Basis Swaps

A bank whose funding and investments reference different rates sees its earnings move with the gap between them. A basis swap exchanges that gap, and staff who build interest rate curves for systems meet it when they handle gaps between rates of different tenors.

Exchanging two floating rates

A basis swap is an interest rate swap that exchanges one floating rate for another¹. The basic mechanism of exchanging a fixed rate for a floating rate is covered in Interest Rate Swaps.

Consider a hypothetical example in which Bank E and Securities Firm F enter into a basis swap exchanging compounded TONA (Tokyo Overnight Average Rate) for 3-month TIBOR (Tokyo Interbank Offered Rate), with a notional amount of 10 billion yen and a term of 3 years. In Japanese yen, examples include the exchange of this pair and the exchange of TIBOR of different tenors². Bank E pays 3-month TIBOR and receives compounded TONA plus 0.2% a year. Securities Firm F does the opposite. Both sides pay every 3 months. In an interest rate swap, the notional amount is used only to calculate interest and is never exchanged³.

A margin such as the 0.2% a year added to one of the floating rates is called the basis spread. Different reference rates have different values, so a spread is added to make the two sides balance⁴. TIBOR and TONA also contain different amounts of credit risk. TIBOR is a term rate at which banks lend to other banks without collateral, so it includes bank credit risk. TONA is an overnight rate, and the credit risk it contains is very limited⁵. Basis spreads also show up in how discount and projection curves are built under the multi-curve framework.

TONA and TIBOR payments

The two floating rates are set at different times. The TIBOR rate is set before the interest period starts, while the compounded TONA rate is set at the end of the period by compounding TONA over the period⁵. So Bank E knows the TIBOR interest it pays at the start of the period, but does not know the TONA interest it receives until the end. This way of setting the rate is called compounding in arrears⁶.

The interest for one period is given below. $N$ is the notional amount and $d$ is the number of days in the period. $L$ is the 3-month TIBOR set before the period. $R$ is the annualized rate obtained by compounding TONA over the period, and $b$ is the basis spread. Japan Securities Clearing Corporation (JSCC) is a central counterparty. For trades it clears, the day count convention is Actual/365 (Fixed) on both the TONA and the TIBOR side⁷.

$$\text{TIBOR interest} = N \times L \times \frac{d}{365}$$ $$\text{TONA interest} = N \times (R + b) \times \frac{d}{365}$$

Consider a hypothetical example in which the 3-month TIBOR set before the period is 0.9% a year. To keep the arithmetic simple, treat 3 months as 0.25 years. The TIBOR interest Bank E pays is 10 billion yen × 0.9% × 0.25, or 22.5 million yen. The table below shows Bank E's payments for three assumed compounded TONA rates over the period. A positive net amount is received by Bank E, and a negative one is paid by Bank E.

If the gap between TIBOR and compounded TONA for the period is 0.3%, Bank E pays 2.5 million yen, and if it is 0.1%, Bank E receives 2.5 million yen. If the gap equals the basis spread of 0.2%, the payments balance. Securities Firm F's amounts have the opposite sign to Bank E's.

Tenor basis

TIBOR has a different value for each tenor. Japanese Yen TIBOR has five tenors: 1 week, 1 month, 3 months, 6 months and 12 months⁸. A trade that exchanges TIBOR of different tenors, such as 3-month and 6-month, is called a tenor swap². On this page, the basis spread in an exchange of floating rates with different tenors is called the tenor basis. Fujii, Shimada and Takahashi (the paper) discuss the tenor basis, using LIBOR examples, in a paper on the multi-curve framework. According to the paper, even an exchange of two LIBOR rates that differ only in tenor does not balance unless a nonzero spread is added to one side⁴. In yen LIBOR, one form of tenor swap had one party pay the 3-month rate plus a spread every 3 months and receive the 6-month rate every 6 months. That spread was often more than 10bp (0.1%)⁹.

