Suppose you have decided to do some savings. You will deposit $700 this year into an account that earns 2% per year and increase the amount deposited each year by 20% in every year that follows. How much will you have in the account after 26 years?

**Solution:**

*Brute Force Method:*

Using an Excel document, you can use a brute force method to quickly calculate the amount in the savings account after 26 years.

Here are the variables for the geometric series:**Note: starting point is at n=0 where n is the time period of focus

Variables | Values |
---|---|

A_{1} (Starting Value) |
700 |

i (Interest %) | .02 |

g (Growth %) | .20 |

N (total time period) | 26 |

Here is the formula to find the amount of cash for a time period n: F = (A_{1})*(1+i)^{N-n}*(1+g)^{n}

So, for this case, the Excel document will have values as such:

Time Period | Cash |
---|---|

0 | 1171.39 |

1 | 1378.11 |

2 | 1621.31 |

3 | 1907.42 |

4 | 2244.02 |

5 | 2640.02 |

6 | 3105.91 |

7 | 3654.01 |

8 | 4298.84 |

9 | 5057.46 |

10 | 5949.95 |

11 | 6999.94 |

12 | 8235.22 |

13 | 9688.49 |

14 | 11398.23 |

15 | 13409.69 |

16 | 15776.10 |

17 | 18560.12 |

18 | 21835.43 |

19 | 25688.75 |

20 | 30222.05 |

21 | 35555.36 |

22 | 41829.83 |

23 | 49211.57 |

24 | 57895.96 |

25 | 68112.90 |

26 | 80132.82 |

SUM |
527580.90 |

The sum $527,580.90 is the total amount in the savings account after 26 years.

*More Elegant Solution:*

This problem is half plug and chug and half spotting that you are looking for future value and not present worth. Let's take it in a few steps.

If you were looking for the present worth from period one to 27 you would have,$P=700(P|A_1, g=20\%, i=2\%, 27)$. The problem is that the questions specifies that you are starting NOW, which means period zero. That means you have to time shift it back to time period zero.

The present worth of the sequence is then, $P=\frac{700(P|A_1, g=20\%, i=2\%, 27)}{(1+.02)^{-1}}$. If you follow the "N plus the exponent" rule on time shifting you now have the present worth of a geometric series that start in time period zero and ends in time period, $27+ (-1)=26$, which is what was described. This would give you how much to deposit now, time period zero, to fund these withdrawals. That last bit was just one of the normal interpretations of present worth.

To convert that into future value you need to time shift again, this time taking all the value that you concentrated in time zero and moving it to time period 22. That is the first time value of money trick you were shown $F=P(1+.02)^{26}=\frac{700(P|A_1, g=20\%, i=2\%, 27) (1+.02)^{26} }{(1+.02)^{-1}}$.