C3.6 Investigate how changing conditions affects an annuity

Ontario Grade 11 Mathematics

Discrete Functions

MCR3U · Financial Mathematics · Expectation C3.6

Suppose you put CAD 200 into a savings account every month for five years. At the end, how much will you have? Now suppose you bump that monthly deposit up to CAD 250, or the bank raises its interest rate — does your total grow a little, or a lot? This lesson explores exactly that question. You will see how four key conditions of an annuity — the regular payment, the interest rate, the compounding frequency, and the total number of payments — each pull the final amount in a different direction. Understanding these relationships helps you make smarter decisions about saving for a goal or choosing between loan offers.

What you will learn

Identify the four key conditions of an annuity: payment amount, interest rate, compounding frequency, and number of payments.
Predict and explain how increasing or decreasing each condition changes the future value of an annuity.
Calculate the future value of an ordinary simple annuity and compare results when one condition is changed.
Interpret the effect of each change in a real-world saving or borrowing context.

Prerequisite Bridge: What Is an Annuity?

An annuity is a series of equal payments made at regular time intervals. A car loan you repay monthly, a retirement fund you contribute to every two weeks, and a college savings plan you top up each year are all annuities.

In this course we focus on an ordinary simple annuity. 'Ordinary' means each payment is made at the end of the period, not the beginning. 'Simple' means the compounding period matches the payment period — for example, monthly payments with monthly compounding.

The four conditions that define any ordinary simple annuity are: the regular payment amount

R

, the interest rate per compounding period

i

, the number of periods

n

, and whether we are asking about a future value

FV

(saving) or a present value

PV

(borrowing). Changing any one of these conditions changes the result.

An annuity is a series of equal, equally spaced payments.
Ordinary simple annuity: payment at end of period; compounding matches payment frequency.
The four conditions are payment $R$ , rate per period $i$ , number of periods $n$ , and the value type ( $FV$ or $PV$ ).

The Future Value Formula and What Each Part Does

When you save by making regular deposits, the goal is usually to find the future value — the total amount accumulated, including all interest earned. The formula for the future value of an ordinary simple annuity is shown below.

Every term in this formula is tied to one of the four conditions. Raising

R

(paying more each period) scales the whole result up proportionally. Raising

i

(higher interest rate) makes the annuity factor grow faster than linearly, because interest compounds on earlier payments. Raising

n

(more payment periods) also increases the annuity factor, but the effect accelerates over time — a hallmark of geometric growth.

Because interest compounds, even a small change in

i

can have a surprisingly large effect when

n

is large. This is one of the most important ideas in this lesson: the four conditions do not all scale the result in a simple, proportional way.

FV = R ·

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

Future value measures total savings accumulated at the end of all payments.
Increasing $R$ increases $FV$ proportionally.
Increasing $i$ or $n$ increases $FV$ by more than a simple proportion because of compounding.
The expression $\frac{(1+i)^n-1}{i}$ is called the annuity factor; it captures the combined effect of compounding and multiple payments.

How Each Condition Pulls the Result

It helps to think of each condition as a dial you can turn up or down. The table in this lesson shows the direction and rough strength of each dial's effect on future value. Here is a plain-language summary before you look at the table.

Payment amount

R

: doubling

R

exactly doubles

FV

. This is the only strictly proportional relationship among the four conditions. If you can afford to save CAD 400 per month instead of CAD 200, you will end up with exactly twice as much, assuming everything else stays the same.

Interest rate

i

: increasing

i

increases

FV

by more than a simple proportion, especially when

n

is large. Even moving from 4% to 6% annual interest compounded monthly over 20 years can add tens of thousands of dollars to a retirement fund. This is the power of compound interest.

Number of periods

n

: increasing

n

increases

FV

, but the growth is geometric, not linear. The last few years of a long savings plan contribute far more than the first few years, because the accumulated balance is larger and earns more interest on itself.

