Annuity — the math of recurring cash flows · Finance Fundamentals Quest

Money that arrives in pieces, not all at once

So far we've discounted single future payments. But most real-world money arrives in streams. Salary every month. Mortgage payments every month. Bond coupons every six months. Pension every year. Each is a payment, repeated, on a schedule.

A stream of equal payments at equal intervals is called an annuity. Doesn't matter what the actual length or amount is — once it's regular, the math works the same way.

Annuity = sum of PVs — that's literally all it is

If a savings or pension plan pays you ₩100 every year for 5 years, what's the whole stream worth today? Each payment is its own PV. Year 1's ₩100 is worth ₩100 / (1+r). Year 2's is worth ₩100 / (1+r)^2. And so on.

The total PV is just the sum:

P V = \frac{100}{1 + r} + \frac{100}{( 1 + r ) ^{2}} + \frac{100}{( 1 + r ) ^{3}} + \frac{100}{( 1 + r ) ^{4}} + \frac{100}{( 1 + r ) ^{5}}

That's an annuity in plain form. Just five separate present values, added together. No magic. Track 1's integral lesson — area under a curve, summed up — is exactly this picture, in discrete form.

The closed-form formula (for when you're tired of summing)

Mathematicians don't like writing five terms when one will do. The sum above has a closed-form expression:

P V = C \times \frac{1 - ( 1 + r ) ^{- n}}{r}

Where C is the periodic payment and n is the number of payments. You don't need to derive this. You don't even need to remember it. Spreadsheets and calculators do it for you (the function is usually called PV or NPV). What you need is the picture: a stream of equal payments, each discounted to today, summed up.

Here's the example with the formula. ₩100 per year for 5 years at 5%:

P V = 100 \times \frac{1 - ( 1.05 ) ^{- 5}}{0.05} \approx 100 \times 4.329 \approx ₩432.95

So a stream of five ₩100 payments is worth ₩432.95 today, not ₩500. Each future payment lost a bit of value to discounting; the further out it is, the more it lost.

Real cases — where annuities actually live

Annuities are everywhere once you know what you're looking at:

Mortgages: equal monthly payment for 30 years. The bank gave you a chunk today (the loan); you pay back an annuity. The PV of your future payments at the bank's rate must equal what they lent you.
Pensions: retirement plan pays you ₩X per month from age 65 until you die (or for a fixed term). What's that worth in today's terms? Annuity math.
Bonds: a coupon bond pays you fixed interest twice a year. The coupon stream is an annuity; the face value at maturity is a separate single PV. Track 7 returns to this.
Lottery payouts: "₩1,000,000,000 prize, paid out as ₩50,000,000/year for 20 years." That headline ignores discounting — the actual lump-sum-equivalent today is much less.

The takeaway

An annuity is just sum of equal payments, each discounted. The closed-form formula saves you from typing out the sum, but it doesn't add new physics. Whenever you see "regular payments over time," translate it into "annuity → discount each one → add up." That picture is enough for almost any TVM problem you'll meet.

Annuity — the math of recurring cash flows