Recall from Week 1 that we valued single cash flows — one lump sum, moved forward or backward through time. That's a powerful starting point, but most financial decisions don't involve just one payment. Think about a car loan: you make the same payment every month for five years. Think about retirement savings: you contribute the same amount to your 401(k) every paycheck. Think about a business that signs a lease and commits to paying $15,000 a year for the next five years.
In each of these cases, we're dealing with a stream of equal payments made at regular intervals. Finance has a name for this: an annuity. Formally, an annuity is a series of equal cash flows occurring at equal intervals for a fixed number of periods. The word comes from the Latin annus (year), but annuities can involve any regular interval — monthly car payments, quarterly dividend payments, or annual lease payments all qualify.
The key insight is that you could value each payment individually — discount or compound each one using the single cash flow formulas from Week 1, then add them up. And that would give you the right answer. But it would be tedious, especially when you're dealing with 360 monthly mortgage payments or 30 years of retirement contributions. Annuity formulas are shortcuts that collapse all those individual calculations into one expression.
The most common type is an ordinary annuity, where payments occur at the end of each period. When you make a car payment, it's due at the end of the month — that's an ordinary annuity. When a company pays annual rent at year-end, that's an ordinary annuity. Unless you're told otherwise, assume the annuity is ordinary.
Suppose you'll receive a payment of PMT at the end of each period for t periods, and you can earn a rate of r per period. What is the present value of that entire stream? You'd discount the first payment back one period, the second payment back two periods, and so on. When you sum all those discounted values and simplify the algebra, you get:
The annuity factor does all the heavy lifting. It tells you how many "dollar-equivalents" each dollar of payment is worth today, given the discount rate and the number of periods. When you multiply the factor by the payment amount, you get the lump-sum value today that's equivalent to the entire stream.
Two things worth noting about how this factor behaves. First, a higher discount rate shrinks the factor — the more aggressively you discount, the less a future stream is worth today. Second, more periods increase the factor — more payments means more value, all else equal. But each additional payment adds a little less than the one before it, because it's further away and gets discounted more heavily. We'll see this concretely in the next two examples.
Notice that five payments of $15,000 total $75,000 in raw cash, but the present value is only about $63,185. That gap of nearly $12,000 is the time value of money at work — you're giving up the ability to invest those funds at 6% while you wait for each payment to arrive.
What happens if Green Acre's discount rate were higher — say 10% instead of 6%? The payments are the same, but each one is worth less in today's dollars because the opportunity cost of waiting is greater.
This comparison illustrates one of the most important relationships in finance: the present value of any future cash flow stream falls when the discount rate rises. We'll see this principle again — dramatically — when we study bond pricing in Week 3.
The PV formula answers the question "what is this stream of payments worth today?" But there's an equally important mirror image: "if I make regular contributions and let them compound, how much will I have at the end?" This is the future value of an annuity, and it's the fundamental tool behind every retirement savings calculation.
The logic is straightforward: the first payment has the longest to compound (it sits for t − 1 periods), the second-to-last payment compounds for just one period, and the very last payment doesn't compound at all — it arrives at the same moment you're measuring the future value. The formula sums all of those compounded amounts.
You contributed a total of $60,000 over 20 years ($3,000 × 20), but you end up with nearly $123,000. The extra $63,000 is pure compounding — interest earned on your contributions, and then interest earned on that interest. The longer the horizon, the more dramatic this effect becomes. Recall from Week 1 how compound growth accelerates over time; here, that same principle applies to each payment in the stream.
What happens to the annuity formula if the payments go on forever? That is, what if t → ∞? This might sound like a theoretical curiosity, but it's actually quite practical. Certain financial instruments — like preferred stock and some endowments — promise to make the same payment indefinitely. A stream of equal cash flows that continues forever is called a perpetuity.
Mathematically, something elegant happens when you let t go to infinity in the PV annuity formula. The term (1 + r)−t shrinks toward zero, which means the numerator in the annuity factor approaches 1, and the whole factor simplifies to just 1/r. The formula becomes remarkably clean:
This is one of the simplest and most useful formulas in all of finance. You'll see it again when we value preferred stock in Week 4 (a perpetuity of dividend payments) and when we study the Gordon Growth Model (a growing perpetuity).
