Time Value of Money: What $1,030 Is Worth Today

0:03Sunday evening: $1,000 now, or $1,030 next year ▶

It's Sunday evening, your phone lights up on the counter, and it's Nora about the money you lent her. I can pay you back a thousand now, she writes, or a thousand and thirty next year. Then she adds the line that makes it your problem: you pick. You put the phone down, and the kitchen is very quiet, and on the counter stands the jar where you keep your spare cash. Say that jar pays three percent a year. Leave her money in there, and a thousand turns into a thousand and thirty. So on the counter, the two offers balance exactly. And that is strange, because one is money you can hold tonight and the other is a sentence in a text message. So which one do you take, and what would have to be true for the answer to flip?

0:56Why money now is worth more — the three reasons ▶

Money now is worth more than the same money later, and there are three reasons why. The first is opportunity cost: money in your hand can be put to work, and money promised cannot. That is what the time value of money is actually about. The second is inflation: prices creep up, so the same money buys less later. That was the last video, and it comes back here with a number. The third is uncertainty: a promise can break, and a payment that never arrives is worth nothing at all. All three push the same way, so we will put a price on each one, on this exact question.

1:40The tool: going back in time you divide, you don't subtract ▶

First we need a way to move money through time. Forward you already know: a dollar in the jar becomes a dollar and three cents, so you multiply by one point zero three. So to bring a dollar back a year, you do the opposite: you divide by one point zero three. Most people take three percent off instead and call it ninety-seven cents, which is close but not the same thing. A dollar next year is ninety-seven point zero nine cents today, so waiting a year costs two point nine one cents. Give the line its letters: F V for the money later, P V for its worth today, r for the rate, n for the years. Here n is one, but the same line carries any number of years, which is what makes it worth writing down.

2:30At 3% the two offers are exactly tied ▶

Now put her offer through it and get its present value: a thousand and thirty, divided by one point zero three. It lands on a thousand dollars exactly, the same as taking the money tonight, and that is not a coincidence. At three percent, her two offers are worth exactly the same to you. Take the quick route instead, three percent off a thousand and thirty, and you get nine hundred and ninety-nine dollars and ten cents. Ninety cents too low, and on a dead tie ninety cents is the whole answer. Because when two offers are exactly equal, the smallest thing you have not counted yet decides it.

3:13Reason 1 — opportunity cost: the rate isn't a property of the offer ▶

Here is the thing about that three percent: it is not a property of her offer but of what you would otherwise have done. That is what opportunity cost means, and the jar is not your only option. Suppose another account pays you five percent, the one from the very first video. Bring her thousand and thirty back through five percent instead, and it is nine hundred and eighty dollars and ninety-five cents. Her later money is now nineteen dollars and five cents short of the thousand she can hand you tonight. Turn it around: at five percent she would have to offer a thousand and fifty to make waiting worth it. So the rate you discount with is not a fact you look up, it is the best thing you could otherwise do. Hold on to that nineteen dollars, because the tie at three percent gets tested twice more.

4:06Reason 2 — inflation: keeping up is not getting ahead ▶

The second reason is the one the last video was about: say prices also creep up three percent over that year. Then her thousand and thirty, arriving next year, buys exactly a thousand dollars of today's groceries. So the jar's three percent is not growth at all, it is exactly enough to stand still. Now do it in groceries: five percent against three percent prices is not two percent, it is one point nine four. Discount her thousand of real money at that rate, and you land on nine hundred and eighty dollars and ninety-five cents. The same answer as before, which is the point: inflation does not add a second cost, it tells you what the first one means.

4:49Reason 3 — uncertainty: on a tie, any chance decides it ▶

The third reason is the one people leave out, and here it is the loudest: probably good for it is not certainly. Before the number lands, guess: how big does that chance have to be before it decides the whole thing? Say there is one chance in twenty that something goes wrong and the money never comes. You get a thousand and thirty or you get nothing, but the average is what you can plan with: nine hundred and seventy-eight fifty. Measure that against the jar's own three percent, where her two offers were exactly tied. It comes back to nine hundred and fifty dollars today, fifty short of the money she can hand you tonight. And here is the sharp edge of a tie: it takes any chance at all to break it. At one in a thousand the gap is about a dollar, small but still on the same side. So certainty, on a dead heat, is worth more than the interest rate you were arguing about.

5:51The answer, and where that rate comes from ▶

So you pick up the phone and answer Nora: pay me now. Not because a thousand beats a thousand and thirty, but because that tie only holds if nothing else is true. Opportunity cost breaks it by nineteen dollars and uncertainty by fifty. Those numbers do not add up, by the way: they are separate tests of that one tie. Two of the three decide it outright, and the third takes away the only reason you had to wait. Present value puts the whole argument in one number, and the rate is where your judgement lives — and that's the math behind the money.

6:30What's next: Annuities ▶

Next time we do this to a whole row of payments. Ten payments of a thousand, one a year, are not worth ten thousand dollars. And for you: what rate do you use to weigh later against now? I'm the MoneyMathician, and if you subscribe, the next one will find you. Until then: do the math!