Annuities, Perpetuities, and Growing Payments

Finance April 15, 2026 18 min read Listen

Annuities, Perpetuities, and Growing Payments

Key Takeaways

• Annuities are regular payments for a set period of time — your rent, your car loan, a bond's coupon payments
• Perpetuities are regular payments that last forever — preferred stock dividends, endowments, certain valuations
• Both come in normal and growing flavors, and the timing of the first payment changes which formula you use

Finding the Right Formula

The hardest part of payment problems is not the math — it is figuring out which formula to use. There are five types of payment streams, and each one has its own formula. Pick the wrong one and you get the wrong answer, even if your math is perfect.

The good news: you only need to answer three questions to identify the right formula every time. Use the tool below — click through the questions and it will tell you exactly which formula applies.

Interactive — Formula Finder

Which formula do I use?

Answer three questions to find the right one

1. Do the payments last forever?

No — they stop at some point Yes — they go on forever

2. Do the payments grow over time?

No — same amount each period Yes — they increase by a fixed %

3. When does the first payment occur?

End of the period Beginning of the period

Now let us go through each type one by one, with the formula, the intuition, and a worked example.

The Timing Trap

Before we dive into formulas, there is one concept that trips people up more than anything else in this topic: when exactly does the payment happen?

Here is the thing most people miss. These three statements all describe the exact same moment in time:

"A payment at the end of period 1" = "a payment at time period 1" = "a payment at the beginning of period 2."

They are the same point on the timeline. The end of one period is literally the beginning of the next. Once this clicks, half the confusion in annuity problems disappears.

Why does this matter? Because the distinction between a normal annuity (payments at the end of each period) and an annuity due (payments at the beginning of each period) changes the formula and the answer. And the wording in problems can be sneaky — saying "payments begin at the start of year 2" is the same as "payments at the end of year 1." Same moment, different words.

There is also a critical detail about where the PV formula "drops" you on the timeline. The PV of a normal annuity gives you the value one period BEFORE the first payment. The PV of an annuity due gives you the value at the same time as the first payment. Keep this in mind — it affects whether you need to discount one more period to get to "today."

Normal Annuity

A normal annuity is the most common type. It is a series of equal payments made at the end of each period for a set number of periods. Your car loan payment. Your monthly rent. A bond paying semi-annual coupons. These are all normal annuities.

$$PV = \frac{PMT}{k} \left[ 1 - \frac{1}{(1+k)^n} \right]$$ Present Value of a Normal Annuity

$$FV = PMT \times \frac{(1+k)^n - 1}{k}$$ Future Value of a Normal Annuity

Worked Example — Car Loan

You take out a car loan and will pay $400/month for 5 years (60 months). The monthly interest rate is 0.5%. What is the present value of all those payments — in other words, how much did the car cost?

PV = (400 / 0.005) × [1 − 1/(1.005)⁶⁰]

PV = 80,000 × [1 − 1/1.34885]

PV = 80,000 × [1 − 0.74137]

PV = 80,000 × 0.25863

PV = $20,690.40

Your 60 monthly payments of $400 are equivalent to a lump sum of $20,690.40 today. That is the price of the car in present value terms.

Annuity Due

An annuity due is identical to a normal annuity except for one difference: payments happen at the beginning of each period instead of the end. Rent is the classic example — you pay at the start of the month, not the end.

$$PV = \frac{PMT}{k} \left[ 1 - \frac{1}{(1+k)^n} \right] \times (1+k)$$ Present Value of an Annuity Due = Normal Annuity PV × (1+k)

$$FV = PMT \times \frac{(1+k)^n - 1}{k} \times (1+k)$$ Future Value of an Annuity Due = Normal Annuity FV × (1+k)

Notice something? The annuity due formula is just the normal annuity formula multiplied by (1+k). That is it. One extra multiplication. And the reason is beautifully simple.

Since every payment in an annuity due happens one period earlier than in a normal annuity, each payment has one extra period to compound. Shifting the entire payment stream forward by one period is the same as multiplying the whole thing by (1+k). That is why the annuity due is always worth more than the equivalent normal annuity — each payment has more time to grow.

Worked Example — Rent

You sign a 12-month lease at $1,500/month paid at the beginning of each month. The monthly discount rate is 0.6%. What is the present value of all your rent payments?

Step 1 — Normal annuity PV:

PV = (1,500 / 0.006) × [1 − 1/(1.006)¹²]

PV = 250,000 × [1 − 1/1.07442]

PV = 250,000 × [1 − 0.93075]

PV = 250,000 × 0.06925

PV = $17,312.50

Step 2 — Multiply by (1+k) for annuity due:

PV = $17,312.50 × 1.006

PV = $17,416.38

The annuity due is worth slightly more than the normal annuity ($17,416 vs $17,313) because each payment happens one period earlier and therefore has one extra period of compounding value.

Test Yourself

All else equal, an annuity due is always worth more than a normal annuity because:

• A. The payments are larger in an annuity due
• B. Each payment occurs one period earlier, giving it one extra period to compound
• C. There are more payments in an annuity due
• D. The discount rate is lower for an annuity due

Correct: B. The payments are the same size and there are the same number of them. The only difference is timing — each payment in an annuity due happens one period sooner. That one-period head start means each payment has one extra period of compounding, which is captured mathematically by multiplying by (1+k).

Growing Annuity

A growing annuity is a series of payments that increase by a fixed percentage each period for a set number of periods. Think of a salary that grows at 3% per year for 30 years before retirement. Or a lease agreement where rent increases by 2% annually.

$$PV = \frac{PMT_1}{k - g} \left[ 1 - \left( \frac{1+g}{1+k} \right)^n \right]$$ Where PMT₁ is the first payment and g is the growth rate

The formula looks intimidating, but the logic is the same as a normal annuity — you are just adjusting for the fact that each payment is slightly larger than the last. The key difference is the (k − g) term in the denominator instead of just k, and the growth adjustment inside the brackets.

