Annuities: The Present Value of an Income Stream

0:00Saturday morning: $1,200,000, or the cash option ▶

It's Saturday morning, and Nora walks into your kitchen holding a letter with both hands. A single ticket in the state lottery, and the letter says she won. She hands it to you, and you read it twice, because the first time it does not sound real. One point two million dollars, paid out in ten yearly checks. Underneath, another line, and she sits down: or skip the annuity, it says, and take the cash option today. She does the sum everyone does: ten checks of a hundred and twenty thousand, one point two million. You watch her write it down on the back of the letter and underline it twice. So the cash should be one point two million too, she says, and you are not so sure. You know that a hundred and twenty thousand ten years from now is not a hundred and twenty thousand this morning. So what is that row of checks worth to you today?

1:04One tool, and one warning: the word annuity ▶

Say your money could sit in a savings account at five percent a year. In that account money grows by one plus the rate each year, so one year back is one division by it. Ten equal checks, one at every equal step of time: in finance that row has a name, an annuity. One warning, because the word is a trap: on the insurance market an annuity is a contract you buy from an insurer. Underneath that contract sits this math, and this math is the video.

1:37The long way: ten checks, each brought back to today ▶

Put the ten checks on the counter in a row, one for every year. Say the first one comes a year from today, say they all stay the same size, and leave the tax office out of it. A real jackpot's checks grow a little each year, but that is a different video. Before anything moves, guess what part of the one point two million survives the trip back to today. The first check comes back one year, one division by one point zero five: just above a hundred and fourteen thousand. The tenth comes back ten years, ten of those divisions: just under seventy-four thousand. Every check in between lands a little lower than the one before it. Add the ten up, and the row is worth nine hundred and twenty-six thousand dollars and change today. Over a quarter of a million short of the letter, and it gets worse: what if the checks never stopped coming?

2:34Forever: a row that never ends is worth $2.4 million ▶

Suppose the letter had said forever instead of ten years: a check every year, without end. Adding those up never ends, so ask instead what pile of money, at five percent, pays out one check every year and never runs down. Five percent of the pile must be one check, so the pile is the check divided by five percent. That is two point four million dollars: a check every year until the end of time, worth that much today, not infinity. And that pile is what the endless row is worth today: own the pile, and the account turns it into the row. Forever is easy, and ten years is just forever with the tail cut off.

3:20The cut: forever minus the tail is the annuity formula ▶

Picture the endless row and cut it right after the tenth check. What is left is another endless row, exactly the same, except it starts ten years later. So the tail is worth two point four million too, but as of ten years from now. Bring that back ten years, ten divisions, and today it is worth about one point four seven million. Your ten checks are forever minus the tail: two point four million minus one point four seven, the number the ten slow divisions gave. Write it in letters and you get the formula every textbook prints: P over r, minus that pile shrunk back n years. Take P over r out in front, and it reads P over r, times one minus one over one plus r to the n.

4:13The check: the formula against the ten divisions ▶

Now the check, because a formula that came out of a story still has to earn it. Fill in the letters: the check, five percent, ten years. The line lands on nine hundred and twenty-six thousand, six hundred and eight dollars and nineteen cents. The same number, to the cent: the formula has earned its keep. So at five percent the row has one honest price; what happens when you change that five?

4:42Change the rate: the same row at 3% ▶

Nora points at the jar on the counter: but my money only makes three percent. Fair, and that changes the price: five percent was an assumption, and three is a different one. Run the same row at three percent and it is worth just over a million today. The gap shrinks but does not close: a check next year is never worth a check today. The formula does not pick your rate; it prices the row once you have.

5:14Thirty checks: why the cash option is about half ▶

Nora asks you where the rest of her money went, so make the row longer, like a real jackpot. The big national ones pay you thirty installments, each a little larger than the last. Keep the checks level, the first one still a year out, and give her thirty. The letter would say three point six million. Bring your thirty checks back to today at five percent, and the row is worth one point eight four million. About half the headline is gone, and nobody took it from you: the calendar did, one division at a time. So the cash option you see under a billboard sits that far below it: it is the row's value today. And every check after year thirty, all of them, forever, would add only five hundred and fifty-five thousand more.

6:03What the math says, and what it doesn't ▶

Nora puts the letter down and asks the only question left: so which one do I take? The math does not answer that, and neither will I. What the math says is narrower: at five percent, her ten checks are worth nine hundred and twenty-six thousand in cash today. Whatever cash number a letter prints, you measure it against the row's value at your own rate, not against the headline. Shrink the checks and only the answer shrinks: ten checks of a thousand come to seven thousand seven hundred and twenty-two the same way. Next time a headline is a row of payments, you can read what it is worth today, and that's the math behind the money.

6:45What's next: The Asymmetry of Losses ▶

Next time: the asymmetry of losses, because losing half of your money and then gaining half back leaves you a quarter short. And for you: which rate is yours when you price a row of payments, the jar's, the account's, or something else? I'm the MoneyMathician, and if you subscribe, the next one will find you. Until then: do the math!