B. Exponential Functions · Financial Mathematics: Compound Interest and Annuities

Changing the Conditions: What Makes an Annuity Grow or Cost More?

B3.6About 75 minutes

Learning goals

I can investigate, with a TVM Solver, spreadsheet or formula, how changing the payment, payment frequency, interest rate or compounding period changes an ordinary simple annuity.
I can explain why time and rate have such a strong effect on long-term savings.
I can compare loan options by payment and total interest, and see the effect of paying a little extra.
I can change one condition at a time and describe its effect in a clear sentence.

Words to know

condition (of an annuity)
One of the inputs: payment amount, payment frequency, interest rate, compounding period, or term.

sensitivity
How much an answer changes when one input changes.

controlled comparison
Changing only one condition at a time, so you know what caused the change.

long-term savings plan
Regular deposits over many years, for example for retirement or a child's education.

prepayment
Paying more than the required payment on a loan, which reduces the principal faster.

Which lever moves the most?

Imagine a control panel for a savings plan with four dials: how much you deposit, how often, the interest rate, and how long. Which dial makes the biggest difference? Most people guess "how much". The answer may surprise you. In this lesson you run the experiment, the way a scientist would: change one condition at a time and keep the others fixed.

The base case

$100 deposited at the end of every month for 30 years at 6% compounded monthly.

FV = 100[(1.005)³⁶⁰ − 1] ÷ 0.005 ≈ $100 451.50 (deposits: $36 000; interest: $64 451.50).

TVM Solver: N = 360, I% = 6, PV = 0, PMT = −100, FV = ?, P/Y = 12, C/Y = 12.

Now change one thing at a time.

Change (everything else as in the base case)	Future value	Compared with base
Base: $100/month, 6%, 30 years	$100 451.50	—
Double the payment: $200/month	$200 903.01	exactly 2 times
Lower the rate to 4%	$69 404.94	about 31% less
Raise the rate to 8%	$149 035.94	about 48% more
Shorten to 20 years	$46 204.09	less than half
Lengthen to 40 years	$199 149.07	almost 2 times

One change at a time. Doubling the payment doubles the result. But adding 10 years, or 2 percentage points of rate, does almost as much, without increasing the monthly deposit.

The same $100 a month at three rates. For the first few years the lines are close together. After 30 years, 8% gives more than twice what 4% gives (values rounded to the dollar). A small difference in rate becomes a big difference over a long time.

Changing the payment frequency (and compounding period)

In an ordinary simple annuity, the payment interval and the compounding period are the same, so changing how often you pay also changes how often interest is compounded. Keep the yearly total the same, $1 200, and the rate at 6%:

$1 200 once a year, compounded annually: n = 30, i = 0.06
$100 every month, compounded monthly: n = 360, i = 0.005
$1 200 ÷ 52 ≈ $23.08 every week, compounded weekly (the figure uses the exact value 1200 ÷ 52): n = 1560, i = 0.06 ÷ 52

Paying more often means each deposit starts earning interest sooner, and interest is compounded more often too.

Same $1 200 a year, same 6%: paying (and compounding) more often helps, but modestly. Monthly beats annually by about $5 582 after 30 years; weekly adds only about $416 more than monthly.

Worked example

Example 1: Investigating a loan

Sophie needs a $20 000 loan for a used car. The base offer is 6% compounded monthly over 5 years. Investigate (a) the rate at 4% and 8%, (b) the term at 4 and 6 years, (c) paying $50 extra each month.

Base: R = 20 000 × 0.005 ÷ [1 − (1.005)⁻⁶⁰] ≈ $386.66; total interest ≈ 60 × 386.66 − 20 000 = $3 199.60.
(a) 4%: R ≈ $368.33, interest ≈ $2 099.80. 8%: R ≈ $405.53, interest ≈ $4 331.80.
(b) 4 years: R ≈ $469.70, interest ≈ $2 545.60. 6 years: R ≈ $331.46, interest ≈ $3 865.12.
(c) Paying $436.66 a month: a TVM Solver with PMT = −436.66 gives N ≈ 52.15, so the loan is paid off with the 53rd payment (a small final payment of $64.00). Total interest ≈ $2 770.32 instead of about $3 199, saving about $429 and 7 months.

Lower rates and shorter terms reduce total interest. A longer term lowers the payment but adds interest. Paying a little extra each month shortens the loan and saves interest.

Worked example

Example 2: Start early or save more?

Jasmine deposits $25 a month from age 20 to 65 (45 years). Marcus deposits $50 a month from age 30 to 65 (35 years). Both earn 6% compounded monthly. Who has more at 65? Who deposited more?

Jasmine: n = 540. FV = 25[(1.005)⁵⁴⁰ − 1] ÷ 0.005 ≈ $68 899.82. Deposits: 540 × 25 = $13 500.
Marcus: n = 420. FV = 50[(1.005)⁴²⁰ − 1] ÷ 0.005 ≈ $71 235.51. Deposits: 420 × 50 = $21 000.
Marcus has only about $2 336 more, even though he deposited $7 500 more.

