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

Finding the Regular Payment: Savings Goals and Loans

C3.7About 70 minutes

Learning goals

I can rearrange the annuity formulas to find the regular payment R.
I can find the payment needed to reach a savings goal.
I can find the payment needed to repay a loan, and the total interest paid.
I can explain how the length of a loan affects the payment and the total interest.

Words to know

regular payment (R or PMT)
The equal payment made at the end of each period of an annuity.

sinking fund
An annuity used to save for a known future amount (a savings goal).

term of a loan
The length of time over which the loan is repaid.

total cost of borrowing
All the interest paid over the life of a loan: total of payments minus amount borrowed.

What will it take?

Most real money questions are about the payment: "How much do I need to save each month to have $8 000 for college or university in 4 years?" or "What will my monthly car payment be?" You already have the two annuity formulas. Today you will solve them for R.

Rearranging the formulas

Savings goal (future value known):

FV = R[(1 + i)ⁿ − 1] ÷ i → R = FV × i ÷ [(1 + i)ⁿ − 1]

Loan (present value known):

PV = R[1 − (1 + i)⁻ⁿ] ÷ i → R = PV × i ÷ [1 − (1 + i)⁻ⁿ]

Payments are money, so round R to the cent. (Lenders usually round the payment and adjust the last payment slightly; you will see this in Lesson 12.)

Decide whether the known single amount is at the end of the timeline (a savings goal, FV) or at the start (a loan, PV), then solve that formula for R.

Worked example

Example 1: Saving for post-secondary

Malik wants $8 000 in 4 years for his first year of college. His account pays 3.6% compounded monthly. How much must he deposit at the end of every month?

The $8 000 is at the end: a future value. FV = 8000, i = 0.003, n = 48.
R = 8000 × 0.003 ÷ [(1.003)⁴⁸ − 1].
(1.003)⁴⁸ ≈ 1.154635, so the bottom is about 0.154635.
R = 24 ÷ 0.154635 ≈ $155.20.
Check: 48 deposits of $155.20 total $7 449.60; the other $550.40 or so comes from interest.

About $155.20 a month.

Worked example

Example 2: A car loan payment

Emma borrows $24 000 for a car at 6.24% compounded monthly, to be repaid over 5 years with payments at the end of each month. (a) Find the monthly payment. (b) Find the total interest. (c) Compare the interest with the principal.

The $24 000 is received at the start: a present value. PV = 24 000, i = 0.0624 ÷ 12 = 0.0052, n = 60.
(a) R = 24000 × 0.0052 ÷ [1 − (1.0052)⁻⁶⁰] ≈ $466.67.
(b) Total paid: 60 × 466.67 = $28 000.20. Interest ≈ $4 000.20.
(c) The interest is about 16.7% of the $24 000 borrowed.

Payment $466.67; total interest about $4 000.20.

Worked example

Example 3: Shorter or longer loan?

A $15 000 loan at 7.2% compounded monthly can be repaid over 3 years or 5 years. Compare the payments and the total interest.

i = 0.006.
3 years (n = 36): R = 15000(0.006) ÷ [1 − (1.006)⁻³⁶] ≈ $464.53. Total interest = 36 × 464.53 − 15 000 ≈ $1 723.08.
5 years (n = 60): R = 15000(0.006) ÷ [1 − (1.006)⁻⁶⁰] ≈ $298.44. Total interest = 60 × 298.44 − 15 000 ≈ $2 906.40.
The longer loan has a payment about $166.09 lower each month but costs about $1 183.32 more in interest.

The 5-year loan is easier each month but more expensive overall.

Stretching the same loan over more years lowers the monthly payment (labels above the bars), but the yellow interest part keeps growing.

Worked example

Example 4: Two steps, saving for a stream of payments

Taylor wants to be able to withdraw $1 000 at the end of every month for 20 years, starting 30 years from now, from an account paying 4.8% compounded monthly during the withdrawals. To get there, Taylor will save at the end of every month for 30 years at 6% compounded monthly. How much must Taylor save each month?

Step 1: amount needed at the start of the withdrawals = PV of the withdrawals: 1000[1 − (1.004)⁻²⁴⁰] ÷ 0.004 ≈ $154 093.30.
Step 2: that amount is the future value goal of the savings annuity: i = 0.005, n = 360.
R = 154 093.30 × 0.005 ÷ [(1.005)³⁶⁰ − 1] ≈ $153.40.

