# 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

• $FV$ = Future value or maturity value
• $PV$ = Present value of principal
• $PMT$ = Annuity payment amount
• $I/Y$ = Nominal interest rate
• $P/Y$ = Number of payments per year or payment frequency
• $C/Y$ = Number of compounds per year or compounding frequency
• $n$ or $N$ = Total number of annuity payments

#### Formulas Used

• ##### Formula 2.1 – Total Number of Payments (Annuity)

$n=P/Y \times t$

• ##### Formula 2.2 – Future Value of Ordinary Annuity

$FV=PMT \times \left[\frac{(1+i_2)^n-1}{i_2}\right]$

• ##### Formula 2.3 – Future Value of Annuity Due

$FV=PMT \times (1+i_2) \times \left[\frac{(1+i_2)^n-1}{i_2}\right]$

• ##### Formula 2.4 – Present Value of Ordinary Annuity

$PV=PMT \times \left[\frac{1-(1+i_2)^{-n}}{i_2}\right]$

• ##### Formula 2.5 – Present Value of Annuity Due

$PV=PMT \times (1+i_2) \times \left[\frac{1-(1+i_2)^{-n}}{i_2}\right]$

## 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 $\75$ 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.

### Things to Watch Out For

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 $11$.

## Finding the Term

The value of $N$ is the number of payment periods for the annuity, after the value obtained from the calculator is rounded up to the next whole number. For example, $N=8.3174$ means that there are $9$ payments in the annuity. $N$ does not represent the time period or term. Because, $N=P/Y \times t$, where $t$ is the time in years, the number of years can be found by

$\displaystyle{\mbox{Number of Years}=\frac{\mbox{rounded up value of }N}{P/Y}}$

The whole number portion of the number of years (the part on the left-side of the decimal) represents the number of years. As needed, take the decimal number portion (the part on the right-side of the decimal point) and multiply it by $12$ to convert it to months.

### Key Takeaways

1. To determine the term of the annuity, the value of $N$ must be calculated first.
2. In annuity calculations, the value of $N$ is rounded UP to the next whole number before converting to years and months.
3. Because the value of $N$ is rounded up to the next whole number, the last payment in the annuity will typically be smaller than the other payments. In a later section, you will learn how to calculate the size of the last payment.

### Example 2.5.1

Samia has $\500,000$ accumulated in her retirement savings when she decides to retire at age $60$. If she wants to receive beginning-of-month payments of $\3,000$ and her retirement annuity can earn $5.2\%$ compounded monthly, how old is Samia when the fund is depleted?

Solution

The timeline for the retirement annuity appears below.

Step 1: Calculate the value of $N$.

 PMT Setting BGN $N$ ? $PV$ $-500,000$ $FV$ $0$ $PMT$ $3,000$ $I/Y$ $5.2$ $P/Y$ $12$ $C/Y$ $12$

$N=293.6601...$

Rounding up, Samia receives $294$ monthly payments.

Step 2: Convert (rounded UP) $N$ to years and months.

$\begin{eqnarray*} \mbox{Number of Years} & = & \frac{\mbox{rounded up }N}{P/Y} \\ & = & \frac{294}{12} \\ & = & 24.5 \\ & \rightarrow & 24 \mbox{ years} \\ \\ \mbox{Number of Months} & = & 0.5 \times 12 \\ & = & 6 \end{eqnarray*}$

Step 3: Write as a statement.

Samia can receive payments for $24$ years and $6$ months. If Samia is currently $60$ years old and the annuity endures for $24$ years and $6$ months, then she will be $84.5$ years old when the annuity is depleted.

### Example 2.5.2

Brendan is purchasing a brand new Mazda MX$-5$ GT. Including all options, accessories, and fees, the total amount he needs to finance is $\47,604.41$ at the dealer’s special interest financing of $2.4\%$ compounded monthly. If he makes payments of $\1,000$ at the end of every month, how long will it take to pay off his car loan?

Solution

The timeline for the car payments appears below.

Step 1: Calculate the value of $N$.

 PMT Setting END $N$ ? $PV$ $47,604.41$ $FV$ $0$ $PMT$ $-1,000$ $I/Y$ $2.4$ $P/Y$ $12$ $C/Y$ $12$

$N=50.0755...$

Rounding up, Brendan makes $51$ monthly payments.

Step 2: Convert (rounded UP) $N$ to years and months.