Marcus ends slightly ahead, but Jasmine earned far more interest ($55 399.82 versus $50 235.51) with much smaller deposits. Ten extra years almost made up for half the payment.

How to run a fair investigation

Write the base case and its result.
Change one condition. Recalculate. Record the new value and the change (in dollars or as a percent).
Put the base case back before changing the next condition.
Write a sentence for each: "Increasing the rate from 6% to 8% increased the future value by about 48%."

A TVM Solver or spreadsheet makes this fast: change one box and solve again.

Just for fun

Ask a retired person what financial advice they would give their younger self. A very common answer is some version of "start earlier". Another favourite is "buy that house in 1985". Only one of those is still available to you.

Hands-on activity

The annuity experiment

You need: a TVM Solver (graphing calculator or a free website), a spreadsheet, or a scientific calculator; paper for a results table.

Base case: $50 at the end of every month for 10 years at 5% compounded monthly.

Find the base future value and record it.
Change one condition at a time: payment $100; rate 3% and 7%; term 5 years and 20 years; payments of $150 every quarter compounded quarterly (same yearly total).
Record each result and the percent change from the base.
Rank the changes from biggest to smallest effect.
Write a short piece of advice (3 sentences) for a friend starting their first job, backed by your numbers.

What to look for: base ≈ $7 764.11. $100/month ≈ $15 528.23 (double). 3% ≈ $6 987.07; 7% ≈ $8 654.24. 5 years ≈ $3 400.30; 20 years ≈ $20 551.68. Quarterly $150 at 5% compounded quarterly ≈ $7 723.43. Time (term) usually has the biggest effect, followed by the payment, then the rate.

Your turn

Base case: $100 a month, 30 years, 6% compounded monthly, FV ≈ $100 451.50. If the payment is $150, what is the FV? (No new calculation needed.)

Find the FV of $100 a month for 25 years at 6% compounded monthly. How does 5 fewer years compare with the base?

Find the monthly payment on a $10 000 loan at 9% compounded monthly for 3 years, and then for 4 years. Compare total interest.

$1 000 at the end of each year for 20 years at 5% compounded annually, versus $500 at the end of every 6 months at 5% compounded semi-annually. Find both FVs.

Which change increases the future value of a savings annuity: (a) a lower rate, (b) more frequent payments with the same yearly total, (c) a shorter term?

Explain in one sentence why lowering a loan's interest rate lowers the payment.

More practice: warm-up

If you triple the regular payment (same rate and term), what happens to the FV?

True or false: lengthening the term of a savings annuity increases its future value.

True or false: lengthening the term of a loan increases the total interest paid.

For $200 every 3 months, compounded quarterly, for 5 years at 4%: find i and n.

In a fair investigation, how many conditions do you change at once?

More practice: on your own

Find the FV of $75 a month for 15 years at 3%, 5% and 7% compounded monthly. Describe the pattern.

A $30 000 loan at 6.6% compounded monthly: compare 5-year and 8-year terms (payment and total interest).

A family can afford $400 a month for a car at 7.2% compounded monthly. How much more car can they afford with a 6-year term than a 4-year term?

Payment frequency: $2 400 a year saved for 10 years at 4%. Compare once a year (compounded annually) with $200 a month (compounded monthly).

Write a one-paragraph recommendation for a 16-year-old about long-term saving, using at least two numbers from this lesson.

Challenge

For Sophie's $20 000 loan at 6% compounded monthly, how much extra per month would pay it off in about 4 years (48 payments) instead of 5? What would she save in interest?

A savings plan of $100 a month for 30 years at 6% gives about $100 451.50. What monthly payment at 4% would give the same future value? (Use technology or rearrange the formula.)

Tomas says, "Weekly deposits with weekly compounding always beat monthly deposits with monthly compounding by a lot." Use the figure in this lesson to argue for or against.

Check your understanding

Quiz

Doubling the regular deposit of a savings annuity (same rate and term) makes the future value:

Extending a savings annuity from 30 to 40 years (same deposit and 6% rate) makes the FV about:

For a loan of fixed size and rate, a longer term gives:

Keeping the yearly total and the rate the same, switching from annual to monthly payments (with matching compounding) makes the FV:

Paying $50 extra each month on a car loan will:

In a fair investigation of annuity conditions, you should:

For parents and teachers

Big idea (B3.6): Students determine, through investigation using technology (such as a TVM Solver or online tools), the effects of changing the conditions of an ordinary simple annuity (the payments, the frequency of the payments, the interest rate, the compounding period), for long-term savings plans and loans.

How to help: Do the activity together and talk about the results. Many adults wish they had learned the "start early" lesson at 16. If you have a loan, ask your teen to calculate what an extra $25 or $50 a month would save.

Common difficulties:

Changing more than one condition at once, then not knowing which caused the change.
Confusing "lower payment" with "cheaper loan".
Forgetting that in a simple annuity, changing the payment frequency also changes the compounding period (and i and n).

Rates are examples for investigation, not current offers.