A coin in your hand today is worth more than the same coin promised tomorrow. This module puts numbers to that idea — the formulas, the calculators, and the Excel functions that turn an intuition into a working tool. Includes a free Excel toolkit you can keep.
Imagine a wealthy aunt in Singapore offers you a choice. She will hand you 10,000 Singapore dollars today, or she will mail you a cheque for the same amount on this date five years from now. Same currency, same numbers. Which do you take?
Almost everyone instinctively reaches for today. We rationalize the decision after the fact — perhaps Auntie might forget, or markets might collapse — but the deeper reason is something economists call the time value of money. The 10,000 dollars in your hand today can be deposited, invested, or used. It can grow. The 10,000 dollars promised in five years can do none of those things until then.
This is not a financial trick. It is a fundamental property of money under any positive interest rate, in any currency, anywhere on Earth.
From this single observation, the entire architecture of modern finance unfolds: how mortgages are priced in Lagos, how Italian government bonds are valued, how a Brazilian pension promises retirement income, how a Tokyo café decides whether to take out a loan for a new espresso machine.
The chapter ahead introduces the two motions of money through time — forward, into the future value, and backward, into the present value — along with the engine that powers both: compounding.
Before any formulas, look at this table. It shows what a single deposit of 10,000 today is worth after various time horizons, at two different interest rates.
| Year | at 3% | at 7% | Gap |
|---|---|---|---|
| today | 10,000 | 10,000 | — |
| +5 years | 11,593 | 14,026 | 21% |
| +10 years | 13,439 | 19,672 | 46% |
| +20 years | 18,061 | 38,697 | 114% |
| +30 years | 24,273 | 76,123 | 214% |
| +40 years | 32,620 | 149,745 | 359% |
Notice how the gap between 3% and 7% is barely visible at year 5 — and astronomical by year 40. Time amplifies rate. This is the most important sentence in this entire module.
Future value answers a forward-facing question: If I plant this sum today, at this rate, what will it grow into by then? The formula is simple, but it conceals one of the most powerful forces in finance.
FV future value · PV present value · r rate per period · n number of periods
The exponent n is what makes this expression dangerous. A linear input — number of years — produces a non-linear output. Money does not grow in a straight line. It curves upward, gently at first, then steeply, then almost vertically. We call this curve compounding, and it is the closest thing finance has to magic.
If interest compounds more often than once per year — most savings accounts and mortgages compound monthly or daily — the formula adapts:
where m is the number of compounding periods per year (12 for monthly, 365 for daily). The more frequent the compounding, the slightly larger the result.
A single deposit of 1,000, two destinies. The dotted line earns interest only on the principal. The bold curve earns interest on its interest. Slide the rate and term to see how the gap opens.
A street trader in Istanbul, a banker in Frankfurt, and a teenager opening her first savings app in Manila can all answer the same question without a calculator: how long until my money doubles? Divide 72 by the rate.
Accurate to within a fraction of a year for any rate between 4% and 12%. Slightly off at the extremes.
| Setting | Annual rate | 72 ÷ r | True answer |
|---|---|---|---|
| Japanese postal savings | 0.5% | 144 yrs | 139 yrs |
| Eurozone time deposit | 3% | 24 yrs | 23.4 yrs |
| US S&P 500, long-run | 10% | 7.2 yrs | 7.3 yrs |
| Brazilian Selic, 2024 | 12% | 6.0 yrs | 6.1 yrs |
| Argentine peso bond | 35% | 2.1 yrs | 2.3 yrs |
The shortcut breaks down at extreme rates, but for a thirty-second head-calculation it is astonishingly close.
Run the gears in reverse and you get the more useful question: given a sum I will receive (or owe) in the future, what is it worth right now? This is how lottery winnings are priced, how bonds are valued, how a startup's "annual recurring revenue" gets translated into a sale price.
The same equation as Future Value, simply solved for the other unknown.
The rate r here is called the discount rate. The higher it is, the more aggressively the future is shrunk back into the present. A million dollars promised in thirty years, discounted at 8%, is worth less than $100,000 today.
This is also why high-inflation environments produce such different financial behavior: when discount rates are high, the future is worth almost nothing in present terms. A 30-year fixed pension means something completely different in Tokyo than in Buenos Aires, even before any currency translation.
In practice, no analyst computes (1 + r)n by hand. Excel — and every spreadsheet program that has copied it — ships with a small family of financial functions that handle the algebra for you. Five of them cover almost every time-value-of-money question you will ever ask.
The two essentials are FV() and PV() — direct counterparts to the formulas above. Three more — NPER(), RATE(), and PMT() — solve the same equation for any of the other unknowns: how many years, what rate, what payment.
Excel's one peculiarity is its sign convention. The functions model cash flows from your perspective. Money you pay out (a deposit, a loan disbursement) is entered as a negative number; money you receive (interest earned, a payout) is positive. When you deposit £1,000 today, that's -1000. The function returns the future value as a positive number — money flowing back to you.
If your output ever has the wrong sign, you have simply flipped the perspective. Multiply by −1 and move on. You will see this pattern in the workbook: =-PV(...) negates the result so the displayed number is positive.
| Function | Syntax | Example |
|---|---|---|
| FV |
=FV(rate, nper, pmt, [pv], [type]) Future value of a present sum and / or future payments. |
=FV(0.07, 30, 0, -1000) → $7,612 |
| PV |
=PV(rate, nper, pmt, [fv], [type]) Present value of a future amount and / or future payments. |
=-PV(0.05, 10, 0, 50000) → $30,696 |
| NPER |
=NPER(rate, pmt, pv, [fv], [type]) How many periods to reach a target. |
=NPER(0.07, 0, -1000, 2000) → 10.24 years |
| RATE |
=RATE(nper, pmt, pv, [fv], [type]) What annual rate is implied by these cash flows. |
=RATE(20, 0, -5000, 20000) → 7.18% |
| PMT |
=PMT(rate, nper, pv, [fv], [type]) Regular payment to amortize a loan or hit a savings goal. |
=-PMT(0.06/12, 360, 250000) → $1,498.88 / mo |
A six-sheet Excel workbook built to accompany this module. Every concept above, plus a year-by-year growth schedule and six practice problems with automatic answer-checking.
Time value of money is not an abstraction. It is the silent partner in every meaningful financial decision — the rent you pay, the wage you accept, the bond you buy, the home you mortgage. Below, six small problems set in six places. Try each one before tapping for the answer.
A Shibuya café owner is offered ¥800,000 today, or ¥1,000,000 in five years from a supplier rebate. Local CDs yield 1.5% annually.
Which offer is worth more — and by how much?
A Tesouro Selic bond promises R$10,000 in three years. Brazilian government bonds currently yield 11% annually.
What is the bond worth today?
A 25-year-old engineer invests ₹50,000 once into an Indian equity fund earning a long-run 12% per year. She forgets about it until age 60.
What is the lump sum worth at retirement?
A National Lottery winner can take £500,000 today, or £700,000 in eight years. A safe gilt fund returns 4.5% annually.
Which option carries the higher present value?
A retiree must choose: €200,000 immediately, or €15,000 per year for the next twenty years. Long-term Eurozone rates sit at 3.5%.
Which stream of cash has the larger present value?
A market vendor borrows ₦100,000 to expand her stall. The microfinance lender charges 4% per month, compounded monthly. The loan runs for 12 months.
What does she repay at the end?
Reading is not understanding. The questions below test whether you can see the world the way time-value-of-money trains you to see it. The math is already done — choose the answer that follows from the principle.