Annuities and Present Value

Ask anyone whether they would rather have $1000 today or $1000 in five years, and almost everyone chooses "today" — but few can say precisely how much more today's money is worth. That instinct, made exact, is the single most important idea in all of finance: money has a time value. A dollar in your hand can be invested to grow, so a dollar promised later must be worth less now. Annuities and present value are the mathematical machinery that turns this intuition into numbers you can trust — the same numbers that price your mortgage, size your retirement savings, and value a company's future cash flows.

This page teaches you to move money forward and backward through time with confidence, and to handle streams of payments (annuities), not just single lump sums. Master this and you hold the key to loans, pensions, bonds, leases, and investment decisions alike.

Learning Objectives

• Explain the time value of money and why a future dollar is worth less than a present dollar.
• Compute the future value and present value of a single lump sum.
• Distinguish an ordinary annuity from an annuity due and choose the right formula.
• Derive and apply the annuity present-value and future-value formulas.
• Solve real problems: loan payments, retirement savings, and comparing offers.
• Recognise and correct the most common mistakes in discounting cash flows.

Quick Answer

The time value of money says a sum today is worth more than the same sum later because it can earn interest. To move a single amount forward $n$ periods at rate $i$ per period, multiply by $(1+i)^n$ (future value); to move it backward, divide by the same factor (present value). An annuity is a series of equal payments $PMT$ made each period. For an ordinary annuity (payments at the end of each period) the present value is $PV = PMT \cdot \frac{1-(1+i)^{-n}}{i}$ and the future value is $FV = PMT \cdot \frac{(1+i)^n - 1}{i}$. An annuity due pays at the start of each period, so both values are simply multiplied by an extra $(1+i)$. These four factors handle most everyday financial calculations.

Where It Came From

The need behind present value is ancient: lenders have charged interest since Babylonian times, and merchants always knew that credit had a cost. But the systematic mathematics grew out of two pressures — commerce and mortality.

In the 17th century, European governments and cities raised money by selling life annuities: you paid a lump sum, and the seller promised you a fixed income for the rest of your life. The seller faced a hard question — how much should we charge? Charge too little and you go bankrupt paying the long-lived; charge too much and no one buys. Answering it required combining two ideas: how likely a buyer was to still be alive to collect each future payment, and how much each of those uncertain future payments was worth today.

The first mortality tables came from John Graunt (1662), who analysed London death records. The Dutch statesman and mathematician Johan de Witt, working with Christiaan Huygens, published a 1671 treatise pricing annuities by weighting each future payment by both its survival probability and a discount factor — arguably the birth of actuarial mathematics. Soon after, Edmond Halley (of comet fame) built a cleaner life table from the city of Breslau in 1693 and computed annuity values from it. Compound-interest tables themselves had been laid out by Richard Witt in 1613 and refined throughout the era.

The stakes were enormous. Mispriced government annuities helped bankrupt states; correctly priced ones funded navies and pensions. When formal pension systems and life-insurance companies arose in the 18th and 19th centuries, they were built entirely on these formulas. So the algebra below is not academic decoration — it is the tool that let humanity make credible promises about money across decades.

The Time Value of Money and Single Sums

The core engine is compound interest. If you invest a present amount $PV$ at an interest rate $i$ per period, after one period it becomes $PV(1+i)$; the next period earns interest on that larger balance, and so on. After $n$ periods:

$FV = PV \,(1+i)^n$

Rearranging gives the present value of a single future sum — the amount you would need now to end up with $FV$ later:

$PV = \frac{FV}{(1+i)^n} = FV\,(1+i)^{-n}$

The act of dividing by $(1+i)^n$ is called discounting, and $i$ is the discount rate. Notice the symmetry: growing money forward multiplies; pulling it back divides.

Worked example. You will receive $5000 in 3 years. Money can safely earn $6\%$ per year. What is that promise worth today?

$PV = \frac{5000}{(1.06)^3}$

Compute $(1.06)^3 = 1.191016$. Then

$PV = \frac{5000}{1.191016} \approx 4198.10$

So $5000 in three years is worth about $4198.10 today. Check by growing forward: $4198.10 \times 1.191016 \approx 5000$. Correct.

This one relationship already lets you compare offers across time — always drag every cash flow to the same date before comparing.

Ordinary Annuities: Level Streams of Payments

Most financial life is not a single sum but a stream: rent, salary, loan repayments, pension cheques. An annuity is a sequence of equal payments $PMT$ at equal time intervals. In an ordinary annuity, each payment lands at the end of its period (the standard for loans and bonds).

