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

The Present Value of an Ordinary Simple Annuity

B3.7About 70 minutes
Learning goals
  • I can explain what the present value of an annuity means, for a fund that pays out and for a loan.
  • I can use PV = R[1 − (1 + i)⁻ⁿ] ÷ i with a scientific calculator.
  • I can find the present value with a TVM Solver or a spreadsheet PV function.
  • I can use present value to find how much I can borrow, and the total interest on a loan.

Words to know

present value of an annuity (PV)
The single amount that, invested now, can pay for all the future payments exactly. For a loan, it is the amount borrowed.
withdrawal
Money taken out of an account.
loan principal
The amount borrowed: the present value of the loan payments.
total interest on a loan
Total of all payments minus the amount borrowed: nR − PV.
lump sum
A single payment, as opposed to a series of payments.

How much do you need today?

Aisha is heading to college in Thunder Bay. She wants to withdraw $600 at the end of each month for the 8 months of the school year to cover groceries and her bus pass. Her savings account pays 3% compounded monthly. How much must be in the account at the start so that the eighth withdrawal empties it exactly?

That amount is the present value of an annuity: one amount now that is equivalent to a stream of payments later.

The formula

For an ordinary simple annuity with regular payment R, rate i per period and n payments:

PV = R[1 − (1 + i)⁻ⁿ] ÷ i

It comes from discounting every payment back to today and adding:

PV = R(1 + i)⁻¹ + R(1 + i)⁻² + … + R(1 + i)⁻ⁿ

This is another geometric series. The first payment is discounted for 1 period; the last for n periods.

Two big uses:

  • Savings that pay out: PV is what you need now to fund future withdrawals.
  • Loans: PV is the amount you can borrow if you will make n payments of R. The total interest is nR − PV.
Present value: $1 000 at the end of each year for 5 years, 5% compounded annuallynowyear 1year 2year 3year 4year 5$1 000$1 000$1 000$1 000$1 0001000(1.05)⁻¹ = $952.381000(1.05)⁻² = $907.031000(1.05)⁻³ = $863.841000(1.05)⁻⁴ = $822.701000(1.05)⁻⁵ = $783.53PV = $4 329.48Each payment is discounted back to today.
Each future $1 000 payment is worth less today, the further away it is. Adding the five present values gives $4 329.48: invested today at 5%, it can pay $1 000 at the end of each of the next 5 years.
Worked example

Example 1: Aisha's school-year fund

Find the amount needed now to withdraw $600 at the end of each month for 8 months, at 3% compounded monthly.

  1. R = 600, i = 0.03 ÷ 12 = 0.0025, n = 8.
  2. PV = 600[1 − (1.0025)⁻⁸] ÷ 0.0025.
  3. Calculator: 1.0025 ^ (−8) ≈ 0.9802231; 1 − 0.9802231 = 0.0197769; ÷ 0.0025 ≈ 7.910745; × 600 ≈ 4746.45.
  4. She withdraws 8 × 600 = $4 800 in total, so interest supplies 4800 − 4746.45 = $53.55.

Aisha needs $4 746.45 in the account at the start. Interest supplies the other $53.55.

Technology

  • TVM Solver: N = 8, I% = 3, PV = ? (solve), PMT = 600, FV = 0, P/Y = 12, C/Y = 12. PV comes out negative (−4746.45): you pay that in now, and receive $600 each month (positive).
  • Spreadsheet: =PV(0.03/12, 8, 600, 0) gives −4746.45 for the same reason.
  • Calculator: type 600 × (1 − (1 + 0.03 ÷ 12) ^ (−8)) ÷ (0.03 ÷ 12). Do not round partway through: rounding 0.9802231 to 0.98 would give a very wrong answer.
Worked example

Example 2: How much car can I afford?

Owen can afford car payments of $350 at the end of each month for 5 years. The dealer offers financing at 6.9% compounded monthly. What is the largest loan he can take? How much interest would he pay?

  1. R = 350, i = 0.069 ÷ 12 = 0.00575, n = 60.
  2. PV = 350[1 − (1.00575)⁻⁶⁰] ÷ 0.00575 ≈ 17 717.88.
  3. Total of payments: 60 × 350 = $21 000.
  4. Total interest: 21 000 − 17 717.88 = $3 282.12.

Owen can borrow at most $17 717.88. He would pay $3 282.12 in interest over the 5 years. (Insurance, gas and maintenance come on top of the payment, so a smart budget leaves room for them.)

Worked example

Example 3: Lump sum or payments?

A contest offers $350 000 now or $2 000 at the end of every month for 20 years. If the winner could invest at 4.8% compounded monthly, which is worth more today?

  1. Find the present value of the payments: R = 2000, i = 0.004, n = 240.
  2. PV = 2000[1 − (1.004)⁻²⁴⁰] ÷ 0.004 ≈ 308 186.61.
  3. Compare: $308 186.61 (payments, in today's dollars) versus $350 000 (lump sum).

