What a Swap Cash-Flow Calculation Does
A plain-vanilla interest-rate swap describes an exchange of interest cash flows calculated on a notional principal amount. One party pays interest at a fixed rate and receives interest at a floating rate; the other party takes the opposite positions. The notional amount is used to calculate the payments, but it is not ordinarily exchanged between the parties.
For a cash-flow question, the essential task is to calculate each leg’s payment for each accrual period and compare them using one clearly stated sign convention. The floating rate is generally set at the start of its accrual period and paid at the end. Thus, the rate applicable to a payment date is the rate specified for the period just ended, not automatically the rate for the next period.
For consistency, this tutorial uses the fixed-rate payer’s perspective: a positive net amount means the fixed-rate payer receives money; a negative net amount means that payer pays money. The net settlement at the end of period \(j\) is therefore the floating-leg payment less the fixed-leg payment.
A positive result means the floating receipt exceeds the fixed payment, so the fixed-rate payer receives the difference. A negative result means the fixed payment is larger, so the fixed-rate payer pays the absolute value of the difference. The floating-rate payer has the opposite net settlement.
Why the Net-Settlement Formula Holds
Before netting, the fixed-rate payer owes \(NK\alpha_j\) on the fixed leg and is owed \(NL_j\alpha_j\) on the floating leg. Subtracting the amount owed from the amount received gives
This is a cash-flow identity: it follows from comparing two payments with the same notional and accrual fraction. It does not by itself determine the market value of the swap. A valuation calculation would require additional information and a separate method; here the goal is to calculate the contractual cash flow due on each payment date.
Worked Cash-Flow Calculations
Worked Example: One Quarterly Net Settlement
A hypothetical swap has a notional of $4,000,000. The fixed-rate payer pays 4.8% per year and receives a floating rate of 5.2% per year for a quarter-year accrual period, so \(\alpha_j=0.25\). Calculate both coupons and the net settlement.
Method. Calculate each leg separately, then subtract the fixed payment from the floating receipt. This makes the sign and direction of the settlement explicit.
Fixed leg. The fixed payment is
Floating leg. The floating payment is
Net settlement. From the fixed-rate payer’s perspective,
The positive result means the fixed-rate payer receives $4,000; the floating-rate payer pays $4,000. The gross coupons differ by exactly the net amount, which checks the calculation. A common mistake is to report a payment of $4,000 by the fixed-rate payer simply because that party pays the fixed leg; the floating receipt must also be included.
Worked Example: Settlements Across Three Reset Periods
A hypothetical fixed-rate payer has a $8,000,000 swap notional, pays a fixed annual rate of 4.6%, and receives floating rates that are reset for three successive half-year periods. The floating rates are 4.2%, 5.0%, and 4.4%. Each period has \(\alpha_j=0.5\). Calculate the net settlement in each period and state the direction of payment.
Method. The fixed coupon is the same each period because the notional, fixed rate, and accrual fraction are unchanged. Calculate each floating coupon using that period’s reset rate, then subtract the fixed coupon.
Fixed coupon per period.
First period. At 4.2%, the floating coupon is \(\$8{,}000{,}000(0.042)(0.5)=\$168{,}000\). The net settlement is \(\$168{,}000-\$184{,}000=-\$16{,}000\), so the fixed-rate payer pays $16,000.
Second period. At 5.0%, the floating coupon is \(\$8{,}000{,}000(0.050)(0.5)=\$200{,}000\). The net settlement is \(\$200{,}000-\$184{,}000=\$16{,}000\), so the fixed-rate payer receives $16,000.
Third period. At 4.4%, the floating coupon is \(\$8{,}000{,}000(0.044)(0.5)=\$176{,}000\). The net settlement is \(\$176{,}000-\$184{,}000=-\$8{,}000\), so the fixed-rate payer pays $8,000.
The settlements are therefore a payment of $16,000, a receipt of $16,000, and a payment of $8,000 from the fixed-rate payer’s perspective. The net across these three dates is a payment of $8,000. This total is a useful check, but it does not replace identifying the settlement due on each date. A common mistake is to use one floating rate for all periods instead of matching each payment to its own reset rate.
Worked Example: Applying an Actual/360 Accrual Fraction
A hypothetical swap has a $6,000,000 notional. For a 91-day period, the fixed-rate payer pays 4.25% per year and receives 4.75% per year. The contract specifies Actual/360. Calculate each coupon and the net settlement from the fixed-rate payer’s perspective.