Think about why this works: if you invest $50,000 at 5%, you earn $2,500 each year — exactly the scholarship amount. You never touch the principal, so the $50,000 stays invested and generates $2,500 again next year, and the year after that, forever. The perpetuity formula captures this self-sustaining logic in a single division.
Everything we've done so far assumes payments at the end of each period — the ordinary annuity convention. But some payment streams start immediately. Rent is the classic example: your landlord wants payment on the first of the month, not the last. Insurance premiums, lease payments, and retirement annuity payouts often work the same way.
An annuity due is identical to an ordinary annuity except that each payment occurs at the beginning of the period rather than the end. Because every payment arrives one period sooner, each payment has one fewer period of discounting (for PV) or one additional period of compounding (for FV). The adjustment is simple: compute the ordinary annuity value first, then multiply by (1 + r).
The annuity due is worth slightly more ($14,012.43 vs. $13,942.72) because each payment arrives one month sooner. The $69.71 difference might seem small on a monthly lease, but the same principle on larger amounts — like a business lease or retirement payout — can produce significant differences.
So far we've computed PV or FV given a known payment. But in practice, the payment is often what you're trying to find. How much do I need to save each year to reach my retirement goal? What will my monthly car payment be? These are "solve for PMT" problems, and they're among the most practical applications of annuity math.
The algebra is straightforward — just rearrange the annuity formula. If you know the present value, rate, and number of periods, you can isolate PMT:
Let's apply both. The first is a loan payment problem — one of the most common calculations in personal finance. The second is a savings goal.
Over 60 months, you'll pay a total of $469.49 × 60 = $28,169.40 — meaning you pay $3,169.40 in interest on top of the $25,000 principal. Notice that the first step was converting the annual rate to a monthly rate and the annual term to months. When payments are monthly, everything — rate, number of periods — must be expressed in monthly terms. This is a common source of errors, so watch for it.
Ten annual contributions of $3,793.40 total $37,934 in deposits. The remaining $12,066 comes from compound interest. That's the payoff for starting early and letting compounding work — a theme we introduced in Week 1 and that keeps showing up.
When a bank advertises a savings account at "5.2% APR, compounded quarterly" or a credit card at "18% APR, compounded monthly," the quoted rate doesn't tell you exactly how much interest you'll actually earn or owe over a year. That's because the annual percentage rate (APR) — also called the stated rate or quoted rate — is simply the periodic rate multiplied by the number of periods per year. It does not account for compounding within the year.
For example, if a credit card charges 1.5% per month, the APR is 1.5% × 12 = 18%. But you don't actually owe just 18% per year, because the interest charged in January starts compounding in February, and so on. The actual annual rate, accounting for this within-year compounding, is higher. This true rate is called the effective annual rate (EAR), and it's what you actually experience.
The EAR is always greater than or equal to the APR. They're equal only when m = 1 (annual compounding). The more frequently interest compounds, the larger the gap between APR and EAR.
This matters for comparison shopping. When you're choosing between two financial products — two savings accounts, two loans — the APR alone can be misleading if the compounding frequencies differ. The EAR is the true apples-to-apples comparison.
This example drives home the lesson: more frequent compounding helps, but it can't overcome a meaningfully lower stated rate. When comparing financial products, always convert to EAR first, then decide.
When you take out a loan — a mortgage, a car loan, a business term loan — your payment each period is fixed. But what's happening inside that payment changes over time. Each payment has two components: an interest portion (which compensates the lender for the use of their money) and a principal portion (which actually reduces what you owe). Understanding this split is called loan amortization.
Here's the key mechanism: interest each period is calculated on the remaining balance, not the original loan amount. In the early periods, the balance is large, so the interest portion of your payment is large and the principal portion is small. As you pay down the balance, the interest portion shrinks and the principal portion grows — even though your total payment stays the same. By the end of the loan, almost all of your payment goes toward principal.
