C3.7 Solve ordinary simple annuity problems using technology

Ontario Grade 11 Mathematics

Discrete Functions

Finding the future value and present value of regular equal payments

Before this lesson, you learned about compound interest, where a single lump sum grows over time because interest is added and then earns more interest itself. In real life, though, people rarely save or borrow with just one deposit. They pay a fixed amount every month for a car loan, or they deposit a fixed amount every month into a savings plan. This lesson is about that pattern: a series of equal payments made at regular time intervals. This kind of series is called an annuity. We will look at ordinary simple annuities, which are the most basic and common type, and you will learn how to use the future value and present value formulas together with a calculator to solve real problems.

What you will learn

Explain what makes a series of payments an ordinary simple annuity
Identify the payment amount, interest rate per period, and number of payments in a real situation
Use the future value formula to find the total amount an annuity grows to
Use the present value formula to find the lump sum equivalent to a series of payments
Solve annuity problems with a calculator, checking that the answer makes sense

What Is an Ordinary Simple Annuity?

An annuity is a series of equal payments made at regular time intervals, such as CAD 100 deposited every month, or CAD 500 paid every year. Each payment earns compound interest from the moment it is made until the end of the annuity term.

The word simple in simple annuity means the payment interval and the compounding interval are the same. For example, if interest compounds monthly and you also pay monthly, the payment and compounding periods line up perfectly. This matches the compound interest ideas you already studied, where interest is calculated once per compounding period.

The word ordinary in ordinary annuity means each payment is made at the end of every payment interval, not at the beginning. Think of a monthly savings deposit made on the last day of the month, or a loan payment due at the end of each month. This is the most common real-world pattern, such as mortgage payments, car loan payments, and many retirement savings plans.

Because the payment and compounding periods match in a simple annuity, we can reuse the compound interest idea of a periodic interest rate, which is the annual interest rate divided by the number of compounding periods per year.

i = \frac{r}{n}

An annuity is a series of equal payments at regular intervals
Simple annuity means payment interval equals compounding interval
Ordinary annuity means payments happen at the end of each interval
The periodic rate is the annual rate divided by the number of periods per year

Future Value of an Ordinary Simple Annuity

Future value means the total amount of money the annuity is worth at the end of the term, after every payment has been made and has earned interest for however long it was invested. The first payment earns interest for almost the whole term, while the last payment earns no interest at all because it is made right at the end.

Instead of adding up every payment separately with its own interest, we use a single formula that combines all of these growing amounts at once. In this formula,

R

is the regular payment amount,

i

is the interest rate per compounding period written as a decimal, and

n

is the total number of payments.

This formula works because it is really the sum of a special pattern of terms, one for each payment, where earlier payments are multiplied by a larger growth factor than later payments. You do not need to add each term by hand. A scientific or graphing calculator, or a financial app, computes the formula directly once you enter

R

,

i

, and

n

.

Always find

i

and

n

first, matching the compounding period to the payment period, before you substitute into the formula. This step is where most errors happen if the annual rate or the number of years is used without converting to periods.

FV = R [

\frac{(1+i)^{n} - 1}{i}

]

Future value FV is the total value of the annuity at the end of the term
R is the payment amount, i is the rate per period, n is the number of payments
Convert annual rate and years into periodic rate and number of payments first
A calculator evaluates the formula once the three values are known

Present Value of an Ordinary Simple Annuity

Present value means the single lump sum today that is equivalent to receiving or paying the whole series of future payments. This is useful for questions like: what loan amount can be paid off using a certain monthly payment, or how much money must be set aside today to fund a certain series of future withdrawals.

The present value formula has the same variables as the future value formula,

R

for the payment,

i

for the periodic rate, and

n

for the number of payments, but it is structured to discount, or reduce, future payments back to today's value, since money in the future is worth less than the same amount today.

Present value and future value are connected: if you take the present value of an annuity and let it grow with compound interest for the whole term, you get the same result as the future value formula. This connection is why both formulas use the same three inputs.

As with future value, always determine

i

and

n

carefully before calculating, and use a calculator to evaluate the expression, since the numbers involved usually have several decimal places.

