# 5.5 Calculating the Term of an Annuity

## LEARNING OBJECTIVES

- Calculate the number of payments in an annuity.
- Calculate the term of an annuity.

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 $75 per month into your vacation fund, how long will it take to save up the $1,000 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 N (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 (PV, FV, PMT, I/Y, P/Y and C/Y), paying close attention to the cash flow sign convention for PV, PMT, and FV.
- After all of the known quantities are loaded into the calculator, press CPT and then N to solve for the number of payments.

### NOTE

Because N is the number of payments made during the annuity, N must be a whole number. However, it is unlikely that the calculation of N will result in a whole number. After N is calculated, the value of N must be rounded UP to the next whole number. For example, if N=10.1397 after working out the value of N on the calculator, then this value is rounded up to 11.

**Exercises**

- You make $3,000 quarterly payments into a savings account earning 6.35% compounded quarterly. How long will it take you to accumulate $100,000?

**Click to see Answer**6 years, 9 months

- You make $2,500 contributions at the beginning of every six months into your RRSP. If the RRSP earns interest at 7% effective, how long will it take your RRSP to reach an accumulated value of $175,000?

**Click to see Answer**18 years

- You took a $50,000 loan at 7.2% compounded quarterly. The loan agreement requires you to make monthly payments of $500 until the loan is paid off. How long will it take you to repay the loan?

**Click to see Answer**12 years, 9 months

- Your have $1,000,000 in your RIF. You want to receive $40,000 beginning-of-quarter payments for as along as possible from the RIF. If the RIF earns 9% compounded monthly, how long will it take to exhaust the RIF?

**Click to see Answer**9 years, 3 months

- An investment of $100,000 today will make advance quarterly payments of $4,000. If the annuity can earn 7.3% compounded semi-annually, how long will it take for the annuity to be depleted?

**Click to see Answer**8 years, 3 months

- The neighbourhood grocery store owned by Raoul needs $22,500 to upgrade its fixtures and coolers. If Raoul contributes $3,000 at the start of every quarter into a fund earning 5.4% compounded quarterly, how long will it take him to save up the needed funds for his store’s upgrades?

**Click to see Answer**2 years

- Andre has stopped smoking. If he takes the $80 he saves each month and invests it into a fund earning 6% compounded monthly, how long will it take for him to save $10,000?

**Click to see Answer**8 years, 2 months

- How much longer will a $500,000 investment fund earning 4.9% compounded annually last if beginning-of-month payments are $3,500 instead of $4,000?

**Click to see Answer**3 years, 2 months

- Consider a $150,000 loan with month-end payments of $1,000. How much longer does it take to pay off the loan if the interest rate is 6% compounded monthly instead of 5% compounded monthly?

**Click to see Answer**3 years, 6 months

- You make $250 month-end contributions to your RRSP, which earns 9% compounded annually.
- How much less time will it take to reach $100,000 if you increase your payments by 10%?
- Which alternative requires less principal and by how much? (Assume all payments are equal.)

**Click to see Answer**a. 9 months; b. regular payments require $2250 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 25-year mortgage for $200,000 at a fixed rate of 5% compounded semi-annually.
- How many fewer payments does it take to pay off your mortgage if you increased your monthly payments by 10%?
- How much money is saved by “topping up” the payments? Assume that all payments are equal amounts in your calculations.

**Click to see Answer**a. 48 fewer payments; b. $26,521.44

#### Attribution

“11.5: Number of Annuity Payments” from Business Math: A Step-by-Step Handbook Abridged by Sanja Krajisnik; Carol Leppinen; and Jelena Loncar-Vines is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

“11.5: Number of Annuity Payments” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.