Consider a hypothetical example in which Bank C and Securities Firm D enter into a tenor swap with a notional amount of 10 billion yen. Bank C pays 3-month TIBOR plus 0.1% a year every 3 months and receives 6-month TIBOR every 6 months. Securities Firm D does the opposite. For the first 6 months, assume the 6-month TIBOR set on the start date is 1.0% a year, and the 3-month TIBOR set on the start date for the first 3 months is 0.8% a year. The 3-month TIBOR for the next 3 months, set 3 months later, is not yet known.

Figure description

These are the payments over the first 6 months for Bank C, which pays 3-month TIBOR plus 0.1% a year and receives 6-month TIBOR. Nothing changes hands on the start date. After 3 months, Bank C pays 22.5 million yen of interest for the first 3 months (10 billion yen × (0.8% + 0.1%) × 0.25). After 6 months, it receives 50 million yen of 6-month TIBOR interest (10 billion yen × 1.0% × 0.5) and pays interest for the next 3 months. That interest depends on the 3-month TIBOR set after 3 months, and the length of its arrow in the figure uses 27.5 million yen, the amount if 3-month TIBOR is 1.0%, as a guide.

The table below shows Bank C's payments over the first 6 months for three assumed 3-month TIBOR rates for the next 3 months. The 6-month TIBOR interest treats 6 months as 0.5 years.

If 3-month TIBOR for the next 3 months falls to 0.9%, Bank C receives 2.5 million yen, and if it rises to 1.1%, Bank C pays 2.5 million yen. Securities Firm D's amounts have the opposite sign to Bank C's. Because the two sides pay at different frequencies, the payment after 3 months goes out before the amount received after 6 months comes in.

Basis spread changes and gains and losses

Gains and losses on a basis swap depend on the gap between the two floating rates, not on the level of rates. Bank E in the TONA and TIBOR example receives in periods when the gap between TIBOR and compounded TONA is smaller than the 0.2% basis spread, and pays in periods when it is larger. Securities Firm F, which receives TIBOR, is the other way around and receives when the gap is larger.

When the market basis spread for the same remaining term moves after the trade is agreed, the market value of the trade moves. The paper estimates the loss from paying the tenor basis over the remaining term with a formula of roughly the following form¹⁰. The change in market value from a change in the basis spread can be estimated in the same form.

$$\text{Change in market value} \approx N \times \Delta b \times A$$

Here $\Delta b$ is the change in the basis spread, and $A$ is the sum, over the remaining payment dates, of each discount factor multiplied by the length of its period in years. Consider a hypothetical example in which the remaining term is 3 years and $A$ is 2.95. If the market basis spread widens from 0.2% to 0.3%, Bank E's market value falls by about 29.5 million yen (10 billion yen × 0.1% × 2.95), and Securities Firm F's rises by the same amount. If the spread narrows to 0.1%, Bank E's market value rises by the same amount. Because gains and losses come from changes in rates rather than their level, the effect is not small even though the basis spread is small compared with the level of rates¹⁰.

Hedging a mismatch between funding and investment rates

One use of basis swaps is hedging by financial institutions whose funding and investment rates are linked to different floating rates. An institution that funds at a cost linked to the overnight rate and invests in assets linked to TIBOR has floating rates on both sides, but it is still exposed to changes in the gap between the two rates. A basis swap exchanging compounded TONA for TIBOR (in the document's terms, an OIS/TIBOR basis swap) can hedge this risk. The Study Group on Risk-Free Reference Rates (the Study Group), for which the Bank of Japan acts as secretariat, describes this use in its document¹¹.

Consider a hypothetical example in which Bank E lends 10 billion yen at 3-month TIBOR plus 0.5% a year, funded by borrowing overnight and rolling the loan over every day. Assume its borrowing rate equals the compounded TONA rate over the period. If Bank E enters into the basis swap from the first example (paying 3-month TIBOR and receiving compounded TONA plus 0.2%), its margin for a period in which 3-month TIBOR is 0.9% is as follows.