Compounding frequency: if the annual rate stays fixed but compounding happens more often, the rate per period

i

decreases — but you also have more periods

n

. The net effect is a slight increase in

FV

. For example, monthly compounding on a 6% annual rate gives

i = 0.5\%

per month and

n

counted in months, which earns slightly more than annual compounding at 6%.

Doubling $R$ doubles $FV$ exactly.
Increasing $i$ raises $FV$ faster than proportionally due to compounding.
Increasing $n$ raises $FV$ geometrically — later periods contribute the most.
More frequent compounding slightly raises $FV$ when the annual rate is held fixed.

Comparing Scenarios Side by Side

The clearest way to see how a condition affects an annuity is to change exactly one condition at a time and recalculate. Hold everything else constant, change one thing, and observe what happens. This approach is sometimes called a sensitivity analysis.

For example, start with a base scenario: CAD 300 deposited at the end of each month, 4.8% annual interest compounded monthly, for 10 years. Calculate

FV

. Then change only the rate to 7.2% annual and recalculate. The difference between the two future values tells you the dollar impact of that rate change alone.

Working through the two examples in the next section will give you practice with this technique. After each calculation, ask yourself: is this change bigger or smaller than I expected, and why?

Change one condition at a time to isolate its effect.
Recalculate $FV$ for each scenario and compare the results.
Ask whether the change is proportional, more than proportional, or less than proportional.

Interpreting Results in Context

Numbers mean more when they are tied to a real decision. Suppose you are saving for a car. You find that raising your monthly deposit by CAD 50 adds CAD 3 200 to your total over four years. That extra CAD 50 per month costs you CAD 2 400 in additional deposits but earns CAD 800 in extra interest — a clear benefit.

Now suppose a second bank offers a rate that is 1% higher per year. You calculate that this adds CAD 1 500 to the same four-year plan without you paying anything extra. Choosing the higher-rate bank is clearly better, all else being equal.

This kind of comparison — translating formula results into plain-language decisions — is the real goal of studying how conditions affect annuities. The formula is a tool; the reasoning is the skill.

Compare the extra deposits you make against the extra interest you earn.
A higher rate costs you nothing extra, so it is always preferable when everything else is equal.
Longer saving periods amplify the effect of every other condition.

Effect of Increasing Each Condition on Future Value

Condition Changed	Direction of Change	Type of Effect	Key Reason
Payment R increased	FV increases	Proportional (linear)	R is a direct multiplier; annuity factor unchanged
Rate per period i increased	FV increases	More than proportional	Compounding accelerates; each payment earns more interest
Number of periods n increased	FV increases	Geometric growth	More payments plus longer compounding on earlier deposits
Compounding frequency increased (fixed annual rate)	FV increases slightly	Small positive effect	More frequent compounding raises the effective annual yield

Worked example

Example 1 — Changing the Payment Amount

Aaliya deposits CAD 200 at the end of every month into a savings account that earns 6% per year, compounded monthly. She plans to save for 3 years. (a) Find the future value of her annuity. (b) She then decides to increase her monthly deposit to CAD 300. Find the new future value. (c) Explain why the future value increases by exactly 50% even though nothing else changed.

Identify the conditions for part (a)
The payment is $R = 200$ . The annual rate is 6%, compounded monthly, so the rate per period is $i = \frac{0.06}{12} = 0.005$ . The term is 3 years, so the number of monthly periods is $n = 3 \times 12 = 36$ .
$R = 200, \quad i = 0.005, \quad n = 36$
Compute the power $(1.005)^{36}$
This power is evaluated first because it appears inside the annuity factor. Using a calculator gives the value below.
$(1.005)^{36} \approx 1.19668$
Complete the future value calculation for part (a)
Subtract 1 from the power result, divide by $i = 0.005$ to get the annuity factor, then multiply by $R = 200$ . The annuity factor is $\frac{1.19668 - 1}{0.005} = \frac{0.19668}{0.005} = 39.336$ . Multiplying gives $FV = 200 \times 39.336 \approx 7867.22$ .
$FV = 200 \times 39.336 \approx 7867.22$
Identify the conditions for part (b)
Only the payment changes. Now $R = 300$ , while $i = 0.005$ and $n = 36$ stay the same. Because the annuity factor depends only on $i$ and $n$ , it remains 39.336.
$R = 300, \quad i = 0.005, \quad n = 36$
Calculate the new future value for part (b)
Multiply the unchanged annuity factor by the new payment of 300.
$FV = 300 \times 39.336 \approx 11804.83$
Explain why the increase is exactly proportional
The payment increased by 50% (from 200 to 300). Because $R$ is a direct multiplier in the formula and the annuity factor did not change, the future value also increased by exactly 50%. Dividing the two future values confirms this: $\frac{11804.83}{7867.22} \approx 1.500$ . This proportional relationship holds only for changes to $R$ ; changing $i$ or $n$ gives a non-proportional effect.
$\frac{11804.83}{7867.22} \approx 1.500$

