Compounding in Arrears

When an OIS settles or a loan that compounds in arrears pays interest, the amounts are calculated at the end of the period from the daily TONA, following the contract terms. Knowing how the daily rates are compounded and how days are counted helps you check whether a discrepancy in the calculated amounts comes from the rates, the day counts, or the holidays.

Period rate built by compounding daily overnight rates

The rate for compounding in arrears is the annualized growth you would get by investing funds overnight on the first day of the interest period and reinvesting them every business day together with the interest received. Because the overnight rate changes every day, the rate for the period is not set until the rate for the last business day of the period is known. That is why it is called "in arrears." A method that sets the rate at the start of the period is called "in advance"¹. A Bank of Japan and Financial Services Agency paper covers Japanese yen benchmark rates. It states that the rate for TONA (Tokyo Overnight Average Rate) compounded in arrears is fixed just before the interest payment date. By contrast, the term rates TORF (Tokyo Term Risk Free Rate) and TIBOR (Tokyo Interbank Offered Rate) are fixed in advance². TONA, TORF, and TIBOR were identified as candidates to replace yen LIBOR (London Interbank Offered Rate).

A Japanese yen OIS exchanges a fixed rate for a floating rate calculated by compounding TONA over the period. The floating amount is fixed at the end of the period, when the TONA values during the period are compounded³. For loans, in December 2020 the Cross-Industry Committee on Japanese Yen Interest Rate Benchmarks published its conventions for calculating interest on loans that use TONA compounded in arrears⁴. The definition and calculation of TONA are covered in TONA.

How to calculate the compounded rate

The rate compounded in arrears is calculated in three steps. First, for each business day in the period, multiply that day's TONA by the number of days the rate applies to, divide by 365, and add 1. Next, multiply these values together over all the business days. Finally, subtract 1 from the product to get the growth over the period, and multiply by 365 ÷ the number of days in the period to annualize it. As a formula⁵:

$$R = \left( \prod_{i=1}^{d_b} \left( 1 + \frac{r_i \times n_i}{365} \right) - 1 \right) \times \frac{365}{D}$$

$R$ is the rate for the period (annualized), and $d_b$ is the number of business days in the period. Here, $r_i$ is TONA for the $i$th business day of the period, and $n_i$ is the number of calendar days that TONA applies to. Finally, $D$ is the number of calendar days in the period. The interest is the principal multiplied by $R$ and by $D \div 365$.

$$\text{Interest} = \text{Principal} \times R \times \frac{D}{365}$$

Under the Committee's conventions, the period rate $R$ is rounded to five decimal places (expressed as a percentage), and the interest amount is rounded to 1-yen units⁶. A spread added to the rate in a loan is not included in the compounding. It is added to the compounded rate afterward⁷.

Deep diveCalculating the compounded rate with the SOFR Index

The Federal Reserve Bank of New York (New York Fed), which publishes the US dollar overnight rate SOFR (Secured Overnight Financing Rate), also publishes an index that compounds SOFR every business day (the SOFR Index). If $I_x$ is the index value on the first day of the period, $I_y$ is the value on the end date of the period, and $d_c$ is the number of calendar days in the period, the rate compounded in arrears for the period is⁸:

$$R = \left( \frac{I_y}{I_x} - 1 \right) \times \frac{360}{d_c}$$

The ratio of the index values corresponds to the product in the formula above, with 365 replaced by 360. However, because the index is published rounded to eight decimal places, the New York Fed explains that a rate calculated from it may differ at the fifth decimal place from published values such as the 30-day SOFR Average for the same period⁸.

Counting days and handling non-business days

The number of days in the period, $D$, includes the first day and excludes the last day. The Committee's paper calls this counting one end only, with the last day excluded (our translation). The day count convention used to annualize the rate is ACT/365 (Fixed), with a 365-day year to match TONA¹⁰. The SOFR Averages and Index, on the other hand, divide the actual number of days by 360, following US dollar money market convention¹¹. Non-business days are not compounded one by one. Instead, the rate for the preceding business day is applied for the number of calendar days until the next business day¹². So, unless the following Monday is a holiday, Friday's TONA counts for 3 days, including Saturday and Sunday, and you multiply by 1 + rate × 3 ÷ 365¹³. Business days are determined by holidays in Tokyo, where the floating rate is calculated¹³. On the business day convention for dates that fall on a non-business day, the loan subgroup of the Committee reported in its consultation that an overwhelming majority supported adopting Modified Following¹⁰.

