Growth over several years: compounding and average annual rates

Several years, one multiplier

A rise of $g$ a year for $n$ years multiplies the starting value by $(1 + g)^n$. Three years at 10% is $1.1 \times 1.1 \times 1.1 = 1.331$: a rise of 33.1%, not 30%. Each year's growth is taken on a level that already includes the earlier growth, so growth earns growth. The lesson on percentages and growth in data chains years with different rates; this one is about many years at once, and about the single rate that sums them up.

Benchmarks to know by heart

No calculator is allowed on the test, so powers come from a few remembered values. Whole numbers make them easy to recall: $11^2 = 121$, $11^3 = 1\,331$, $12^3 = 1\,728$ and $105^2 = 11\,025$ give the decimals below.

Yearly rate	2 years	3 years	4 years
5%	1.1025	about 1.158	about 1.216
10%	1.21	1.331	1.4641
15%	1.3225	about 1.521	about 1.749
20%	1.44	1.728	about 2.07
$-10\%$	0.81	0.729	about 0.656

Longer spans combine shorter ones: six years at 10% is $1.331^2$, or $1.21^3 \approx 1.772$.

The average annual growth rate

When a value goes from $S$ to $E$ in $n$ years, the compound average annual growth rate (often shortened to CAGR) is the constant yearly rate $g$ that would make the same journey:

$$(1 + g)^n = \frac{E}{S}$$

By hand, form the ratio $\frac{E}{S}$ and match it to a benchmark: a ratio of 1.44 over two years means 20% a year, since $1.2^2 = 1.44$. If the ratio is not a benchmark, bracket it between two that are, and try the rate in the middle.

Count $n$ as the number of years of growth, the gaps between the dates: from 2021 to 2024 is three years, although a table shows four columns.

Not the total divided by the years, not the mean of the rates

Two shortcuts give the wrong rate.

• The total rise divided by $n$. A rise of 33.1% over three years is 10% a year, not about 11%. Because growth compounds, the yearly rate behind a rise is always below the total rise divided by the number of years.
• The plain mean of the yearly rates. Up 25% one year and down 20% the next is $1.25 \times 0.8 = 1$: the value ends where it started, so the average annual growth is 0%, although the mean of the two rates is $+2.5\%$. The compound rate is never above the plain mean of the yearly rates, and equals it only when every year has the same rate.

Some reports use the words "average annual growth" for the plain mean of the yearly rates, so read how a question defines it. A question that asks for a constant rate giving the same end value wants the compound rate.

Why adding understates growth

Expanding the power shows what the sum of the rates leaves out:

$$(1 + g)^n = 1 + ng + \frac{n(n-1)}{2}g^2 + \dots$$

The term $ng$ is the yearly rates added up. Every later term is growth on growth, and for a rise each one is positive, so adding the yearly rates understates the total. For a fall the effect runs the other way: three years down 10% leave $0.9^3 = 0.729$ of the start, a fall of 27.1%, not 30%. Compounding makes rises bigger, and falls smaller, than the sum of the rates.

A shortcut for small rates

Keeping the first three terms gives a good estimate when $g$ is small:

$$(1 + g)^n \approx 1 + ng + \frac{n(n-1)}{2}g^2$$

The factor $\frac{n(n-1)}{2}$ counts the pairs of years whose growth compounds: 1 for two years, 3 for three, 6 for four. For two years the formula is exact. For 8% over three years it gives $1 + 0.24 + 3 \times 0.0064 = 1.2592$, against an exact value of about 1.2597. For a rise the estimate is always slightly low, and the error grows with $ng$: at 20% a year for five years it gives 2.4, while the exact value is about 2.49. Use it while $ng$ is no more than about 0.5, and use benchmarks beyond that.

The rule of 72

A quick estimate of doubling time: at $r\%$ a year, a value doubles in roughly $\frac{72}{r}$ years. At 6% that is about 12 years, and $1.06^{12}$ is indeed about 2.01; at 9%, about 8 years. It also runs backwards: doubling in 6 years needs roughly 12% a year.

The rule is an approximation, not an identity. It is close for rates of a few percent up to the low teens and drifts at higher rates: at 20% it suggests 3.6 years, but $1.2^3 = 1.728$ and $1.2^4 \approx 2.07$, so doubling takes nearly 4 years. When the options are close together, confirm with a power.

Comparing growth over different spans

Totals over different spans cannot be compared directly. A rise of 21% in two years and a rise of 33.1% in three years are both 10% a year. Convert each to a yearly rate first, or compare over the same number of years.

A routine for multi-year growth

Count the years of growth. Form the ratio of the end value to the start. Match it to a benchmark, bracket it, or use the shortcut. Before choosing, check the direction: for a rise, the yearly rate must be below the total rise divided by the years; and when every year is a rise, the total must be above the yearly rates added up.