Rather than discount each payment separately, we sum a geometric series. The present value collapses to:

$PV = PMT \cdot \frac{1-(1+i)^{-n}}{i}$

and the future value (the total accumulated just after the last payment) is:

$FV = PMT \cdot \frac{(1+i)^n - 1}{i}$

Worked example (PV — a loan). You borrow to buy a car and will repay $400 at the end of each month for 4 years. The lender charges $12\%$ per year, compounded monthly, so $i = 0.12/12 = 0.01$ per month and $n = 48$ months. How much are you actually borrowing?

$PV = 400 \cdot \frac{1-(1.01)^{-48}}{0.01}$

Here $(1.01)^{48} = 1.612226$, so $(1.01)^{-48} = 0.620260$. Then

$PV = 400 \cdot \frac{1-0.620260}{0.01} = 400 \cdot \frac{0.379740}{0.01} = 400 \times 37.9740 \approx 15,189.58$

You are borrowing about **$15,189.58** today, and you will repay$ 400 \times 48 = $19,200$ in total — the $4010-ish difference is interest.

Worked example (FV — saving). You deposit $200 at the end of every month into a fund earning $6\%$ per year ($i = 0.005$ monthly) for 10 years ($n = 120$). What will you have?

$FV = 200 \cdot \frac{(1.005)^{120} - 1}{0.005}$

Now $(1.005)^{120} = 1.819397$, so

$FV = 200 \cdot \frac{0.819397}{0.005} = 200 \times 163.8794 \approx 32,775.87$

About **$32,775.87**, of which only$ 200 \times 120 = $24,000$ came out of your pocket — the rest is compounding at work.

Annuities Due: Paying at the Start

An annuity due shifts every payment to the beginning of its period. Rent and insurance premiums usually work this way: you pay for the month before you live in it. Because every payment now sits one full period earlier, each has one extra period to earn (or be discounted over) interest. The correction is a single clean factor of $(1+i)$:

$PV_{\text{due}} = PV_{\text{ordinary}} \times (1+i), \qquad FV_{\text{due}} = FV_{\text{ordinary}} \times (1+i)$

Worked example. Take the same saving plan — $200 monthly at $i = 0.005$ for $n = 120$ — but deposit at the start of each month. Multiply the earlier result:

$FV_{\text{due}} = 32,775.87 \times 1.005 \approx 32,939.75$

Paying just a fraction earlier each month yields roughly $164 more over ten years. The lesson: timing matters, and paying early into an investment (or being paid early on a debt) is always advantageous to the payment receiver.

Real-World Applications

• Loans and mortgages. Solving the ordinary-annuity PV formula for $PMT$ gives your monthly payment: $PMT = PV \cdot \frac{i}{1-(1+i)^{-n}}$. Every mortgage amortisation schedule is this formula unrolled.
• Retirement and pensions. How big a pot do you need at 65 to draw $40,000 a year for 25 years? That is an ordinary-annuity present value. Pension funds run this in reverse to size contributions.
• Bond pricing. A bond is an annuity of coupon payments plus a lump-sum face value at maturity; its fair price is the present value of both streams at the market yield.
• Leasing vs. buying. Discount the lease payments (an annuity due) and compare to the purchase price to see which is genuinely cheaper.
• Lottery and settlement choices. "Lump sum or 20 annual payments?" is answered by comparing the annuity's present value to the cash offer.

Common Mistakes

1. Mismatching the rate and the period. Students plug an annual rate into a monthly problem. If payments are monthly, you must use $i = \text{annual rate}/12$ and $n$ in months. Correction: always make the rate, the period count, and the payment frequency agree.

2. Confusing ordinary annuity with annuity due. Using the end-of-period formula for start-of-period payments (or vice versa) is off by a factor of $(1+i)$ every time. Correction: ask when the payment occurs. End of period → ordinary; beginning → due (multiply by $(1+i)$).

3. Adding cash flows from different dates. Writing "$400 \times 48 = 19,200$, so I borrowed $19,200" ignores discounting entirely. Money at different times cannot be added directly. Correction: discount (or grow) every cash flow to one common date first, then combine.

Comparison and Connections

Present value is the inverse operation of future value, and an annuity is just many single-sum calculations bundled by a geometric series. The table below lines up the four workhorse formulas.

