C. Discrete Functions · Financial Applications: Interest, Present Value and Annuities

What If? Changing the Payment, Frequency, Rate or Term

C3.6About 70 minutes

Learning goals

I can use technology (or a calculator) to investigate how changing the payment, the payment frequency, the interest rate or the term affects an annuity.
I can explain why doubling the payment doubles the future value, but doubling the rate or the time does much more.
I can describe how a larger payment or a shorter term reduces the total interest on a loan.
I can make and justify a recommendation about a savings plan or a loan.

Words to know

payment frequency
How often payments are made: annually, semi-annually, quarterly, monthly, weekly, and so on.

sensitivity
How much an answer changes when one input changes.

what-if analysis
Changing one input at a time (in a spreadsheet or TVM Solver) to see its effect on the result.

Four dials

Every ordinary simple annuity is controlled by four "dials": the payment R, the payment frequency (which, for a simple annuity, is also the compounding frequency), the interest rate, and the term. Financial planners turn one dial at a time and watch what happens. That is exactly what the curriculum asks you to do with technology. A spreadsheet or TVM Solver makes it quick, but every result in this lesson can also be checked with the formulas and a scientific calculator.

Setting up a what-if spreadsheet

Put each input in its own cell and refer to the cells in the formula. Then you can change one cell and the answer updates.

B1: payment (100) · B2: annual rate (0.05) · B3: periods per year (12) · B4: years (20)
B5: **=B1((1+B2/B3)^(B3B4)-1)/(B2/B3)** (the future value formula)

Most spreadsheets also have a built-in function: **=FV(B2/B3, B3B4, -B1)* gives the same answer. (The payment is entered as negative because it is money you pay out; Lesson 12 explains this.)

Change (one at a time)	Future value	Compared with base
Base: $100/month, 5%, 20 years	$41 103.37	total deposits $24 000
Double the payment: $200/month	$82 206.73	exactly 2 times
Rate up to 6%	$46 204.09	+$5 100.72
Rate down to 4%	$36 677.46	−$4 425.90
Term up to 30 years	$83 225.86	about 2.02 times (with 1.5 times the deposits)
$1 200 at the end of each year (5% compounded annually)	$39 679.14	−$1 424.22 (same $24 000 deposited)
$300 at the end of each quarter (5% compounded quarterly)	$40 835.64	−$267.73

The effect of changing each dial on a $100-a-month plan. Every row except "Double the payment" deposits $1 200 a year.

Doubling the payment doubles the result. Adding 10 years to the term (only $12 000 more deposited) also roughly doubles it. Paying yearly instead of monthly lowers the result a little.

Worked example

Example 1: Why does doubling the payment exactly double the future value?

Explain using the formula, then explain why changing the rate or the term does not act this way.

FV = R × [(1 + i)ⁿ − 1] ÷ i. The bracket part does not involve R, so FV is directly proportional to R. Double R and FV doubles; triple R and FV triples.
The rate and the term sit inside the exponential part (1 + i)ⁿ. Changing them changes how much every payment compounds, so the effect grows faster than proportionally.
For example, 1.5 times the term (30 years instead of 20) gives about 2.02 times the future value.

R is a constant multiplier; i and n are inside an exponential.

Worked example

Example 2: Same deposits per year, different frequency

Compare saving $1 200 a year for 20 years at 5% in three ways: $1 200 at the end of each year (5% compounded annually), $300 at the end of each quarter (5% compounded quarterly), and $100 at the end of each month (5% compounded monthly).

Annual: 1200[(1.05)²⁰ − 1] ÷ 0.05 ≈ $39 679.14.
Quarterly: 300[(1.0125)⁸⁰ − 1] ÷ 0.0125 ≈ $40 835.64.
Monthly: 100[(1 + 0.05/12)²⁴⁰ − 1] ÷ (0.05/12) ≈ $41 103.37.
More frequent payments put money in the account sooner, and (because these are simple annuities) the interest also compounds more often. Both make the total larger.

Monthly > quarterly > annually, but the differences are fairly small.

Turning the dials on a loan

For a loan, a bigger payment pays the principal down faster, so less interest is charged. A $20 000 loan at 6% compounded monthly:

Term	Monthly payment	Total paid	Total interest
3 years	$608.44	$21 903.84	$1 903.84
4 years	$469.70	$22 545.60	$2 545.60
5 years	$386.66	$23 199.60	$3 199.60
6 years	$331.46	$23 865.12	$3 865.12

The shorter the term, the higher the payment but the lower the total interest. (Payments rounded to the cent.)

Higher monthly payments mean fewer payments and less interest. Going from 6 years to 3 years cuts the interest roughly in half.

Worked example

Example 3: Paying a little extra

On the $20 000 loan at 6% compounded monthly, the 4-year payment is $469.70. What if you pay $500 a month instead?

