B1.6 Investigate how time, rate, and compounding affect future value

Ontario Grade 11 Mathematics

Personal Finance

An MBF3C investigation of how savings grow

If you save money, the amount in the account may grow as interest is added. The amount you will have later is called the future value. In this lesson, you will investigate how that value changes when the money is invested for more time, earns a different rate, or has interest added more often. The examples use fixed rates and do not include deposits or withdrawals after the initial amount.

What you will learn

Explain future value and identify the starting amount, interest rate, time, and compounding frequency in a savings situation.
Use a compound-interest model to estimate future value.
Compare how changing time, rate, or compounding frequency affects future value.

1. Build from what you know

A percent is a rate out of one hundred. For example, 4% means 4 out of 100, or 0.04 as a decimal. To find 4% of CAD 800, multiply 800 by 0.04. The result is CAD 32.

Interest is money added to savings because the account earns a return. The original amount saved is the principal. If interest is added to the account, it can earn interest in later periods too. This is called compounding.

Future value is the account amount at a chosen later time. It includes the principal and any interest earned. To make a fair comparison, keep all but one feature the same when you investigate the effect of time, rate, or compounding.

Convert a percent to a decimal before using it in a calculation.
Future value includes the original principal and accumulated interest.
A comparison is most informative when you change one factor at a time.

2. A model for compound growth

A compounding period is the time between additions of interest. With annual compounding, interest is added once each year. With monthly compounding, it is added twelve times each year.

For a fixed annual rate compounded once per year, multiply by one plus the rate each year. If the principal is CAD 800 and the annual rate is 4%, the amount after one year is CAD 800 multiplied by 1.04. In the next year, interest is calculated on the new account amount, not just on the original principal.

A useful model is

A=P(1+r/m)^{mt}

. Here,

A

is the future value,

P

is the principal,

r

is the annual rate written as a decimal,

m

is the number of compounding periods per year, and

t

is the time in years. The rate for one compounding period is

r/m

. The total number of periods is

mt

.

This model assumes the rate stays fixed and that no extra money is added or taken out. The exponent counts how many times the account grows by the period multiplier. Round money to the nearest cent at the end of a calculation.

A=P(1+r/m)^{mt}

Use matching time units: if time is in years, count periods using periods per year.
More frequent compounding means interest is added more often.
The model describes growth on the changing account balance.

3. Investigate the three factors

Time: If the principal, rate, and compounding frequency stay fixed, a longer investment generally has a greater future value. There are more periods in which the balance can grow. The increase from one extra year is not simply the same fixed amount each year, because later interest is calculated on a balance that already includes earlier interest.

Rate: If principal, time, and compounding frequency stay fixed, a higher rate gives a greater future value. A higher rate makes the multiplier for each period larger. Even a small rate difference can matter when it applies over many periods.

Compounding: If principal, annual rate, and time stay fixed, adding interest more often generally gives a slightly greater future value. Interest added earlier can itself earn interest in later periods. The difference may be small for a short time or a modest rate, so calculate both amounts before deciding how important it is.

A table can help organize a comparison. In each row, all conditions except the factor being tested should stay the same. A calculator can evaluate the model, but check that the rate is entered as a decimal and the exponent counts the correct number of periods.

Longer time, a higher rate, and more frequent compounding each tend to increase future value when other conditions are fixed.
The size of the change depends on the other conditions too.
Use the same principal and consistent assumptions when comparing options.

4. Read results with care

A future value is a model-based estimate, not a promise that an account will actually pay that amount. The model assumes the stated rate and compounding schedule continue unchanged.

When comparing choices, first identify what changes. If both the rate and time change, the final amounts alone do not show which change caused the difference. To investigate one factor, hold the others constant.

Keep unrounded values in the calculator while working, then round the final dollar amount to cents. Rounding each period can slightly alter a result.

State the assumptions behind a comparison.
Do not attribute a difference to one factor when several factors changed.
Round only the final amount.

