When TONA (Tokyo Overnight Average Rate) is positive on every day of the period, which statement correctly describes how the rate for compounding in arrears compares with the simple average for the same period (the daily rates averaged and weighted by the number of days)?
Quiz: Compounding in Arrears
Last reviewed: September 25, 2026
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4 questions to check your understanding of the chapter “Compounding in Arrears”. Choose an answer and select “Check answer” to see the answer and explanation. You can also open “Answer and explanation” to see the answer first.
Question 1
Answer and explanation
Answer: B
Compounding in arrears multiplies 1 + rate × days ÷ 365 for each business day. The product contains interest on interest in addition to what a sum would give, so when rates are positive the compounded rate is higher than the simple average. Interest on interest is larger when rates are higher. As a hypothetical example, over an 88-day period, a rate of 0.5% every day gives a compounded rate of 0.5003%, a gap of 0.0003% from the simple average (0.5%). At 5% every day, the compounded rate is 5.0296%, and the gap widens to 0.0296%.
The choice saying the gap is larger when rates are lower gets the relationship backwards, because interest on interest grows as rates rise. The choice saying the simple average is higher gets the direction backwards. When rates are low, the gap is small and the two can look almost the same, but with positive rates they are never equal.
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Question 2
As a hypothetical example, take an interest period of 4 days from a Thursday to the following Monday. There are no holidays, the business days in the period are Thursday and Friday, and TONA was 0.365% on Thursday and 0.730% on Friday. If the rate is not rounded along the way and the interest is rounded to the nearest yen, what is the interest on a principal of 100 million yen?
Answer and explanation
Answer: A
For each business day, compute 1 + rate × days ÷ 365. Thursday counts 1 day: 1 + 0.365% × 1 ÷ 365 = 1.00001. Friday counts 3 days including Saturday and Sunday: 1 + 0.730% × 3 ÷ 365 = 1.00006. Multiplying gives 1.0000700006, and subtracting 1 gives 0.0000700006, the growth over the 4 days. Multiplying by 365 ÷ 4 gives a period rate of about 0.638755%. The interest is 100 million yen × 0.638755…% × 4 ÷ 365 = 7,000.06 yen, or 7,000 yen after rounding.
3,000 yen uses Friday's rate for only 1 day (1.00001 × 1.00002 − 1 = 0.0000300002). 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 day. The 6,000 yen figure uses 0.5475%, the average of the two rates without weighting by days (100 million yen × 0.5475% × 4 ÷ 365). The 638,755 yen figure multiplies the principal by the annual rate only and leaves out the 4 ÷ 365 for the length of the period.
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Question 3
A lockout is added to a loan that compounds in arrears. Suppose rates fall during the last few days of the period, after the cut-off date (a date a number of business days before the end of the period). Compared with having no lockout, what happens to the interest the borrower pays?
Answer and explanation
Answer: D
A lockout is a convention that repeats the rate for the cut-off date through the last business day of the period. Rate moves during a lockout are not reflected in the rate. Because the rate for the cut-off date, from before the fall, keeps being used, the fall is not reflected and the borrower pays more interest. Conversely, if rates rise at the end of the period, the rise is not reflected and the borrower pays less interest. As a hypothetical example, over a 28-day period in which rates rise from around 0.477% to around 0.727% and the higher rates apply on the last 2 business days, a 3-business-day lockout makes the interest on a principal of 100 million yen 2,732 yen lower. Whether rates go up or down during a lockout, the result can be a gain or a loss for either the lender or the borrower.
The choice saying it does not change confuses rate moves not being reflected in the rate with the amount of interest not changing. The choice saying the interest changes only if rates rise overlooks that a fall is also not reflected, which changes the interest as well. The choice saying it decreases describes the case where rates rise.
Learn more: Lookback, Lockout and Payment Delay
Question 4
What changes when a "payment delay" is added to a loan that compounds in arrears?
Answer and explanation
Answer: C
The rate for compounding in arrears is not set until the rate for the last business day of the period is published. A lookback uses rates from a number of business days earlier to bring forward the date on which the rate is known, and a lockout fixes the rates for the last few days to do the same. A payment delay, by contrast, leaves the rate unchanged. The rate is set on the end date of the period, and the parties use the business days before payment to confirm the amount and pay. For example, in a Japanese yen OIS, the floating rate amount is calculated on the end date of the period after TONA is published, and the net amount is settled 2 business days later.
Because payment is later, investors may not earn income on the money in the meantime. The choice about replacing rates with those from a number of business days earlier describes a lookback, the choice about shifting the observation period describes the observation shift variant of a lookback, and the choice about fixing rates describes a lockout.
Learn more: Lookback, Lockout and Payment Delay