$\begin{eqnarray*} \mbox{Number of Years} & = & \frac{\mbox{rounded up }N}{P/Y} \\ & = & \frac{51}{12} \\ & = & 4.25 \\ & \rightarrow & 4 \mbox{ years} \\ \\ \mbox{Number of Months} & = & 0.25 \times 12 \\ & = & 3 \end{eqnarray*}$

Step 3: Write as a statement.

To own his vehicle, Brendan will make payments for $4$ years and $3$ months.

### Example 2.5.3

Trevor wants to save up $\1,000,000$. He will contribute $\2,500$ every six months to an investment earning $5.2\%$ compounded quarterly. How long will it take for Trevor to reach his goal?

Solution

Step 1: Calculate the value of $N$.

 PMT Setting END $N$ ? $PV$ $0$ $FV$ $1,000,000$ $PMT$ $-2,500$ $I/Y$ $5.2$ $P/Y$ $2$ $C/Y$ $4$

$N=94.436...$

Rounding up, Trevor will need to make $95$ payments.

Step 2: Convert (rounded UP) $N$ to years and months.

$\begin{eqnarray*} \mbox{Number of Years} & = & \frac{\mbox{rounded up }N}{P/Y} \\ & = & \frac{95}{2} \\ & = & 47.5 \\ & \rightarrow & 47 \mbox{ years} \\ \\ \mbox{Number of Months} & = & 0.5 \times 12 \\ & = & 6 \end{eqnarray*}$

Step 3: Write as a statement.

Trevor will need to make the semi-annual payments for $47$ years and $6$ months to reach his goal.

### Try It

1) Amarjit wants to save up for a down payment on his first home. A typical starter home in his area sells for $\250,000$ and the bank requires a $10\%$ down payment. If he starts making $\300$ month-end contributions to an investment earning $4.75\%$ compounded monthly, how long will it take for Amarjit to have the necessary down payment?

Solution
 PMT Setting END $N$ ? $PV$ $0$ $FV$ $25,000$ $PMT$ $-300$ $I/Y$ $4.75$ $P/Y$ $12$ $C/Y$ $12$

$N=72.1612...\rightarrow 73 \mbox{ payments}$

$\begin{eqnarray*} \mbox{Number of Years} & = & \frac{\mbox{rounded up }N}{P/Y} \\ & = & \frac{73}{12} \\ & = & 6.0833... \\ & \rightarrow & 6 \mbox{ years} \\ \\ \mbox{Number of Months} & = & 0.833... \times 12 \\ & = & 1 \end{eqnarray*}$

Final Statement: It will take $6$ years and $1$ month.

### Try It

2) Hi-Tec Electronics is selling a $52$” LG HDTV during a special “no sales tax” event for $\1,995$ with end of month payments of $\100$ including interest at $15\%$ compounded semi-annually. How long will it take a consumer to pay off her new television?

Solution
 PMT Setting END $N$ ? $PV$ $1,995$ $FV$ $0$ $PMT$ $-100$ $I/Y$ $15$ $P/Y$ $12$ $C/Y$ $2$

$N=22.9783...\rightarrow 23 \mbox{ payments}$

$\begin{eqnarray*} \mbox{Number of Years} & = & \frac{\mbox{rounded up }N}{P/Y} \\ & = & \frac{23}{12} \\ & = & 1.9166... \\ & \rightarrow & 1 \mbox{ years} \\ \\ \mbox{Number of Months} & = & 0.9166... \times 12 \\ & = & 11 \end{eqnarray*}$

Final statement: It will take $1$ year and $11$ months.

## Section 2.5 Exercises

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

$6$ years, $9$ months

2. 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 $\175,000$?
Solution

$18$ years

3. 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?
Solution

$12$ years, $9$ months

4. 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?
Solution

$9$ years, $3$ months

5. 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?
Solution

$8$ years, $3$ months

6. 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?
Solution

$2$ years

7. 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 $\10,000$?
Solution

$8$ years, $2$ months

8. 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 $\4,000$?
Solution

$3$ years, $2$ months

9. 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?
Solution

$3$ years, $6$ months

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

a. $9$ months; b. regular payments require $\2250$ less principal

11. 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.
1. How many fewer payments does it take to pay off your mortgage if you increased your monthly payments by $10\%$?
2. How much money is saved by “topping up” the payments? Assume that all payments are equal amounts in your calculations.
Solution

a. $48$ fewer payments; b. $\26,521.44$