Without the swap, a rise in TONA that narrows the gap to TIBOR reduces the margin, to 15 million yen when compounded TONA is 0.8%. With the swap, Bank E passes the TIBOR it receives on the loan to the swap counterparty, and receives from the swap the TONA it pays on the borrowing. The margin is therefore 17.5 million yen (10 billion yen × (0.5% + 0.2%) × 0.25) however the gap moves. On the other hand, if the gap widens and compounded TONA falls to 0.6%, the margin stays at 17.5 million yen instead of the 20 million yen it would have been without the swap. A basis swap fixes the margin, but the institution gives up the extra margin it would earn if the gap widened.

A tenor swap can be used to align the tenors of the floating rates received. In the paper's example, a financial institution that holds equal amounts of trades receiving 3-month and 6-month LIBOR enters into a tenor swap paying the 3-month rate and receiving the 6-month rate, replacing its 3-month receipts with 6-month receipts. To do this, it has to pay the tenor basis on top¹⁰.

Clearing and trading practice

JSCC clears interest rate swaps that exchange one floating rate for another. Its table of eligible products lists exchanges of rates with different tenors as tenor swaps, and exchanges of OIS for Japanese Yen TIBOR as curve swaps². Clearing through a central counterparty helps reduce the credit risk of the other party to an interest rate swap (counterparty risk)¹².

The TIBOR that JSCC clears is Japanese Yen TIBOR published by JBA TIBOR Administration, with a floating rate tenor of 1 month, 3 months or 6 months. Interest payments use the Tokyo calendar⁷. Euroyen TIBOR, the other JBA TIBOR rate, ceased publication at the end of December 2024, while Japanese Yen TIBOR continues to be published⁸¹³.

How it differs from similar trades

Interest rate swaps come in several types, depending on which rates are exchanged¹. A basis swap differs from other interest rate swaps in that it has no fixed rate, and its gains and losses depend only on the gap between two floating rates.

A trade that exchanges floating rates in different currencies is called a cross-currency basis swap. In a dollar/yen trade, the parties exchange principal and also exchange the US dollar 3-month reference rate for the yen 3-month reference rate plus the cross-currency basis¹⁵. The mechanism, including the exchange of principal, is covered in Cross-Currency Swaps.

Notes

1. Study Group on Risk-Free Reference Rates, "日本円ＯＩＳ（Overnight Index Swap）─取引の概要と活用事例─" (Yen OIS: overview of transactions and examples of use, April 2018, in Japanese), "（OIS の定義）" (Definition of OIS), page 3 (page 5 of the PDF). It describes an interest rate swap as "異なる金利（固定金利と変動金利または変動金利同士）を一定期間毎に交換する取引" (a trade that exchanges different rates (a fixed rate and a floating rate, or two floating rates) at set intervals; our translation). ↩ ↩²

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), the table of eligible products ("Basis Swap-Tenor Swap（JPY）", and "OIS vs D TIBOR" under "Basis Swap-Curve Swap") and item b, "変動金利と変動金利を交換するIRS" (IRS exchanging a floating rate for a floating rate), of "3．スワップの種類" (Types of swaps) in the conditions. An English version is available: "Clearing Products", the table "List of Clearing Products" and "3．Types of Swaps". Checked on September 25, 2026. ↩ ↩² ↩³

3. Society of Actuaries, "Interest Rate Swaps" (FM-25-17), page 3 (page 4 of the PDF). It says, "The notional principal amount under an interest rate swap is never paid by either counterparty." ↩

4. Fujii, Shimada, Takahashi, "A Note on Construction of Multiple Swap Curves with and without Collateral" (revised January 2010; Fujii et al.), Section 1, page 1. A tenor swap exchanging two LIBOR rates of different tenors needs a nonzero spread on one side. ↩ ↩²

5. Study Group on Risk-Free Reference Rates, "日本円ＯＩＳ（Overnight Index Swap）─取引の概要と活用事例─" (April 2018, in Japanese), "（１）IBOR と OIS レートの性質の違い" ((1) Differences in the nature of IBOR and OIS rates), pages 7–8 (pages 9–10 of the PDF). IBORs include bank credit risk and are set before the period; OIS has very limited credit risk and is fixed at maturity. ↩ ↩²