Quantity	Formula	When to use
Future value, single sum	$FV = PV(1+i)^n$	Grow one amount forward
Present value, single sum	$PV = FV(1+i)^{-n}$	Discount one amount back
Ordinary annuity PV	$PMT \cdot \dfrac{1-(1+i)^{-n}}{i}$	Value of end-of-period payments now
Ordinary annuity FV	$PMT \cdot \dfrac{(1+i)^n-1}{i}$	Accumulated value of end-of-period payments

A perpetuity is the limiting case of an annuity that never ends: as $n \to \infty$, the ordinary-annuity PV simplifies to $PV = PMT/i$. This connects to infinite geometric series in algebra. The whole subject also links tightly to exponential growth (the $(1+i)^n$ factor) and, through continuous compounding, to the number $e$ and calculus.

Practice Questions

Recall

State the formula for the present value of an ordinary annuity, and describe in words what each symbol means.

Answer: $PV = PMT \cdot \frac{1-(1+i)^{-n}}{i}$, where $PMT$ is the level payment, $i$ is the interest rate per period, and $n$ is the number of payments (all measured per the same period).

Understanding

Why is an annuity due always worth more than an otherwise identical ordinary annuity?

Answer: Every payment occurs one period earlier, so each has an extra period to earn interest (for FV) or is discounted over one fewer period (for PV). Mathematically, both values are multiplied by $(1+i) > 1$.

Application

You deposit $1500 at the end of each year for 5 years at $8\%$ annually. How much will you have just after the last deposit?

Answer: $FV = 1500 \cdot \frac{(1.08)^5 - 1}{0.08}$. Since $(1.08)^5 = 1.469328$, $FV = 1500 \cdot \frac{0.469328}{0.08} = 1500 \times 5.86660 \approx 8799.90$.

Analysis

A lottery offers you either $100,000 now or $8000 at the end of each year for 20 years. If money is worth $6\%$ per year, which is better, and by how much?

Answer: Value the annuity: $PV = 8000 \cdot \frac{1-(1.06)^{-20}}{0.06}$. With $(1.06)^{20} = 3.207135$, $(1.06)^{-20} = 0.311805$, so $PV = 8000 \cdot \frac{0.688195}{0.06} = 8000 \times 11.46992 \approx 91,759$. The $100,000 lump sum is worth about $8241 more today, so take the cash — unless you can earn less than about $5.1\%$, the rate at which the two break even.

FAQ

Is the "interest rate" the same as the "discount rate"? They are the same number viewed from opposite directions. When growing money forward you call $i$ the interest rate; when pulling money back you call it the discount rate. The choice of rate should reflect your true opportunity cost or the risk of the cash flow.

What if compounding is more frequent than the payments, or vice versa? Convert to a consistent per-period rate first. If a rate is $12\%$ compounded monthly but payments are annual, find the equivalent annual rate: $(1.01)^{12} - 1 \approx 12.68\%$, then use that. Always align the rate's period with the payment period.

How do I find the payment on a loan? Solve the annuity-PV formula for $PMT$: $PMT = PV \cdot \frac{i}{1-(1+i)^{-n}}$. Plug in the amount borrowed, the per-period rate, and the number of payments.

Why does my calculator's answer differ slightly from mine? Almost always rounding. Keep full precision in the $(1+i)^n$ factor until the final step; rounding that factor early can throw the answer off by dollars over long horizons.

Do these formulas assume payments never change? Yes — a standard annuity has level (equal) payments. For payments that grow at a constant rate $g$, use a growing annuity formula; for irregular cash flows, discount each one individually and add. The level-payment case is just the simplest and most common.

Quick Revision

• Time value of money: a dollar today is worth more than a dollar later.
• Single sum: $FV = PV(1+i)^n$; $PV = FV(1+i)^{-n}$.
• Ordinary annuity (end of period): $PV = PMT\frac{1-(1+i)^{-n}}{i}$, $FV = PMT\frac{(1+i)^n-1}{i}$.
• Annuity due (start of period): multiply either result by $(1+i)$.
• Loan payment: $PMT = PV\frac{i}{1-(1+i)^{-n}}$.
• Perpetuity: $PV = PMT/i$.
• Golden rule: rate, period count, and payment frequency must all match; move all cash flows to one date before comparing.

Related Topics

Prerequisites

• Financial Mathematics overview
• Exponents and geometric series (Algebra)

Related Topics

• Compound interest and rates of growth
• Bond valuation and yields

Next Topics

• Loan amortisation and mortgage schedules
• Retirement planning and pension funding