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I have a question regarding the business day convention. Suppose I have a bond that matures on the 17th of September 2023 and pays an annual coupon of $1%$. It has a $30/360$ day-count convention and a forward business day adjustment.

Now the 17th of September 2021 is a Friday. The 17th of September 2022 is a Saturday and the 17th of September 2023 is a Sunday. In the $30/360$ day count convention I get

days from 17th of September 2021 to 17th of September 2022 (daycount conenvtion): $360$

days from 17th of September 2022 to 17th of September 2023 (daycount convention): $360$

But since 17th of September 2022 is a Saturday, the forward adjustment kicks in and coupon payment is exchanged on the 19th of September. Now my question:

Since payment date is adjusted, does the same apply to the accrual period? That means on the 19th of September the bond holder receives $N\cdot0.01\frac{362}{360}$ for notional $N$? Moreover, is the next accrual period then also adjusted, i.e. from the 19th of September 2022 to the 18th September 2023?

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There is no universal answer. The coupon may or may not be daycounted. You should figure this out from the prospectus, rather than guess.

The coupon is less likely to be daycounted if the top of the period fraction is a number ("30" rather than Actual), if the coupon is "fixed", or if the product is "bond".

But there are many counterexamples to these heuristics. U.S. Treasies that pay fixed coupon use ACT/ACT daycount to calculate accrued coupon during the coupon period, but full coupons are not daycounted. Mexican treasuries (MBONO Govt and MUDI Govt on Bloomberg) pay fixed coupon, but are daycounted. Etc ad infinitum.

The coupon is very likely to be daycounted if the coupon is "floating" or if the product is "swap" or "loan" (or some loan-linked instrument, like a loan participation note (LPN)).

Some bonds (usually perpetuals, but not all perps do this, and some non-perps do this) have a "stacking" feature - if the coupon date is bumped due to a non-trading day, e.g. from 17th to 18th, then the accrual is adjusted, but the following coupons roll on the 18th's until they get bumped again. You can't guess this - you must read the prospectus.

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