How Forward-Rate-Agreement Algorithms Turn Curves into Settlement Cash: Forward Rates, Discount Factors, Day Counts, Fixing Risk and Settlement Conventions

Reader question: If nobody lends or borrows the FRA notional, how can a forward-rate agreement still lock in the economics of a future interest rate—and how does a computer calculate the cash amount that eventually changes hands?

A forward-rate agreement, or FRA, is a cash-settled interest-rate derivative. The parties agree today on a fixed rate for a future accrual period. Later, a specified floating benchmark is observed or otherwise determined under the contract. The algorithm compares the floating rate with the fixed FRA rate, applies the difference to a notional and day-count fraction, and settles the resulting value under the agreed payment convention.

The computational core has three layers: derive the fair forward rate, value the contract before fixing, and calculate the final cash settlement after fixing. Each layer depends on dates, curves, day counts, benchmark definitions and sign conventions.

What this page owns — and what it does not

This page owns the single-period forward-rate derivative: curve-implied forward rate, FRA mark-to-market and settlement cash.

It does not replace yield-curve construction, day-count fractions, financial date engines, interest-rate swap valuation, or futures convexity adjustment.

This is derivatives mathematics and public computational education. It is not a recommendation to enter an FRA or hedge any reader’s exposure.

The contract timeline

A simple FRA can be described with four times:

• trade date: when the FRA rate is agreed;
• fixing date: when the floating reference is determined under the contract;
• T₁: start of the underlying interest period;
• T₂: end of that period.

The notional N is used to scale the interest difference. It is generally not exchanged as principal in the way a cash loan would exchange principal.

A notation such as “3×6 FRA” historically describes a forward-starting rate covering approximately the period from month 3 to month 6, but the actual contractual dates, calendars and benchmark conventions control the calculation.

Derive the forward rate from discount factors

For a simple-compounded forward rate over the period from T₁ to T₂, let:

• P(0,T₁) = discount factor to the period start;
• P(0,T₂) = discount factor to the period end;
• τ = contractual day-count fraction for the accrual period.

Under a single-curve no-arbitrage setup, the simple forward rate is:

F(0;T₁,T₂) = [P(0,T₁)/P(0,T₂) − 1] / τ.

The logic is a replication argument. Investing from today to T₂ directly should be economically consistent with investing to T₁ and then reinvesting from T₁ to T₂ at the implied forward rate.

A small forward-rate example

Suppose:

• P(0,T₁) = 0.9800;
• P(0,T₂) = 0.9650;
• τ = 0.25.

Then:

F = (0.9800/0.9650 − 1)/0.25 ≈ 6.2176%.

If a new FRA is struck at approximately that rate under the same curve and conventions, its initial value is near zero before transaction costs and other adjustments.

Modern multi-curve valuation separates projection from discounting

Collateralized derivatives valuation often distinguishes:

• a projection curve used to forecast the floating benchmark or compounded reference;
• a discount curve used to present-value the resulting cash flow.

The old single-curve identity remains useful for understanding the mechanism, but a production system must know which curve supplies the forward and which curve supplies discount factors.

This is the same architectural separation used in the swap-valuation engine. An FRA is one period rather than a strip of many periods, but the curve dependencies are closely related.

Value the FRA before fixing

Let:

• K = fixed FRA rate;
• F = current forward rate for the underlying period;
• N = notional;
• τ = day-count fraction.

For a position that receives floating and pays fixed, an end-of-period equivalent cash difference is approximately:

N τ (F − K).

If this amount is economically paid at T₂, its present value under discount factor P_d(0,T₂) is:

V ≈ N τ (F − K) P_d(0,T₂).

The exact production formula depends on the FRA’s settlement convention, benchmark definition, collateral agreement and whether the payment is made at period start or period end.

Why traditional FRA settlement is often discounted

Many classic IBOR-style FRAs settle the interest difference near the beginning of the underlying deposit period rather than waiting until the end.

If the floating fixing is L, the undiscounted interest difference payable at T₂ for receive-floating/pay-fixed is:

N τ (L − K).

If the equivalent value is paid at T₁, it is discounted over the underlying period. A familiar settlement expression is:

Settlement = N τ (L − K) / [1 + Lτ].

