2.5 Calculating the Term of an Annuity
Learning Objectives
 Calculate the number of payments in an annuity
 Calculate the term of an annuity
Formula & Symbol Hub
Symbols Used
 [latex]FV[/latex] = Future value or maturity value
 [latex]PV[/latex] = Present value of principal
 [latex]PMT[/latex] = Annuity payment amount
 [latex]I/Y[/latex] = Nominal interest rate
 [latex]P/Y[/latex] = Number of payments per year or payment frequency
 [latex]C/Y[/latex] = Number of compounds per year or compounding frequency
 [latex]n[/latex] or [latex]N[/latex] = Total number of annuity payments
Formulas Used

Formula 2.1 – Total Number of Payments (Annuity)
[latex]n=P/Y \times t[/latex]

Formula 2.2 – Future Value of Ordinary Annuity
[latex]FV=PMT \times \left[\frac{(1+i_2)^n1}{i_2}\right][/latex]

Formula 2.3 – Future Value of Annuity Due
[latex]FV=PMT \times (1+i_2) \times \left[\frac{(1+i_2)^n1}{i_2}\right][/latex]

Formula 2.4 – Present Value of Ordinary Annuity
[latex]PV=PMT \times \left[\frac{1(1+i_2)^{n}}{i_2}\right][/latex]

