Swap Valuation Between Reset Dates

Why a Reset Date Matters

At a reset date, the rate for a floating-rate period is set using market information. Between reset dates, that period’s floating coupon is already known, even though it has not yet been paid. This changes the valuation: the next floating payment is not an unknown rate to forecast. It is a known contractual cash flow, discounted to the valuation date along with the remaining payments.

The key distinction is between the full coupon payable on the next payment date and the portion of that coupon that may have accrued since the last reset. The contractual payment remains based on the entire accrual period. Accrued interest is not a reason to replace it with a smaller payment.

Model and assumptions: Consider a single-currency, plain-vanilla swap with notional principal $N$, no exchange of notional, no floating spread, and valuation date $t$ strictly between reset dates $T_0$ and $T_1$. The floating rate for the period $T_0$ to $T_1$ was set at $T_0$ and is denoted by $L_0$. Let $\alpha_j$ be the year fraction for the relevant accrual period and $D_j$ the discount factor from $t$ to payment date $T_j$. Assume the forward rates used for later floating coupons are consistent with these discount factors. The formulas below value the remaining contractual cash flows on this discount curve; they do not include credit adjustments, collateral effects, or transaction costs.

Valuing the Floating Leg Between Reset Dates

Let $T_1,\ldots,T_n$ be the remaining floating payment dates. The next floating coupon is already fixed: its amount is $N L_0\alpha_0$, where $\alpha_0$ covers the full accrual period from $T_0$ to $T_1$. Coupons after $T_1$ depend on rates set at later reset dates.

For a later accrual period from $T_j$ to $T_{j+1}$, let $F_j$ be its forward rate and $\alpha_j$ its year fraction. Consistency between that forward rate and the discount factors means $1+F_j\alpha_j=D_j/D_{j+1}$. The present value of that period’s floating coupon is therefore $N(D_j-D_{j+1})$. Summing these coupons telescopes:

\sum_{j=1}^{n-1}N F_j\alpha_jD_{j+1} = N\sum_{j=1}^{n-1}(D_j-D_{j+1}) = N(D_1-D_n).

Adding the already-fixed coupon, whose present value is $N L_0\alpha_0D_1$, gives the floating-leg value:

V_{\mathrm{float}} = N\left[(1+L_0\alpha_0)D_1-D_n\right].

This formula values the coupon payments only. The subtraction of $ND_n$ is essential: the telescoping expression $N(D_1-D_n)$ represents later floating coupons, while the swap does not exchange principal at maturity. If principal were exchanged, that cash flow would need to be included explicitly.

Valuing the Fixed Leg and the Swap

Let $K$ be the fixed rate, and let $\alpha_j^{\mathrm{fix}}$ be the fixed-leg accrual factor for the coupon paid at $T_j$. Every remaining fixed coupon, including the one due at $T_1$, is discounted from its payment date:

V_{\mathrm{fixed}} = N K\sum_{j=1}^{n}\alpha_j^{\mathrm{fix}}D_j.

For a party receiving floating and paying fixed, the swap value is $V_{\mathrm{float}}-V_{\mathrm{fixed}}$. The other party has the opposite value. This sign rule follows from valuing the cash flows received as positive and those paid as negative.

Theorem: Under the stated assumptions, the value to a receive-floating, pay-fixed party between reset dates is $$ V_{\mathrm{swap}} = N\left[(1+L_0\alpha_0)D_1-D_n\right] - NK\sum_{j=1}^{n}\alpha_j^{\mathrm{fix}}D_j. $$ A pay-floating, receive-fixed party’s value is $-V_{\mathrm{swap}}$. The formula does not apply unchanged if the swap has a floating spread, exchanges principal, or uses different discounting or projection assumptions.

A Practical Valuation Sequence

Locate the current period.
Identify the last reset date, the next payment date, the rate fixed at the last reset, and the full accrual factor for that period.

Value the floating coupon stream.
Use the known current coupon and the telescoping formula for later floating coupons. Subtract the final discount factor because the swap does not exchange principal.

Value the fixed coupon stream.
Discount every remaining fixed coupon using its own accrual factor and payment-date discount factor.

Apply the party’s sign.
Subtract the leg paid from the leg received, then check that the other party’s value is its negative.

Worked Example: The Fixed Coupon Is Already Known

A party receives floating and pays fixed on a swap with notional $N=\$2{,}000{,}000$. The valuation date falls between resets. The current floating rate was set at $5.2\%$, and the full current accrual factor is $0.25$. The fixed rate is $4.7\%$. The remaining payment dates are $T_1,T_2,T_3$, with discount factors $D_1=0.996$, $D_2=0.985$, and $D_3=0.970$. The fixed accrual factors are $0.25, 0.50, 0.50$. Find the swap value.

Method. Value the current known floating coupon together with the later floating coupons, then subtract the present value of the fixed coupons. This avoids treating the next floating rate as if it were still unknown.

The current coupon rate multiplied by its full accrual factor is $0.052(0.25)=0.013$, so $1+L_0\alpha_0=1.013$. The floating-leg value is

\begin{aligned} V_{\mathrm{float}} &=2{,}000{,}000\left[(1.013)(0.996)-0.970\right]\\ &=2{,}000{,}000(1.008948-0.970)\\ &=2{,}000{,}000(0.038948)\\ &=\$77{,}896. \end{aligned}

For the fixed leg, the discounted accrual-factor sum is

$$ (0.25)(0.996)+(0.50)(0.985)+(0.50)(0.970) =0.249+0.4925+0.485 =1.2265. $$

Thus,

\begin{aligned} V_{\mathrm{fixed}} &=2{,}000{,}000(0.047)(1.2265)\\ &=2{,}000{,}000(0.0576455)\\ &=\$115{,}291. \end{aligned}

The party receives floating and pays fixed, so its value is

V_{\mathrm{swap}}=\$77{,}896-\$115{,}291=-\$37{,}395.

