How the CAGR formula works

A compound annual growth rate is not a description of any single year. It is the one constant yearly rate that would have turned the starting value into the ending value, had growth been perfectly smooth.

$$\text{CAGR} = \left(\frac{V_t}{V_0}\right)^{\frac{1}{t}} - 1$$

$V_0$ is the value at the start. $V_t$ is the value at the end. $t$ is the number of years between them. The answer is a decimal, so 0.1029 is 10.29 percent a year.

On $10,000 growing to $18,000 over 6 years, the growth multiple is $18000/10000 = 1.8$. The sixth root of 1.8 is 1.10292357. Subtract 1 and the CAGR is 10.29 percent a year. Grow $10,000 at that rate for 6 years and you land back on $18,000. That round trip is the whole claim.

Count years, not readings. A value at the start of year one and a value at the end of year six is 6 years of growth, not 7. An exponent that is one year out changes this sheet by more than a percentage point.

The CAGR calculator on this page is that identity. Run it the other way and you have how compound interest works: a rate in, an ending value out. The two pages are inverses of one formula.

The CAGR explorer is the same identity as a picture: drag the ending value and watch the rate move.

Hold the two values still and change only the years. $10,000 to $18,000 over 6 years is 10.29 percent a year. The same two values over 3 years are 21.64 percent a year. Same start, same finish, and the shorter run has to grow faster every year to get there.

Halving the time more than doubles the rate, because each year has to carry more of the 1.8 multiple. That is not a quirk of these numbers. It is what a root does: the exponent $1/t$ gets larger as $t$ gets smaller.

So a CAGR quoted without its period is not a number anyone can use. A fund reporting 21.64 percent over 3 years and 10.29 percent over 6 is not contradicting itself. It is reporting two windows on the same money. When you compare two holdings, compare equal periods ending on the same date. A window that stops just before a bad quarter is a different claim from one that includes it.

Annualised return is the family name. CAGR is the geometric member of that family, the one that reproduces the ending value.

Add a list of yearly returns and divide by the number of years, and you have an arithmetic mean. It is always at least as large as the CAGR, and the more the yearly returns bounce around, the wider the gap.

The clean case is on this page. A holding gains 60 percent in year one and loses 37.5 percent in year two. Those two returns average 11.25 percent a year. Turn them into growth factors and they multiply: $1.6 \times 0.625 = 1$. A total factor of 1 means the ending value equals the starting value. From $10,000 you still have $10,000. The CAGR is 0 percent.

Quoting 11.25 percent here is not a small overstatement. It is a rate that never happened. The money did not grow. The list of yearly percentages averaged to a number the money never earned.

The fix is one habit. Turn each year into a growth factor, multiply the factors, then take the root. 1.6 times 0.625 is 1, and the square root of 1 is 1, so the rate is zero. A geometric average is the only average that reproduces the ending value, which is the entire reason for quoting a growth rate.

That gap between the average of the returns and the return of the average is volatility drag. Arithmetic against geometric return is the same pair in a table.

The formula sees two balances and a clock. It has no column for money you put in or took out. A deposit between the start and the finish raises the ending value, and the root treats that extra cash as if the original holding had earned it.

Hold the opening $10,000 still. Pay in the rest of the rise the day before you take the ending reading, and the account shows $18,000. The CAGR is still 10.29 percent over 6 years, because 1.8 to the power of one sixth is still 1.1029. Nothing grew at 10.29 percent. You supplied the rise yourself.

That is why a CAGR taken from an account you were funding is not a performance figure. It is a mixture of your deposits and whatever the holding did. The honest stretch is one with no cash flows in it, or a rate built for dated cash flows. NPV and IRR is that other rate: IRR sets the present value of a dated series to zero, which is a different object from a root on two balances.

A withdrawal in the middle does the opposite. Take out everything the holding earned, leave $10,000 at the end, and the CAGR prints zero even if the holding had a good run, because the two balances match. The formula cannot tell a loss from a withdrawal. It can only see the two numbers it was given.

The one check that the arithmetic was done right is to grow the start at the CAGR and land on the finish. On the first sheet, $10,000 at 10.29 percent for 6 years is $18,000. If that round trip fails, the exponent or the two values are wrong, not the market.

Run the same two values backwards and the CAGR is negative. $18,000 falling to $10,000 over 6 years is a growth multiple of $10000/18000 = 0.5556$. The sixth root is 0.9067, and subtracting 1 leaves minus 9.33 percent a year. Grow $18,000 at that rate for 6 years and you land on $10,000. A negative CAGR is not a special case. It is the same root, on a ratio below 1.

The arithmetic average of a bumpy path can stay positive while the CAGR is negative. That is volatility drag again: the list of yearly percentages can average to a number the money never earned. Only the compound rate reproduces the ending value, which is the entire reason for quoting it.

CAGR assumes the money went in once at the start and came out once at the end. Deposits and withdrawals in between break that assumption, because a rate worked out from an opening and a closing balance counts your own deposits as investment growth. Use a stretch with no cash flows in it, or a measure built for dated cash flows. NPV and IRR is that other measure: IRR is the rate that sets the present value of a dated series to zero, which is a different object from a CAGR on two balances.

For interest rather than investment growth the same idea already has a name. An effective annual rate is the CAGR of an interest-bearing balance, which is what how APR and APY work works out.

A CAGR can be negative. When the ending value is below the starting value the ratio is less than 1, its root is less than 1, and subtracting 1 leaves a negative number. That is the steady annual rate at which the money shrank, read exactly like a positive one.

This page is the one root, the window that is part of the number, and the mistake of averaging yearly returns instead of compounding them. It is not a cash-flow IRR, not a forecast, and not a ranking of funds. The three sheets are $10,000 to $18,000 over 6 years (10.29 percent), the same money over 3 years (21.64 percent), and up 60 percent then down 37.5 percent (a CAGR of 0 percent against an 11.25 percent average). This is educational material, not financial advice.