PV = R [

\frac{1 - (1+i)^{-n}}{i}

]

Present value PV is the lump sum today equivalent to the full series of payments
PV uses the same R, i, and n as the future value formula
PV answers questions about loan amounts or funding future payments
Rounding should happen only at the final step to avoid accumulated error

Matching Annual Terms to Periodic Values

Compounding frequency	Periods per year	How to get i	How to get n
Monthly	12	annual rate divided by 12	years multiplied by 12
Quarterly	4	annual rate divided by 4	years multiplied by 4
Semi-annually	2	annual rate divided by 2	years multiplied by 2
Annually	1	annual rate divided by 1	years multiplied by 1

Worked example

Future Value of Monthly Savings Deposits

Amara deposits CAD 150 at the end of every month into a savings account that earns 6 percent per year, compounded monthly. How much will be in the account after 4 years?

Identify the payment and term details
The payment is $R = 150$ dollars, made monthly. The account compounds monthly, so this is an ordinary simple annuity because the payment and compounding periods match, and payments happen at the end of each month.
$R = 150$
Find the periodic interest rate
The annual rate is 6 percent, and there are 12 compounding periods per year since compounding is monthly. Divide the annual rate by 12 to get the rate per month as a decimal.
$i = \frac{0.06}{12} = 0.005$
Find the total number of payments
The term is 4 years, and payments happen 12 times per year, so multiply 4 by 12 to get the total number of monthly payments.
$n = 4 × 12 = 48$
Substitute into the future value formula
Now place $R = 150$ , $i = 0.005$ , and $n = 48$ into the future value formula for an ordinary simple annuity. FV = 150 [ $\frac{(1.005)^{48} - 1}{0.005}$ ]
Evaluate using a calculator
A calculator gives $(1.005)^{48} \approx 1.270489$ . Subtract 1 to get $0.270489$ , divide by $0.005$ to get $54.0978$ , then multiply by 150.
$FV \approx 150 × 54.0978 \approx 8114.67$

Answer: The account will hold about CAD 8114.67 after 4 years.

Check: Amara deposited 48 payments of CAD 150, which totals CAD 7200 with no interest. Since the account earned interest on top of that, an answer of about CAD 8114.67, which is more than CAD 7200, makes sense.

Worked example

Present Value of a Loan Payment Plan

A car loan is being repaid with payments of CAD 320 at the end of every month for 3 years. The loan charges interest at 8.4 percent per year, compounded monthly. What was the original loan amount, that is, the present value?

Identify the payment and term details
The payment is $R = 320$ dollars per month, matching the monthly compounding, so this is again an ordinary simple annuity.
$R = 320$
Find the periodic interest rate
The annual rate is 8.4 percent, or 0.084 as a decimal, and there are 12 periods per year, so divide by 12.
$i = \frac{0.084}{12} = 0.007$
Find the total number of payments
The term is 3 years with 12 payments per year, so multiply to find the total number of monthly payments.
$n = 3 × 12 = 36$
Substitute into the present value formula
Place $R = 320$ , $i = 0.007$ , and $n = 36$ into the present value formula for an ordinary simple annuity. PV = 320 [ $\frac{1 - (1.007)^{-36}}{0.007}$ ]
Evaluate using a calculator
A calculator gives $(1.007)^{-36} \approx 0.774629$ . Subtract this from 1 to get $0.225371$ , divide by $0.007$ to get about $32.196$ , then multiply by 320.
$PV \approx 320 × 32.196 \approx 10302.7$

Answer: The original loan amount was about CAD 10302.72.

Check: Over 36 months at CAD 320 per payment, the total paid is CAD 11520, which is more than the loan amount of about CAD 10302.72. This extra amount represents the interest charged on the loan, which is expected.

Common mistakes and how to avoid them

Using the annual interest rate directly in the formula instead of the rate per compounding period.

Correction: Always divide the annual rate by the number of compounding periods per year to get

i

before substituting into the formula.

Using the number of years instead of the total number of payments for

n

.

