Payment Frequency and Accrual Factors in Swap Pricing

Why Payment Schedules Matter

A swap’s fixed and floating legs need not pay on the same dates or use the same accrual periods. A fixed leg might pay quarterly while a floating leg pays every six months. Even if both legs use the same notional principal, their cash flows must be calculated and discounted separately before they are netted. The frequency stated for a leg does not replace the contractual accrual factor for an individual payment period.

The central discipline is to keep three dates distinct: the start and end of a coupon’s accrual period, and its payment date. The accrual dates determine the coupon amount; the payment date determines when the cash flow is received and which discount factor applies. This distinction is important when payment dates are lagged or adjusted under a contract.

Definition: For a swap leg with notional principal $N$, annualized rate $R_j$, and accrual factor $\alpha_j$ for period $j$, the coupon is $N R_j \alpha_j$. The accrual factor is the contractual year fraction for that period, expressed in years. If the coupon is paid at time $p_j$, its present value is $N R_j \alpha_j D(p_j)$, where $D(t)$ is the discount factor to time $t$. Use the rate and accrual factor specified for that leg; do not assume the accrual factor is the reciprocal of the number of annual payments.

Fixed and Floating Coupons on Their Own Schedules

For a fixed leg, the annual fixed rate $K$ is multiplied by each period’s accrual factor. Thus, the fixed coupon for period $j$ is $N K\alpha_j$. If payment dates differ across the periods, each coupon is discounted separately at its own payment date.

For a floating leg, a future coupon is commonly projected from a forward rate for the coupon’s accrual period. Under simple interest over a period from $a_j$ to $b_j$, with accrual factor $\beta_j$, the forward rate implied by discount factors on a consistent curve is $F_j=(D(a_j)/D(b_j)-1)/\beta_j$. The projected coupon, before any contractual spread, is $N F_j\beta_j$. A known rate fixed earlier is used in place of a projection for that coupon.

If the floating leg includes a spread $s$, its coupon is $N(F_j+s)\beta_j$. The spread is annualized in the same way as the reference rate and is multiplied by the period’s accrual factor. These formulas assume the floating rate is quoted and compounded on the stated simple-accrual basis. If the contract specifies a different rate convention, use that convention instead.

\text{Fixed-leg PV}=\sum_j N K\alpha_jD(p_j)

\text{Floating-leg PV}=\sum_j N(F_j+s)\beta_jD(q_j)

Here $p_j$ and $q_j$ are the fixed and floating payment times, respectively. They need not coincide with one another or with the corresponding accrual end dates. For a position that receives fixed and pays floating, the value is fixed-leg PV minus floating-leg PV. For a position that pays fixed and receives floating, reverse the subtraction.

Why the Floating-Leg Shortcut Sometimes Works

When the floating rate for each period is projected from the same discount curve, there is no spread, and each coupon is paid at its accrual end date, the present value of a projected floating coupon simplifies:

N F_j\beta_jD(b_j) =N\left(\frac{D(a_j)}{D(b_j)}-1\right)D(b_j) =N\bigl(D(a_j)-D(b_j)\bigr).

Summing over consecutive accrual periods makes the intermediate discount factors cancel. For periods covering time 0 through time $T$, the floating-leg value is then $N[1-D(T)]$. This is a telescoping result, not a universal floating-leg formula. It does not directly give the value when coupons are paid on dates different from their accrual ends, when rates are already fixed at different values, or when additional cash flows such as a spread must be included.

Key takeaway: Apply the accrual factor to the rate to calculate each coupon, then discount that coupon at its actual payment date. Use a telescoping floating-leg shortcut only when its assumptions hold.

A Reliable Valuation Process

The following sequence keeps accruals, payment dates, and signs separate. It applies whether the legs have different payment frequencies or simply have irregular first or final periods.

List the leg schedules.
For every coupon, identify its accrual start and end, accrual factor, and payment date. Record the fixed and floating schedules separately.

Calculate or project coupons.
Use $N K\alpha_j$ for a fixed coupon. For a future floating coupon, use the applicable forward rate and floating accrual factor; use a rate already fixed when applicable.

Discount each cash flow.
Multiply each coupon by the discount factor for its payment date, not by a factor chosen merely because it is the accrual end date.

Subtract in the position’s direction.
For receive-fixed/pay-floating, subtract floating-leg PV from fixed-leg PV. For pay-fixed/receive-floating, subtract fixed-leg PV from floating-leg PV.

