Calculating interest on lump sum to match regular savings account interest?

Question

In the United Kingdom some banks offer a type of savings account called a Regular Saver, with these attributes:

• every month you can deposit a limited amount of money M;
• the interest rate i is calculated daily but paid after 12 months.

Here is an example. This particular offering has a variable interest rate, but for this problem let's assume the interest is going to be fixed.

If I were to stick the whole lump sum I would pay in of $M*12$ in a traditional savings account with a different interest rate of j compounded yearly, how can I calculate the value of j that will earn me the same interest payment of the Regular Saver with interest i?

Answer 1

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.

Answer 2

So following the example you have referred:

Assuming you make monthly deposits of $\$250$ will give you a balance of $\$3,081.25$ at the end of the year.

Using the simple compound interest formula:

(Assuming you will deposit $\$250 \times 12 = \$3000$ in one lump sum)

$$3081.25=3000(1+r)^1$$

where $r$ - annual interest rate

Therefore:

$$ \frac{3081.25}{3000}=1+r $$

$$ r=0.02708 = 2.708 \% $$

So basically, if you deposit $\$3000$ in an account that pays interest every $12M$ at $2.708\%$ will be the exact same as if you had deposited monthly in the Regular Saver you have mentioned in your question.