Cumulative return and a yearly compound rate
Here KRW means Korean won. Start with KRW 1,000,000 and finish with KRW 1,500,000 exactly five years later. Assume no extra deposits, withdrawals, taxes, fees or payments taken out along the way. The table calls money you add or take out external cash flows. Cumulative return compares the final value with the initial money: 1,500,000 ÷ 1,000,000 − 1 = 50%. That 50% alone does not tell you how long the change took.
Ask what constant yearly percentage could produce the same ending value. Each year it must apply to the balance including all earlier gains or losses. This repeated calculation is compounding. The required rate is called the compound annual growth rate, or CAGR. It is an equivalent pace connecting the endpoints; it does not claim the actual investment rose by that exact percentage every year.
The calculation is CAGR = (ending value ÷ starting value)^(1 ÷ years) − 1. Here, 1.5^(1 ÷ 5) − 1 ≈ 8.45%. Multiply the result by 100 to express it as a percentage. The ^ symbol means raising a number to a power; here the formula turns the overall growth factor into an equal yearly factor. Five years of exactly 10% compound growth would instead multiply the balance by 1.1^5 = 1.61051, a cumulative gain of 61.051%.
A 50% gain needs a date beside it
Keep the same KRW 1,000,000 start and KRW 1,500,000 finish but change the time taken. Each row is an independent investment with no money added or taken out. The table shows the whole-period change and the equivalent rate that, applied repeatedly for that number of complete years, gives the same result.
On a small screen, scroll within each table to read the remaining columns.
| Complete years | Starting value | Ending value | Cumulative return | CAGR |
|---|---|---|---|---|
| 1 | 1,000,000 | 1,500,000 | 50.00% | 50.00% |
| 3 | 1,000,000 | 1,500,000 | 50.00% | 14.47% |
| 5 | 1,000,000 | 1,500,000 | 50.00% | 8.45% |
The same gain spread over more years needs a lower yearly rate. Annualization puts different holding periods into yearly units, but it cannot make their dates, risks or market conditions identical. Use matching start and end dates where possible. Turning a few months’ pace into a yearly rate would also be a hypothetical repetition, rather than an observed year of results; these tables use complete years.
Why averaging yearly returns gives a different result
A separate investment rises 20%, falls 10%, then rises 20%. The second year’s loss is a percentage of the balance after the first gain, not of the original money. The third rise applies to the balance left after that loss. Follow the amounts in each row before combining the percentages.
| Year | Opening value | Yearly return | Ending value |
|---|---|---|---|
| 1 | 1,000,000 | 20.00% | 1,200,000 |
| 2 | 1,200,000 | −10.00% | 1,080,000 |
| 3 | 1,080,000 | 20.00% | 1,296,000 |
For the simple arithmetic average, add the yearly rates and divide by the number of years. Here it is (20 − 10 + 20) ÷ 3, or 10.00%. The balance, however, grows by 1.20 × 0.90 × 1.20 = 1.296. Cumulative return is 29.60%, and CAGR is 9.03%.
Use the unrounded rate to reconstruct the ending balance. The displayed 9.03% is rounded to two decimal places.
CAGR uses only the start, finish and elapsed time. Another investment can reach the same final amount after a much deeper fall along the way, giving the same CAGR. The number therefore does not show the worst loss, how sharply values fluctuated or what would have been available if you needed to withdraw money before the end.
Your account can grow even when the investment earns nothing
Now suppose the investment value never changes. At the very end of one year, you deposit another KRW 100,000. The account rises from KRW 1,000,000 to KRW 1,100,000, but the extra money came from you.
| What changed? | Starting value | Your deposit | Ending balance | Investment gain | Investment return |
|---|---|---|---|---|---|
| Contribution only | 1,000,000 | 100,000 | 1,100,000 | 0 | 0.00% |
| Investment gain only | 1,000,000 | 0 | 1,100,000 | 100,000 | 10.00% |
In the contribution-only case, investment gain is 1,100,000 − 1,000,000 − 100,000, or KRW 0, and the return is 0.00%. Inserting the raw account balances into the CAGR formula would incorrectly label the deposit as a 10% return.
A deposit at the start of a year experiences that year’s investment changes; one at the very end does not. Repeated deposits and withdrawals therefore need their dates as well as their amounts. The total account increase alone cannot distinguish new money from money earned.
There are different ways to answer this question. Time-weighted return divides the history at deposits and withdrawals, calculates the investment’s return within each segment and compounds those segments together. It helps describe how the investment performed apart from the size of your deposits. Money-weighted return also reflects how much of your money was invested and when. Simply subtracting all deposits from the final amount generally cannot replace either dated annual-return calculation.
This distinction matters when comparing a fund's published performance with your own regular contributions.
Losses, zero values and the limits of CAGR
The same calculation works for a loss when the starting amount is above zero and the ending amount is zero or above. These two examples each run for two complete years without external cash flows.
| Outcome | Starting value | Ending value | Years | Cumulative return | CAGR |
|---|---|---|---|---|---|
| Partial loss | 1,000,000 | 800,000 | 2 | −20.00% | −10.56% |
| Total loss | 1,000,000 | 0 | 2 | −100.00% | −100.00% |
The partial loss has cumulative return of −20.00% and CAGR of −10.56%. Dividing the 20% loss by two would again miss compounding. For a zero ending value, the formula returns −100%; it does not reveal when the loss happened.
The division needs a starting amount above zero and a positive period of time. A zero start makes the formula divide by zero. Borrowing or other obligations can also make the amount left in an account negative. Such starting or ending values do not fit this simple model of positive invested money growing or shrinking by yearly factors. A calculator result for them should not be presented as an ordinary investment CAGR.
Dates, distributions and costs behind the return
A company can give cash to its owners as a dividend; a fund can pay a distribution. A price-only return leaves those payments out. A total return includes them, while a reinvested total-return series assumes the money was used to buy more of the investment when paid. If you took the payments out, the ending holding value alone misses some of your result. Adding several years’ payments at the end also does not reproduce the growth they could have had if reinvested. Our ETF distributions and total returns guide follows these separate cases.
Check the dates, currency and treatment of fees and taxes too. Converting a foreign investment into your own currency can change the measured result as exchange rates move. These examples are nominal: they measure the money change without adjusting for purchasing power, meaning how many goods or services that money can buy as prices change. They exclude fees and taxes. CAGR does not automatically correct differences in any of those inputs.
A short-period gain expressed as an annual rate is not a promise that the same pace will continue. Even over several years, a past CAGR says what connects two values, not what next year's return will be.
About the examples
The account histories, amounts and tables were created for this guide. They do not describe a real security, fund or investor. Currency amounts are in Korean won to keep the Korean and English examples directly comparable. The horizon, changing-return, contribution and loss examples are separate scenarios.
Each period contains complete years. Unless a table explicitly introduces a contribution, there are no external deposits or withdrawals; no cash distributions are paid out. Fees, taxes, inflation and currency movements are excluded. Calculations retain full precision before rounding the displayed percentages. Our editorial policy explains how we check the guides.