FRM® swap par rate: how is the fixed leg set at inception?

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The fixed rate on a new plain vanilla interest rate swap is chosen so the present values of its fixed and floating payment legs match at inception. Under a simple single-curve, annual-payment setting, that rate is the par rate for the same maturity and payment dates. It is a coupon-equivalent rate for a stream of payments, not the spot rate used to discount one terminal cash flow.

Start with the cash flows

A swap exchanges interest payments calculated on an agreed notional. A fixed payer sends a known rate applied to the notional for each accrual period and receives a floating amount determined by the contract’s reference rate and reset convention. In an ordinary interest rate swap, the notional is a calculation base rather than cash paid to the other party. The BIS account of swap valuation describes this distinction. The side called “payer” matters: pay fixed means receive floating, while receive fixed means pay floating. A candidate who reverses the legs will reverse the economic effect of a rate move even if every discount factor is correct.

The rate cannot be selected by looking only at the final payment date. The fixed leg makes payments throughout the contract. A rate that looks reasonable for one maturity may give a nonzero present value once earlier coupons are included. At inception, absent an upfront exchange in the simplified at-market case, neither leg should be worth more than the other. The agreed fixed rate solves that equality. This is the meaning of “par” here: it makes the contract begin with zero net value under the assumed curve and conventions. A later change in rates can make an existing contract valuable to one counterparty and costly to the other without changing its contractual fixed rate.

Derive the rate from discount factors

Let each annual payment date have a discount factor measured from today, with the last factor corresponding to maturity. In the simple one-curve framework, the present value of the floating leg at a reset is one unit of notional less the discounted principal-equivalent amount at maturity. The fixed leg pays the same annual coupon at each date, so its value is the coupon rate multiplied by the sum of those discount factors. Equating the legs gives the fixed rate. The BIS discussion of swap bootstrapping explains the zero-value par-swap starting point and the curve connection.

The sum of discount factors is an annuity factor. It represents what a one-unit annual payment stream is worth today. It is smaller than a simple count of payments when all relevant discount factors are below one. The numerator represents the present-value difference between one unit now and one unit at maturity. Dividing by the annuity factor spreads that difference over every fixed payment date. For unequal accrual periods, each factor must be multiplied by its period’s accrual fraction. The annual formula below is therefore a teaching case, not a universal contract specification. Before calculating, identify payment frequency, day-count basis, settlement timing and which curve prices each leg.

An invented annual-payment example

Consider an illustrative, newly struck three-payment swap with discount factors of 0.96, 0.92 and 0.88 for successive annual dates. These are invented round inputs, not an observed market curve. The annuity factor is 0.96 plus 0.92 plus 0.88, or 2.76. The terminal difference is one less 0.88, or 0.12. The annual fixed par rate is 0.12 divided by 2.76, which equals about 0.04348, or 4.348%. Every percentage here follows only from the invented inputs. The FRM program’s learning objectives provide the official framework for deciding which rate concepts a candidate must understand.

Check the result by pricing both sides using one unit of notional. The fixed payments have present value 0.04347826 multiplied by 2.76, which is 0.12 to the shown precision. The simplified floating leg also has present value one less 0.88, or 0.12. Their difference is zero apart from rounding. If the notional were an illustrative 100 units, each present value would be 12 units and the zero-value conclusion would be unchanged. This cross-check is more reliable than trying to remember a rate formula in isolation. It also exposes a common arithmetic slip: discounting only the last fixed coupon would ignore the first two payments and produce a quite different answer.

Why the par rate is not a spot rate

A spot rate belongs to a single cash flow at a specified horizon. A par rate belongs to a coupon stream that spans several horizons. Both are derived from the discount curve, but they answer different pricing questions. In the invented example, the third discount factor determines the present value of a payment only at the last date. The swap’s fixed leg, by contrast, uses all three discount factors. Therefore the par rate should not replace the third-horizon spot rate when discounting a single terminal payment. Nor should the final spot rate be used as the fixed coupon without checking the other dates. The precise relationship between par and spot rates depends on the full curve and payment conventions.

An exam prompt may supply discount factors directly, quote several par swap rates, or ask which rate makes a new swap worth zero. With discount factors, add the discounted payment weights and solve directly. With par rates at consecutive maturities, work from the shortest maturity onward to infer the next discount factor, a process known as bootstrapping. That inference requires matching payment dates and consistent curve assumptions. If the question instead asks for the value of an existing swap, its original fixed rate is already locked. Reprice today’s remaining legs under the current curve; do not silently replace the contractual rate with a new par quote.

What changes in a real contract

Real-world conventions complicate the neat one-curve classroom identity. Floating rates can reset on dates that differ from fixed-leg payment dates. Accrual fractions may differ, payments may be semiannual or quarterly, and collateral or discounting arrangements affect valuation. A multi-curve treatment can project floating cash flows from one curve and discount them with another. In that setting, the practical rule remains to choose a fixed rate that equalises the properly valued legs; the short formula using only the terminal discount factor may no longer be the right mechanical shortcut. The BIS valuation paper examines how swap conventions and pricing relationships matter.

For a first reading, keep three layers separate. The legal contract tells you who pays each stream and when. Discount factors turn those future payments into present values. The par condition then picks the fixed rate for a new at-market trade. If a problem supplies a notional, it scales both legs equally in this simplified set-up and cancels from the par-rate calculation; it does not mean principal is exchanged. If a problem supplies a quoted spread over a reference rate, an upfront payment, or mismatched cash-flow dates, read those terms before applying a formula. The same words “swap rate” do not excuse ignoring the specific contract.

A compact decision routine

Write “pay fixed, receive floating” or the reverse before doing arithmetic. Map each fixed payment date to a discount factor and an accrual fraction. If the swap is new and at market, set the present values equal and solve for its fixed rate. If the swap already exists, keep the old fixed rate and calculate the net value using today’s assumptions. Then ask what the resulting number means: a par rate is quoted as a rate; the value of an existing swap is an amount or amount per unit of notional. Confusing these units is as revealing an error as a wrong sign.

A useful final check is to build the answer backward. Apply your calculated fixed rate to the notional and accrual periods, discount every fixed payment, and compare with the floating-leg value from the same assumptions. If the values fail to match at inception, find the missing payment, wrong payment frequency or misread discount factor. For further context, the FRM programme overview locates this concept among the risk topics, while Omni’s methodology describes its teaching approach. The exam task is to identify the cash flows and pricing condition before reaching for a remembered expression.