Answer: (a)

FV \approx

CAD 7 867.22. (b)

FV \approx

CAD 11 804.83. (c)

FV

increased by exactly 50% because

R

is a direct multiplier — the annuity factor stayed constant.

Check: Total deposits in (a):

200 \times 36 =

CAD 7 200; interest earned

\approx

CAD 667. Total deposits in (b):

300 \times 36 =

CAD 10 800; interest earned

\approx

CAD 1 005. Both future values exceed their respective total deposits, confirming interest was earned. The ratio

11804.83 \div 7867.22 \approx 1.500

confirms the 50% proportional increase.

Worked example

Example 2 — Changing the Interest Rate and the Number of Periods

Marcus saves CAD 150 at the end of every month. Scenario A: 4.8% annual interest, compounded monthly, for 5 years. Scenario B: 7.2% annual interest, compounded monthly, for 5 years. Scenario C: 4.8% annual interest, compounded monthly, for 8 years. Compare the three future values and explain which condition — rate or time — has a larger effect in this situation.

Set up Scenario A
Payment $R = 150$ . Annual rate 4.8% compounded monthly gives $i = \frac{0.048}{12} = 0.004$ . Five years gives $n = 5 \times 12 = 60$ months.
$R = 150, \quad i = 0.004, \quad n = 60$
Calculate the future value for Scenario A
First find $(1.004)^{60} \approx 1.27049$ . The annuity factor is $\frac{1.27049 - 1}{0.004} = \frac{0.27049}{0.004} = 67.6225$ . Multiplying gives $FV_A = 150 \times 67.6225 \approx 10143.37$ .
$FV_A = 150 \times 67.6225 \approx 10143.37$
Set up and calculate Scenario B
Only the rate changes to 7.2% annually, so $i = \frac{0.072}{12} = 0.006$ . Payment and term stay the same: $R = 150$ , $n = 60$ . Find $(1.006)^{60} \approx 1.43204$ . Annuity factor: $\frac{1.43204 - 1}{0.006} = \frac{0.43204}{0.006} = 72.0067$ . So $FV_B = 150 \times 72.0067 \approx 10801.00$ .
$FV_B = 150 \times 72.0067 \approx 10801.00$
Set up and calculate Scenario C
The rate returns to 4.8% monthly ( $i = 0.004$ ), but the term is now 8 years, giving $n = 8 \times 12 = 96$ months. $R = 150$ . Find $(1.004)^{96} \approx 1.46933$ . Annuity factor: $\frac{1.46933 - 1}{0.004} = \frac{0.46933}{0.004} = 117.3325$ . So $FV_C = 150 \times 117.3325 \approx 17599.88$ .
$FV_C = 150 \times 117.3325 \approx 17599.88$
Compare the three results
Scenario A gives approximately CAD 10 143. Scenario B (higher rate, same time) gives approximately CAD 10 801 — an increase of about CAD 658 over Scenario A. Scenario C (same rate, more time) gives approximately CAD 17 600 — an increase of about CAD 7 457 over Scenario A. Extending the saving period by 3 years has a much larger effect than raising the rate by 2.4 percentage points in this situation.
$FV_C - FV_A \approx 17599.88 - 10143.37 = 7456.51$
Explain why time has a larger effect here
Extending from $n = 60$ to $n = 96$ is a 60% increase in the number of periods, and each extra period earns interest on an ever-growing balance. The rate increase from 4.8% to 7.2% is a 50% increase in the rate, but $i$ also appears in the denominator of the annuity factor, which partially offsets the numerator's growth. For much longer terms, the rate effect becomes more significant. Both conditions matter — the relative importance always depends on the specific numbers, so calculate before concluding.

