments, so look-through should not be applied indiscriminately.

Interest-rate swap

An interest-rate swap exchanges interest payments calculated on a reference notional. A common swap exchanges fixed interest for floating interest, allowing the parties to change their exposure to future rates.

Interpolation

Interpolation supplies values between known curve points. Linear interpolation in rates and linear interpolation in logarithms of discount factors need not produce the same intermediate prices or sensitivities.

Liquidity and solvency

Liquidity concerns the ability to meet obligations when due. Solvency concerns the ability of assets and loss-absorbing resources to cover liabilities and losses; a solvent institution can still face a timed cash shortage.

Long and short positions

A long position generally benefits from ownership or a rise in the relevant instrument’s value. A short position has the opposite exposure. For complex derivatives, the precise payment and sign convention should always be checked.

Maturity

Maturity is the date or remaining period until a contract ends or principal becomes due. It can differ from an instrument’s next interest-rate reset or an option’s expiry.

Multiple curves

A multiple-curve framework can use one curve to project a floating-rate cash flow and another to discount it. This allows different benchmark and collateral relationships to be represented consistently.

Notional

Notional is the reference amount used to scale a financial contract’s payments or exposure measures. It is not generally the contract’s market value or the amount that would be lost on default.

OIS

An overnight indexed swap exchanges a fixed rate for a payment linked to compounded overnight rates. OIS curves are often relevant to collateralised discounting under the applicable currency and collateral conventions.

Option

An option gives its holder a contractual choice, such as buying or selling an asset on specified terms. Optionality also occurs inside other products, for example when a borrower can repay early.

Par rate

A par rate is the coupon or fixed rate that makes an instrument’s value equal to its stated par value, or makes a swap have zero initial value, under the specified valuation conventions.

Principal

Principal is the amount borrowed or invested before interest. A derivative’s notional principal is a reference amount used to calculate payments and may never be exchanged.

Receive-fixed and pay-fixed

A receive-fixed swap position receives fixed interest and pays floating interest; a pay-fixed position does the reverse. Their value sensitivities generally have opposite signs under the same contract and market assumptions.

Sensitivity

A sensitivity measures how a value changes when a specified input changes, under a stated bump and unit convention. Greeks are familiar option sensitivities such as delta, gamma and vega.

Settlement

Settlement is the completion of a payment or delivery obligation. A settlement lag separates the trade date from completion and can create cash needs and exposure before the exchange is final.

SONIA

SONIA is the Sterling Overnight Index Average, a benchmark for overnight sterling borrowing transactions. Its publication and calculation conventions are maintained by the Bank of England.

Spot price

The spot price is the price or exchange rate for a transaction with the usual near-term settlement convention. It differs from a forward price agreed for a later transaction.

Yield

Yield is a rate of return implied by a price and a specified set of cash flows under a stated convention. Different yield definitions and compounding conventions can produce different numerical rates.

Yield

A yield is a rate of return implied by a price and a specified cash-flow convention. Yield to maturity is the single discount rate that equates a bond’s promised payments to its price; realised returns can differ.

Yield curve

A yield curve or term structure describes how rates or discount factors vary with maturity. Projection curves and discount curves can serve different purposes within one valuation.