Check. The pay-fixed party’s value is negative, so the opposite party’s value is $+\$37{,}395$. The two values sum to zero. The common mistake is to round or truncate the fixed-leg value before taking the difference; using the computed fixed-leg value gives the net value shown.

Accrued Interest Is Not a Smaller Coupon

Suppose a coupon accrues over a half-year period and the valuation date is one quarter-year after the reset. The coupon due at the end of the period is still calculated using the full half-year accrual factor. A quarter-year may be relevant to an accrued-interest measure, but it does not replace the contractual half-year factor in the coupon amount.

The valuation above is the present value of the remaining contractual payments, sometimes called a dirty value when accrued amounts are included in the quoted convention. If a problem explicitly asks for a clean value obtained by subtracting an accrued-interest adjustment, that is a separate reporting convention. Do not subtract accrued interest unless the question specifies that convention and defines the adjustment.

Worked Example: Full Coupon Despite a Mid-Period Valuation

A receive-floating, pay-fixed party has notional $\$1{,}500{,}000$. The valuation date is halfway through a current accrual period: one quarter-year has elapsed, and one quarter-year remains. The full period’s accrual factor is $0.50$, and its floating rate, fixed at the last reset, is $4.0\%$. The fixed rate is $3.6\%$. There are three remaining payment dates with discount factors $0.980,0.955,0.927$; each remaining fixed accrual factor is $0.50$. Find the value.

Method. Use the full $0.50$ accrual factor for the already-fixed current coupon. The fact that only one quarter-year remains until payment does not change the coupon amount.

The floating-leg value is

\begin{aligned} V_{\mathrm{float}} &=1{,}500{,}000\left[(1+0.040(0.50))(0.980)-0.927\right]\\ &=1{,}500{,}000\left[(1.020)(0.980)-0.927\right]\\ &=1{,}500{,}000(0.9996-0.927)\\ &=\$108{,}900. \end{aligned}

For the fixed leg, the discounted accrual-factor sum is

$$ (0.50)(0.980)+(0.50)(0.955)+(0.50)(0.927) =0.490+0.4775+0.4635 =1.431. $$

Therefore,

\begin{aligned} V_{\mathrm{fixed}} &=1{,}500{,}000(0.036)(1.431)\\ &=1{,}500{,}000(0.051516)\\ &=\$77{,}274, \end{aligned} \qquad V_{\mathrm{swap}}=\$108{,}900-\$77{,}274=\$31{,}626.

Check. The receive-floating, pay-fixed party has positive value; the opposite party has value $-\$31{,}626$. The common mistake is to use the remaining time, $0.25$, as the current coupon’s accrual factor. That would understate the known coupon and misvalue the floating leg.

When the Shortcut Does Not Apply

The floating-leg expression is a shortcut for a particular cash-flow structure, not a universal rule. It relies on standard floating coupons with no spread, no principal exchange, and forward rates consistent with the discount factors used in the valuation. If any of those assumptions changes, write down the actual remaining payments and discount them directly.

Floating spread: a spread adds cash flows that do not telescope in the same way as the forward-rate portion.
Principal exchange: include any contractual principal payment rather than assuming the swap exchanges only coupons.
Different curves: if forecasting floating rates and discounting payments use different curves, the simple forward-rate identity used in the derivation need not hold.
Sign convention: calculate a leg’s positive present value first, then subtract the leg the party pays.

This topic is aligned with the SOA Exam FM (Financial Mathematics) syllabus. In exam calculations, the most reliable checks are to confirm that the next floating rate has already been set, use the full contractual accrual factor, include all remaining fixed coupons, and verify the sign from the party’s perspective.

Check Your Understanding

For each problem, use the stated discount factors and the standard assumptions of a single-currency swap with no principal exchange or floating spread. Values are from the perspective of the party receiving floating and paying fixed.

A swap has notional $\$1{,}000{,}000$, current fixed floating rate $3.6\%$, and current full accrual factor $0.25$. The final payment date is $T_3$, with $D_1=0.990$ and $D_3=0.950$. The fixed rate is $4.2\%$; the three fixed accrual factors are each $0.25$, with discount factors $0.990,0.970,0.950$. Find the swap value. (Answer: $+\$18{,}355$)
A swap has notional $\$2{,}400{,}000$, current floating rate $5.0\%$, and current full accrual factor $0.50$. The last remaining floating payment is at $T_3$, with $D_1=0.984$ and $D_3=0.941$. The fixed rate is $4.6\%$; fixed accrual factors are each $0.50$, with discount factors $0.984,0.963,0.941$. Find the swap value. (Answer: $+\$2{,}822.40$)
A swap has notional $\$750{,}000$, current floating rate $4.8\%$, and current full accrual factor $0.25$. Use $D_1=0.995$ and $D_3=0.960$ for the floating-leg formula. The fixed rate is $5.1\%$; its accrual factors are $0.25,0.50,0.50$, with corresponding discount factors $0.995,0.978,0.960$. Find the swap value to the cent. (Answer: $-\$11{,}373.94$)
A party pays floating and receives fixed on a swap with notional $\$5{,}000{,}000$. The already-set current floating rate is $6.2\%$, its full accrual factor is $0.50$, and $D_1=0.981,\ D_3=0.925$. The fixed rate is $5.8\%$, with accrual factors $0.50,0.25,0.25$ and discount factors $0.981,0.965,0.925$. Find the party’s value. (Answer: $-\$152{,}785$)

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