Worked Valuations

Worked Example: Quarterly Fixed Coupons and Semiannual Floating Coupons

A one-year swap has notional $N=\$1{,}000{,}000$. The holder receives fixed at an annual rate $K=4\%$, with quarterly accrual factors of 0.25 and payment dates at 0.25, 0.50, 0.75, and 1.00 years. The holder pays floating, with two semiannual accrual periods, each of length 0.50, paid at 0.50 and 1.00 years. There is no spread or payment lag. The discount factors at 0.25, 0.50, 0.75, and 1.00 years are 0.990, 0.980, 0.965, and 0.950. Calculate the swap’s value.

Method. Calculate each leg on its own schedule and then take fixed-leg PV minus floating-leg PV. The floating rate for each period is projected from the given discount curve using simple accrual.

Fixed leg. Each quarterly coupon is $ \$1{,}000{,}000(0.04)(0.25)=\$10{,}000 $. Therefore,

\text{Fixed-leg PV} =\$10{,}000(0.990+0.980+0.965+0.950) =\$10{,}000(3.885) =\$38{,}850.

Floating leg. The first forward rate is $(1/0.980-1)/0.50=0.04081633$, or about 4.081633%. Its projected coupon is $\$1{,}000{,}000(0.04081633)(0.50)=\$20{,}408.16$, and its present value is $\$20{,}408.16(0.980)=\$20{,}000.00$.

The second forward rate is $(0.980/0.950-1)/0.50=0.06315789$, or about 6.315789%. Its projected coupon is $\$1{,}000{,}000(0.06315789)(0.50)=\$31{,}578.95$, and its present value is $\$31{,}578.95(0.950)=\$30{,}000.00$. Thus, floating-leg PV is $\$20{,}000+\$30{,}000=\$50{,}000$.

Value and check. The receive-fixed value is $\$38{,}850-\$50{,}000=-\$11{,}150$. The negative sign means that, at these discount factors, the fixed coupons are worth less than the floating payments. As a check, the floating PV also equals $\$1{,}000{,}000(1-0.950)=\$50{,}000$, because the floating periods are consecutive, have no spread, and pay at their accrual ends.

Common mistake avoided: Applying the 4% annual fixed rate as though it were the rate for each quarter would produce a coupon of $\$40{,}000$. The correct quarterly coupon uses the accrual factor: $4\%(0.25)=1\%$ of notional, or $\$10{,}000$.

Worked Example: Accrual End Dates and Payment Dates Do Not Match

Use the same notional, fixed rate, and receive-fixed position as in the preceding example. Fixed coupons accrue quarterly over 0.25-year periods, but each is paid 0.01 years after its accrual end. The fixed-leg payment-date discount factors at 0.26, 0.51, 0.76, and 1.01 years are 0.990, 0.979, 0.963, and 0.948. The floating leg has two 0.50-year periods ending at 0.50 and 1.00 years, with forward rates projected from $D(0)=1$, $D(0.50)=0.980$, and $D(1.00)=0.950$. Its payments are also made 0.01 years after each accrual end, with discount factors 0.979 at 0.51 and 0.948 at 1.01. There is no spread. Find the swap value.

Fixed leg. The quarterly coupon remains $\$1{,}000{,}000(0.04)(0.25)=\$10{,}000$. Discount at the payment dates:

\text{Fixed-leg PV} =\$10{,}000(0.990+0.979+0.963+0.948) =\$38{,}800.

Floating leg. The forward rates depend on the accrual-end discount factors, not the lagged payment-date discount factors. For the first period, $F_1=(1/0.980-1)/0.50=0.04081633$. The coupon is $\$20{,}408.16$, but its payment-date present value is $\$20{,}408.16(0.979)=\$19{,}979.59$.

For the second period, $F_2=(0.980/0.950-1)/0.50=0.06315789$. The coupon is $\$31{,}578.95$, and its present value at the lagged payment date is $\$31{,}578.95(0.948)=\$29{,}936.84$. The floating-leg PV is therefore $\$19{,}979.59+\$29{,}936.84=\$49{,}916.43$.

Value and check. The receive-fixed swap value is $\$38{,}800-\$49{,}916.43=-\$11{,}116.43$, rounded to the nearest cent. The floating-leg PV is less than the $\$50{,}000$ no-lag telescoping value, consistent with discounting its positive coupons at later payment dates.

Common mistake avoided: Discounting at 0.50 and 1.00 years would use the accrual-end dates rather than the payment dates. The coupon rates still come from the accrual periods, but the discount factors must correspond to the actual cash-flow dates.

