Compound interest and future value: how money grows over time
Interest that is added to the balance starts earning interest of its own, and over long periods that second layer does most of the work. This lesson builds the compound interest formula, shows how compounding frequency, inflation and time change the result, and uses the same arithmetic to read the growth of Apple's net sales.
- Level
- Beginner
- Reading time
- About 25 minutes
- Know this first
- DuPont analysis
By the end you will be able to:
- work out the future value of a sum using the compound interest formula
- separate interest on the original sum from interest on earlier interest
- convert a nominal rate into an effective annual rate and compare offers fairly
- turn a nominal return into a real return, and a start and end value into a CAGR
Why a dollar today is worth more than a dollar later
Offered $1,000 today or $1,000 a year from now, almost everyone takes the money today, and they are right to. Money in hand can be lent out, saved or invested, and by next year it will have grown. It can also be spent before prices rise. There is a third reason too: the later promise might not be kept. These three things, the chance to earn, the loss of buying power and the risk of waiting, are why finance treats the date of a payment as part of its size. That idea is the time value of money, and the rest of this module is built on it.
The tool that measures it is interest, the price paid for the use of money over time. Three quantities describe any interest calculation. The principal is the sum that starts the process. The rate is the interest charged or earned per period, as a percentage of the principal. The term is how long the money is left in place. Everything in this lesson is a way of connecting those three to a final amount, the future value.
Simple interest and compound interest
With simple interest the interest is always worked out on the original principal. Deposit $10,000 at 6% and you earn $600 each year, whether it is the first year or the twentieth. After 20 years that is $12,000 of interest and a balance of $22,000.
With compound interest each year's interest is added to the balance, and the next year's interest is worked out on the larger sum. The first year still earns $600. The second earns 6% of $10,600, which is $636. The third earns 6% of $11,236, which is $674.16. The interest earns interest, and the gap over simple interest widens every year. The formula captures it in one line:
Grow a sum and watch where the gain comes from
The calculator below applies the formula. Set the starting sum, the yearly rate, how many years the money is left alone, how often interest is added to the balance, and the yearly rate of inflation. The left page shows what the sum grows to and how that gain divides between interest on the original sum and interest on earlier interest. The right page shows the rate you actually earn per year after compounding, the buying power of the result, and how long the sum takes to double. Nothing is withdrawn and no tax or fee is taken, so the figures are the arithmetic, not a forecast.
Left to grow 20 years at 6.0% paid once a year, $10,000 becomes $32,071. Of the $22,071 gained, $12,000 is interest on the original sum and $10,071 is interest earned on earlier interest. After 2.5% a year of inflation, that buys what $19,572 buys today.
Four experiments. First, leave the defaults and slide the years up to 40. The balance grows to $102,857, 10.3 times the start, against 3.2 times after 20 years. Compounding is back-loaded: the last ten years add $45,422, almost as much as the first thirty years put together ($47,435). Second, set the rate to 3% with 20 years. The balance is $18,061, and the gain is only 36.5% of the gain at 6%, so halving the rate takes away far more than half of the gain. Third, return to 6% and 20 years and move the interest slider from once a year to every day. The balance rises from $32,071 to $33,198, a difference of $1,127. Frequency matters, but much less than rate or time. Fourth, raise inflation to 6% with the rate also at 6%: the real yearly rate falls to zero and the sum buys no more than it did at the start.
Simple against compound, year by year
The table repeats the example of $10,000 at 6% a year, added once a year. The two columns start together and separate. The gap after a long time is mostly interest on interest.
| After | Simple interest | Compound interest | Gain from interest on interest |
|---|---|---|---|
| 5 years | $13,000 | $13,382 | $382 |
| 10 years | $16,000 | $17,908 | $1,908 |
| 20 years | $22,000 | $32,071 | $10,071 |
| 30 years | $28,000 | $57,435 | $29,435 |
| 40 years | $34,000 | $102,857 | $68,857 |
Notice the 40-year row. Simple interest has added $24,000 to the starting sum, while compounding has added $92,857. Of that compound gain, $68,857 is interest earned on interest, which is 74% of the whole. This is what people mean when they say that time is the most important input. The rate is fixed by the market, but the term is partly in your hands.