Guess and check with the present value formula: we need the number of $500 payments whose present value is $20 000.
n = 44: 500[1 − (1.005)⁻⁴⁴] ÷ 0.005 ≈ $19 704.12 (less than $20 000, so 44 payments are not enough).
n = 45: ≈ $20 103.60 (more than $20 000, so the 45th payment can be smaller).
A TVM Solver gives N ≈ 44.74: 44 full payments and a smaller 45th payment of about $370.33.
Total interest ≈ 44 × 500 + 370.33 − 20 000 ≈ $2 370.33, compared with $2 545.60 for 48 payments of $469.70.

The loan is paid off 3 months sooner and saves about $175.27 in interest.

One dial at a time

In a what-if investigation, change only one input and keep the others fixed. Otherwise you cannot tell which change caused the effect. Record each result in a table, as above.

Just for fun

A financial planner walks into a bakery and asks for a dozen doughnuts.

"Would you like them all now, or one at the end of every month?"

"Monthly, please. And can you tell me the present value of a doughnut?"

Hands-on activity

Your own what-if table

You need: a free spreadsheet app (or a calculator and paper).

Set up the four input cells and the future value formula from this lesson.
Start with a plan you might really use: for example $50 at the end of every month for 10 years at 4% compounded monthly.
Make a table with these rows: base; double the payment; rate + 1%; rate − 1%; term + 5 years; yearly payments of 12 × your monthly amount (with yearly compounding).
Write three sentences: which change helped most, which helped least, and which you could actually control in real life.

What to look for (model for $50/month, 4%, 10 years): base ≈ $7 362.49; double payment ≈ $14 724.98; 5% ≈ $7 764.11; 3% ≈ $6 987.07; 15 years ≈ $12 304.52; $600 a year at 4% compounded annually ≈ $7 203.66. The payment and the term are the dials you control most.

Your turn

Find the future value of $100 at the end of every month for 20 years at 5% compounded monthly.

Without recalculating, what is the future value if the payment is $300 instead?

Find the future value if the rate is 6% instead of 5% (still $100 monthly for 20 years).

Find the future value if the term is 10 years instead of 20.

Find the monthly payment on a $20 000 loan at 6% compounded monthly for 3 years and the total interest.

Find the monthly payment on the same loan over 5 years at 8% compounded monthly instead of 6%.

More practice: warm-up

If the payment of an annuity is tripled, what happens to its future value?

Which usually gives more: $1 200 at the end of each year, or $100 at the end of each month (same annual rate, compounding matching the payments)?

For a loan, does a longer term increase or decrease the monthly payment?

For a loan, does a longer term increase or decrease the total interest?

Why should you change only one input at a time in a what-if table?

More practice: on your own

Compare $1 200/year (5% compounded annually), $300/quarter (5% compounded quarterly) and $100/month (5% compounded monthly) for 20 years.

A $20 000 loan at 6% compounded monthly: find the total interest for terms of 4, 5 and 6 years.

$100 a month for 20 years: find the future value at 3%, 4%, 5%, 6% and 7% compounded monthly. Do equal rate increases give equal gains?

The 4-year payment on a $20 000 loan at 6% compounded monthly is $469.70. How many $500 payments are needed instead? Show a check.

A family can save either $150 a month for 15 years or $250 a month for 9 years, both at 4.8% compounded monthly. Which plan gives more at the end?

Challenge

Show that doubling the term of a savings annuity more than doubles the future value. Use $100/month at 5% compounded monthly for 10 and 20 years, and explain why.

Rates on a $20 000, 5-year loan (compounded monthly) are 4%, 6%, 8% and 10%. Find each payment and total interest. Is the total interest proportional to the rate?

A loan of $20 000 at 6% compounded monthly is paid with $500 a month. Find the final, smaller payment and the interest saved compared with 48 payments of $469.70.

Check your understanding

Quiz

The payment of a savings annuity is doubled and nothing else changes. The future value:

Which change to a 20-year, $100-a-month savings plan at 5% increases the future value the most?

For a loan with a fixed rate and principal, a shorter term means:

Which plan gives more after 20 years at 5%: $1 200 at the end of each year (compounded annually) or $100 at the end of each month (compounded monthly)?

In a what-if spreadsheet, why are inputs placed in their own cells?

Paying $500 a month instead of $469.70 on a $20 000 loan at 6% compounded monthly:

For parents and teachers

Big idea: Expectation C3.6 asks students to use technology to investigate how changing the payments, the payment frequency, the interest rate or the compounding period affects ordinary simple annuities, such as long-term savings plans and loans.

How to help: This lesson is a great place for a real conversation about saving and borrowing. Ask your teen which dial they could control in their own life (usually the payment and the start date) and which they cannot (the rate).

Common difficulties:

Changing several inputs at once and drawing conclusions.
Expecting every change to act like the payment (proportional). The rate and the term are inside an exponential.
Comparing plans with different total deposits without noticing. Always note the total deposited.
Remember that in this course the payment and compounding periods always match (simple annuities).