6. Loans, Investments, & Annuities

Lesson

Most graphics and CAS calculators come with a built-in financial application which can be used to solve problems involving compound interest. These applications simply require you to enter in the known quantities (such as principal, interest, and number of compounding periods per year), and then the compound interest formula is applied or rearranged in the background to calculate the desired unknown quantity.

These financial applications typically use the following notation:

N | \text{total number of payments} |

I\% | \text{interest rate as a percentage per annum} |

PV | \text{the present value, or the principal} |

PMT | \text{the value of any additional regular payment} |

FV | \text{the future value, or the final amount} |

P\text{/}Y \text{ or } PpY | \text{number of payments per year} |

C\text{/}Y \text{ or } CpY | \text{number of compounding periods per year} |

One point of difference between these solvers and the way we have been using the compound interest formula is that if you enter a positive value for PV then the solver will return a negative value for FV. This corresponds to borrowing: when you borrow you have a positive present value (the bank gives you money) but in the future you owe money to the bank, which is what the negative number represents. Conversely, if you enter a negative number for PV then the solver returns a positive FV - this corresponds to investing.

Borrowing money - the bank is giving you money - use positive value for PV

Investing money - you are giving your money to the bank - use negative value for PV

Another difference is that the solvers are set up to deal with regular payments in addition to the accumulation of interest. If there is no payment then we set PMT to 0. We also set P \text{/}Y=C \text{/}Y (the number of compounds per year) and then N (number of payments of zero) is equal to the total number of compounding periods.

Nadia borrows \$12\,000 at an interest rate of 3.5\% p.a. compounded weekly. If she makes no repayments, find the amount of interest that is owed after 3 years in dollars. Assume there are 52 weeks in a year. Round your answer to the nearest cent.

Worked Solution

Neil invests \$900 in a term deposit with a rate of 2.3\% p.a. compounded daily. How many years will it take for the investment to at least double in value? Assume there are 365 days in a year.

Worked Solution

Idea summary

Financial solver input values:

N | \text{total number of payments} |

I\% | \text{interest rate as a percentage per annum} |

PV | \text{the present value, or the principal} |

PMT | \text{the value of any additional regular payment} |

FV | \text{the future value, or the final amount} |

P\text{/}Y ext{ or } PpY | \text{number of payments per year} |

C\text{/}Y ext{ or } CpY | \text{number of compounding periods per year} |

with the aid of a calculator or computer-based financial software, solve problems involving compound interest loans or investments; for example, determining the future value of a loan, the number of compounding periods for an investment to exceed a given value, the interest rate needed for an investment to exceed a given value