How often interest is added
Lenders and banks quote a nominal rate and then say how often it is applied. A 6% nominal rate compounded monthly does not add 6% a year. It adds 6% ÷ 12 = 0.5% each month, twelve times. The balance after one year is then 1.005 to the power 12, which is 1.0617, so the sum rises 6.17% in a year. That 6.17% is the effective annual rate, the rate that would give the same result if interest were added once a year.
| Interest added | Effective annual rate | Balance after one year, from $10,000 |
|---|---|---|
| Once a year | 6.00% | $10,600 |
| Twice a year | 6.09% | $10,609 |
| Every quarter | 6.14% | $10,614 |
| Every month | 6.17% | $10,617 |
| Every day | 6.18% | $10,618 |
| Continuously | 6.18% | $10,618 |
The effective rate climbs as compounding gets more frequent, but with a ceiling. Compounding every instant, called continuous compounding, gives 6.18%. Going from yearly to monthly adds 0.17 percentage points, and going from monthly to daily adds only 0.02. The practical point is to compare loans and deposits by their effective rates. In the United States, deposit accounts are advertised with an annual percentage yield (APY), which includes compounding, while loans and cards quote an annual percentage rate (APR). They are different measures, so make sure you are comparing like with like.
The rule of 72
A quick way to estimate the time to double is to divide 72 by the rate in percent. At 6% that is 12 years; the exact answer is 11.9 years. The shortcut works because the doubling time is the logarithm of 2 divided by the logarithm of (1 + r), and for rates between about 4% and 12% that is close to 72 divided by the rate. It is less accurate at very low or very high rates, as the table shows.
| Yearly rate | Rule of 72, years | Exact, years | Rule's error |
|---|---|---|---|
| 1% | 72.0 | 69.7 | +3.4% |
| 2% | 36.0 | 35.0 | +2.8% |
| 4% | 18.0 | 17.7 | +1.9% |
| 6% | 12.0 | 11.9 | +0.9% |
| 8% | 9.0 | 9.0 | −0.1% |
| 10% | 7.2 | 7.3 | −1.0% |
| 12% | 6.0 | 6.1 | −1.9% |
| 18% | 4.0 | 4.2 | −4.5% |
The rule also runs backwards. If a quantity doubled in 9 years, its growth rate was about 72 ÷ 9 = 8% a year. That is the reasoning behind the next section.
Nominal and real returns
A 6% return is a nominal return, measured in dollars. If prices rise 2.5% in the same year, each dollar buys less, and the gain in buying power is smaller. The real return removes the effect of inflation, and it is not found by simple subtraction. The exact relation is:
With a nominal 6% and inflation of 2.5%, the real return is 1.06 ÷ 1.025 − 1 = 3.41%, a little below the 3.5% that subtraction suggests. Over 20 years the difference becomes visible: $10,000 grows to $32,071 in dollars, but that is the buying power of $19,572 at today's prices. The calculator shows both. The same logic explains why a savings account that pays 1% while prices rise 3% is losing value even though the balance is going up.
Growth is compounding too: Apple's net sales
Compounding is not only about savings accounts. Any quantity that grows by a percentage each year follows the same curve, so the formula can be turned round to find the steady rate that links a starting figure to an ending one. That rate is the compound annual growth rate, or CAGR:
Apple Inc. files an annual report (Form 10-K) with the US Securities and Exchange Commission, and these are free to read. Its fiscal year ends in late September. The table shows total net sales for the last three fiscal years, divided into products and services as Apple reports them, in millions of dollars.
| Fiscal year (ends late September) | Products | Services | Total net sales |
|---|---|---|---|
| 2023 | 298,085 | 85,200 | 383,285 |
| 2024 | 294,866 | 96,169 | 391,035 |
| 2025 | 307,003 | 109,158 | 416,161 |
| Compound annual growth, 2023 to 2025 | +1.48% | +13.19% | +4.20% |
Net sales rose from $383,285 million in fiscal 2023 to $416,161 million in fiscal 2025. That is two years of growth, so the CAGR is (416,161 ÷ 383,285) to the power 1/2, minus 1, which is 4.20%. The two single years were 2.0% and 6.4%, so the compound rate lies between them, close to their average. Break the total in two and the picture changes. Services grew from $85,200 million to $109,158 million, a CAGR of 13.19%. Products, which are iPhone, Mac, iPad and wearables, went from $298,085 million to $307,003 million, a CAGR of 1.48%, and fell in the first year before rising in the second. Services were 22.2% of net sales in fiscal 2023 and 26.2% in fiscal 2025.
At a steady 13.19% a year, services would double in about 5.6 years, which the rule of 72 puts at 5.5. Carried forward five years at that same pace, $109,158 million would become about $202,813 million. That is arithmetic, not a forecast: no company can count on a rate continuing, and the larger the base, the harder it is to hold a high percentage. The exercise shows how quickly even a moderate rate piles up, and why investors look hard at which part of a business is growing.
