Amortizing Swaps

A company that wants to fix the interest on a loan with a repayment schedule, and the bank that takes the other side, arrange an interest rate swap whose notional amount falls with the loan balance.

Reducing the notional amount in line with the loan balance

An amortizing swap is an interest rate swap whose notional amount decreases over time. Most interest rate swaps keep the same notional amount for their whole term, but some swaps reduce the notional amount to follow the balance of a loan that is repaid in installments¹. As in a swap with a constant notional amount, the parties exchange a fixed rate for a floating rate and never exchange the notional amount itself. The basic mechanics are covered in Interest Rate Swaps.

Consider a hypothetical example in which Company B borrows 10 billion yen from Bank C for 4 years at a floating rate (with no spread added) and repays 2.5 billion yen at the end of each year. To fix its interest payments, Company B enters into an interest rate swap with Bank A. The swap's notional amount falls in step with the loan balance. Company B pays a fixed rate of 2% a year and receives the floating rate. Bank A does the opposite.

Figure description

This figure shows how the loan and the swap connect through Company B. Company B pays Bank C both the principal and the floating-rate interest, while the swap with Bank A involves interest only and no principal. The floating interest that Company B receives from Bank A matches the floating interest it pays Bank C, so the two cancel. What remains for Company B is the fixed-rate payment to Bank A.

For each interest period, the swap's notional amount is set equal to the loan balance for that period.

Cash flows for the fixed-rate payer

The fixed interest Company B pays on the swap falls each year with the notional amount, even though the fixed rate stays at 2%. In year 1 it is 10 billion yen × 2% = 200 million yen, and in year 4 it is 2.5 billion yen × 2% = 50 million yen. The floating interest it receives depends on that year's floating rate and notional amount. Assume the floating rate turns out to be 1% in year 1, 2% in year 2, 3% in year 3, and 4% in year 4. Company B's cash flows on the swap are then as follows.

Figure description

These are the swap payments of Company B, the fixed-rate payer, assuming the floating rate is 1%, 2%, 3%, and 4% in years 1 to 4. Company B pays fixed-rate interest of 200 million yen at the end of year 1, 150 million yen at the end of year 2, 100 million yen at the end of year 3, and 50 million yen at the end of year 4. It receives floating-rate interest of 100 million yen, 150 million yen, 150 million yen, and 100 million yen at the same dates. The fixed payment falls by the same amount each year, in line with the notional amount.

The table below shows the payments on each payment date. A positive net amount is received by Company B, and a negative one is paid by Company B. Each interest amount is calculated for a full year. As in a swap with a constant notional amount, actual trades count the length of each period with a day count convention and set payment dates with a business day convention².

In year 4 the floating rate reaches 4%, but because the notional amount is only 2.5 billion yen, the floating interest is 100 million yen, less than the 150 million yen in year 3. This example assumes a round fixed rate of 2%. The next section covers how it compares with the fixed rate that makes the market value zero when the trade is agreed.

Notional amounts set the weights behind the fixed rate

The fixed rate $K$ that makes the market value zero when the trade is agreed is an average of the forward rates $F_i$ for each period. Each weight is the product of the notional amount, the length of the period, and the discount factor³. With $n$ periods, $Q_i$ as the notional amount for period $i$, $\tau_i$ as the length of the period in years, and $DF(t_i)$ as the discount factor for the payment date, the formula is as follows.

$$K = \frac{\sum_{i=1}^{n} Q_i \, \tau_i \, F_i \, DF(t_i)}{\sum_{i=1}^{n} Q_i \, \tau_i \, DF(t_i)}$$

If the notional amount is the same in every period, $Q_i$ cancels out of the numerator and denominator, and the formula becomes the par rate formula for an interest rate swap with a constant notional amount⁴. How to derive forward rates and discount factors, and the formula for a constant notional amount, are covered in Valuing an Interest Rate Swap.

As in the previous section, suppose the forward rates derived from today's curve are 1%, 2%, 3%, and 4% for years 1 to 4. Every period is one year long. The discount factors are built up from the forward rates in order: 1 ÷ 1.01 for year 1, that value divided again by 1.02 for year 2, and so on. The table compares the weight of each period in the amortizing swap and in a 4-year swap with a constant notional amount of 10 billion yen⁵.