The denominator reflects the fact that a cash amount paid at the start of the period can earn or fund at the reference-period rate through the end of the period.

CFTC materials describing FRA trading conventions and public bank product documents use this same economic structure. However, modern RFR-linked and bespoke contracts can define settlement differently, so the governing definitions always outrank a generic textbook formula.

Sign conventions are a major source of silent errors

One desk may say “buy FRA” to mean pay fixed and receive floating. Another system may label the same exposure “payer FRA.” A vendor can report value from the opposite counterparty perspective.

A robust engine should therefore store the economic leg explicitly:

• pay fixed / receive floating;
• receive fixed / pay floating.

Then the sign of the result follows from that definition rather than from an ambiguous word such as “buyer.”

Day count changes the economics

The accrual factor τ is not merely a time-display convention. It scales the interest difference.

If the period contains 91 actual days:

• ACT/360 gives τ = 91/360;
• ACT/365 Fixed gives τ = 91/365.

The same notional, fixed rate and floating rate then produce different settlement amounts.

The day-count engine therefore supplies a mathematical input to the FRA rather than cosmetic metadata.

Dates and publication calendars can change the fixing

The fixing date may be offset from the period start. The benchmark may have a publication calendar distinct from the payment calendar. Weekends, holidays and modified-following rules can move relevant dates.

This creates a dependency on the financial date engine. A correct rate formula fed the wrong fixing date is still a wrong FRA valuation.

Forward rate is not the same thing as a futures-implied rate

A short-term interest-rate futures contract is marked to market daily. An FRA is an OTC forward-style derivative with different cash-flow timing.

When rates and discount factors are stochastic, daily margining creates a convexity difference between a futures rate and a true forward rate. The SOFR-futures convexity-adjustment page owns that conversion.

Using a futures-implied rate as an FRA forward without checking convexity can therefore introduce a model error.

Inputs and outputs

A production-shaped FRA engine can require:

• trade date;
• fixing date;
• period start and end dates;
• notional;
• fixed rate;
• floating benchmark definition;
• day-count convention;
• business-day calendars;
• projection curve;
• discount curve;
• settlement convention;
• counterparty perspective/sign convention;
• observed fixing after the benchmark becomes known.

Outputs can include:

• current forward rate;
• initial fair FRA rate;
• present value before fixing;
• settlement amount after fixing;
• discount factor and day-count fraction used;
• DV01 or rate sensitivity;
• diagnostic flags for missing fixings or inconsistent dates.

Evidence polarity: what supports confidence?

Evidence for confidence includes a fair-rate round trip to near-zero initial value, agreement between discount-factor and direct-forward calculations, correct sign when floating rises above fixed, an independently verified day-count fraction, exact benchmark fixing provenance, and reconciliation between projected pre-fixing value and realized post-fixing settlement.

Evidence against confidence includes a non-zero value for a freshly struck fair FRA under the same curves, value sign opposite to the economic leg, disagreement between two systems caused by different curve IDs, a settlement amount that omits the advance-payment discount factor where the contract requires it, or use of a benchmark observation from the wrong fixing date.

Counterexample: notional is not principal exchanged

A 100 million notional FRA does not normally mean 100 million of principal changes hands. The notional scales the interest difference.

If a system creates a principal cash flow because it treats the FRA like a deposit, the transaction can be off by orders of magnitude while every rate formula appears correct.

Counterexample: identical quoted rates can produce different cash settlements

Two FRAs can quote the same fixed rate and observe the same floating rate but use different day counts or settlement timing.

One can therefore settle a different cash amount from the other. A rate is not sufficient to specify the derivative.

Counterexample: a correct forward can still give the wrong PV

Suppose the projection curve is correct, but the system discounts the expected cash flow using an obsolete or wrong collateral discount curve.

The forward rate can be perfect while the present value is wrong.

This falsifies the idea that FRA valuation is “just calculate one forward rate.” Projection and discounting are separate jobs.

Counterexample: the textbook IBOR settlement formula is not universal

Benchmark reform has produced contracts linked to overnight risk-free rates, compounded observations and different settlement definitions. A system that hard-codes one legacy LIBOR-style formula for every FRA-like product can be wrong even when the algebra is internally consistent.