Limits of the compound interest formula
- It assumes a constant rate. Real deposits and investments change rate over time, and a market investment can fall as well as rise. The formula describes a fixed rate, which only a few instruments, such as a fixed-term deposit, offer.
- It ignores tax and fees. Both reduce the rate you really earn, and over decades a small yearly fee compounds into a large difference.
- It assumes nothing is withdrawn or added. Regular deposits are covered in the lesson on annuities.
- Past growth is not a forecast. A CAGR describes what happened between two dates. It says nothing of the path between them or what comes next.
- Endpoints matter. A CAGR from a weak year to a strong one looks better than one measured between two ordinary years. Always say which years were used.
Growth that compounds: three global companies
Compounding applies to anything that grows by a percentage of what it already is, not just to bank balances. The table takes the revenue growth of Apple, Microsoft and Toyota over their latest year and asks what it would mean if the same rate were repeated. This is an exercise in arithmetic, not a forecast, since no company keeps one growth rate for a decade. It shows how strongly a difference in the rate changes the outcome.
| Revenue growth, latest year over prior | Growth rate | Years to double, rule of 72 | Years to double, exact | Ten years at this rate |
|---|---|---|---|---|
| Apple | 6.4% | 11.2 | 11.1 | 1.86× |
| Microsoft | 14.9% | 4.8 | 5.0 | 4.02× |
| Toyota | 6.5% | 11.0 | 11.0 | 1.88× |
Microsoft's revenue grew 14.9% in the year, which if repeated would double it in 5.0 years, while Apple's 6.4% and Toyota's 6.5% would take about 11.1 and 11.0 years. Over ten years, Microsoft's revenue would multiply 4.02 times, from $281.7 billion to about $1,133 billion, and the other two would multiply by less than two. A growth rate that is only about 8 points higher gives a result more than twice as large. This is why investors pay so much attention to growth rates, and why a growth rate that cannot be sustained, which is almost every one, must not be extended across a long horizon without scrutiny.
When the return is in another currency
An investor who buys an asset in another country earns the return in that country's currency, and then converts it back. The currency can add to the return or subtract from it. The table is hypothetical: it takes an investment that earns 3% in a foreign currency and shows the return in dollars if that currency has moved against the dollar by the amount in the first column over the year.
| Change in the foreign currency against the dollar | Dollar return on a 3% foreign-currency investment |
|---|---|
| −10% | −7.30% |
| −5% | −2.15% |
| 0% | 3.00% |
| +5% | 8.15% |
| +10% | 13.30% |
The formula is a compounding one: the dollar return is one plus the foreign return, multiplied by one plus the currency change, less one. A currency that weakens by 5% turns a 3% gain into a loss in dollars, and one that strengthens by 5% turns it into more than 8%. For long-term investors in foreign assets, the currency is a second source of return, and for a company like Toyota that earns most of its profit abroad, it affects the figures directly, as the lesson on what finance is showed.
How interest rates are quoted around the world
The compounding frequency matters when two rates are compared, and the rate people see in advertisements is often not the one that determines what they earn. Most countries have a standard way to show the real yearly return on a deposit: in the United States it is the annual percentage yield (APY), and in the United Kingdom the annual equivalent rate (AER). Both are effective annual rates. A loan or deposit quoted as a nominal annual rate, 6% for example, compounds at some frequency, and the effective annual rate is higher: 6.090% if compounded twice a year, 6.136% quarterly and 6.168% monthly.
Before comparing two rates, from different banks or different countries, put both on the same basis: effective annual, in the same currency, after inflation if you want to compare purchasing power. Two offers that look far apart, or close together, on their headline rates can look different once both are on this basis.
Common mistakes
Adding rates instead of compounding them
A 6% return for 20 years does not make 120%. It makes 220.7%, because each year's gain is earned on a larger base. Use the power, not the product.
Comparing nominal rates that compound differently
A 6% rate compounded daily beats a 6% rate compounded yearly. Compare effective rates.
Subtracting inflation instead of dividing
The shortcut is close at low rates and misleading at high ones. At a 20% nominal rate and 15% inflation, subtraction gives 5%, while the true real return is 4.35%.
Using the rule of 72 outside its range
It is a mental shortcut for rates between roughly 4% and 12%. At 1% it says 72 years when the true figure is 69.7.
Treating a CAGR as a smooth path
Apple's 4.20% is the steady rate that links two years. The company did not grow at that rate in either year.
