College Algebra Project: Saving for the Future
College Algebra Project
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
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