Method. Under Actual/360, the year fraction is the actual number of accrual days divided by 360. Use the same stated fraction for each leg.
Accrual fraction.
Fixed coupon.
Floating coupon.
Net settlement.
Using the unrounded coupon amounts, the difference is \(\$6{,}000{,}000(0.0475-0.0425)(91/360)=\$7{,}583.333\ldots\), which rounds to $7,583.33. The fixed-rate payer receives $7,583.33. The one-cent discrepancy from subtracting the separately rounded displayed coupons occurs because each coupon was rounded before subtraction; the net amount should be rounded from the unrounded calculation. The check is that the floating rate exceeds the fixed rate, so a fixed-rate payer must receive, not pay, the net amount.
A common mistake is to use \(91/365\) or \(91/12\) despite the stated Actual/360 convention. Another is to round both coupons and then treat their displayed difference as the exact settlement. Use the contract’s day-count basis and round the final settlement only after calculating it.
Reading the Contract and Avoiding Sign Errors
A swap description may say “pay fixed, receive floating” or “receive fixed, pay floating.” Translate that wording into the cash flows before calculating. The payer of the fixed leg owes \(NK\alpha_j\); the receiver of the fixed leg receives that amount. The floating leg works the same way with \(NL_j\alpha_j\). The following table summarizes the positions.
| Position | Fixed leg | Floating leg | Net amount received |
|---|---|---|---|
| Fixed-rate payer | Pays \(NK\alpha_j\) | Receives \(NL_j\alpha_j\) | \(N(L_j-K)\alpha_j\) |
| Fixed-rate receiver | Receives \(NK\alpha_j\) | Pays \(NL_j\alpha_j\) | \(N(K-L_j)\alpha_j\) |
The same arithmetic can be expressed from either party’s perspective, but the sign reverses. State the perspective first, and keep it throughout the calculation. If the problem asks for the amount paid by one party, give a nonnegative amount and name the payer; if it asks for a signed cash flow, state the sign convention.
Determine which party pays fixed and which party receives fixed. Choose the perspective for the sign of the net amount.
Match the payment date to the applicable floating reset, the stated day-count convention, and the accrual period.
Use notional times annualized rate times accrual fraction for each leg. Include a stated floating spread in the floating rate.
Subtract the payment owed from the payment received for the chosen party, then translate the sign into who pays whom.
Do not add an exchange of principal merely because a notional amount appears in the contract: in the usual plain-vanilla interest-rate swap, the notional is a calculation base, not a principal payment. Likewise, do not multiply an annualized rate by the number of days without converting the period to the specified year fraction. If the contract uses Actual/360, for example, 91 days corresponds to \(91/360\), not to a quarter by assumption.
Check Your Understanding
For each problem, identify the settlement direction as well as the amount. Round money to the nearest cent where needed.
- A fixed-rate payer on a $4,000,000 swap pays 3.6% and receives 4.1% for a 92-day period under Actual/360. Calculate both coupons and the net settlement. (Answer: fixed coupon $36,800.00, floating coupon $41,911.11, fixed-rate payer receives $5,111.11.)
- A party receives fixed at 5.2% and pays floating at 4.9% on a $12,000,000 notional for a half-year period with \(\alpha_j=0.5\). Calculate both coupons and the net settlement. (Answer: fixed coupon $312,000, floating coupon $294,000, fixed-rate receiver receives $18,000.)
- A fixed-rate payer pays 4.7% and receives 4.4% on a $5,500,000 notional for 87 days under Actual/360. Calculate both coupons and the settlement. (Answer: fixed coupon $62,470.83, floating coupon $58,483.33, fixed-rate payer pays $3,987.50.)
- A fixed-rate payer has a $3,000,000 notional, a 4.0% fixed rate, and three quarterly floating rates of 3.5%, 4.1%, and 4.6%. Each accrual fraction is 0.25. Find the settlement at each date and the total net cash flow over all three dates. (Answer: pays $3,750, receives $750, receives $4,500; overall receives $1,500.)
- A fixed-rate payer pays 4.3% and receives a floating index rate plus a 0.15% spread on a $20,000,000 notional for a 180-day period under Actual/360. The index rate is 4.0%. Calculate the net settlement. (Answer: floating coupon $415,000, fixed coupon $430,000, fixed-rate payer pays $15,000.)