Many people have savings accounts in banks and receive interest on a per-day basis. Meaning, the interest accrued is calculated according to the number of days the money is the account, such that, in effect, the individual makes money for every day he has money in the account.
As far as the laws of Ri’bit (interest) are concerned, this arrangement is certainly permissible, because Halacha allows taking interest from gentiles. A more complicated question, however, arises regarding the interest paid by the bank for Shabbat. The prohibition of "Sechar Shabbat" forbids earning money on Shabbat. Seemingly, then, one should be required to return the money earned from his savings on Shabbat, in light of the prohibition against earning profits on Shabbat.
Rav Moshe Feinstein (Russia-New York, 1895-1986) addresses this issue in his work Iggerot Moshe, and he rules that generally, it is permissible to earn per-day interest on savings, even on Shabbat. He explains that when the banks calculate the number of days for the purposes of paying interest, they define a "day" as beginning and ending at 12 midnight. Thus, they pay Friday’s interest in exchange for having the money from 12 midnight Thursday night until 12 midnight Friday night, and Saturday’s interest is paid for the period between 12 midnight Friday night and 12 midnight Saturday night. Of course, this definition of a "day" does not correspond to the Halachic definition, according to which the day begins and ends at sundown. Therefore, one never earns interest exclusively on Shabbat. The interest paid for Friday includes weekday hours and some Shabbat hours, as does the Saturday interest. Accordingly, Rav Moshe writes, this is a situation known in Halacha as "Habla’a," where Shabbat earnings are blended together with weekday earnings. Halacha allows us to consider the Shabbat earnings as "swallowed" by the weekday earnings, and thus one may receive a payment that includes earnings made on Shabbat and earnings made on the weekday. Hence, one may collect interest for the time on Shabbat when the money was in the bank.
However, Rav Moshe adds, this rationale yields an important exception, namely, a case where Yom Tob falls on Friday or Sunday. In such a case, there is an entire day (as defined by the bank) during which one may not earn profits. If Yom Tob falls on Friday, then the entire day of Friday – from midnight Thursday night until midnight Friday night – is a time when earning money is forbidden. And if Yom Tov is on Sunday, then the entire day of Saturday – from midnight Friday night through midnight on Saturday night – is a time when profits are forbidden. Therefore, in such cases, one must return to the bank the interest he earned for the period of Shabbat and Yom Tob.
Summary: It is permissible to collect interest from a gentile-owned bank, and even if the interest is accrued daily, one may collect interest for Saturday. However, if Yom Tov falls on Friday, then one must return the interest earned on Friday, and if Yom Tov falls on Sunday, then he must return the interest he earned on Saturday.
