Lesson 2.4: Annuities

Lesson Overview

Many real-world money decisions involve repeating payments, not one-time lump sums. Think of a monthly mortgage payment, a car loan, a subscription, a pension, or a retirement contribution.

In finance, a stream of equal payments made at regular intervals is called an annuity. This lesson shows you how to value annuities using the same TVM logic you already know: discounting (present value) and compounding (future value).

By the end, you’ll be able to answer practical questions like:

• How big a loan can I afford if I can pay $1,800 per month?
• How much will I have if I invest $300 per month for 20 years?
• Why does paying at the beginning of the month matter?

Learning Objectives

• Define an annuity and recognize common examples.
• Distinguish between an ordinary annuity and an annuity due.
• Calculate the present value (PV) of an annuity.
• Calculate the future value (FV) of an annuity.
• Use timeline thinking to avoid off-by-one period mistakes.
• Connect annuities to loan payments and savings plans.

What Is an Annuity?

An annuity is a series of equal cash flows that occur at regular intervals (monthly, quarterly, annually, etc.) for a finite number of periods.

Three things must be true for the standard annuity formulas:

• Payments are the same amount each period (constant payment).
• Payments occur at equal time intervals (regular timing).
• The discount/interest rate per period is constant.

If payments grow over time, skip around, or change, you can still value them but you’ll use DCF methods later.

Ordinary Annuity vs. Annuity Due

The difference is simple but important: when the payment happens.

• Ordinary annuity: payments occur at the end of each period.
  Example: with many loans your first payment is usually one month after you borrow.
• Annuity due: payments occur at the beginning of each period.
  Example: rent is commonly paid at the start of the month.

An annuity due is more valuable (higher PV and FV) because every payment happens one period earlier.

Timeline Thinking: Put the Cash Flows on a Line

Before using any formula, sketch a quick timeline:

• t = 0 is today
• Payments occur at t = 1, 2, 3... (ordinary annuity) or at t = 0, 1, 2... (annuity due)

Most annuity mistakes come from discounting the wrong number of periods. A timeline fixes that fast.

Present Value of an Ordinary Annuity

The present value of an ordinary annuity is the value today of a fixed set of payments received in the future.

Formula:

PV = PMT × [1 - (1 + r)^-n] / r

• PMT = payment each period
• r = interest/discount rate per period
• n = number of payments (periods)

Intuition: PV is the sum of many discounted payments but the formula gives you the total in one step.

Worked Example: PV of a Payment Stream

You are offered $200 at the end of each year for 5 years. If your discount rate is 6%, what is this annuity worth today?

• PMT = 200
• r = 0.06
• n = 5

PV = 200 × [1 - (1.06)^-5] / 0.06

The exact numeric result depends on rounding, but the setup is the skill: identify PMT, r, and n correctly.

Future Value of an Ordinary Annuity

The future value of an ordinary annuity tells you how much a series of equal contributions grows to by the end. This is the savings-plan version of annuities.

Formula:

FV = PMT × [(1 + r)ⁿ - 1] / r

Intuition: each contribution compounds for a different number of periods; the formula adds them efficiently.

Worked Example: Monthly Saving Plan

You deposit $300 at the end of each month into an account earning 6% APR compounded monthly. How much will you have after 3 years?

• PMT = 300
• Monthly rate: r = 0.06 / 12
• Number of months: n = 3 × 12 = 36

FV = 300 × [(1 + r)³⁶ - 1] / r

Again, the key is period matching: monthly payments require a monthly rate and monthly periods.

Annuity Due Adjustment (Payments at the Beginning)

If payments happen at the beginning of each period, you can start with the ordinary annuity formula and adjust by one extra period of growth:

PV (Annuity Due) = PV (Ordinary) × (1 + r)
FV (Annuity Due) = FV (Ordinary) × (1 + r)

Why this works: every payment in an annuity due occurs one period earlier, so the whole stream is worth one extra period.

Loan Payments: The Annuity You Live With

Most installment loans (mortgages, auto loans) are built on annuities. The idea:

• You borrow a lump sum today (the loan amount).
• You repay it with equal payments over time.
• The loan amount is the present value of those payments discounted at the loan rate.

In later lessons, you’ll see how payments are split between interest and principal (amortization).

How to Set Up Any Annuity Problem (A Simple Checklist)

1. Identify PMT: what is the repeating payment amount?
2. Identify timing: end of period (ordinary) or beginning (due)?
3. Match periods: monthly payments → monthly rate and number of months.
4. Choose PV or FV: are you valuing it today or at a future endpoint?
5. Plug into the correct formula and be consistent with r and n.

Common Mistakes (and Quick Fixes)

• Mistake: Using APR as if it is the rate per month.
  Fix: For monthly compounding, use r = APR/12.
• Mistake: Counting years when payments are monthly.
  Fix: Convert years to months (n = years × 12).
• Mistake: Forgetting ordinary vs due timing.
  Fix: If payments are at the beginning, multiply by (1+r).
• Mistake: Treating n as “years” by default.
  Fix: n is the number of payments (periods), not necessarily years.

Practice: Check Your Understanding

1. What makes a cash-flow stream an annuity?
2. What is the difference between an ordinary annuity and an annuity due?
3. For a monthly annuity, what must be true about r and n?
4. In a loan, what does the PV of payments represent?
5. Why is an annuity due worth more than an ordinary annuity (all else equal)?

Key Takeaways

• An annuity is a series of equal payments at regular intervals.
• PV of an annuity values payments in today’s dollars; FV values what they grow into.
• Ordinary (end-of-period) vs due (beginning-of-period) timing matters.
• Most loans and savings plans are annuities in disguise.
• Period matching (monthly vs annual) prevents most calculation errors.

What’s Next?

In Lesson 2.5: Perpetuities, you’ll learn how to value payments that continue indefinitely, and why the discount rate becomes the whole story when the timeline never ends.