Saving for the Future

In this project you will investigate compound interest, specifically how it applies to the typical retirement plan. For instance, many retirement plans deduct a set amount out of an employee’s paycheck. Thus, each year you would invest an additional amount on top of all previous investments including all previously earned interest. If you invest P dollars every year for t years in an account with an interest rate of r (expressed as a decimal) compounded n times per year, then you will have accumulated C dollars as a function of time, given by the following formula.

Compound Interest Formula, with Annual Investments:

FG1 + r IJ LM1 − FG1 + r IJ P H n K MN H n K C (t ) = F rI 1 − G1 + J H nK n n

nt

OP PQ

I will derive this formula to give you a broader understanding of where it came from and how it is based upon the single deposit compound interest formula.

If you invest P dollars every year for t years at an interest rate r, expressed as a decimal, compounded n times per year, then you will have accumulated the cumulative amount of C dollars given by the formula derived bellow: Each annual investment would grow according to the compound interest formula:

F rI A(t ) = PG 1 + J H nK

nt

Here A(t) is the amount that each deposit would be worth at the end of t years. Thus the first deposit of P dollars would draw interest for the full t years, the second deposit would only draw interest for t-1 years, the third deposit would only draw interest for t-2 years… and the last deposit would only draw interest for a single year. Thus we need to add up each deposit and their respectively gained interest values, resulting in the following:

F rI C (t ) = P G 1 + J H nK

nt

FG r IJ LMFG1 + r IJ + FG1 + r IJ + FG1 + r IJ +K+FG1 + r IJ b g + FG1 + r IJ b g OP H n K MNH n K H n K H n K H n K H n K PQ F r I LF r I + FG1 + r IJ + FG1 + r IJ +K+FG1 + r IJ + 1OP, note: FG1 + r IJ = PG 1 + J MG 1 + J H n K