Formula 2.5 – Present Value of Annuity Due
[latex]PV=PMT \times (1+i_2) \times \left[\frac{1(1+i_2)^{n}}{i_2}\right][/latex]
Introduction
When saving up for future goals, many people and businesses simply determine how much they can afford to invest each time period and then try to be patient until they meet their savings goal. What they do not know is how long it will take them. For example, if you can put [latex]\$75[/latex] per month into your vacation fund, how long will it take to save up the [latex]\$1,000[/latex] needed for a spring break vacation in Puerto Vallarta? How long it takes to fulfill the goal of your annuity depends on your annuity payment, interest rate, and the amount of money involved.
You must calculate the number of annuity payments, and the corresponding term, in a variety of scenarios such as:
 Savings planning.
 Debt extinguishment.
 Sustaining withdrawals from an investment.
Using a Financial Calculator
Although it is possible to find the number of payments by using the appropriate future value or present value formula, it is much more practical to use a financial calculator. You use the financial calculator in the same way as described previously, but the only difference is that the unknown quantity is [latex]N[/latex] (the number of payments in the annuity). You must still load the other six variables into the calculator and apply the cash flow sign conventions carefully.
Using the TI BAII Plus Calculator to Find the Number of Payments for an Annuity
 Set the calculator to the correct payment setting (END or BGN).
 Enter values for the known variables ([latex]PV[/latex], [latex]FV[/latex], [latex]PMT[/latex], [latex]I/Y[/latex], [latex]P/Y[/latex] and[latex]C/Y[/latex]), paying close attention to the cash flow sign convention for [latex]PV[/latex], [latex]PMT[/latex], and [latex]FV[/latex].
 After all of the known quantities are loaded into the calculator, press [latex]CPT[/latex] and then [latex]N[/latex] to solve for the number of payments.
Things to Watch Out For
Because [latex]N[/latex] is the number of payments made during the annuity, [latex]N[/latex] must be a whole number. However, it is unlikely that the calculation of [latex]N[/latex] will result in a whole number. After [latex]N[/latex] is calculated, the value of [latex]N[/latex] must be rounded UP to the next whole number. For example, if [latex]N=10.1397[/latex] after working out the value of [latex]N[/latex] on the calculator, then this value is rounded up to [latex]11[/latex].
Section 2.5 Exercises
 You make [latex]\$3,000[/latex] quarterly payments into a savings account earning [latex]6.35\%[/latex] compounded quarterly. How long will it take you to accumulate [latex]\$100,000[/latex]?
Solution
[latex]6[/latex] years, [latex]9[/latex] months
 You make [latex]\$2,500[/latex] contributions at the beginning of every six months into your RRSP. If the RRSP earns interest at [latex]7\%[/latex] effective, how long will it take your RRSP to reach an accumulated value of [latex]\$175,000[/latex]?
Solution
[latex]18[/latex] years
 You took a [latex]\$50,000[/latex] loan at [latex]7.2\%[/latex] compounded quarterly. The loan agreement requires you to make monthly payments of [latex]\$500[/latex] until the loan is paid off. How long will it take you to repay the loan?
Solution
[latex]12[/latex] years, [latex]9[/latex] months
 Your have [latex]\$1,000,000[/latex] in your RIF. You want to receive [latex]\$40,000[/latex] beginningofquarter payments for as along as possible from the RIF. If the RIF earns [latex]9\%[/latex] compounded monthly, how long will it take to exhaust the RIF?
Solution
[latex]9[/latex] years, [latex]3[/latex] months
 An investment of [latex]\$100,000[/latex] today will make advance quarterly payments of [latex]\$4,000[/latex]. If the annuity can earn [latex]7.3\%[/latex] compounded semiannually, how long will it take for the annuity to be depleted?
Solution
[latex]8[/latex] years, [latex]3[/latex] months
 The neighbourhood grocery store owned by Raoul needs [latex]\$22,500[/latex] to upgrade its fixtures and coolers. If Raoul contributes [latex]\$3,000[/latex] at the start of every quarter into a fund earning [latex]5.4\%[/latex] compounded quarterly, how long will it take him to save up the needed funds for his store’s upgrades?
Solution
[latex]2[/latex] years
 Andre has stopped smoking. If he takes the [latex]\$80[/latex] he saves each month and invests it into a fund earning [latex]6\%[/latex] compounded monthly, how long will it take for him to save [latex]\$10,000[/latex]?
Solution
[latex]8[/latex] years, [latex]2[/latex] months
 How much longer will a [latex]\$500,000[/latex] investment fund earning [latex]4.9\%[/latex] compounded annually last if beginningofmonth payments are [latex]\$3,500[/latex] instead of [latex]\$4,000[/latex]?
Solution
[latex]3[/latex] years, [latex]2[/latex] months
 Consider a [latex]\$150,000[/latex] loan with monthend payments of [latex]\$1,000[/latex]. How much longer does it take to pay off the loan if the interest rate is [latex]6\%[/latex] compounded monthly instead of [latex]5\%[/latex] compounded monthly?
Solution
[latex]3[/latex] years, [latex]6[/latex] months
 You make [latex]\$250[/latex] monthend contributions to your RRSP, which earns [latex]9\%[/latex] compounded annually.
 How much less time will it take to reach [latex]\$100,000[/latex] if you increase your payments by [latex]10\%[/latex]?
 Which alternative requires less principal and by how much? (Assume all payments are equal.)
Solution
a. [latex]9[/latex] months; b. regular payments require [latex]\$2250[/latex] less principal
 Most financial institutions tout the benefits of “topping up” your mortgage payments—that is, increasing from the required amount to any higher amount. Assume a [latex]25[/latex]year mortgage for [latex]\$200,000[/latex] at a fixed rate of [latex]5\%[/latex] compounded semiannually.
 How many fewer payments does it take to pay off your mortgage if you increased your monthly payments by [latex]10\%[/latex]?
 How much money is saved by “topping up” the payments? Assume that all payments are equal amounts in your calculations.
Solution
a. [latex]48[/latex] fewer payments; b. [latex]\$26,521.44[/latex]
Attribution
“11.5: Number of Annuity Payments” from Business Math: A StepbyStep Handbook Abridged by Sanja Krajisnik; Carol Leppinen; and Jelena LoncarVines is licensed under a Creative Commons AttributionNonCommercialShareAlike 4.0 International License, except where otherwise noted.
“11.5: Number of Annuity Payments” from Business Math: A StepbyStep Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons AttributionNonCommercialShareAlike 4.0 International License unless otherwise noted.