Educational material from 
What you will learn
- Calculate each year’s change using that year’s starting value.
- Distinguish an arithmetic average from a compound annual change.
- Explain why the same annual summary measure can hide different paths.
The same start and end, with different years in between
Path A starts at SEK 10,000, rises 50% in the first year to 15,000 and falls 20% in the second year to 12,000. Path B rises 20% in year one to 12,000 and then remains unchanged. All amounts are fictional. We leave fees, tax and dividends out and compare complete years of equal length.
Both paths produce +20% in total: (12,000 − 10,000) ÷ 10,000 × 100. That does not mean their development looked the same each year. In path A the second year’s −20% applies to a larger base, SEK 15,000, than the first year’s 10,000.
If the table does not fit, scroll it sideways.
| Point in time | Path A | Path B |
|---|---|---|
| Start | SEK 10,000 | SEK 10,000 |
| After year 1 | SEK 15,000 (+50%) | SEK 12,000 (+20%) |
| After year 2 | SEK 12,000 (−20%) | SEK 12,000 (0%) |
The arithmetic average answers a question about the yearly figures
The arithmetic average for path A is (50 + (−20)) ÷ 2 = 15%. For path B it is (20 + 0) ÷ 2 = 10%. You add the yearly percentage figures and divide by two. These averages differ even though both paths reach the same ending value.
Try applying 15% each year to the starting amount: 10,000 × 1.15 × 1.15 = SEK 13,225. That does not reproduce path A’s 12,000. The average of yearly figures therefore cannot automatically be used as the constant growth rate that recreates the account’s ending value.
Find the annual factor that produces the right ending value
For two years you need a factor that, multiplied by itself, equals 12,000 ÷ 10,000 = 1.20. The square root of 1.20 is about 1.095445. The compound annual return is therefore (√1.20 − 1) × 100 ≈ 9.54%. Keep more decimal places when checking the ending amount.
This factor is the same for both paths because their starting value, ending value and number of years match. For n complete years without external payments, use the corresponding nth root: ((ending value ÷ starting value) raised to 1/n − 1) × 100. FINRA explains why compound return differs from simply dividing the total change by the number of years.
An annual summary does not describe the whole path
About 9.54% is a constant calculation factor summarizing these two years. It does not say that the account actually rose by that amount in each individual year. Also inspect the yearly values, time period and items included in the result.
When money is deposited or withdrawn, the method must handle the timing of the payments. This lesson’s root formula is not a shortcut for such an account. It is also not a forecast for next year. Use the example when comparing a learning activity’s ending value with the decisions and assumptions written before the outcome.
Calculate compound annual return with your own numbers
Choose starting value, ending value and number of full years. The calculator shows both the change over the whole period and the constant annual factor that would produce the same ending value. The annual measure is also called CAGR. Its percentage formula is ((ending value ÷ starting value)^(1 ÷ number of years) − 1) × 100.
Start and end must use the same currency and valuation basis, with no deposits or withdrawals between them. The calculator makes no adjustments for dividends, fees, tax or inflation. Its result summarizes two values; it does not reveal the actual yearly changes, their arithmetic average or the risk along the way. It does not forecast next year’s result.
Calculated from 10,000 SEK to 12,000 SEK over 2 full years.
- Change over the whole period
- +20 %
- Compound annual return (CAGR)
- +9.54 %
((12,000 ÷ 10,000) ^ (1 ÷ 2) − 1) × 100
CAGR is the constant annual calculation factor, rather than a claim about what happened each year. Percentages display at most two decimal places. A small change below 0.01 percent is explicitly labelled as an increase or decrease.
Try it yourself
A fictional value begins at SEK 10,000, rises 25% in year one and falls 20% in year two. Calculate the ending value, arithmetic average of the yearly percentages and compound annual return.
Show the solution and explanation
Ending value: SEK 10,000. Arithmetic average: +2.5%. Compound annual return: 0%.
10,000 × 1.25 × 0.80 = 10,000. (25 − 20) ÷ 2 = 2.5. Starting and ending values match, so √(10,000 ÷ 10,000) − 1 = 0. The different bases explain why a positive arithmetic average does not mean a gain over the full period.
Check your understanding
Does the same compound annual return show that the paths were equally steady?
No. Our two paths have the same start and end but different changes during the years.
Is 20% in total over two years always 10% compounded each year?
No. Applying 10% twice gives 1.10 × 1.10 = 1.21, or 21% in total. For a total of 20%, the two-year annual factor is √1.20.
Practise with virtual money
Try what you just learned in a practice portfolio with virtual money. You don't deposit any real money.
How the practice portfolio worksKeep learning and practising
More lessons on this topic
Returns, inflation and company results
Sources and examples
The sources support the factual explanations. Calculations and exercises were created for this lesson. Amounts and price changes are invented unless stated otherwise.
This material is educational and does not recommend securities to buy. A simulated result does not guarantee a future result.