Correction: Multiply the number of years by the number of payments per year to get the correct

n

, since

n

must count individual payments.

Rounding

i

or intermediate results too early, which changes the final answer.

Correction: Keep several decimal places during the calculation and round only the final dollar answer, usually to two decimal places.

Mixing up the future value and present value formulas.

Correction: Use future value when asked what an amount grows to in the future, and present value when asked what lump sum today is equivalent to future payments.

Lesson summary

An ordinary simple annuity is a series of equal payments made at the end of regular intervals that match the compounding period.
The periodic rate $i$ is the annual rate divided by the number of compounding periods per year, and $n$ is the total number of payments.
The future value formula finds the total amount an annuity grows to by the end of the term.
The present value formula finds the lump sum today that is equivalent to the whole series of payments.
Always find $i$ and $n$ carefully before using a calculator, and check that the final answer is reasonable compared to the total of all payments.

Check your understanding

Question 1

Deshawn deposits CAD 200 at the end of every month for 2 years into an account earning 12 percent per year, compounded monthly. What are the correct values of

i

and

n

to use in the future value formula?

i = 0.12 and n = 2
i = 0.01 and n = 24
i = 0.12 and n = 24
i = 0.01 and n = 12

Show answer and explanation

i = 0.01 and n = 24

The periodic rate is the annual rate divided by 12 periods per year, giving

i = 0.12/12 = 0.01

. The number of payments is 2 years times 12 payments per year, giving

n = 24

.

Question 2

Which situation best describes an ordinary simple annuity?

A single deposit that grows with compound interest for 5 years
Equal payments made at the end of every month, with interest compounded monthly
Equal payments made at the start of every month, with interest compounded yearly
Payments of different amounts made at random times during the year

Show answer and explanation

Equal payments made at the end of every month, with interest compounded monthly

An ordinary simple annuity requires equal payments at the end of each interval, and the payment interval must match the compounding interval, which happens when both are monthly.

Question 3

A present value calculation for an annuity is used to answer which type of question?

How much a single deposit grows to after several years
What lump sum today is equivalent to a series of future equal payments
How many years it takes to double an investment
What the interest rate must be to reach a savings goal

Show answer and explanation

What lump sum today is equivalent to a series of future equal payments

Present value finds the lump sum today that is equivalent to receiving or paying a whole series of future equal payments, such as the amount of a loan given fixed monthly payments.

Question 4

In the future value formula

FV = R\left[\frac{(1+i)^n - 1}{i}\right]

, if the calculated future value comes out smaller than the total of all payments made, what does this tell you?

The calculation is likely correct because interest always reduces the total
The calculation likely has an error, since future value should exceed the total of payments when interest is earned
This is normal only when the interest rate is very low
This means the annuity is not ordinary

Show answer and explanation

The calculation likely has an error, since future value should exceed the total of payments when interest is earned

Since each payment earns positive interest before the end of the term, the future value should always be greater than the simple total of all payments. A smaller result signals an error, often from using the wrong

i

or

n

.

Key terms

Annuity: A series of equal payments made at regular time intervals.
Ordinary annuity: An annuity where each payment is made at the end of every payment interval.
Simple annuity: An annuity where the payment interval and the compounding interval are the same length.
Periodic interest rate: The interest rate that applies to one compounding period, found by dividing the annual rate by the number of periods per year.
Future value: The total amount of money an annuity is worth at the end of its term, including all payments and the interest they earned.
Present value: The single lump sum today that is mathematically equivalent to a whole series of future equal payments.
Term: The total length of time over which an annuity's payments are made.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation C3.7. It is a study resource, not an official curriculum publication.

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C3.7 · Solve ordinary simple annuity problems using technology

Ontario Grade 11 Mathematics

Finding the future value and present value of regular equal payments

What you will learn

What Is an Ordinary Simple Annuity?

Future Value of an Ordinary Simple Annuity

Present Value of an Ordinary Simple Annuity

Matching Annual Terms to Periodic Values

Future Value of Monthly Savings Deposits

Present Value of a Loan Payment Plan

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MCR3U

About this lesson