0.8 Mathematics of finance  (Page 4/9)

Suppose we put $1 in an account that pays 100% interest. If the interest is compounded once a year, the total amount after one year will be $\$1\left(1+1\right)=\$2$ .

If the interest is compounded semiannually, in one year we will have $\$1{\left(1+1/2\right)}^2=\$2\text{.}\text{25}$

If the interest is compounded quarterly, in one year we will have $\$1{\left(1+1/4\right)}^4=\$2\text{.}\text{44}$ , etc.

We show the results as follows:

We have noticed that the $1 we invested does not grow without bound. It starts to stabilize to an irrational number 2.718281828... given the name " $e$ " after the great mathematician Euler.

In mathematics, we say that as $n$ becomes infinitely large the expression ${\left(1+\frac{1}{n}\right)}^n$ equals $e$ .

Therefore, it is natural that the number $e$ play a part in continuous compounding. It can be shown that as $n$ becomes infinitely large the expression ${\left(1+\frac{r}{n}\right)}^{\text{nt}}=e^{\text{rt}}$ .

Therefore, it follows that if we invest $\$P$ at an interest rate $r$ per year, compounded continuously, after $t$ years the final amount will be given by $A=P\cdot e^{\text{rt}}$ .

Next we learn a common-sense rule to be able to readily estimate answers to some finance as well as real-life problems. We consider the following problem.

By doing a few similar calculations we can construct a table like the one below.

The pattern in the table introduces us to the law of 70.

The Law of 70:

The number of years required to double money = 70 ÷ interest rate

It is a good idea to familiarize yourself with the law of 70, as it can help you to estimate many problems mentally.

We summarize the concepts learned in this chapter in the following table: