Discounting Fixed and Floating Swap Cash Flows

Why Swap Leg Valuation Requires Two Rate Inputs

A swap’s fixed payments are specified by the contract, while future floating payments depend on rates that will be set or observed later. To value either leg, identify its payment dates and cash amounts, then discount each amount to the valuation date. The key distinction is that a projected floating rate determines a coupon amount; a discount factor determines how much a payment on a particular date is worth today.

This tutorial values dated swap cash flows from the perspective of a specified valuation date. It does not derive the fair fixed rate. All rates, payment dates, and discount factors below are hypothetical inputs. The methods are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Let $D(0,t)$ be the discount factor at the valuation date 0 for a payment made at date $t$. A cash flow $C_t$ due at $t$ has present value $C_tD(0,t)$. For payments on dates $t_1,\ldots,t_n$, the present value is the sum of the dated cash flows, each multiplied by its own discount factor.

A swap usually exchanges interest payments calculated on a notional principal; the notional is not itself exchanged. For a period with notional $N$, accrual fraction $\alpha_i$, and annualized rate $r_i$, the interest payment is $N\alpha_i r_i$. The accrual fraction reflects the contract’s day-count convention and the length of that interest period. Use the fraction supplied or implied by the problem rather than assuming it equals a calendar fraction.

For a fixed leg, the rate is contractual, so each coupon can be calculated directly. For a floating leg, future coupon rates must be supplied as projections or otherwise specified by the valuation model. In a deterministic-curve setup, these may be forward rates implied by the stated curve. In a more general pricing setup, the expected cash amounts must be appropriate to that setup. In either case, the discount factor is selected by payment date; it is not obtained by reusing the coupon rate.

Present-Value Formulas and Their Basis

Suppose a fixed leg has notional $N$, annual fixed rate $K$, and accrual fractions $\alpha_i$, with coupons paid at dates $t_i$. If the contract has no exchange of notional, its value to the party receiving fixed is the present value of those coupons:

PV_{\text{fixed}}=\sum_{i=1}^{n}N\alpha_iK\,D(0,t_i).

For a floating leg, let $F_i$ denote the rate used to project the coupon for period $i$, and let $C_i^{\text{float}}$ denote the corresponding projected payment. Then:

C_i^{\text{float}}=N\alpha_iF_i, \qquad PV_{\text{float}}=\sum_{i=1}^{n}C_i^{\text{float}}D(0,t_i).

Justification: A discount factor converts one dated payment to its present value. For a collection of payments, present value is additive: the value of their sum is the sum of their values. Applying the single-payment rule $C_{t_i}D(0,t_i)$ to each date and adding gives $\sum_i C_{t_i}D(0,t_i)$. Substituting the contractual fixed coupon $N\alpha_iK$, or the projected floating coupon $N\alpha_iF_i$, gives the formulas above. This argument assumes the relevant discount factors and cash-flow amounts are defined consistently with the valuation model; it does not justify discounting at a coupon rate.

Use a sign convention before combining leg values. A party receiving a leg has a positive value for that leg; a party paying it has a negative value. Thus, the value to a receive-floating, pay-fixed position is $PV_{\text{float}}-PV_{\text{fixed}}$. A negative result means that the position, as described, has negative value at the valuation date.

Valuing a Fixed Leg

Worked Example: Three Annual Fixed Coupons

A hypothetical swap has notional $N=\$1{,}000{,}000$, a 4.5% annual fixed rate, and three annual coupon dates at $t=1,2,3$ years. There is no exchange of notional. The maturity-matched discount factors are $D(0,1)=0.970$, $D(0,2)=0.935$, and $D(0,3)=0.890$. Find the value of the fixed leg to the party receiving fixed.

Method. Calculate the contractual coupon for each one-year period, then multiply each coupon by the discount factor for its own payment date.

Each accrual fraction is 1, so each coupon is:

\$1{,}000{,}000(1)(0.045)=\$45{,}000.

