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B3.6 · Investigate how changing conditions affects an annuity

Learn to investigate how changing conditions affects an annuity through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Exponential Functions

Exploring how payment amount, interest rate, and number of periods change the future value of an annuity

In earlier lessons, you learned what an annuity is and how to calculate its future value using a formula. Now you will investigate something different: what happens to that future value when one condition changes. Conditions include the size of each payment, the interest rate, and the number of payment periods. Understanding how these changes affect an annuity helps you make smart decisions about saving for a car, a trip, or retirement, and it helps you understand loan payments too.

What you will learn

Explain, using words, what an annuity is and identify its changeable conditions.
Predict how increasing or decreasing the payment amount, interest rate, or number of periods affects the future value of an annuity.
Use a table of values and a graph to compare annuities under different conditions.
Calculate the future value of an annuity before and after a condition changes, using the given formula.
Explain why interest rate changes affect an annuity exponentially rather than in a straight line.

Prerequisite Bridge: What You Already Know

Before we look at changing conditions, let's review two ideas from earlier grades and earlier lessons. First, you learned about exponential growth in Grade 10 and earlier in this course. Exponential growth means a quantity increases by a percentage each period, not by a fixed amount. For example, if you have CAD 100 growing at 5 percent per period, you gain 5 dollars in the first period, but slightly more than 5 dollars in the next period, because interest is now earned on the new total.

Second, you learned that an annuity is a series of equal payments made at equal time intervals, and that its future value is the total amount built up after all payments and all interest are combined. The future value formula for an annuity is written using the payment amount, the interest rate per period, and the number of periods.

In this lesson, we do not derive this formula. Instead, we use it as a tool to investigate what happens when one part of the annuity changes while the other parts stay the same. This kind of investigation is sometimes called changing one variable at a time, which lets us clearly see the effect of that one condition.

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

Exponential growth means increasing by a percentage each period, not a fixed dollar amount.
An annuity is a series of equal payments at equal time intervals.
Future value is the total built up after all payments and interest.
We change one condition at a time to clearly see its effect.

Plain Language: The Three Conditions That Change an Annuity

An annuity has three main conditions that can change. The first is the payment amount, which is how much money is deposited or paid each period. The second is the interest rate per period, which controls how fast the money grows. The third is the number of periods, which is how many times a payment is made and interest is applied.

When you change the payment amount, the future value changes by roughly the same scale, because every payment in the series changes together. For example, if you double every payment, the total future value roughly doubles too, because interest still works the same way on each deposit.

When you change the interest rate, the effect is different. Because interest compounds, a higher rate does not just add a little more money. It multiplies the growth again and again across every period. This means small changes in interest rate can create large changes in future value, especially over many periods.

When you change the number of periods, you are changing two things at once: how many payments are made, and how many times interest is applied to the earlier payments. Adding more periods almost always increases future value more than people expect, because the earliest payments have the most time to grow.

Changing the payment amount scales the future value in a similar proportion.
Changing the interest rate has a compounding effect, so its impact grows over time.
Changing the number of periods affects both the number of payments and the amount of compounding.
Small interest rate changes can cause large future value changes over long time spans.

Multiple Representations: Table and Graph Thinking

To see these effects clearly, it helps to build a table that keeps two conditions fixed and changes only one. Below is a table showing an annuity with a payment of CAD 500 per year, and periods held constant at 10 years, while the interest rate per period changes.

Look at how the future value increases faster and faster as the interest rate goes up. This is not a straight-line increase. If you plotted these points on a graph with interest rate on the horizontal axis and future value on the vertical axis, the points would curve upward, showing exponential growth rather than a straight line.

This curved pattern is the key idea of this lesson. A straight line would mean equal increases in rate always add the same extra future value. Instead, because interest compounds, equal increases in rate add more and more future value each time.

Holding two conditions constant lets you isolate the effect of the third.
A table of values reveals the pattern before you use a formula.
A graph of future value against interest rate curves upward, showing exponential growth.
Equal rate increases produce unequal, growing increases in future value.

