The Time Value of Money and Compound Interest: How Time Grows Money

1.$1,000,000 Today vs. $1,000,000 a Year from Now

If someone asked whether you would rather get $1,000,000 now or $1,000,000 a year from now, most people would choose now. Money you have now can go into a deposit and earn interest, it can cover an emergency, and if prices rise over the year, the same $1,000,000 buys less. The idea that the same amount is worth different things depending on when you receive it is called the time value of money.

The tool for putting a number on time value is the interest rate. It is what the lender earns in return for waiting and taking risk, and what the borrower pays for the use of money now. Throughout this course, the interest rate is the common language that ties together deposits, loans, investments, and inflation. This lesson covers the foundation: simple and compound interest, and present value, which runs time in reverse.

2.Simple vs. Compound Interest: Does Interest Earn Interest?

With simple interest, interest is paid only on the original principal. Leave $1,000,000 at 5% a year simple interest and it grows by the same $50,000 every year. With compound interest, the interest earned so far is added to the principal, and interest is then paid on that total. So compound interest grows by multiplying, not by adding, each year. At 5% a year compounded, you multiply by 1.05 every year.

All calculations in this lesson use annual compounding, with interest added once a year, and results are rounded to the nearest dollar. Real financial products give slightly different results depending on how often interest is added and how it is taxed, so here we focus on the principle. The table below is a hypothetical example assuming $1,000,000 left at 5% a year. The gap is small in the first few years but widens over time.

3.The Rule of 72: Estimating Doubling Time

Under compounding, you can roughly find how long it takes money to double by dividing 72 by the annual rate (%). At 6% a year it is 72 ÷ 6 = 12 years; at 3% a year, 24 years. The exact value comes from the logarithms in Math Lesson 5, but this estimate is handy for quick mental comparisons.

The rule of 72 is only an approximation. It works well for rates of roughly 2–10% a year, and the error grows when the rate is very high or very low. Run it the other way, too: when you hear "doubles in so many years," divide 72 by that number of years to see what annual rate the claim assumes. For example, "doubles in 3 years" means growing about 24% every year (about 26% when calculated exactly with logarithms), so when you hear a promise like that, start by asking how much risk it carries (Lesson 9).

4.Present Value: Bringing Future Money Back to Today

Using the compound interest formula in reverse, you can find "what a future amount is worth today." This is called present value, and dividing a future amount by (1 + r)^n is called discounting. Here r is the rate you could expect to earn over that period if you had the money today, and it is called the discount rate.

Present value lets you compare money received at different times on equal footing. When the timing differs, as with "$8,000,000 now" versus "$10,000,000 in 5 years," you cannot compare by the amounts alone. A fair comparison requires converting both to today's value. The result, however, depends on the discount rate you choose, so note alongside it that the discount rate itself is an assumption.

5.A Feel for Regular Compound Saving: What You Put In vs. What It Grew

The principle is the same when you put in a fixed amount every year instead of one lump sum. Each year's deposit compounds for the time it has left, and you add them all up. Money deposited earlier is multiplied more times, so it grows more.

With regular saving, always separate "the money you put in" from "the growth." As an example, assume you deposit $1,000,000 at the end of every year for 10 years at 5% a year compounded: you put in $10,000,000, and the final amount is about $12,577,893. The difference of about $2.57 million is the growth. Ads that show only a big final number sometimes blend the two together, so build the habit of looking at them separately.

6.What to Watch For When Reading Compound Interest Calculations

The compound interest formula is exact, but the rate you plug into it is usually an assumption. Unless the rate is fixed in advance, as with a deposit, a calculation of "X% a year for 30 years" rests on the assumption that the rate holds for all 30 years. Real returns go up and down from year to year, and a single year with a loss can change the outcome a lot (Lesson 8).

• The rate is an assumption: first check whether it is a guaranteed rate or an expected or example rate
• Subtract taxes and fees: interest is taxed, and some products deduct fees every year, and even a small percentage builds up like compounding (Lessons 2 and 7)
• Think about inflation: even if your total grows, your purchasing power can shrink if prices rise faster (Lesson 2)
• Compounding works on loans too: when unpaid interest is added to the principal, debt also grows by multiplying (Lesson 4)