Compound annual growth rate is the straight line drawn between two points. Two investments with identical CAGRs can be completely different experiences to live through.
11 min read · Level 4
Smoothing is normally treated as a presentational sin, and there is a case that it is closer to the opposite - that the smoothed figure is the honest one and the path is the misleading part. The argument runs like this: the path is a record of prices other people were willing to pay on particular days, most of which had nothing to do with the asset, and a long-term holder who never transacted at those prices was never actually affected by them. On that reading, the 48.97% fall along path B is noise that a disciplined holder correctly ignored, and quoting it gives it a significance it does not deserve. The argument is genuinely strong and it has one fatal gap, which is that the path determines who is still holding at the end. Drawdowns cause selling, redemptions, margin calls and career endings, and an investor who did not survive year four never collected the 7.1773%. So the path is not merely a record of other people's opinions - it is the mechanism by which returns are or are not received, which makes it part of the return rather than a decoration on it.
The compound annual growth rate is the single constant rate that would take a starting value to an ending value over a given number of years.
CAGR = (end / start)^(1 / years) - 1
It uses exactly two observations and the number of years between them. Everything that happened in between is discarded by construction - not approximated, discarded - and that is the correct way to think about it. A CAGR does not summarise the path. It replaces it.
It is a geometric mean, which is what makes it the right average for anything that compounds. The arithmetic mean of a series of annual returns answers a question nobody asked, because returns multiply rather than add, and the gap between the two grows with how much the series moved around.
Because compounding is multiplicative and averaging is usually taught as addition. Gain 50% and then lose 50% and the arithmetic mean is zero, while 100,000 has become 75,000 - a two-year CAGR of -13.40%. Nobody would describe that as breaking even, yet the arithmetic mean says exactly that, and the arithmetic mean is what an unguarded instinct reaches for.
And because a smoothed figure is the form in which almost all long-run returns are quoted. Understanding what the smoothing removed is what stops someone reading a historical CAGR as a description of what holding the thing would have felt like - which matters, because the experience is what determines whether a person is still holding it at the end.
Educational, not personalised financial advice. Nothing in the Investor Academy is a recommendation to buy, sell or hold anything — what is right for you depends on your own circumstances and the rules where you live.
Two investments. Both start at 100,000 and both end at 200,000 after ten years. The CAGR of each is 2 to the power of one-tenth, minus one, which is 7.1773%.
Path A compounds at that rate every year:
107,177 114,870 123,114 131,951 141,421 151,572 162,450 174,110 186,607 200,000
Path B takes a different route to the same place:
145,000 92,000 118,000 74,000 96,000 130,000 112,000 158,000 175,000 200,000
The annual returns along path B are approximately +45.0%, -36.6%, +28.3%, -37.3%, +29.7%, +35.4%, -13.8%, +41.1%, +10.8% and +14.3%. Multiply all ten growth factors together and the product is exactly 2.000000, which is why path B genuinely ends at 200,000 and genuinely has a CAGR of 7.1773%.
Two things come out of that.
First, the arithmetic mean of path B's ten annual returns is 11.68%, against a CAGR of 7.1773%. A 4.51 percentage point gap, on a series that ended exactly where the CAGR says it did. That gap is volatility drag, and it is not an error in either figure: the arithmetic mean is simply not the average that describes compounding, and it is always at least as large as the geometric one.
Second, the deepest fall along path B runs from the 145,000 reached at the end of year one to the 74,000 at the end of year four. That is a decline of 48.97%, it took three years, and the 145,000 was not seen again until year eight. Path A never fell at all. A reader told only that both returned 7.1773% a year for a decade has been told the truth and has been told nothing about which of the two they could have held.
And this is exactly why CAGR is the figure marketing reaches for. It is honest, it is checkable, and it removes the only part of the record that would have made someone hesitate.
The first mistake is averaging returns arithmetically. Add ten annual percentages and divide by ten and you have computed something that does not describe what happened to any amount of money.
The second is choosing the start and end dates after seeing the data. A CAGR uses two observations, so moving either one moves the answer, and a decade beginning the month after a crash is a different claim from one beginning the month before it.
The third is reading a long-run CAGR as a description of the experience. The number contains no drawdown, no duration of decline and no ordering, and those are the properties that determine whether a holding is actually held.
The fourth is applying CAGR to something with money going in and out. It is defined on two values and a number of years; a position that was added to in year three has no meaningful start value, and the measure for that situation is in the next lesson.
The fifth is quoting more decimal places than the inputs deserve. 7.1773% is computed from two numbers that are themselves estimates in most real cases, and four decimal places on an estimate is a claim about precision that nothing supports.
End over start, to the power of one over years, minus one. Never sum annual returns and divide. The two answers differ by more the more the series moved, and the difference is entirely in the direction that flatters.
Choose them for a reason you can state - a full market cycle, the date of purchase, a fixed decade - and write the reason down. Dates selected after seeing the outcome are the single easiest way to produce a defensible-looking figure that means nothing.
Peak to trough, as a percentage, and how long it lasted. On path B that is 48.97% over three years. Two numbers instead of one, and the second is the one that determines whether a person would still have been holding at the end.
11.68% against 7.1773% on path B. The gap is a direct measure of how bumpy the road was, it costs nothing to compute, and it is the cleanest single indicator that a smoothed figure is hiding something.
CAGR is defined on a start value and an end value. Contributions and withdrawals make both undefined, and forcing the formula on them produces a figure that is not wrong so much as meaningless.
For comparing the long-run outcomes of things held continuously with no money added or removed, and for any statement about the growth of a single quantity between two dates.
Avoid it as a description of experience, avoid it on anything you added to or withdrew from, and avoid it over periods short enough that the start and end dates dominate the answer - a three-year CAGR is mostly a statement about two particular days.
A shape that recurs across performance material is a long-run rate measured from a start date that sits at or near a market low. Nothing is fabricated, and the arithmetic is checkable; the selection is what does the work, and it is rarely deliberate - the period chosen is usually the one the presenter has clean data for, or the date a strategy began, which is itself often shortly after a decline made the opportunity attractive. Because CAGR uses only two observations, a start date moved by a few months can change a decade's stated rate by several percentage points.
Recompute any long-run rate from two or three alternative start dates before relying on it. If the answer moves a lot, the figure is mostly a statement about two particular days.
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