Annuities and Perpetuities

What if you get the same amount of money every year for 20 years — how much is that stream worth to you today? And what if the payments never stop? This chapter gives you the tools to answer those questions quickly, without adding up lots of separate cash flows.

The Big Picture#

Many financial arrangements — car loans, mortgages, lottery prizes, pensions, even stock prices — involve a stream of cash flows that repeats. Adding them up one by one would be slow and easy to get wrong. In this chapter we’ll build simple formulas that value a whole series of equal or growing payments in one go. By the end, you’ll be able to move comfortably between present value, future value, and the special cases where payments start right away or go on forever.

The Present Value of a Level Annuity (PVIFA)#

Imagine a deal that promises to pay you 1,000 is worth in today’s money (at 5% interest) and then add them up:

That works, but it becomes a real hassle when there are 30 or 300 payments. Because the cash flows are all the same size and spaced evenly, we can collapse the whole sum into a single factor.

Annuity: A stream of equal cash flows paid at the end of every period for a fixed number of periods. (When payments happen at the end of each period, we call it an ordinary annuity.)

Let the cash flow be , the interest rate per period be , and the number of periods be . The present value of an ordinary annuity is

The fraction on the right is the Present Value Interest Factor of an Annuity, often abbreviated PVIFA. It is simply the sum of the discount factors for the first periods:

So we can write the present value as

Why does the formula work? Think of it as taking a perpetual stream of per year (a perpetuity, which we’ll see soon) and then cutting off the payments after year . You don’t need to memorise the derivation, but the intuition is useful: an annuity is a perpetuity with the tail chopped off.

Example. A lottery offers $50,000 per year for 20 years, with the first payment a year from now. If you use a 6% discount rate, the prize is worth

So even though the total payments add up to 573,500.

📝 Section Recap: The present value of a level annuity is the cash flow times PVIFA, a factor that packs all the discounting into one number. It saves you from discounting each payment individually.

The Future Value of a Level Annuity (FVIFA) and Its Link to PVIFA#

Sometimes you want to know how much a stream of deposits will grow to by the end of the investment horizon. For instance, if you save $200 at the end of every month for 10 years in an account that pays 0.5% per month, what will the balance be right after the last deposit?

You could work out what each $200 deposit will grow to one by one, but the equal‑payment pattern gives a quicker way. The future value of an ordinary annuity is

The factor is the Future Value Interest Factor of an Annuity, or FVIFA.

FVIFA: The multiplier that, when applied to the periodic cash flow, gives the future value of a level annuity immediately after the last payment.

Example. You deposit n = 120r = 0.005$). The future value is

Now, notice the relationship between the present value and the future value of the same annuity. If you take the present value we computed earlier and compound it forward periods at the same rate , you must get the future value:

Substituting the annuity formulas:

Cancel , and you get a simple link:

This makes perfect sense: the future value factor is just the present value factor grown by periods of interest. You can always move between PV and FV of the same annuity by multiplying or dividing by .

📝 Section Recap: The future value of a level annuity is , and FVIFA is simply PVIFA compounded forward. Understanding this link means you only need to remember one core factor and can derive the other.

When Payments Come at the Start: Annuities Due#

So far we’ve assumed payments come at the end of each period — an ordinary annuity. But many real‑world payments come at the start. Rent is usually due on the first of the month, insurance premiums are paid upfront, and lease contracts often require the first payment immediately.

Annuity Due: An annuity where each cash flow occurs at the beginning of the period, rather than at the end.

Since each payment comes a period sooner, an annuity due is worth more than an ordinary annuity with the same cash flows, rate, and length. The extra value comes from the fact that every payment earns (or avoids) one extra period of interest.

The simplest way to value an annuity due is to start with the ordinary annuity formula and shift the whole stream forward by one period. Mathematically, you multiply the ordinary annuity value by :

Example. Revisit the lottery prize of 573,495. The annuity‑due present value is

Why the increase? The first $50,000 is received immediately and therefore is not discounted at all. The remaining 19 payments each get discounted by one year less than before.

If you were saving $200 at the beginning of each month instead of the end, the future value after 10 years would be

The extra $164 comes from each deposit spending an additional month earning interest.

📝 Section Recap: An annuity due is worth exactly times its ordinary counterpart because every cash flow arrives one period earlier. This simple multiplier works for both present and future values.

Perpetuities: Infinite Cash Flows, Finite Value#

What if the payments never stop? A stream of equal cash flows that continues forever is called a perpetuity. At first glance, an infinite number of payments might seem to have infinite value, but because distant cash flows are discounted so heavily, the present value is finite — and surprisingly simple.

Perpetuity: A level stream of cash flows that occurs at the end of every period and has no end date.

If you receive every period forever, and the discount rate is , the present value is

Intuition. Suppose you want to create a scholarship that pays 5,000 in annual interest without touching the principal. That lump sum is exactly 125,000$. The perpetuity formula is simply the amount of capital that yields the desired cash flow as interest.

Example. A preferred stock promises a fixed dividend of $2 per year, paid at year‑end, in perpetuity. With a required return of 8%, the stock’s value is

Now, many real‑world cash flows grow over time. A growing perpetuity pays one period from now, then the next period, the period after that, and so on forever. Think of a company whose dividends are expected to increase by a steady percentage each year.

The present value of a growing perpetuity, with first cash flow one period from now, growth rate , and discount rate , is

This formula only works if the growth rate is less than the discount rate (). If , the denominator becomes zero or negative, which would imply an infinite or undefined value — economically, the cash flows would grow at least as fast as they are discounted, so their present value would explode. The condition is therefore essential; we call it the convergence condition.

Example. A stock just paid a dividend of 1.04 if dividends grow at 4%). With a required return of 10%, the present value of all future dividends is

📝 Section Recap: A level perpetuity is worth , and a growing perpetuity is worth provided . These compact formulas give a finite present value even though the payments never end, because distant cash flows become negligible in today’s terms.

Growing Annuities#

Often payments increase at a steady rate for a fixed number of periods — a salary that rises 2% each year for 30 years, or a lease where the rent goes up by a fixed percent each year. A growing annuity combines the finite horizon of an ordinary annuity with a constant growth rate .

Growing Annuity: A stream of cash flows lasting periods, where the first cash flow occurs one period from now and each subsequent cash flow is times the previous one.

The present value formula is a natural extension of the level annuity and the growing perpetuity:

Notice what happens when : the term becomes , and the formula simplifies to the ordinary level‑annuity PVIFA. If and , the numerator approaches 1, leaving exactly the growing perpetuity formula . So the growing annuity formula sits neatly between the two extremes.

Example. You will receive a bonus that starts at $10,000 one year from now and grows by 3% each year for a total of 15 payments. Your discount rate is 7%. The present value is

If the growth rate were zero, the same 15‑year level annuity at 7% would be worth 91,07910,000, even though later payments surpass it.

📝 Section Recap: A growing annuity’s present value blends the finite horizon of an ordinary annuity with a constant growth rate. The formula reduces to the level annuity when and to the growing perpetuity when goes to infinity, provided .

Summary#

You now have a handful of handy shortcuts. Instead of working through each cash flow one by one, you can price a whole stream with a single multiplication. We began with the basic level annuity and linked its present and future values. Then we looked at payments that start right away (annuities due) and payments that never stop (perpetuities). Finally, we added growth, giving you growing annuities and growing perpetuities. These tools will show up again and again — in bonds, stocks, loans, and retirement planning — so mastering them now really pays off.