In last month’s issue, we looked at the concept of compound interest and saw how compounding accrued interest resulted in higher amounts of interest. We looked at the difference over a year between interest accruing annually and monthly.

It is, of course, possible that interest could accrue more frequently than monthly and, in such a scenario, still higher amounts of interest would arise.

#### Using the formula from last month:

\( Annual \: interest = Principal \: \times \: 1 \: + \frac{i}{n}n\)Where i

Increasing the number of interest payments a year will, as we saw last month, increase the annual proceeds from the investment. In most cases the maximum would occur if interest were calculated on a daily basis. However interest could, theoretically, be paid more frequently than this – by the hour, minute or even second. The logical conclusion is that there could be an infinite number of interest payments.

Derivatives are often valued on this basis using the concept of an infinite numbers of interest payments.

Where there are an infinite number of interest payments in a single year, the annual proceeds of the investment are from our formula:-

\( Principal \times 1 \; + \: \frac{i}{\infty}\infty\)where

Mathematically, this formula can be solved using natural logarithms. But before doing this we can simplify the calculation since the proceeds of investment are:-

Principal x 1 + r

Where

”

r = e^{i}-1

where i

#### What is e?

The number e was first studied by the Swiss mathematician Leonhard Euler in the 1720s, although its existence was more or less implied in the work of John Napier, the discoverer of logarithms, in 1614. e is an irrational number, so its decimal expansion never terminates, nor is it eventually periodic. Thus no matter how many digits in the expansion of e you know, the only way to predict the next one is to compute “e”. It is used extensively in calculus and is the basis of logarithms. In some respects it is like p, it works for us. One maths teacher described e and p as the Burger King and McDonalds of the mathematics world.

Now we can simplify our formula since the total amount that accrues interest and principal is:-

Principal x1 + r = Principal x1 + e^{i }-1= Principal xe^{i}

#### Looking more at e

Using our formula we can begin to see how e emerges if we take different interest rates. This is most simply demonstrated by using an example of an investment with an annual interest rate of 100%. From the equation above, the proceeds of the investment if interest is continuously compounded is Principal x e^{i}. When the annual rate of interest is 100, the proceeds of the investment would be calculated as Principal x e^{i} = Principal x e.

We know that the annual proceeds from an investment of 100 will be greater, the more frequent the interest payments accrue. Should interest accrue only once a year, the annual proceeds will be 100 (principal) + 100 (interest) = 200 (twice the initial investment).

If interest accrues twice a year, the annual proceeds will be 100 (principal) + 50 (first interest payment) + 75 (second interest payment – 50 earned on the principal and 25 earned on the reinvested interest) = 225 (2.25 times the initial investment).

The more times interest accrues and is compounded, the higher the annual proceeds. For example, if interest accrues three times a year (33.333% each time), annual proceeds will equal 2.37 times the principal. For four payments of 25%, annual proceeds will be 2.44 times the principal. Five payments will increase the investment by 2.49 times.

This pattern shows that the investor is always better off, the more often interest accrues. This improvement is, however, successively less. The best case scenario is when an infinite number of interest payments are made, in other words when interest is continuously compounded.

This case is the limit where an initial investment would multiply by 2.718281828… which is ‘e’! Although we have shown the case where the annual interest rate is 100%, this principle can be applied to any interest rate. This is done by using e as the base for what is called an exponential calculation such that the annual proceeds = Principal x e^{i}.

#### Illustration of difference

To show the effect of the different methods of calculating interest, we can compare investments of 100 over a year, assuming an interest rate of 6% per annum:

Where interest is paid annually, the annual proceeds = \(100\:\times\:1\:+\:\frac{0.06}{1}\:1\:=\:106\) Where interest is paid monthly, the annual proceeds = \(100\:\times\:1\:+\:\frac{0.06}{12}\:12\:=\:106.17\) Where interest is continuously compounded, the annual proceeds= 100 x e ^{0.06 }= 106.18

The difference for continuously compounded interest is not as great as might be expected.

#### Calculation of continuously compounding interest

Because of the use of exponentials, this calculation cannot be performed using a conventional calculator (without an exponential function).

##### Scientific calculator

Using the scientific calculator on a Windows computer (Start, Programs, Accessories, Calculator, View, Scientific), you would need to enter the following:

100 * 0.06 check the box marked Inv ln = this will give the result 106.1837 …

##### HP12C

Using an HP12C (or similar calculator using Reverse Polish Notation):

0.06 e ^{x}100 x this will give the result 106.1837 …

### Summary: Continuous compounding produces higher interest

The greater the frequency that interest is earned, and therefore reinvested, the greater the final proceeds from any investment or interest charge on any loan. Continuous compounding therefore provides the highest possible earning potential for any given nominal rate of interest.