Worked example over one week

Take a week with no national holidays. The period runs for 7 days, from Monday, March 23, 2026, to Monday, March 30, 2026, and has 5 business days, from the 23rd to Friday the 27th. Consider a hypothetical example in which the rates are the sample overnight rates in the simulator below. The sample values are not actual TONA. The figure below shows how the dates map to the number of days each rate is used.

Figure description

The figure lines up the dates from March 23 to March 30, 2026, from oldest to newest, with the number of days each date's rate is used. For the 4 business days from the 23rd to the 26th, each day's rate is used for 1 day. The rate for Friday the 27th is used for 3 days, including Saturday and Sunday. The 30th is the end date of the period and is not counted as a day. The days add up to 7, the number of calendar days in the period. The spacing in the figure does not represent lengths of time, and the filled circles mark the first day and the end date of the period.

Subtracting 1 from the final factor gives 0.0001395414, the growth over the 7 days. Multiplying this by 365 ÷ 7 gives a period rate of 0.7276%. The interest on a principal of 100 million yen is 100 million yen × 0.7276% × 7 ÷ 365, or 13,954 yen¹⁴.

How the rate varies with the period and rate moves

The period rate compounds each day's rate weighted by the number of days, so if the rate rises during the period, the period rate falls between the rates before and after the rise. For the 88-day period from February 2 to May 1, 2026, the same sample rates (around 0.477% through March 18 and around 0.727% from March 19) give a period rate of 0.5997%. The interest on a principal of 100 million yen is 100 million yen × 0.5997% × 88 ÷ 365, or 144,587 yen¹⁴. The higher rate applies for about half of the period, so the period rate falls roughly midway between the two rates. The sample treats national holidays (February 11, February 23, March 20 and April 29) as business days and uses rates for them, so the result differs from a calculation based on actual Tokyo business days.

Difference from the simple average

Averaging the rates over the period weighted by days, without compounding, gives the simple average. As a formula:

$$\text{Simple average} = \frac{1}{D} \sum_{i=1}^{d_b} r_i \times n_i$$

The Committee's paper does not preclude the use of simple interest in loans where practical needs call for it¹⁵.

When rates are positive, compounding in arrears gives a higher rate than the simple average. This is because multiplying together the values of 1 + rate × days ÷ 365 adds the interest earned on interest to the sum of the daily interest. In the 88-day example, the compounded rate is 0.5997% against a simple average of 0.5993%, and the difference in interest on 100 million yen is 102 yen. For the same period, the gap widens as the rate level rises. Over the same 88 days, a rate of 0.5% every day gives a compounded rate of 0.5003%, while a rate of 5% every day gives 5.0296%, about 0.03% above the simple average of 5%¹⁴.

The rate is not known until the end of the period

With compounding in arrears, the interest amount is not known until the end of the period. The Bank of Japan publishes the final (confirmed) TONA for a business day at around 10:00 a.m. on the next business day¹⁶, so TONA for the last business day of the period becomes known only on the end date of the period. In the Japanese yen OIS example in the April 2018 paper of the Study Group on Risk-Free Reference Rates, the floating amount is calculated on the end date of the period after TONA is published, and the net amount is settled 2 business days later¹⁷.

For loans and bonds that pay interest on the end date of the period, this leaves no time to confirm the interest amount and prepare the payment. Conventions such as lookback, which uses the rates from a set number of business days earlier, address this problem¹⁸. How lookback, lockout and payment delay work, and how they affect the rate, is covered in Lookback, Lockout and Payment Delay.

Notes

1. Cross-Industry Committee on Japanese Yen Interest Rate Benchmarks, "日本円金利指標の適切な選択と利用等に関する市中協議" (Public Consultation on the Appropriate Choice and Usage of Japanese Yen Interest Rate Benchmarks, in Japanese), July 2019, page 10, "選択肢（１）O/N RFR 複利（前決め）" (Option 1: O/N RFR compounded, fixed in advance) and "選択肢（２）O/N RFR 複利（後決め）" (Option 2: O/N RFR compounded, fixed in arrears). An English version is available: "Public Consultation on the Appropriate Choice and Usage of Japanese Yen Interest Rate Benchmarks". ↩

2. Bank of Japan Financial Markets Department and Financial System and Bank Examination Department, and Financial Services Agency Strategy Development and Management Bureau and Supervision Bureau, "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), the row "金利決定のタイミング" (Timing for fixing the interest rate). An English version is available: "Review of JPY LIBOR Transition and Future Initiatives" (Bank of Japan Review 2022-E-4, May 2022), Chart 2, page 3. ↩

3. 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), page 8, "（変動金利の受払額が金利計算期間の満期日に確定）" (The floating amount is fixed on the maturity date of the calculation period). ↩

4. Cross-Industry Committee on Japanese Yen Interest Rate Benchmarks, "貸出における TONA（後決め）のコンベンション（利息計算方式）について" (Conventions for TONA compounded in arrears in loans (interest calculation methods), in Japanese), main text, page 1. The month of publication is from Chart 3 on page 3 of the Bank of Japan's "円LIBOR移行対応の振り返りと今後の取り組み" (May 2022, in Japanese). ↩

5. Cross-Industry Committee on Japanese Yen Interest Rate Benchmarks, "貸出における TONA（後決め）のコンベンション（利息計算方式）について" (in Japanese), attachment, page 1, "１．記号・数式等" (1. Symbols and formulas). The document writes $R$ as $ACR_{db}$, $r_i$ as $TONA_i$, $d_b$ as $db$, and $D$ as $tn_{db}$. ↩

6. The same document, "端数処理" (Rounding) in the main text, page 2, and "【ステップ２】利息額（Interest Amount）の計算" (Step 2: calculating the interest amount) in the attachment, page 1. ↩

7. The same document, main text, page 3, "スプレッドの取扱い" (Treatment of spreads). ↩

8. Federal Reserve Bank of New York, "Additional Information about Reference Rates Administered by the New York Fed," "Calculation Methodology for the SOFR Averages and Index." ↩ ↩²

9. Cross-Industry Committee on Japanese Yen Interest Rate Benchmarks, "貸出における TONA（後決め）のコンベンション（利息計算方式）について" (in Japanese), "利息計算手法" (Interest calculation methods) in the main text, page 2, and "２．各方式の計算式" (2. Formulas for each method) in the attachment, pages 1–2. ↩ ↩²

10. The same document, main text, page 2, "日数計算" (Day count), "計算期間の初日と最終日の取扱い" (Treatment of the first and last days of the calculation period) and "営業日" (Business days). ↩ ↩²

11. Federal Reserve Bank of New York, "Additional Information about Reference Rates Administered by the New York Fed," the second paragraph of "Calculation Methodology for the SOFR Averages and Index." ↩

12. Cross-Industry Committee on Japanese Yen Interest Rate Benchmarks, "貸出における TONA（後決め）のコンベンション（利息計算方式）について" (in Japanese), main text, page 2, "休日処理" (Treatment of holidays). ↩

13. 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), page 5, "基本的な取引方式" (Basic trading terms). ↩ ↩²

14. The worked examples on this page and the simulator calculate interest without rounding the rate along the way, and show interest rounded to the nearest yen. If the rate is rounded to five decimal places (as a percentage), as in the Committee's conventions, the interest in the 88-day example is 144,587.6 yen. ↩ ↩² ↩³

15. Cross-Industry Committee on Japanese Yen Interest Rate Benchmarks, "貸出における TONA（後決め）のコンベンション（利息計算方式）について" (in Japanese), main text, page 2, "単利" (Simple interest). ↩

16. Bank of Japan Financial Markets Department, "「コール市場関連統計」の解説" (Explanation of call market statistics, in Japanese), the table in "３．公表方法" (Publication method), page 2. ↩

17. 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), page 5, "基本的な取引方式" (Basic trading terms), and page 6, "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 one year with interest paid only at maturity). ↩

18. Cross-Industry Committee on Japanese Yen Interest Rate Benchmarks, "貸出における TONA（後決め）のコンベンション（利息計算方式）について" (in Japanese), main text, page 1, "Lookback 期間（調整日数）" (Lookback period (adjustment days)). ↩