Worked Example — Rising Salary

You just got a job that pays $60,000 in the first year, with a 3% raise every year. You plan to work for 25 years. If the discount rate is 8%, what is the present value of your total career earnings?

PV = 60,000 / (0.08 − 0.03) × [1 − ((1.03)/(1.08))²⁵]

PV = 60,000 / 0.05 × [1 − (0.95370)²⁵]

PV = 1,200,000 × [1 − 0.30832]

PV = 1,200,000 × 0.69168

PV = $830,016

Your entire career earnings — 25 years of salary growing at 3% per year — are worth about $830,000 in today's dollars. That is the present value of your working life, which is a pretty powerful number to know.

Perpetuity

A perpetuity is the simplest formula in all of finance. It is a series of equal payments that last forever. No end date. No maturity. Just the same payment, every period, into infinity.

$$PV = \frac{PMT}{k}$$ Present Value of a Perpetuity — that is it

That is the whole formula. Payment divided by discount rate. It sounds too simple to be real, but it works because of a mathematical property: as the number of periods approaches infinity, the normal annuity formula simplifies down to this.

Where does this show up in real life? Preferred stock is the classic example. A preferred share pays a fixed dividend forever with no maturity date. If a preferred share pays $5 per year and the required return is 8%, its value is $5 / 0.08 = $62.50.

Worked Example — Endowment

A university wants to create a scholarship that pays $10,000 per year forever. The endowment earns 5% annually. How much does the university need to deposit today?

PV = $10,000 / 0.05

PV = $200,000

If the university deposits $200,000 today and earns 5% every year, it generates $10,000 in interest annually — which funds the scholarship forever without ever touching the principal. That is the power of a perpetuity.

Test Yourself

A preferred share pays an annual dividend of $4. If investors require a 10% return, what should the share be worth?

• A. $4.00
• B. $44.00
• C. $40.00
• D. $400.00

Correct: C. Preferred stock is a perpetuity — fixed payment forever. PV = PMT / k = $4 / 0.10 = $40. That is the price investors should pay for a $4 annual dividend when they require a 10% return.

Growing Perpetuity

A growing perpetuity is a series of payments that grow at a fixed rate forever. This is the foundation of the Gordon Growth Model (also called the Dividend Discount Model) used to value stocks with stable, growing dividends.

$$PV = \frac{PMT_1}{k - g}$$ Where PMT₁ is the NEXT payment and g is the constant growth rate

Two critical things to understand here.

First: PMT₁ is the next payment, not the current one. If a company just paid a dividend of $2 and dividends grow at 5%, then PMT₁ = $2 × 1.05 = $2.10. You plug in the next dividend, not the one that was just paid.

Second: k must be greater than g. If the growth rate equals or exceeds the discount rate, the denominator becomes zero or negative, and the formula spits out infinity or a negative number — neither of which makes sense. Intuitively, if cash flows grow faster than the rate at which we discount them, the value would be infinite. In reality, no company grows faster than the economy forever, so this constraint is usually satisfied.

Worked Example — Stock Valuation

A company just paid a dividend of $3.00. Dividends are expected to grow at 4% per year forever. The required return is 11%. What is the stock worth?

Step 1 — Find PMT₁:

PMT₁ = $3.00 × (1 + 0.04) = $3.12

Step 2 — Apply the formula:

PV = $3.12 / (0.11 − 0.04)

PV = $3.12 / 0.07

PV = $44.57

The stock is worth $44.57 today. This is the price an investor should be willing to pay for an infinite stream of dividends that start at $3.12 and grow 4% every year, discounted at 11%.

Test Yourself

Why does the growing perpetuity formula require that the discount rate (k) must be greater than the growth rate (g)?

• A. Because if g ≥ k, the cash flows grow faster than they are being discounted, producing an infinite value
• B. Because negative growth rates are not allowed in the formula
• C. Because the payments would stop growing if g exceeded k
• D. It is a mathematical convention with no real-world significance

Correct: A. If cash flows grow at the same rate or faster than the discount rate, each future payment is worth the same or more than the previous one in present value terms. Summing an infinite series of values that do not shrink gives you infinity. In reality, no asset can sustain a growth rate above the discount rate forever — which is why this constraint is both a mathematical necessity and an economic truth.

The Big Picture

Here is how everything connects. All five payment types are variations of the same idea — a stream of cash flows moving through time. The differences are just about duration (finite vs infinite), growth (fixed vs growing), and timing (beginning vs end of period).

If you remember one thing from this post, remember this: start by identifying what type of payment stream you are dealing with, then pick the formula. The three questions from the formula finder at the top of this post will always get you there. Duration, growth, timing. That is the decision tree. Everything else is just plugging in numbers.

The hierarchy: The perpetuity is a special case of an annuity where n goes to infinity. The growing perpetuity is a special case of a growing annuity where n goes to infinity. The annuity due is a normal annuity shifted forward by one period. Once you see the connections, you realize there is really only one concept here — payments through time — expressed five different ways.

Watch It

If you want to see these formulas applied step by step with practice problems, check out these videos.

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Ismail Francillon

Ex-McKinsey · Adjunct Finance Professor at Concordia · Co-Founder at Atwater Partners

On this page

• Finding the Right Formula
• The Timing Trap
• Normal Annuity
• Annuity Due
• Growing Annuity
• Perpetuity
• Growing Perpetuity
• The Big Picture
• Watch It
• Keep Going