About $153.40 a month for 30 years.

Just for fun

Why did the monthly payment go to therapy?

It had serious commitment issues: it had agreed to show up on the same day every month for five years.

Hands-on activity

Plan a real goal

You need: a calculator, paper, and a grown-up to talk with.

Pick a savings goal for 1 to 4 years from now (first-year tuition deposit, a bike, a laptop, a trip). Estimate its price.
Assume a rate of 3% compounded monthly. Find the monthly deposit needed.
Then pick something people often borrow for (a car, furniture). Use a price you find in an ad and a rate of 7.2% compounded monthly. Find the monthly payment over 2, 3 and 4 years, and the total interest for each.
Talk with your grown-up: Which loan term would you choose and why?

What to look for (model): to save $3 000 in 2 years at 3% compounded monthly, R = 3000(0.0025) ÷ [(1.0025)²⁴ − 1] ≈ $121.44 a month. For a $10 000 purchase at 7.2%: about $448.63 (2 years, interest ≈ $767.12), $309.69 (3 years, ≈ $1 148.84) and $240.39 (4 years, ≈ $1 538.72).

Your turn

How much must Malik deposit at the end of every month for 4 years at 3.6% compounded monthly to have $8 000?

Find the monthly payment on a $24 000 loan at 6.24% compounded monthly over 5 years.

Find the total interest on the loan in the last question.

How much must be deposited at the end of every month for 5 years at 6% compounded monthly to have $10 000?

Find the annual payment needed to save $5 000 in 4 years at 3% compounded annually.

Find the monthly payment on an $18 000 loan at 5.4% compounded monthly over 4 years.

More practice: warm-up

A savings goal is known. Which formula do you rearrange: FV or PV?

A loan amount is known. Which formula do you rearrange?

Without interest, how much would you need to save each month to have $1 200 in 1 year?

Without interest, what would the monthly payment on a $1 200 loan over 1 year be?

A loan has 36 payments of $310. The amount borrowed was $10 000. Find the total interest.

More practice: on your own

A $15 000 loan at 7.2% compounded monthly: find the payment and total interest over 3 years and over 5 years.

Save $2 500 in 2 years at 3% compounded monthly. Find the monthly deposit and the interest earned.

A $1 200 balance on a credit card at 19.99% compounded monthly is to be paid off in 12 equal monthly payments (no new purchases). Find the payment and the total interest.

A $3 000 loan at 9% compounded monthly is to be repaid in 24 months. Find the payment.

Explain why a longer loan has a lower payment but more total interest.

Challenge

Taylor wants $1 000 a month for 20 years (4.8% compounded monthly) after saving for 30 years (6% compounded monthly). Find the monthly savings.

A $30 000 loan at 7.2% compounded monthly over 10 years. Find the payment and compare the total interest with the principal.

Show that R = PV × i ÷ [1 − (1 + i)⁻ⁿ] gives a payment bigger than PV ÷ n whenever i > 0. Use the $15 000, 3-year, 7.2% loan as an example.

Check your understanding

Quiz

To find the monthly deposit needed to reach a savings goal, you solve:

Which is R = PV × i ÷ [1 − (1 + i)⁻ⁿ] used for?

A loan has 60 payments of $466.67 and the principal was $24 000. The total interest is about:

If a loan's term is increased from 3 years to 5 years (same rate and principal):

Without interest, saving $2 400 over 2 years would need $100 a month. With interest at 3% compounded monthly, the payment needed is:

For parents and teachers

Big idea: Expectation C3.7 asks students to solve problems involving the regular payment of an ordinary simple annuity. These are the calculations behind savings plans and loan payments.

How to help: Use a real goal your teen cares about. Ask them to estimate the payment without interest first (goal ÷ number of payments) and to say whether the true payment should be higher (a loan) or lower (savings).

Common difficulties:

Using the FV form for a loan, or the PV form for savings.
Calculator brackets: the whole [1 − (1 + i)⁻ⁿ] must be in the denominator.
Rounding R too early when it is used in later steps; round the payment, then use the rounded payment for totals, as lenders do.