Worked Example: A Quarterly Floating Leg with a Spread

A one-year swap has notional $N=\$2{,}000{,}000$. The position pays fixed at $K=3.6\%$ semiannually, with accrual factors 0.50 and payment dates at 0.50 and 1.00 years. It receives floating quarterly, with accrual factors 0.25 and a spread of 0.20% per year. The floating payments occur at 0.25, 0.50, 0.75, and 1.00 years. The discount factors at those dates are 0.990, 0.978, 0.964, and 0.948. Rates are projected from the same discount curve, with simple accrual and no payment lag. Calculate the value to the pay-fixed position.

Fixed leg. Each semiannual fixed coupon is $\$2{,}000{,}000(0.036)(0.50)=\$36{,}000$. Its present value is

\$36{,}000(0.978+0.948) =\$36{,}000(1.926) =\$69{,}336.

Floating leg. Project each quarterly rate from consecutive discount factors. The four forward rates, rounded here for display, are 4.040404%, 4.907975%, 5.809129%, and 6.751055%. Add the 0.20% spread to each rate before calculating its coupon. The resulting coupon amounts are approximately $\$21{,}202.02$, $\$25{,}539.88$, $\$30{,}045.64$, and $\$34{,}755.27$. Discounting at 0.25, 0.50, 0.75, and 1.00 years gives present values of $\$20{,}990$, $\$24{,}978$, $\$28{,}964$, and $\$32{,}948$, respectively.

For a check, the no-spread floating-leg PV telescopes to $\$2{,}000{,}000(1-0.948)=\$104{,}000$. The spread contributes $\$2{,}000{,}000(0.002)(0.25)(0.990+0.978+0.964+0.948)=\$3{,}880$. Thus the floating-leg PV is $\$104{,}000+\$3{,}880=\$107{,}880$, matching the sum of the four discounted coupon values.

Value and check. The position pays fixed and receives floating, so its value is $\$107{,}880-\$69{,}336=\$38{,}544$. The positive value is consistent with the floating leg being worth more than the fixed leg.

Common mistake avoided: The spread is not added as a one-time amount to the swap value. It increases each floating coupon by $N(0.002)(0.25)$, and each resulting increase is discounted at that coupon’s payment date.

Common Schedule and Accrual Traps

Using a payment frequency as an accrual factor. A quarterly payment schedule often has accrual factors near 0.25, but the contract’s stated year fraction controls. Stub periods and day-count conventions can produce different values.
Discounting on the wrong date. A coupon accrues over one interval but may be paid later. Calculate the coupon from its accrual period and discount it at the payment date.
Pairing the wrong forward rate with a coupon. A forward rate must correspond to that coupon’s accrual start and end dates and compounding convention.
Netting before valuing the legs. If fixed and floating dates differ, there may be no single date-specific net cash flow. Discount each leg’s cash flows and then subtract their present values.
Using the telescoping result outside its assumptions. A spread or payment lag changes the simple shortcut. Explicitly value the affected cash flows instead.
Reversing the position’s sign. Receive-fixed/pay-floating is fixed PV minus floating PV; pay-fixed/receive-floating is floating PV minus fixed PV.

Key takeaway: A swap valuation is a schedule-by-schedule present-value calculation. The correct coupon depends on the contractual accrual factor; the correct discount factor depends on the payment date. This schedule discipline is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated accrual factors, payment dates, and sign convention. Round monetary answers to the nearest cent when needed.

A receive-fixed leg has notional $\$800{,}000$, annual fixed rate 3.2%, quarterly accrual factors of 0.25, and payment-date discount factors 0.995, 0.987, 0.976, and 0.960. Find the fixed-leg present value. (Answer: $\$25{,}075.20$)
A floating coupon has notional $\$500{,}000$, accrual factor 0.40, and accrual-end discount factors 0.984 at the start and 0.970 at the end. Under simple accrual, find the implied forward rate and the coupon’s present value if paid at the accrual end. (Answer: Forward rate $3.6082\%$; present value $\$7{,}000.00$)
A floating leg has one period with notional $\$1{,}000{,}000$, accrual factor 0.50, discount factors 0.990 at the start and 0.975 at the end, and a 0.15% spread. If payment occurs at the accrual end, find the floating coupon’s present value. (Answer: $\$15{,}731.25$)
A one-year receive-fixed/pay-floating swap has notional $\$1{,}000{,}000$. The fixed leg pays semiannually at 4.2% with accrual factors 0.50 and discount factors 0.982 and 0.955. The floating leg pays quarterly, has no spread, pays at accrual ends, and has discount factors 1.000, 0.990, 0.982, 0.970, and 0.955 at times 0, 0.25, 0.50, 0.75, and 1.00 years. Find the swap value. (Answer: $-\$4{,}323.00$)

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