Guided Example: Comparing Two Conditions

Now we apply the formula directly to see the size of these effects using real numbers. This connects the plain language ideas and the table pattern to a precise calculation.

Application: Choosing Wisely With Annuities

These ideas apply to real decisions. If you are saving for a goal, increasing your payment amount helps in a fairly predictable way, since the future value scales with the payment. But if you can find an account with a higher interest rate, that change can be more powerful over a long time span, because of compounding.

If you are paying off a loan, the same ideas work in reverse. A higher interest rate on a loan makes the total amount you owe grow faster, which is why comparing interest rates carefully before borrowing money is so important.

Increasing the number of periods, meaning saving for a longer time, is often the most powerful condition of all, because it combines more payments with more compounding time. This is why financial advisors often say that starting to save early is one of the best financial decisions a person can make.

Higher payment amounts scale future value in a predictable way.
Higher interest rates create compounding effects that grow stronger over time.
More periods combine both more payments and more compounding, often producing the largest change.
Starting to save early takes advantage of the number-of-periods effect.

Future Value With Payment and Periods Held Constant (R = CAD 500, n = 10 years)

Interest rate per year	Growth factor (1+i)^10	Future value (CAD)
2 percent	1.219	6084.87
4 percent	1.480	6003.05... (recompute)	-
4 percent	1.480	6003.05
6 percent	1.791	6590.40
8 percent	2.159	7243.90

Worked example

Comparing an Annuity Before and After a Condition Change

Priya deposits CAD 400 at the end of each year into a savings account for 5 years. The account earns 4 percent interest per year, compounded annually. First, find the future value of this annuity. Then, suppose Priya increases her interest rate to 6 percent per year, with everything else the same. Find the new future value, and explain what kind of change this represents.

Identify the known values for the first situation
The payment $R$ is CAD 400, the interest rate per period $i$ is 4 percent written as a decimal, and the number of periods $n$ is 5 years.
$R = 400,\ i = 0.04,\ n = 5$
Substitute into the future value formula for the first situation
Place the known values into the annuity future value formula to prepare for calculation. FV_1 = 400[ $\frac{(1.04)^5 - 1}{0.04}$ ]
Calculate the first future value
Evaluate $(1.04)^5$ first, which is approximately 1.2167. Subtract 1 to get 0.2167, then divide by 0.04 to get approximately 5.4163. Multiply by 400.
$FV_1 \approx 400(5.4163) \approx 2166.53$
Identify the known values for the second situation
Only the interest rate changes to 6 percent. The payment and number of periods stay the same, since we are isolating one condition.
$R = 400,\ i = 0.06,\ n = 5$
Substitute into the future value formula for the second situation
Place the new interest rate into the same formula structure. FV_2 = 400[ $\frac{(1.06)^5 - 1}{0.06}$ ]
Calculate the second future value
Evaluate $(1.06)^5$ first, which is approximately 1.3382. Subtract 1 to get 0.3382, then divide by 0.06 to get approximately 5.6371. Multiply by 400.
$FV_2 \approx 400(5.6371) \approx 2254.84$
Compare the two results
The future value increased from about CAD 2166.53 to about CAD 2254.84, an increase of about CAD 88.31. This increase came only from raising the interest rate by 2 percentage points, while the payment and number of periods stayed fixed, showing the compounding effect of the rate change on its own.

Answer: The future value rises from about CAD 2166.53 at 4 percent to about CAD 2254.84 at 6 percent, an increase of about CAD 88.31 caused only by the interest rate change.

Check: Check the growth factors:

(1.04)^5 \approx 1.2167

and

(1.06)^5 \approx 1.3382

, both reasonable since the base is close to 1 and the exponent is small. The second future value is larger than the first, matching the expectation that a higher interest rate produces a larger future value when payment and periods are unchanged.

Common mistakes and how to avoid them

Assuming that doubling the payment always doubles the future value of the annuity.

Correction: Remember that interest compounds on every payment, so doubling the payment changes total contributions proportionally, but the total future value grows by that same factor only if the rate and number of payments stay the same. Check each part of the situation separately before concluding what changed.

