Say you deposit an sum of $M$ every month, twelve times. Assume the interest rate is $i$, in annual terms, but calculated daily. After twelve months, the regular saver account will have earned interest as follows:
- The first deposited $M$ becomes $M(1+i/365)^{365}$.
- The second deposited $M$ (at the beginning of the second month), becomes $M(1+i/365)^{365-m_1}$, where $m_1$ is the number of days of the first month.
- The third deposited $M$ (at the beginning of the third month), becomes $M(1+i/365)^{365-m_1-m_2}$, where $m_2$ is the number of days of the second month.
- ...
- The last deposited $M$ (at the beginning of the $12$th month), becomes $(1+i/365)^{m_{12}}$, where $m_{12}$ is the number of days in the $12$th month.
Summing it all up, the regular saver account will have paid in total:
$$M(1+i/365)^{365}\left(1+\sum_{k=1}^{11}(1+i/365)^{-m_1-m_2-\cdots - m_k}\right)$$
This expression is rather difficult to work with, and it is highly plausible that any bank would prefer the 30/360 convention (that is: every month has $30$ days) for compounding, yielding a somewhat simpler and workable expression:
$$M(1+i/360)^{360}\left(1+\sum_{k=1}^{11}(1+i/360)^{-30k}\right)$$
So, for instance, if you deposit $250$ every month with $i=2\%$, say, you'll get $3032.726$ by the end of the year. So had you deposited $3000$ at the beginning of the year this would reflect an annual interest rate of $1.09\%$, roughly half of what is quoted in the Regular Saver saving account.
So once again we find that banks have their stealthy ways of quoting attractive interest rates while in effect paying much less: this is the idea of the Regualr Saver savings account: you think you are getting $2\%$, but in effect you are getting much less, because those $2\%$ are not paid over the entire year on the entire sum.
