Time value of money: present and future value

Hold a dollar today and you can put it to work at once. Wait a year for it and you cannot. That single difference is the time value of money, and as soon as any positive return exists anywhere, it stops being an opinion and becomes arithmetic.

Three forces push the same way:

• Earning power. Money in hand can be invested, or used to clear a debt, which is the same return seen from the other side.
• Risk. A promise can be broken, and the longer it has to survive, the more can go wrong before it is kept.
• Preference. People would rather have things now, and will give up something to stop waiting.

Earning power is the one the arithmetic uses directly, because it sets the base return the money could be making anyway. Risk has a market price too, visible in the extra yield a shakier borrower has to offer, and preference has no price you can look up at all. Both of those get folded in by demanding a higher return before you agree to wait.

Notice what is not on that list. Inflation is a genuine reason a future dollar buys less, but the time value of money would survive at zero inflation: with prices frozen forever, a dollar today could still be lent out and a dollar next year still could not. Inflation is a second reason for the same conclusion rather than the cause of it, and it gets handled separately when you choose a rate.

So the question is never whether waiting costs something. It is how much. Everything below turns that cost into a number you can set beside any offer.

Future value is what an amount today becomes once it has been left to grow. Multiply by $(1 + r)$ one time for every period you wait:

$$FV = PV \times (1 + r)^n$$

Here $PV$ is what you start with, $r$ is the return for a single period written as a decimal, and $n$ is the number of periods. At 5 percent a year, added once a year, $1,000 left alone for ten years reaches $1,628.89.

That figure is what the money becomes if the rate is genuinely earned in every one of those years, which is why it is a fair description of a fixed rate and only a projection for anything that moves. Where the yearly return varies, the rate that actually lands is the compound one, and it sits below the simple average of the yearly returns whenever those returns vary at all.

The growth is not 50 percent. Ten years of 5 percent charged on the original amount alone would be, but the second year earns 5 percent on the first year's interest as well, the third year earns on both, and by year ten the balance is 62.9 percent bigger than it started. The distance between 50 and 62.9 is compounding, and it is the same widening gap the compound interest calculator plots between what you paid in and what you hold.

Two details decide whether the formula gives the right answer. The rate and the period have to agree: 5 percent a year with $n$ counted in years, or a monthly rate with $n$ counted in months. And the factor $(1+r)^n$ is worth naming out loud, because it returns immediately in the opposite direction. The number 1.628895 is the multiplier that turns $1,000 into $1,628.89 over ten years, and it is also the divisor that turns $1,000 promised in ten years into $613.91 today. One factor, two directions of travel.

Present value runs the same machine backwards. If multiplying by $(1+r)^n$ carries money forward through time, dividing by it carries money back:

$$PV = \frac{FV}{(1 + r)^n}$$

Turn that divisor upside down and you have the discount factor, $1/(1+r)^n$, which is the price of waiting written as a fraction. At 5 percent a year, $1,000 promised in ten years is worth $613.91 today, because $613.91 is precisely the amount that would grow into $1,000 over those ten years at that rate. Discounting is not a haircut applied out of pessimism. It answers an exact question: how much would I need in hand right now to produce that future amount myself?

Read the last column downwards. One year of waiting costs under 5 percent of the promise, ten years costs just under 39 percent, and thirty years costs more than three quarters of it. The loss compounds exactly as growth does, which is why distant cash flows are where value quietly drains out of a long project or a long-dated promise, and why raising the rate punishes the far years much harder than the near ones.

The discount rate is the one input nobody hands you. It is the return the same money could earn in its next best use at similar risk, which makes it an opportunity cost: the thing you give up by tying the money here rather than there.

That has three consequences worth holding on to.

