3.3 Annuities and Perpetuities

Annuities and perpetuities help you figure out what a stream of regular payments is actually worth, whether those payments last for a set number of years or go on forever. These concepts show up constantly in finance: mortgages, retirement planning, stock valuation, and bond pricing all rely on them. Mastering the formulas here builds directly on the present value and future value foundations from earlier in this unit.

Annuities: Ordinary vs Due

Defining Annuities

An annuity is a series of equal payments (or receipts) that occur at evenly spaced intervals over a fixed period of time. Think mortgage payments, pension checks, or car loan installments. Every annuity has three core components: a fixed payment amount, a consistent time interval between payments, and a defined end point.

The two main types differ only in when each payment happens within the period:

• Ordinary annuity: payments occur at the end of each period
• Annuity due: payments occur at the beginning of each period

That one-period timing shift changes the math, so you need to know which type you're dealing with before you start calculating.

Ordinary Annuities

With an ordinary annuity, each payment lands at the end of the period. Most loan repayments work this way. For example, if you have a car loan, your first monthly payment is due one month after you take out the loan, not on the day you borrow.

All the standard future value and present value annuity formulas assume end-of-period payments by default. If a problem doesn't specify timing, it's almost always an ordinary annuity.

Annuities Due

With an annuity due, each payment happens at the beginning of the period. Lease payments and insurance premiums typically work this way: you pay at the start of the coverage period, not the end.

Because each payment arrives one period earlier than in an ordinary annuity, every payment has one extra period to compound (for future value) or one less period of discounting (for present value). This means annuities due always produce higher future and present values than otherwise identical ordinary annuities.

Importance of Distinguishing Between Annuity Types

Using the wrong type in your calculation will give you the wrong answer. If you treat an annuity due as an ordinary annuity, you'll underestimate its value. In practice, the difference can be meaningful: on a large mortgage or retirement fund, that one-period shift can amount to thousands of dollars.

Future and Present Value of Annuities

Future Value of an Ordinary Annuity

The future value of an ordinary annuity tells you how much a series of equal end-of-period payments will be worth at the end of the last period, assuming each payment earns compound interest.

$FVA = PMT \times \frac{(1 + r)^n - 1}{r}$

• $PMT$ = periodic payment
• $r$ = interest rate per period
• $n$ = number of periods

Example walkthrough: You invest $\$1{,}000$ at the end of each year for 5 years at 5% annual interest.

1. Identify your variables: $PMT = 1{,}000$, $r = 0.05$, $n = 5$

2. Calculate $(1 + 0.05)^5 = 1.27628$

3. Subtract 1: $1.27628 - 1 = 0.27628$

4. Divide by $r$: $0.27628 / 0.05 = 5.52563$

5. Multiply by $PMT$: $1{,}000 \times 5.52563 = \$5{,}525.63$

Notice the total you deposited was only $\$5{,}000$. The extra $\$525.63$ is the interest earned on your earlier payments.

Present Value of an Ordinary Annuity

The present value of an ordinary annuity tells you what a series of future end-of-period payments is worth right now, given a discount rate.

$PVA = PMT \times \frac{1 - (1 + r)^{-n}}{r}$

• $PMT$ = periodic payment
• $r$ = discount rate per period
• $n$ = number of periods

Example: You'll receive $\$1{,}000$ at the end of each year for 5 years, and the discount rate is 5%.

1. Calculate $(1 + 0.05)^{-5} = 0.78353$

2. Subtract from 1: $1 - 0.78353 = 0.21647$

3. Divide by $r$: $0.21647 / 0.05 = 4.32948$

4. Multiply by $PMT$: $1{,}000 \times 4.32948 = \$4{,}329.48$

So receiving $\$1{,}000$ a year for five years is equivalent to receiving about $\$4{,}329$ today. The total undiscounted payments would be $\$5{,}000$, but money in the future is worth less than money today.

Applying Future and Present Value Concepts

These formulas come up in two common scenarios:

• Saving for retirement: You know your target future value and need to solve for $PMT$ to figure out how much to save each period.
• Valuing investments: You know the expected cash flows and need to find the present value to decide if the investment is worth its price.