The engine should route by contractual benchmark and settlement definition rather than by the label “FRA” alone.

Weak links in implementation

Wrong curve pair. Projection and discount curves are swapped or stale.

Wrong day count. ACT/360 is assumed when the contract specifies another basis.

Fixing-date error. A benchmark from an adjacent publication date is used.

Sign inversion. Pay-fixed and receive-fixed perspectives are reversed.

Advance-settlement omission. The interest difference is not discounted when the contract pays at period start.

Futures/forward confusion. A futures quote is used directly as an FRA forward without checking convexity.

Calendar mismatch. Fixing and payment calendars are treated as one calendar.

Premature rounding. Forward rates or day-count fractions are rounded before settlement cash is calculated.

Benchmark fallback mismatch. The reference index changes but the engine continues using the old fixing source.

Diagnostics: how to test the FRA engine

• Fair-rate test: set K = F under the same curves; initial value should be near zero.
• Discount-factor identity test: reconstruct the forward from P(0,T₁) and P(0,T₂).
• Sign test: raise the floating fixing above fixed and confirm a receive-floating position gains value.
• Settlement test: compare end-of-period interest difference with its correctly discounted start-of-period equivalent.
• Day-count test: run the same dates under ACT/360 and ACT/365 and verify the change is intentional.
• Calendar test: place fixing near a weekend or holiday and compare against the contractual schedule.
• Curve test: perturb the projection curve and discount curve separately.
• Futures test: compare a futures-implied rate with a convexity-adjusted forward where relevant.
• Fixing replay: reproduce a historical settlement using the archived benchmark publication.
• missing-fixing test: require an explicit exception rather than silently substituting the latest rate.

What would falsify confidence?

Confidence should be withdrawn if a freshly struck FRA is materially non-zero under the curves used to set its rate; if the settlement sign contradicts the economic leg; if the benchmark fixing cannot be traced to the contractual fixing date; if the day count or payment timing is unknown; if discounting and projection use undocumented curves; or if an RFR-linked contract is forced through a legacy IBOR settlement formula without contractual support.

Alternatives answer different questions

Interest-rate swaps extend fixed-versus-floating exchange across many periods.

Short-term interest-rate futures provide exchange-traded exposure with daily margining, but require attention to futures/forward convexity.

OIS exchanges a fixed rate against compounded overnight rates and is central to modern discounting frameworks.

Caps and floors add optionality: the payoff is nonlinear rather than the FRA’s linear rate difference.

Forward-starting loans or deposits exchange principal and carry funding/credit mechanics that an FRA notional does not.

How this connects to the surrounding knowledge estate

Yield curves provide forward and discount information. Day-count algorithms convert dates into the accrual factor. Date engines determine fixing and settlement dates. Swap valuation generalizes the same logic to multiple periods. Futures convexity explains why exchange-traded rates are not automatically FRA forwards.

Verification and update triggers

Preserve the contract definitions, benchmark version, fixing source, fixing calendar, day-count basis, settlement convention, projection curve, discount curve, collateral assumptions, sign convention and valuation timestamp. Revalidate after benchmark methodology changes, curve migrations, collateral-policy changes, date-library updates, market-data-source changes or any unexplained settlement break against a counterparty or clearing record.

Primary and high-quality references

• Commodity Futures Trading Commission, Forward Rate Agreements — Contract Definition and Trading Conventions, public 2026 rule filing material describing FRA cash-settlement structure and trading conventions.
• ISDA, 2021 ISDA Interest Rate Derivatives Definitions, for current interest-rate derivative period, benchmark and business-day definitions.
• Borsa Italiana, Forward Rate Agreement, describing the fixed-versus-floating cash-difference mechanism and advance settlement.
• Federal Reserve Bank of New York, Additional Information about Reference Rates, for SOFR rate and compounded-index mechanics.
• Federal Reserve Board, The U.S. Treasury Yield Curve: 1961 to the Present, for term-structure and discount-factor foundations.

Educational boundary: This article explains single-period interest-rate derivative mathematics. It does not recommend any hedge, trade or interest-rate position and does not provide personalized financial advice.