• It is personal. A company usually discounts at its cost of capital, and even then only for projects that carry the risk the rest of the business already carries. Someone holding an expensive credit card balance has a much higher honest rate, because clearing that balance returns the card's rate with no market risk attached, for as long as the card goes on charging it.
• Riskier money faces a higher rate. A promise from a shaky payer should be discounted harder than a government bond, and that extra slice of rate is the risk premium. Two separate jobs hide inside it. The chance that a payer never pays belongs in the cash flow, as an amount weighted by how likely each outcome is, while the rate carries what is demanded for the risk left over that no amount of spreading money around removes. Discount a promised amount as though it were an expected one and a default assumption ends up buried in the rate where nobody can see it.
• Small changes move large answers. $12,000 promised in three years clears a $10,000 price by $75.43 at 6 percent and falls short of it at 8 percent. Not one cash flow moved. Only the rate did.

Inflation belongs in this decision too, and the rule is to match. Cash flows written in today's prices need a rate with inflation stripped out; cash flows that already include future price rises need a rate that still contains it. Mixing the two is the quiet error that makes long projects look far worse, or far better, than they are, and the real return calculator converts between the two versions of a rate.

Tax belongs in the same conversation, and it is one reason a rate that is honest for one person is the wrong rate for another. Every rate on this page is a before-tax figure. What a return or a debt is genuinely worth once tax has taken its share depends on the country doing the taxing and on the type of account the money sits in, so a rate lifted from a headline should be converted to its after-tax version before anything is compared with it.

Because the rate is a judgement, two people can do identical arithmetic and reach opposite conclusions. That is not a flaw in the method. It is the method dragging the disagreement into the open, onto one visible number, instead of leaving it buried in a feeling about whether a deal sounds good.

Most real decisions are not one amount against one amount. They are a series: pay something now, receive various amounts over several years. Net present value is the time value of money applied to the whole series at once. Discount every future amount back to today, add the results, then subtract what you pay now.

$$NPV = \sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}$$

Year zero is the trap in that formula. At $t = 0$ the divisor is $(1+r)^0 = 1$, so money moving today is not discounted at all. A positive NPV then means three things at once: the series returns your money, it pays the rate you demanded for every year the money was tied up, and a surplus is left over, measured in today's money.

Push the rate high enough and NPV falls through zero. That crossing point is the indifference rate, better known as the internal rate of return. For the offer worked through below it sits at 6.2659 percent a year: under that rate waiting is worth more, above it cash now is worth more, and the IRR calculator solves for it directly.

That single clean crossing is a property of the shape of the series, not a law. One payment out followed by payments in crosses zero once. A series that switches direction more than once, paying out again partway through, can cross zero at several rates or at none, and then no single rate answers anything and the number to read is the NPV at the rate you actually demand.

The same arithmetic sits under more than it appears to. A bond's price is the present value of its coupons and its face value, discounted at the yield the market is asking for. A pension lump sum weighed against a monthly income is a present value comparison, with the extra wrinkle that an income for life is only paid while the person is alive, so the amounts have to be weighted by how likely each payment is to be reached. Every schedule the loan payment calculator produces is a stream of future payments whose present value, at the loan's own rate and before any fee charged at the start, equals the amount borrowed, which is what makes the payment the right payment.

Four errors account for most wrong answers here. Three are pure arithmetic. The fourth is an assumption that arrives dressed as arithmetic, which is what makes it hard to spot.

• Discounting money that moves today. The amount at year zero is already in today's money. Divide it by $(1+r)$ anyway and you shrink the cost of every project by a few percent, which is enough to turn a rejection into an acceptance on a close call.
• Letting the rate and the period disagree. A 6 percent annual rate applied once per month discounts each amount as though it sat twelve times further into the future. Convert the rate to the period first, then count periods.
• Mixing real cash flows with a nominal rate. Forecast in today's prices, discount at a rate that already contains inflation, and you have charged for inflation on the way out without ever adding it on the way in. Long horizons make that error enormous.
• Borrowing somebody else's rate. A rate that fits a government bond does not fit a start-up. The rate carries a risk assumption, so an inherited rate is an inherited assumption you never made.

One more mistake is not arithmetic at all: judging a series by the total cash it returns. Two offers can hand back the same total and be worth different amounts, because the one that pays later hands you money that has been divided by a bigger number. A total is not a value until every amount in it has a date attached.

One habit catches most of this. Write the series out with a year number against every figure, year zero included, before you divide anything.