Answer: Scenario A:

FV \approx

CAD 10 143. Scenario B:

FV \approx

CAD 10 801. Scenario C:

FV \approx

CAD 17 600. Extending the saving period by 3 years added about CAD 7 457, while raising the rate by 2.4 percentage points added only about CAD 658. For these values, time has the larger effect.

Check: Scenario A total deposits:

150 \times 60 =

CAD 9 000; interest

\approx

CAD 1 143. Scenario C total deposits:

150 \times 96 =

CAD 14 400; interest

\approx

CAD 3 200. All three future values exceed their respective total deposits, confirming that interest was earned in every scenario and that the results are directionally correct.

Common mistakes and how to avoid them

Using the annual interest rate directly as

i

instead of dividing by the number of compounding periods per year.

Correction: Always convert the annual rate first. For 6% compounded monthly, divide by 12 to get

i = 0.005

, not 0.06.

Assuming that doubling the interest rate doubles the future value.

Correction: The rate appears inside an exponent through compounding, so doubling the rate more than doubles the annuity factor. The effect is greater than proportional, not exactly double.

Forgetting to match

n

to the payment frequency — for example, using

n = 5

(years) when payments are monthly.

Correction: Count

n

in the same unit as the payment period. Five years of monthly payments means

n = 60

, not 5.

Concluding that a higher rate always has a bigger impact than a longer term (or vice versa) without calculating.

Correction: The relative impact of rate versus term depends on the specific values. Always calculate both scenarios and compare the results before drawing a conclusion.

Getting a small rounding difference between the interest earned and the difference between

FV

and total deposits, then thinking the formula is wrong.

Correction: Small discrepancies arise from rounding powers like

(1.005)^{36}

during intermediate steps. Use more decimal places while calculating and round only the final answer.

Lesson summary

An ordinary simple annuity has four key conditions: payment amount $R$ , rate per period $i$ , number of periods $n$ , and the type of value (future or present).
The future value formula is $FV = R \cdot \frac{(1+i)^n - 1}{i}$ ; each condition appears in a specific part of this formula and affects the result differently.
Increasing $R$ raises $FV$ proportionally; increasing $i$ or $n$ raises $FV$ by more than a simple proportion because of compounding.
To study the effect of one condition, change only that condition and recalculate — keep all others constant.
The relative impact of changing the rate versus changing the term depends on the specific values; always calculate to compare rather than assuming one dominates.
Connecting formula results to real decisions — such as choosing a higher-rate account or saving for an extra year — is the practical goal of this expectation.

Check your understanding

Question 1

Mia has an annuity with monthly payments of CAD 400, a rate of 0.5% per month, and 24 monthly payments. She changes only the payment to CAD 800. What happens to the future value?

It stays the same, because the rate and term did not change.
It exactly doubles, because $R$ is a direct multiplier in the formula.
It more than doubles, because the interest compounds on the larger payment.
It increases, but by less than double, because the annuity factor decreases.

Show answer and explanation

It exactly doubles, because

R

is a direct multiplier in the formula.

The payment

R

is a direct multiplier in

FV = R \cdot \frac{(1+i)^n-1}{i}

. The annuity factor depends only on

i

and

n

, which did not change. So doubling

R

exactly doubles

FV

.

Question 2

Two saving plans both have

R =

CAD 200 and

n = 60

monthly payments. Plan X has

i = 0.4\%

per month and Plan Y has

i = 0.8\%

per month. Which statement is correct?