6. Bank of Japan and Financial Services Agency, "円LIBOR移行対応の振り返りと今後の取り組み" (Bank of Japan Review 2022-J-8, May 2022, in Japanese), page 3, Chart 2 "代替金利指標の３つの選択肢" (Three options for alternative benchmarks). It lists the timing of rate fixing as "前決め" (fixed in advance) for TIBOR and "後決め" (fixed in arrears) for TONA compounded in arrears. An English version is available: "Review of JPY LIBOR Transition and Future Initiatives" (Bank of Japan Review 2022-E-4, May 2022), page 3, Chart 2. ↩

7. 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), "4．変動金利の種類" (Types of floating rates), "5．変動金利の期間" (Floating period), "11．日数計算式" (Day count fraction) and "13．金利の支払いに係るカレンダー" (Calendar for interest payments) in the conditions. An English version is available: "Clearing Products", "4．Types of Floating Rates", "5．Floating Period", "11．Day Count Fraction" and "13．Financial Center for Interest Payment". Checked on September 25, 2026. ↩ ↩²

8. JBA TIBOR Administration, "金利指標改革を踏まえた全銀協TIBORの現状および今後の展望" (The current state and outlook of JBA TIBOR in light of benchmark reform, June 2024, in Japanese), "①全銀協TIBORとは" (1. What is JBA TIBOR), page 1 (page 2 of the PDF). ↩ ↩²

9. Fujii et al. (same paper), Section 2.3, page 4 (page 5 of the PDF). In a swap that pays 3-month LIBOR plus a spread quarterly and receives 6-month LIBOR semiannually, the observed spread is often more than 10bp. ↩

10. Fujii et al. (same paper), Section 4, pages 17–18 (pages 18–19 of the PDF). The loss is roughly notional × PV01 at the average duration × the tenor basis (the paper's term is tenor spread), with PV01 the sum of the products of discount factors and period lengths. These correspond to $N$, $A$ and $\Delta b$. ↩ ↩² ↩³ ↩⁴

11. Study Group on Risk-Free Reference Rates, "日本円ＯＩＳ（Overnight Index Swap）─取引の概要と活用事例─" (April 2018, in Japanese), "（OIS/IBOR ベーシス・スワップ）" (OIS/IBOR basis swaps) under "②それ以外の活用事例" (Other examples of use), page 10 (page 12 of the PDF). The document says that its examples of use do not recommend any particular trade (page 7). ↩

12. Study Group on Risk-Free Reference Rates, "日本円ＯＩＳ（Overnight Index Swap）─取引の概要と活用事例─" (April 2018, in Japanese), "（清算適格取引の拡充）" (Expanding eligible trades for clearing), page 12 (page 14 of the PDF). ↩

13. JBA TIBOR Administration, "金利指標改革を踏まえた全銀協TIBORの現状および今後の展望" (The current state and outlook of JBA TIBOR in light of benchmark reform, in Japanese), detailed explanatory material (June 2026), page 4 (page 5 of the PDF). It says "2024年12月末をもってユーロ円TIBORは恒久的な公表停止に至りました" (Euroyen TIBOR was permanently ceased at the end of December 2024; our translation) and "一方、日本円TIBORは、引き続き公表を継続していきます" (Japanese Yen TIBOR, on the other hand, will continue to be published; our translation). ↩

14. Study Group on Risk-Free Reference Rates, "日本円ＯＩＳ（Overnight Index Swap）─取引の概要と活用事例─" (April 2018, in Japanese), "例２：１年未満のOIS 清算取引で、満期時のみに利払いを行うケース" (Example 2: a cleared OIS of less than 1 year with interest paid only at maturity), page 6 (page 8 of the PDF). ↩

15. Bank of Japan, "（日銀レビュー）店頭デリバティブ取引データからみた通貨スワップ市場：感染症拡大の影響とその後の回復を中心に" (Bank of Japan Review: The cross-currency swap market seen through OTC derivatives data, focusing on the impact of the spread of COVID-19 and the subsequent recovery, in Japanese; May 2021), "通貨スワップ取引の概要" (Overview of cross-currency swap trades), page 2. The dollar/yen basis swap exchanges principal and also exchanges the dollar and yen 3-month reference rates, the yen rate plus the cross-currency basis. ↩