Thinking a higher interest rate always helps every person the same way.

Correction: A higher rate grows savings faster, but for money you owe (like a loan), a higher rate makes it grow against you. Always ask whether the annuity is money coming to you or money you must pay back.

Forgetting that adding more payment periods increases both the number of deposits and the number of times interest is applied.

Correction: Count both effects: more deposits added and more compounding on earlier deposits. Use the table or graph to see both effects combined.

Comparing two annuities that do not have the same time period.

Correction: Only compare annuities fairly when the total time span matches, or clearly state that the time span is different and why that changes the result.

Lesson summary

An annuity's future value depends on three conditions: payment amount, interest rate per period, and number of periods.
Changing the payment amount changes future value in a roughly proportional way.
Changing the interest rate changes future value exponentially, because interest compounds across all periods.
Changing the number of periods increases both the number of payments and the amount of compounding time.
Tables and graphs show that future value curves upward as interest rate or number of periods increases, rather than rising in a straight line.
These patterns help explain real decisions about saving early, choosing better interest rates, and understanding loan costs.

Check your understanding

Question 1

Which condition of an annuity, when increased, causes the future value to grow because of a compounding effect rather than a simple proportional effect?

The payment amount
The interest rate per period
The currency used for payments
The day of the week payments are made

Show answer and explanation

The interest rate per period

Interest compounds on the growing total each period, so raising the interest rate creates exponential growth in future value, unlike the payment amount, which scales future value roughly proportionally.

Question 2

An annuity has a fixed payment and a fixed interest rate. If the number of periods increases, what happens to the future value, and why?

It decreases, because more periods means less time for growth
It stays the same, because the payment amount did not change
It increases, because there are more payments and more compounding time on earlier payments
It becomes impossible to calculate without a new formula

Show answer and explanation

It increases, because there are more payments and more compounding time on earlier payments

More periods means more total payments are added, and the earlier payments have more time to earn compounding interest, so future value increases.

Question 3

Priya's annuity has payment CAD 400, interest rate 4 percent per year, and 5 years, giving a future value of about CAD 2166.53. If only the payment increases to CAD 800 with everything else the same, what best describes the new future value?

It will be about CAD 4333.06, roughly double, since future value scales close to proportionally with payment
It will be about CAD 2166.53, unchanged, since interest rate controls future value
It will be less than CAD 2166.53, since larger payments take longer to grow
It cannot change unless the interest rate also changes

Show answer and explanation

It will be about CAD 4333.06, roughly double, since future value scales close to proportionally with payment

Since payment amount scales future value roughly proportionally when interest rate and periods stay fixed, doubling the payment roughly doubles the future value, giving about CAD 4333.06.

Question 4

Why does a graph of future value versus interest rate curve upward instead of forming a straight line?

Because payment amounts always increase automatically over time
Because interest compounds, so equal increases in rate produce larger and larger increases in future value
Because the number of periods always changes when the rate changes
Because future value formulas only work for straight-line patterns

Show answer and explanation

Because interest compounds, so equal increases in rate produce larger and larger increases in future value

Compounding means growth is exponential, not linear, so equal steps in interest rate create increasingly larger jumps in future value, producing a curved graph.

Key terms

Annuity: A series of equal payments made at regular time intervals, such as monthly savings deposits or loan payments.
Future value: The total amount of money an annuity grows to after all payments and interest are added together.
Interest rate: The percentage used to calculate how much extra money is added to savings, or owed on a loan, over a set time period.
Compounding: The process where interest is calculated not only on the original money but also on interest already earned.
Payment period: The fixed length of time between one annuity payment and the next, such as every month or every year.
Condition: In this lesson, one changeable part of an annuity, such as the payment amount, the interest rate, or the number of payment periods.
Exponential growth: A pattern of increase where a quantity grows by a consistent percentage each period, causing the amount to rise faster and faster over time.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation B3.6. It is a study resource, not an official curriculum publication.

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