The fixed rate that makes the market value zero is 1.97% for the amortizing swap and 2.46% for the 4-year swap with a constant notional amount. The amortizing rate is lower because the swap puts more weight on the early periods, when the notional amount is large and rates are low. As a rough check by hand, treating the discount factors as 1, the amortizing swap gives (10 × 1% + 7.5 × 2% + 5 × 3% + 2.5 × 4%) ÷ 25 = 2.0%, and the constant-notional swap gives 2.5%, the simple average of the four rates. When forward rates rise with maturity, as in this example, the fixed rate on an amortizing swap is lower than on a constant-notional swap with the same term. When forward rates fall with maturity, it is higher.

The 2% rate in the previous section is slightly above 1.97%, so for Company B, the fixed-rate payer, the market value when the trade is agreed is −6,580,942 yen.

The notional amount also stays outstanding for a shorter time in an amortizing swap. Here, the average life means the sum of each period's notional amount times the length of the period, divided by the initial notional amount. In the example it is (10 + 7.5 + 5 + 2.5) ÷ 10 = 2.5 years, a little over half the 4-year term. Averaging the number of years until each 2.5 billion yen of loan principal is repaid (1 to 4 years) gives the same 2.5 years.

Because the average life is shorter, the market value also changes less when rates move. Suppose the forward rate for every period rises by 0.01 percentage points while the discount factors stay the same. The market value for the fixed-rate payer then increases by the sum of notional amount times period length times discount factor, multiplied by 0.01%. In the example, that is 2,415,861 yen for the amortizing swap and 3,809,363 yen for the 4-year swap with a constant notional amount. The fixed-rate payer gains when rates rise and loses when they fall, and the fixed-rate receiver is in the opposite position⁶. The change in market value when rates move by 0.01 percentage points (1bp) is called DV01.

Using the swap to fix loan interest

A company with a floating-rate loan that it repays in installments can pay a fixed rate on an amortizing swap matched to the loan balance. Its interest payments then take the same shape as those on a fixed-rate loan⁷. In the example, the floating interest Company B receives from Bank A covers the floating interest it pays Bank C. However the floating rate moves, Company B's interest cost is 2% of each year's balance. Its total interest over the 4 years is 200 + 150 + 100 + 50 = 500 million yen.

To see when the swap helps and when it hurts, compare two cases in which the floating rate stays at the same level for all 4 years. The loan balances add up to 10 + 7.5 + 5 + 2.5 = 25 billion yen-years.

If the floating rate stays high, the swap cuts Company B's interest payments by 500 million yen. If the floating rate stays low, Company B pays 500 million yen instead of the 250 million yen it would have paid without the swap, 250 million yen more. Like a swap with a constant notional amount, an amortizing swap fixes the amount of interest, and in return the company gives up the benefit of lower payments when rates fall.

If a company instead tries to fix the interest on an amortizing loan with a constant-notional swap, the swap's notional amount exceeds the loan balance from year 2. In the example, against a swap notional of 10 billion yen, the loan balance is 7.5 billion yen in year 2, 5 billion yen in year 3, and 2.5 billion yen in year 4. The excess has no loan to match it, so gains and losses on that part of the swap remain and move with rates.

Prepayment and the risks that remain

If the loan is repaid ahead of schedule, the loan balance and the swap's notional amount no longer match. The schedule for reducing the notional amount is fixed in the contract at the start of the trade. So even if the loan is repaid early, the swap's notional amount keeps falling on the original schedule unless the contract is changed. For example, if Company B repays the remaining 5 billion yen at the end of year 2, the swap, with a notional amount of 5 billion yen in year 3 and 2.5 billion yen in year 4, remains with no loan behind it. What remains is the same as a pay-fixed interest rate swap with no loan.

Its market value depends on how the 2% fixed rate compares with the fixed rate that makes the market value zero, derived from the forward rates for the remaining periods. If that rate is below 2%, the market value is negative for Company B, and to end the swap early Company B would pay Bank A an amount equal to the market value⁸. If it is above 2%, the market value is positive, and Company B would receive that amount. If the curve in the example does not change, the forward rates for the remaining years 3 and 4 are 3% and 4%, both above 2%, so the market value is positive for Company B.