Plan Y's future value is exactly double Plan X's, because the rate doubled.
Plan Y's future value is more than double Plan X's, because compounding amplifies the rate increase.
Plan Y's future value is greater than Plan X's, but by less than double, because $i$ also appears in the denominator of the annuity factor.
Both plans have the same future value, because $R$ and $n$ are identical.

Show answer and explanation

Plan Y's future value is greater than Plan X's, but by less than double, because

i

also appears in the denominator of the annuity factor.

Doubling

i

does not double the annuity factor

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

. The numerator

(1+i)^n - 1

grows, but

i

also doubles in the denominator, partially offsetting that growth. At

i = 0.004

, the annuity factor is approximately 67.6; at

i = 0.008

, it is approximately 76.7 — an increase of about 13.5%, not 100%. So Plan Y's

FV

is higher than Plan X's but by well less than double.

Question 3

Kofi saves CAD 250 per month at 6% annual interest compounded monthly for 4 years. He then considers saving for 6 years instead, keeping the same rate and payment. What is the best description of how the future value changes?

It increases by exactly 50%, because the term increased by 50%.
It increases by less than 50%, because interest is only earned on new deposits.
It increases by more than 50%, because earlier deposits compound for longer and the balance grows geometrically.
It decreases slightly, because spreading payments over more periods reduces each one's contribution.

Show answer and explanation

It increases by more than 50%, because earlier deposits compound for longer and the balance grows geometrically.

Increasing

n

from 48 to 72 months is a 50% increase in time. However, every deposit from the first 4 years now compounds for extra years on top of that, and the new deposits in years 5 and 6 also earn interest. The combined effect of more deposits and longer compounding on the existing balance makes

FV

grow by more than 50%.

Question 4

An annuity has monthly payments of CAD 500 for 2 years at an annual rate of 6%. A second annuity has the same payment, term, and annual rate, but compounds every two weeks instead of monthly. How does the second annuity's future value compare?

The future value is identical, because the annual rate and payment are the same.
The future value is slightly higher, because more frequent compounding raises the effective annual yield.
The future value is lower, because a shorter compounding period gives a smaller rate each period.
The future value is exactly double, because compounding happens twice as often.

Show answer and explanation

The future value is slightly higher, because more frequent compounding raises the effective annual yield.

When compounding becomes more frequent while the annual rate stays fixed, each period's rate decreases, but interest is applied more often. The net effect is a small increase in the effective annual yield and therefore a slightly higher future value. The gain is real but small — not dramatic.

Key terms

Annuity: A series of equal payments made at regular time intervals, such as monthly deposits into a savings account.
Ordinary simple annuity: An annuity in which each payment is made at the end of the period and the compounding period matches the payment period.
Future value (FV): The total amount accumulated at the end of all payments, including all interest earned.
Present value (PV): The single lump sum today that is equivalent to a series of future annuity payments at a given interest rate.
Rate per period (i): The interest rate for one compounding period, found by dividing the annual rate by the number of compounding periods per year.
Number of periods (n): The total number of payment and compounding periods over the life of the annuity.
Annuity factor: The expression that captures the combined effect of compounding and multiple payments; it equals the quantity (1 plus i) raised to the power n, minus 1, all divided by i.
Sensitivity analysis: The process of changing one condition at a time while holding all others constant, in order to measure each condition's individual effect on the result.

Continue through MCR3U

View the complete Ontario Grade 11 Mathematics learning path

About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation C3.6. It is a study resource, not an official curriculum publication.

Official curriculum reference

Report a correction or ask a question

C3.6 · Investigate how changing conditions affects an annuity

Ontario Grade 11 Mathematics

MCR3U · Financial Mathematics · Expectation C3.6

What you will learn

Prerequisite Bridge: What Is an Annuity?

The Future Value Formula and What Each Part Does

How Each Condition Pulls the Result

Comparing Scenarios Side by Side

Interpreting Results in Context

Effect of Increasing Each Condition on Future Value

Example 1 — Changing the Payment Amount

Example 2 — Changing the Interest Rate and the Number of Periods

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