It is now time to introduce *compound interest*. The main difference between compound interest and simple interest is that in compound interest, the amount that is used to apply the interest, changes each period. With simple interest, this amount is the initial investment and never changes.

In compound interest, the interest earned from previous periods is added to the initial investment (the principal) before applying the interest for the current period.

Let’s continue with the example of investing $1000 at 3% interest per annum (that is each year). In compound interest, the interest can be applied (compounded) even within a period, but let’s start out simple and apply the interest at the end of each year.

At the end of the first year, the 3% interest is applied to the initial $1000, so we now have $1030 in the bank just like with simple interest. However, things are different at the end of the second year. In simple interest, the interest rate is applied to the initial $1000, but with compound interest the $30 earned after the first year is added to our initial investment so that the principal has changed to be $1030. So after the second year, instead of applying the 3% to $1000, it is now applied to $1030. The following table shows how the investment is growing for the first 5 years:

n |
P |
r |
I |
A_{n} |

0 | 1000 | 0 | 0 | 1000 |

1 | 1000 | 3% | 30 | 1030 |

2 | 1030 | 3% | 30.90 | 1060.90 |

3 | 1060.90 | 3% | 31.83 | 1092.73 |

4 | 1092.73 | 3% | 32.78 | 1125.51 |

5 | 1125.51 | 3% | 33.77 | 1159.28 |

Notice how the *P* for a current period is the *A*_{n} of the previous period. Compare this table with the one from my last post which showed the investment growth using simple interest:

n |
P |
r |
I |
A_{n} |

0 | 1000 | 0 | 0 | 1000 |

1 | 1000 | 3% | 30 | 1030 |

2 | 1000 | 3% | 30 | 1060 |

3 | 1000 | 3% | 30 | 1090 |

4 | 1000 | 3% | 30 | 1120 |

5 | 1000 | 3% | 30 | 1150 |

Looks like it’s better to invest using compound interest. In our example, we have $9.28 more after 5 years.

So what is the recursive formula for this, that is, how do we calculate the current year’s amount if we know last year’s amount?

Starting with *A*_{0}:

*A*_{1} = *A*_{0} + *A*_{0} × 3/100= $1000 + $30 = $1030

*A*_{2} = *A*_{1} + *A*_{1} × 3/100= $1030 + $30.90 = $1060.90

and so on until

*A*_{5} = *A*_{4} + *A*_{4} × 3/100 = $1125.51 + $33.77 = $1159.28

Again, let’s make this more general. The “3” in the above is the interest rate *r*. So replacing the “3” with *r*, we get the general recursion formula:

*A*_{n+1} = *A*_{n} + *A*_{n} × *r*/100

or if you use the decimal equivalent of *r*:

*A*_{n+1} = *A*_{n} + *A*_{n} × *r*

This is OK for finding the amount next year, but what if you want to find the amount after 10 years? You do not want to apply this recursion 10 times. In my next post, I will develop a more direct formula.