The date-by-date present values are:

\begin{aligned} t=1:&\quad \$45{,}000(0.970)=\$43{,}650,\\ t=2:&\quad \$45{,}000(0.935)=\$42{,}075,\\ t=3:&\quad \$45{,}000(0.890)=\$40{,}050. \end{aligned}

Adding the discounted payments gives:

PV_{\text{fixed}}=\$43{,}650+\$42{,}075+\$40{,}050=\$125{,}775.

Check. The coupons total $\$135{,}000$ before discounting, so a present value of $\$125{,}775$, below that total, is consistent with all three discount factors being less than 1. The common mistake avoided is discounting all coupons with one factor despite their different payment dates.

Projecting and Discounting a Floating Leg

A floating coupon may not be known on the valuation date. A problem can provide projected rates for the remaining periods and discount factors for their payment dates. Treat these as separate inputs: first use the projected rate to calculate the payment, then use the date-specific discount factor to value it.

Worked Example: Projected Quarterly Floating Coupons

A hypothetical floating leg has notional $\$1{,}200{,}000$ and three quarterly accrual periods, each with $\alpha_i=0.25$. The projected annualized rates are 4.0%, 4.8%, and 5.2%. The corresponding payment dates have discount factors 0.990, 0.975, and 0.955. Find the present value of the projected floating coupons, with no notional exchange.

Method. For each period, calculate $N\alpha_iF_i$ and then multiply by the discount factor for that coupon’s payment date.

\begin{aligned} C_1^{\text{float}}&=\$1{,}200{,}000(0.25)(0.040)=\$12{,}000,\\ C_2^{\text{float}}&=\$1{,}200{,}000(0.25)(0.048)=\$14{,}400,\\ C_3^{\text{float}}&=\$1{,}200{,}000(0.25)(0.052)=\$15{,}600. \end{aligned}

Discount each projected payment at its own date:

\begin{aligned} PV_1&=\$12{,}000(0.990)=\$11{,}880,\\ PV_2&=\$14{,}400(0.975)=\$14{,}040,\\ PV_3&=\$15{,}600(0.955)=\$14{,}898. \end{aligned}

Therefore:

PV_{\text{float}}=\$11{,}880+\$14{,}040+\$14{,}898=\$40{,}818.

Check. The undiscounted projected coupons total $\$42{,}000$, and each discount factor is below 1, so the present value is lower. The common mistake avoided is treating a projected coupon rate as the discount rate.

Netting the Two Legs

If both legs are denominated in the same currency, settle on the same dates, and the contract permits net settlement, value can be calculated either by discounting each leg separately and subtracting the totals or by subtracting the same-date cash flows first and discounting the net amounts. Additivity makes the two approaches equivalent. If payments occur on different dates, discount them at their respective dates before combining values; do not net payments across dates as if they were simultaneous.

Worked Example: Receive Floating and Pay Fixed

A hypothetical swap has notional $\$2{,}000{,}000$, semiannual accrual fractions of 0.5, and three remaining payment dates. The fixed rate is 4.0%. The projected floating rates are 3.5%, 4.25%, and 5.0%. The discount factors for the three payment dates are 0.985, 0.960, and 0.925. Find the value to the party receiving floating and paying fixed. Assume same-date net settlement is permitted.

Method. Calculate both legs period by period, then subtract the fixed coupon from the floating coupon on each payment date.

The fixed coupon is the same in each period:

\$2{,}000{,}000(0.5)(0.040)=\$40{,}000.

