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For a bond that pays interest semi-annually, is a must for a coupon payment date to fall on exactly the same day? For instance, if the bond matures in June 30, 2025, the coupon payment dates shall be on the 30th of June and December till maturity? Or it can be any other days close to these dates?

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No, there are some bonds that can vary the coupon dates slightly. For example, they can be scheduled to be paid on the last day of the month (i.e. Dec 31st) or the last business day, etc. So you need to look at the specific terms of the bond, which should outline the exact payment schedule.

From a rough valuation perspective, though, one or two days generally doesn't make a significant difference in price.

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