Try it yourself
1. You deposit $5,000 at 4% a year, added once a year. How much is there after 10 years?
Show the working
5,000 × 1.0410. Since 1.0410 = 1.4802, the balance is $7,401. Of the gain of $2,401, simple interest would have given $2,000, so $401 is interest on interest.
2. A bank advertises 5% compounded monthly. What is the effective annual rate?
Show the working
(1 + 0.05 ÷ 12)12 − 1 = 5.12%. It is above the nominal 5% because interest is added twelve times, each time on a slightly larger balance.
3. A sum of money doubled in 8 years. About what yearly rate does that imply, and what is the exact rate?
Show the working
Rule of 72: 72 ÷ 8 = 9% a year. Exactly, 2 to the power 1/8, minus 1, is 9.05%.
4. Apple's net sales were $391,035 million in fiscal 2024 and $416,161 million in fiscal 2025. Work out the one-year growth, and compare it with the two-year CAGR from fiscal 2023.
Show the working
416,161 ÷ 391,035 − 1 = 6.4%. The two-year CAGR from 383,285 is 4.20%. The latest year grew faster than the two-year average, because fiscal 2024 grew only 2.0%.
Where this is used
- A saver compares deposit accounts by their annual percentage yield, which is the effective annual rate.
- A retirement planner projects how much a regular saving may be worth in thirty years under a range of return assumptions.
- An analyst states a company's growth as a CAGR so that periods of different length can be compared.
- A lender or borrower reads the effective cost of credit, where compounding makes the real cost higher than the quoted rate.
Key terms
- Time value of money
- The idea that a sum is worth more the sooner it is received, because it can earn a return in the meantime.
- Principal
- The sum on which interest is calculated at the start.
- Simple interest
- Interest calculated on the original principal only.
- Compound interest
- Interest calculated on the principal plus the interest already added.
- Future value
- What a sum grows to after a stated time at a stated rate.
- Nominal rate
- The quoted yearly rate before the effect of compounding frequency or inflation.
- Effective annual rate
- The yearly rate that gives the same result as the nominal rate compounded as often as stated.
- Real return
- The return after removing the effect of inflation.
- CAGR
- Compound annual growth rate: the steady yearly rate that links a starting value to an ending value.
Questions people ask
Is compound interest always good?
It works for whoever is owed the interest. It helps a saver and hurts a borrower who carries a balance, for example on a credit card, where interest is added to a debt that is already growing.
Does daily compounding make a big difference?
Less than people expect. At 6%, daily compounding gives an effective rate of 6.18% against 6.17% for monthly. The rate and the number of years matter far more.
Is the rule of 72 exact?
No. It is a shortcut that works well between about 4% and 12%. For a precise answer use the logarithm formula, or the calculator above.
How is the CAGR different from the average growth rate?
The average adds up yearly rates and divides. The CAGR finds the single rate that, applied every year, produces the actual ending value. The CAGR is the one that reconciles with the numbers.
Sources and review
- Apple Inc., Form 10-K for the fiscal year ended 27 September 2025, filed with the US Securities and Exchange Commission. Source of total net sales and of net sales from services for fiscal 2025, 2024 and 2023.
- US Securities and Exchange Commission, Office of Investor Education, Compound Interest Calculator, a public calculator for checking compound growth.
- US Securities and Exchange Commission, Investor.gov glossary: compounding interest, for the definition of compounding.
- AccountingTools, What is the time value of money?, for the basic concept of future value and discounting.
- Brilliant, Time value of money, for the compound interest and effective rate formulas.
- Apple Inc., Form 8-K, Exhibit 99.1: fourth quarter and full-year fiscal 2025 results, filed with the US Securities and Exchange Commission. Source of Apple's statements.
- Microsoft Corporation, Form 8-K, Exhibit 99.1: fourth quarter and fiscal year 2025 results, filed with the US Securities and Exchange Commission. Source of Microsoft's statements.
- Toyota Motor Corporation, FY2025 Financial Summary (IFRS, unaudited, in millions of yen), furnished to the US Securities and Exchange Commission on Form 6-K. Source of Toyota's statements.
How the numbers were checked: Apple's products and services were added up and matched to the printed total net sales in each of the three years. The calculator was tested against separate arithmetic for more than a thousand combinations of principal, rate, term, compounding frequency and inflation, including that more frequent compounding never gives a smaller balance and that simple interest plus interest on interest equals total interest. Every growth rate and doubling time on this page was recomputed by code.