# Interest in Your Bank Account

Suppose you have £100 and you wish to put it into a savings account for a year, and the interest rate is 5% per annum. What does this mean?

Well, it could mean a few things.

First, it could be that the interest rate is applied at the end of the year. So at the end of the year you get:

$\mbox{\textsterling}100 \times 105\% = \mbox{\textsterling}105$

However, it could also mean that there is a 2.5% interest rate applied twice in a year:

$\mbox{\textsterling}100 \times 102.5\% \times 102.5\% = \mbox{\textsterling}105.06$

Earning you a little bit more! Suppose instead that the interest was applied every day:

$\mbox{\textsterling}100 \times \left(100\% + \frac{5\%}{365}\right)^{365} = \mbox{\textsterling}105.13$

In general, if it is updated h times in a year, with an annual interest rate p, then it is equivalent to the interest being calculated once at a rate of:

$\left(1 + \frac{p}{h}\right)^h$

So what if it was updated all the time? It is well known that the above expression tends to $\exp(p)$ as $h \rightarrow \infty$.

The point is just that these things can come up naturally!  This is actually widely used when trading derivatives.