The floating coupons are:

\begin{aligned} C_1^{\text{float}}&=\$2{,}000{,}000(0.5)(0.035)=\$35{,}000,\\ C_2^{\text{float}}&=\$2{,}000{,}000(0.5)(0.0425)=\$42{,}500,\\ C_3^{\text{float}}&=\$2{,}000{,}000(0.5)(0.050)=\$50{,}000. \end{aligned}

The net payments received by this party are $-\$5{,}000$, $\$2{,}500$, and $\$10{,}000$. Discount them by date:

\begin{aligned} PV_1&=-\$5{,}000(0.985)=-\$4{,}925,\\ PV_2&=\$2{,}500(0.960)=\$2{,}400,\\ PV_3&=\$10{,}000(0.925)=\$9{,}250. \end{aligned}

The position’s value is:

PV_{\text{receive float, pay fixed}} =-\$4{,}925+\$2{,}400+\$9{,}250 =\$6{,}725.

Check by separate legs. The floating leg is worth $\$35{,}000(0.985)+\$42{,}500(0.960)+\$50{,}000(0.925)=\$34{,}475+\$40{,}800+\$46{,}250=\$121{,}525$. The fixed leg is worth $\$40{,}000(0.985)+\$40{,}000(0.960)+\$40{,}000(0.925)=\$39{,}400+\$38{,}400+\$37{,}000=\$114{,}800$. Their difference is $\$121{,}525-\$114{,}800=\$6{,}725$, as before. The common mistake avoided is reversing the position’s sign: the party receives floating and pays fixed, so its value is floating-leg value minus fixed-leg value.

Common Discounting Trap

A projected floating rate is not automatically an annual effective discount rate. For example, suppose a projected coupon of $\$60{,}000$ is due in two years and the supplied maturity-matched discount factor is $D(0,2)=0.90$. Its present value is $\$60{,}000(0.90)=\$54{,}000$. If the 6% rate used to project that coupon is wrongly treated as the discount rate, the calculation becomes:

\frac{\$60{,}000}{(1.06)^2}\approx \$53{,}399.79\approx \$53{,}400.

That numerical result is not the value under the stated discount-factor input. The correction is to use the given factor for the payment’s maturity, producing $\$54{,}000$. The 6% rate belongs in the coupon calculation only if the contract and problem data specify it for that coupon. Also check whether a quoted rate is annualized and apply the correct accrual fraction before discounting.

List dates and accruals.
Record each payment date, its accrual fraction, and whether the payment is fixed or floating.

Calculate cash amounts.
Use the contractual fixed rate or the problem’s specified projected floating rate, multiplied by notional and accrual fraction.

Match discount factors to dates.
Multiply each dated amount by its maturity-matched discount factor, not by its coupon rate.

Apply the perspective.
Subtract the value of payments made from the value of payments received, keeping any netting assumption explicit.

Check Your Understanding

For each problem, identify the payment amounts before discounting and keep each discount factor matched to its payment date.

A fixed leg has notional $\$500{,}000$, annual rate 3.6%, and two annual coupons. The discount factors are 0.96 and 0.91. Find its value to the fixed receiver. (Answer: $\$33{,}660$)
A floating leg has notional $\$800{,}000$, two quarterly periods with accrual fraction 0.25, projected rates 4.0% and 5.0%, and discount factors 0.99 and 0.97. Find its present value. (Answer: $\$17{,}620$)
A receive-floating, pay-fixed position has notional $\$1{,}000{,}000$, two semiannual periods, a fixed rate of 4.6%, projected floating rates of 4.0% and 5.0%, and discount factors 0.98 and 0.94. Find its value. (Answer: $-\$1{,}060$)
A receive-floating, pay-fixed position has notional $\$2{,}000{,}000$, two semiannual periods, a fixed rate of 3.8%, projected floating rates of 3.2% and 4.1%, and discount factors 0.985 and 0.950. Find its value. (Answer: $-\$3{,}060$)
A receive-floating, pay-fixed position has notional $\$1{,}500{,}000$, three quarterly periods, a fixed rate of 4.4%, projected floating rates of 4.0%, 4.6%, and 5.2%, and discount factors 0.992, 0.978, and 0.960. Find its value, assuming same-date net settlement is permitted. (Answer: $\$2{,}125.50$)

What You'll Learn