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Most people know that compound interest can exert powerful effects on the value of money. Stories involving a discovered inheritance generated by a small sum invested a long time ago are more often fiction than reality though! Compound interest is at the heart of nearly all financial calculations and it is vital that you understand exactly how it works.
Compound interest arises when the interest generated by a principal is added to the principal rather than being removed from consideration.
In the case of a loan it corresponds to the interest due being added to the debt - presumably to be paid in full at the end of the loan along with the principal.
In the case of an investment it corresponds to the interest being added to the investment.
In both cases the adding of the interest to the principal results in the interest paid changing at each period - see Figure 2. The only difference between investment and a loan is that in the case of a loan the interest only notionally adds to the amount of money that the borrower actually has. However this is irrelevant from the lender’s point of view because it certainly adds to the size of the debt!
The calculation for compound interest is remarkably simple as long as you have followed and understood the discussion of percentages in Chapter 1.
If the interest is added to the principal this implies that the principal increases by I% in each period. To increase a value by I% you simply multiply by (1+I) and so at the end of the first period the principal, PV, has grown to:
At the end of the second period the principal has grown by another I% and so is given by
and so on.
After n periods the principal has increased to M multiplied by (1+I) n times over.
If you repeatedly multiply by the same value n times this is called calculating a power.
Mathematically this operation is indicated as xn. For example x3 is x*x*x. In the case of spreadsheets calculating a power is indicated either by x^n or x**n.
So after n periods the principal is
For example, an investment of $100 at 1% per month would yield a total of 100*0.01*12, i.e. $12, in one year if the interest were taken each month.
If the interest was reinvested each month the total invested at the end of the year would be 100*(1+0.01)^12 which is $112.68, giving a profit of $12.68. After a single year the difference between simple and compound interest is a mere 68 cents - it hardly seems worth making the effort to do the calculation properly.
However the difference may start out being small but compound interest acts to magnify these small differences over time.
So for example, after 20 years the simple interest investment would have earned a total of $240 but the compound interest investment would have earned $989.25.
The reason for the difference can be seen in Figure 3 which shows a graph of the earnings for the two investments against time. You can see that the compound interest calculation causes the earning due to interest to increase each year and so gradually at first, and then more rapidly, the gap between the two widens.