How Do You Calculate Percentage Change Over Multiple Years?

When a quantity changes over several years, “percentage change” can mean the total change from the first year to the last, or the equivalent rate of change per year. These are not the same. The right calculation depends on whether you want to compare endpoints or describe an annual pace.

Total percentage change across the period

If the starting value is $S$ and the ending value is $E$, the total percentage change is

$$\text{total percentage change} = \frac{E-S}{S}\times 100\%.$$

When the starting value is positive, a positive result means the value increased; a negative result means it decreased. The starting value is the base of the comparison, so changing the starting point can change the percentage.

This formula uses only the first and last values. It does not show what happened in between. For example, a value could rise and then fall while still ending above its starting point.

Equivalent change per year: CAGR

To express the overall change as a single compound rate per year, use the compound annual growth rate, or CAGR. For positive starting and ending values ($S>0$ and $E>0$) and a period of $n>0$ years, then

$$\text{CAGR} = \left(\frac{E}{S}\right)^{1/n}-1.$$

Multiply the result by $100\%$ to write it as a percentage. CAGR answers this question: “What constant yearly rate would take the starting value to the ending value over $n$ years, if the changes compounded?” It is an equivalent rate, not necessarily the rate that actually occurred in each year.

Worked example: a value rises from 100 to 150 in three years

Suppose an investment, population, or other positive quantity starts at 100 and reaches 150 after three years.

First find the total percentage change:

$$\frac{150-100}{100}\times 100\% = 50\%.$$

So the value increased by 50% over the full three-year period.

Now calculate the CAGR:

$$\begin{aligned} \text{CAGR} &= \left(\frac{150}{100}\right)^{1/3}-1 \\ &= 1.5^{1/3}-1 \\ &\approx 0.1447 = 14.47\%. \end{aligned}$$

The equivalent compound growth rate is about 14.47% per year. It is not correct to divide the 50% total increase by three and call the result the compound annual rate: $50\%/3\approx16.67\%$ ignores compounding.

If you know each year’s percentage change

When the annual rates are known, compound them by multiplying the yearly growth factors. If the rates are $p_1,p_2,\ldots,p_n$, written as decimals, then

$$\text{overall multiplier}=(1+p_1)(1+p_2)\cdots(1+p_n).$$

The total percentage change is this multiplier minus 1, expressed as a percentage. For instance, a 10% increase followed by a 10% decrease gives

$$1.10\times 0.90=0.99,$$

so the final value is 99% of the starting value: an overall decrease of 1%. The rates do not cancel because the decrease is applied to the higher value after the increase.

If the annual rates are not all the same, their arithmetic average is not generally the same as the CAGR. For example, a 50% increase followed by a 50% decrease has an arithmetic average rate of 0%, but the multiplier is $1.5\times0.5=0.75$, a total decrease of 25%. Over those two years, the CAGR is $\sqrt{0.75}-1\approx-13.40\%$ per year.

Which measure should you report?

What you want to describe	Use
The net difference between the first and last values	Total percentage change
A constant compound rate equivalent to the whole period	CAGR
The actual pattern of yearly rises and falls	Each year’s percentage change

State the time period alongside the result. “Increased by 50% over three years” and “grew at a CAGR of about 14.47% per year” are clear, distinct statements.

Common mistakes

• Dividing the total change by the number of years: This gives a linearized average relative to the starting value, not the compound annual rate; it does not generally equal the average of the actual yearly percentage changes.
• Adding annual percentage changes: Successive rates apply to changing values, so multiply their growth factors instead.
• Confusing CAGR with the actual yearly path: CAGR summarizes the endpoints. It does not imply the quantity changed at that same rate every year.
• Using the wrong base: Total percentage change is measured relative to the starting value, not the ending value.
• Applying the formula when the starting value is zero: Percentage change from zero is undefined because the calculation divides by the starting value. For negative starting values, the usual percentage interpretation can also be misleading; report the absolute difference or choose a context-appropriate measure instead.