An amortization schedule is a table that shows, for each period, the payment, the interest component, the principal component, and the remaining balance. Building one is mechanical — once you know the payment amount, you just apply the same logic period by period.
Look at how the interest/principal split shifts over time. In Year 1, roughly 27% of the payment is interest. By Year 4, it's down to about 7%. This pattern is universal in amortizing loans — the early payments are interest-heavy, and the balance reduction accelerates over time.
For Green Acre, total payments over four years are $30,192.08 × 4 = $120,768.32. Since the loan was $100,000, the total interest cost is $20,768.32. A manager evaluating this equipment purchase should factor this financing cost into the decision — something we'll formalize when we study net present value in Week 6.
Each $30,192 payment is split between interest and principal. Early payments are interest-heavy (27% in Year 1); by Year 4, interest is only 7% of the payment. The principal portion grows as the outstanding balance shrinks.
A home mortgage works on exactly the same principle as Green Acre's equipment loan — just with monthly payments, a much longer term, and (usually) a much larger balance. The math is identical; only the numbers change. But the scale reveals something striking about how mortgages work in practice.
Read that total interest number again: $446,406. You're paying more in interest than the original loan amount. In the first month, 86% of your payment goes to interest and only $316 actually reduces the balance. This is why financial advisors talk so much about the impact of even small rate differences on mortgages — on a 30-year, $350,000 loan, each quarter-point of interest rate is worth tens of thousands of dollars over the life of the loan.
It's also why making extra principal payments early in a mortgage is so powerful: every extra dollar you pay in Year 1 avoids 30 years of compound interest.
Before we move to the practice problems, let's revisit perpetuities from a different angle. In Example 4, we knew the payment and the rate and found the price. But you can also rearrange the perpetuity formula to find the rate. If you know the payment and the current price, the required return is simply r = PMT / PV. This version shows up frequently in stock valuation — we'll formalize it in Week 4.
This is a preview of how the same TVM toolkit we're building now connects directly to asset valuation. The perpetuity formula for preferred stock is essentially identical to the zero-growth dividend discount model you'll learn in Week 4 — you're just applying it to a different context.
| Formula | Expression |
|---|---|
| PV of Ordinary Annuity | PV = PMT × [(1 − (1 + r)−t) ÷ r] |
| FV of Ordinary Annuity | FV = PMT × [((1 + r)t − 1) ÷ r] |
| PV of Perpetuity | PV = PMT ÷ r |
| Annuity Due Adjustment | Valuedue = Valueordinary × (1 + r) |
| Solve for PMT (PV known) | PMT = PV × [r ÷ (1 − (1 + r)−t)] |
| Solve for PMT (FV known) | PMT = FV × [r ÷ ((1 + r)t − 1)] |
| Effective Annual Rate (EAR) | EAR = (1 + APR ÷ m)m − 1 |
You win a small lottery that pays $40,000 per year for 25 years, with the first payment arriving one year from today. If your discount rate is 8%, what is this prize worth in today's dollars?
You plan to save $5,500 at the end of each year for 30 years in a retirement account earning 9% annually. How much will you have when you retire?
A university wants to fund a $8,000-per-year scholarship that will last forever. If the endowment earns 4% per year, how much must be donated today?
You're taking out a $180,000 mortgage at 5.5% APR, compounded monthly, for a 15-year term. What is your monthly payment?
You're choosing between two savings accounts. Account A offers 6.1% APR compounded semiannually. Account B offers 6.0% APR compounded monthly. Compute the EAR for each account and determine which one actually pays more.
Blue Acre Financial Services makes a $20,000 business loan to a client at 6% annual interest for 3 years with equal annual payments. Compute the annual payment and construct the full amortization schedule showing the interest component, principal component, and remaining balance for each year.
Explain in your own words why an annuity due is always worth more than an otherwise identical ordinary annuity. Your explanation should reference the time value of money — don't just say "because the formula has an extra (1 + r)."
You currently have $10,000 in a savings account that earns 7% per year. You also plan to deposit an additional $2,000 at the end of each year for the next 15 years into the same account. How much will you have at the end of 15 years? (Hint: treat this as two separate pieces — a lump sum and an annuity — and add them together.)