How to investigate each factor

Factor being tested	Change	Keep fixed
Time	Number of years	Principal, rate, compounding frequency
Rate	Annual rate	Principal, time, compounding frequency
Compounding	Periods per year	Principal, annual rate, time

Worked example

Example 1: The effect of time

A savings account starts with CAD 800 and earns 4% per year, compounded annually. Compare the future value after 3 years and after 5 years.

Identify the fixed conditions
The principal is CAD 800, the annual rate is 4%, and interest is added once per year. Only the investment time changes, so this comparison isolates the effect of time.
$P=800,\quad r=0.04,\quad m=1$
Calculate the 3-year value
Convert 4% to 0.04. With annual compounding, there is one period per year, so 3 years gives 3 periods. Substitute these values into the model.
$A=800(1.04)^3\approx 899.89$
Calculate the 5-year value
The principal and rate are unchanged. Five years gives five annual periods, so use an exponent of 5.
$A=800(1.04)^5\approx 973.32$

Answer: After 3 years, the future value is about CAD 899.89. After 5 years, it is about CAD 973.32.

Check: The 5-year value is greater because the account has two more annual growth periods. The difference is about CAD 73.43.

Worked example

Example 2: The effect of compounding frequency

An account starts with CAD 1,200, earns 6% per year, and stays invested for 2 years. Compare annual compounding with monthly compounding.

Set up annual compounding
For annual compounding, interest is added once each year. The annual rate is 0.06 and there are 2 periods in 2 years.
$A=1200(1.06)^2=1348.32$
Set up monthly compounding
There are 12 periods per year, so the rate for each period is the annual rate divided by 12. Over 2 years, there are 24 monthly periods. Use these values in the model.
$A=1200(1+0.06/12)^{24}\approx 1352.59$
Compare the results
Both calculations use the same principal, annual rate, and total time. Only the compounding frequency changes. Subtracting the rounded values shows the size of the difference.
$1352.59-1348.32=4.27$

Answer: Monthly compounding gives about CAD 1,352.59, compared with CAD 1,348.32 for annual compounding. The monthly amount is about CAD 4.27 greater.

Check: The result is consistent with interest being added earlier during the year and then earning interest in later monthly periods.

Common mistakes and how to avoid them

Using 4 instead of 0.04 for a rate of 4%.

Correction: Convert a percent to a decimal by dividing by 100 before substituting it into the model.

Using the number of years as the exponent when compounding happens more than once per year.

Correction: Use the total number of periods, found by multiplying periods per year by years.

Changing the rate and the time together, then claiming the whole difference is caused by one of them.

Correction: Hold the other conditions fixed when investigating a single factor.

Rounding the account balance repeatedly during the calculation.

Correction: Keep the calculator value through the calculation and round the final amount to cents.

Lesson summary

Future value is the amount in an account at a later time, including accumulated interest.
The compound-growth model uses the principal, annual rate, time, and compounding frequency.
When other conditions stay fixed, more time, a higher rate, or more frequent compounding generally increases future value.
Clear comparisons change one factor at a time.

Check your understanding

Question 1

A fixed-rate account has the same principal and annual compounding in both cases. Which change generally increases its future value?

Investing it for more years
Reducing the annual rate
Changing the principal to a smaller amount
correctIndex must integer 0-based. fix format

Show answer and explanation

Investing it for more years

More years provide more growth periods when the other conditions stay fixed.

Question 2

A 3% annual rate is compounded monthly. What rate should be used for each monthly period?

0.03
0.0025
0.25
correctIndex

Show answer and explanation

0.0025

Convert 3% to 0.03, then divide by 12 months: the monthly rate is 0.0025, or 0.25%.

Key terms

Compounding period: The interval of time after which interest is added to an account.
Future value: The account amount at a chosen later time, including accumulated interest.
Principal: The original amount placed in an account.
Rate: The amount of interest described as a fraction or percent of the account amount for a stated time period.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation B1.6. It is a study resource, not an official curriculum publication.

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B1.6 · Investigate how time, rate, and compounding affect future value