As with a constant-notional interest rate swap, there is counterparty credit risk. Because the notional amount itself is not exchanged, the credit risk is smaller than in a trade such as a loan, where principal changes hands. A loss can arise when the swap has a positive market value for you and the counterparty defaults⁶.

Trading in practice

Japan Securities Clearing Corporation (JSCC), a central counterparty, sets conditions for the yen interest rate swaps it clears. One of them is that the notional amount is either constant, or follows a method of decreasing or increasing it that is set at the start of the trade and not changed during the term. Another is that the notional amount is at least 1 yen and less than 4 trillion yen⁹. An amortizing swap whose reduction schedule is set at the start and not changed during the term meets this notional amount condition. To be eligible for clearing, a swap also has to meet other conditions, such as those on the type of swap and the contract term.

How swaps differ by the shape of the notional amount

Interest rate swaps can be grouped by how the notional amount moves, as follows. A swap whose notional amount increases over time is called an accreting swap. One example of its use is a company that expects to increase its borrowing as its business grows¹¹.

Notes

1. Society of Actuaries, "Interest Rate Swaps" (May 2017), section 2, "Definitions", page 4. Section 10, "Glossary", page 31, gives the same definition. ↩

2. Japan Securities Clearing Corporation, "清算対象取引（金利スワップ）" (Eligible trades for clearing: interest rate swaps, in Japanese), "11．日数計算式" (Day count fraction) and "12．営業日調整" (Business day adjustment). As a condition for trades it clears, the source requires the day count fraction and business day adjustment to be ones defined in the ISDA definitions. Checked on September 25, 2026. ↩

3. Society of Actuaries, "Interest Rate Swaps", section 3, "General Formula", formula (3.7), page 10. In the source, $Q_{t_i}$ is the notional amount, $f$ the forward rate, and $P_{t_i}$ the price of a zero-coupon bond, corresponding to $Q_i$, $\tau_i F_i$, and $DF(t_i)$ in the text. The source's $f^{*}$ is the effective rate for the settlement period; when the periods are equal in length, $\tau_i K$ in the text corresponds to the source's fixed rate $R$. ↩ ↩²

4. The same source, after Example 3 in section 3 (page 10), and section 4, "Special Formula" (page 11). It says that with a level notional amount, the notional amounts cancel out of the numerator and denominator of formula (3.7). ↩

5. The discount factors and weights are shown rounded; the fixed rates (1.9728% and 2.4632%), market value, and change in market value are calculated from unrounded values. The discount factors are $DF(t_i) = DF(t_{i-1}) \div (1 + F_i)$ with $DF(t_0) = 1$. The market value is $\sum Q_i (F_i - 2\%) DF(t_i)$, with notional amounts in units of 100 million yen, converted to yen and rounded. ↩

6. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual", Section 4325.1, "Interest-Rate Swaps", "Credit Risk" (April 2001 version, pages 10–11). ↩ ↩²

7. Society of Actuaries, "Interest Rate Swaps" (May 2017), section 8, "Exercises", Exercise 7 (pages 19–20), and the solution in section 9 (page 26). The example pairs an amortizing interest rate swap with a floating-rate loan that is being paid down. ↩

8. Society of Actuaries, "Interest Rate Swaps" (May 2017), section 6, "Market Value" (page 15), footnote 8. The source says a swap can be ended early by one party paying the other the market value. ↩

9. Japan Securities Clearing Corporation, "清算対象取引（金利スワップ）" (in Japanese), "10．想定元本" (Notional amount). Other conditions include "3．スワップの種類" (Types of swaps), "4．変動金利の種類" (Types of floating rates), and "8．契約期間" (Contract term). Checked on September 25, 2026. ↩

10. Accounting Standards Board of Japan, ASBJ Statement No. 10, "金融商品に関する会計基準" (Accounting Standard for Financial Instruments, in Japanese), Note 14, "金利スワップについて" (Interest rate swaps), and paragraph 107 of the basis for conclusions. The version checked on September 25, 2026 (last updated in the ASBJ accounting standards search system on July 17, 2026). ↩

11. Society of Actuaries, "Interest Rate Swaps" (May 2017), section 2, "Definitions" (page 4), and footnote 4. The source calls a swap whose notional amount increases over time an accreting swap, and gives as a reason a business that expects rapid growth over the swap term and needs to increase its borrowing as it grows. ↩