Always confirm whether payments happen at the beginning or end of each period before plugging into a formula.

Future and Present Value of Annuities Due

Calculating the Future Value of an Annuity Due

Since every payment in an annuity due arrives one period earlier, each payment gets one extra period of compounding. Rather than building a whole new formula, you just take the ordinary annuity result and multiply by $(1 + r)$:

$FVAD = FVA \times (1 + r)$

Example: You invest $\$1{,}000$ at the beginning of each year for 5 years at 5%.

1. From the earlier example, $FVA = \$5{,}525.63$
2. Multiply: $5{,}525.63 \times 1.05 = \$5{,}801.91$

The annuity due is worth $\$276.28$ more than the ordinary annuity, purely because each payment had one extra year to earn interest.

Calculating the Present Value of an Annuity Due

The same adjustment applies to present value. Each payment is discounted one fewer period, so:

$PVAD = PVA \times (1 + r)$

Example: You receive $\$1{,}000$ at the beginning of each year for 5 years at a 5% discount rate.

1. From the earlier example, $PVA = \$4{,}329.48$
2. Multiply: $4{,}329.48 \times 1.05 = \$4{,}545.95$

Comparing Annuities Due to Ordinary Annuities

Annuities due always have higher future and present values than ordinary annuities with the same payment, rate, and number of periods. The reason is straightforward: earlier payments mean more compounding time (for FV) and less discounting (for PV).

When deciding which type applies, consider:

• Cash flow timing: When does money actually change hands? Rent paid on the 1st of the month is an annuity due; a bond coupon paid at period-end is an ordinary annuity.
• Interest rate impact: The higher the interest rate, the bigger the gap between the two types, because that extra compounding period matters more.

Perpetuities and Present Value

Understanding Perpetuities

A perpetuity is an annuity that never ends. Payments continue forever at a fixed amount. While no investment truly lasts forever, perpetuities are a useful model for cash flows that continue indefinitely or for so long that the difference from "forever" is negligible.

Because the payments never stop, calculating a future value doesn't make sense (it would be infinite). But the present value is finite, since payments far in the future get discounted to nearly zero.

$PV = \frac{PMT}{r}$

• $PMT$ = periodic payment
• $r$ = discount rate per period

Example: A perpetuity pays $\$1{,}000$ per year and the discount rate is 5%.

$PV = \frac{1{,}000}{0.05} = \$20{,}000$

This means you'd need $\$20{,}000$ today, invested at 5%, to generate $\$1{,}000$ per year forever. That's an intuitive way to check the formula: 5% of $\$20{,}000$ is $\$1{,}000$.

Applications of Perpetuities

Perpetuities are commonly used to value:

• Preferred stock: Pays a fixed dividend with no maturity date. If a preferred share pays $\$4$ per year and investors require a 6% return, its value is $4 / 0.06 = \$66.67$.
• Consols: Government bonds (historically issued by the UK) that pay a fixed coupon forever with no principal repayment.
• Endowments: A university endowment invested to fund scholarships indefinitely behaves like a perpetuity.

Growing Perpetuities

A growing perpetuity has payments that increase at a constant rate $g$ each period. This is more realistic for many real-world situations, since cash flows often grow with inflation or business expansion.

$PV = \frac{PMT}{r - g}$

• $PMT$ = the first payment (received one period from now)
• $r$ = discount rate per period
• $g$ = constant growth rate of payments
• This formula only works when $r > g$. If the growth rate equals or exceeds the discount rate, the present value would be infinite.

Example: A stock pays a $\$1{,}000$ dividend next year, dividends grow at 2% annually, and the discount rate is 7%.

$PV = \frac{1{,}000}{0.07 - 0.02} = \frac{1{,}000}{0.05} = \$20{,}000$

This formula is the foundation of the Gordon Growth Model (also called the Dividend Discount Model), which is one of the most widely used tools for valuing dividend-paying stocks. You'll likely see it again in later units on equity valuation.