At 4.8%, the $350 000 lump sum is worth more today, by about $41 813. (The payments add up to $480 000, which is why comparing totals without discounting is misleading.)

A fund that pays $500 at the end of each month (4.8% compounded monthly)05 00010 00015 00020 0000612182430364248MonthsBalance ($)start: $21 797.12$0 right after the 48th withdrawalafter 2 years: $11 420.25
A fund starting with $21 797.12 (the present value of 48 payments of $500) shrinks a little faster each month: interest is earned on a smaller and smaller balance. It reaches $0 exactly at the 48th withdrawal.
Just for fun

Present value is the only time in math class when "the future isn't worth as much as you think" is a correct answer and not just a gloomy opinion.

Hands-on activity

Envelope fund

You need: 6 envelopes, paper "money" (or play money), a calculator.

  1. Label the envelopes Month 1 to Month 6. Your fund will pay $100 at the end of each month for 6 months. Pretend the account pays a (very generous) 2% per month.
  2. For each envelope, calculate how much you would need to put in today so it grows to exactly $100 by its month: 100(1.02)⁻¹, 100(1.02)⁻², … , 100(1.02)⁻⁶. Write it on the envelope.
  3. Add the six amounts. Then check with PV = 100[1 − (1.02)⁻⁶] ÷ 0.02.
  4. Explain to someone at home why the Month 6 envelope needs the least money now.

What to look for: $98.04, $96.12, $94.23, $92.38, $90.57, $88.80; total ≈ $560.14, which matches the formula. The later the payment, the longer its money can grow, so the less you need today.

Your turn

1

Find the present value of $1 500 at the end of each year for 4 years at 4% compounded annually.

2

How much must be in an account now to withdraw $250 at the end of each month for 3 years at 9% compounded monthly?

3

A loan is repaid with $450 at the end of each month for 4 years at 7.2% compounded monthly. How much was borrowed? Total interest?

4

Write the TVM Solver entries for the last question.

5

Find the present value of $800 at the end of every 3 months for 3 years at 9% compounded quarterly.

6

Explain the difference between the future value and the present value of the same annuity.

More practice: warm-up

1

Evaluate 1.05⁻¹ to 4 decimals.

2

For a loan, what does the present value of the payments represent?

3

A loan of $9 000 is repaid with 24 payments of $400. What is the total interest?

4

In the PV formula, is the exponent on (1 + i) positive or negative?

5

Without calculating, is the PV of 10 payments of $100 more or less than $1 000 (positive rate)?

More practice: on your own

1

A scholarship will pay $5 000 at the end of each year for 4 years. How much must the donor deposit now at 3% compounded annually?

2

You can afford $300 a month for 5 years at 8.4% compounded monthly. What is the maximum loan? Total interest?

3

A phone plan charges $150 at the end of each month for 3 years, including the phone. If money is worth 6% compounded monthly, what is the plan worth today?

4

$120 at the end of each month for 2 years at 15% compounded monthly: find PV, and explain why the total payments ($2 880) are much more.

5

Prize A: $10 000 at the end of each year for 20 years. Prize B: $120 000 now. Which is worth more at 5% compounded annually?

Challenge

1

Prize A in the last question is worth more than Prize B at 5%. Would it still be at 6%? Show the calculation and explain.

2

A fund pays $1 200 at the end of each month for 3 years at 4.2% compounded monthly. Find the PV. Then find the balance left right after the 12th payment. (Hint: it is the PV of the 24 payments still to come.)

3

Show that for the same R, i and n, FV = PV(1 + i)ⁿ, using R = $1 000, i = 0.05, n = 5 from this unit.

Check your understanding

Quiz

1.

Which formula gives the present value of an ordinary simple annuity?

2.

For a loan, the present value of the payments is:

3.

A loan of $12 000 is repaid with 48 payments of $290. Total interest:

4.

PV of $1 000 at the end of each year for 2 years at 10% compounded annually:

5.

As the interest rate rises (same R and n), the present value of an annuity:

6.

In a TVM Solver, the amount needed now to fund withdrawals of $500 per month is solved as PV. Its sign is:

For parents and teachers

Big idea (B3.7): Students solve problems involving the present value of an ordinary simple annuity: the amount needed now to fund future withdrawals, or the amount that can be borrowed for a given payment. Total interest on a loan is total payments minus the present value.

How to help: Use real decisions: "If you can afford $250 a month, how expensive a used car could you finance?" Point out that a loan's monthly payment is only part of the cost of owning something (insurance, fuel, repairs).

Common difficulties:

  • Using the future value formula when the money is needed now (ask: is the unknown at the start or the end?).
  • Typing the negative exponent incorrectly; use the (−) key.
  • Comparing prize options by adding up payments instead of finding present values.

Rates are examples for practice, not current offers.