CAGR

Compound annual growth rate answers one question precisely: at what constant annual rate would this have grown, to get from where it started to where it ended?

The formula

$\text{CAGR} = \left(\frac{V_{\text{end}}}{V_{\text{begin}}}\right)^{1/n} - 1$

where n is the number of years. The Quantitative methods subject derived this from the time value of money; here the concern is what it means and what it conceals.

It is a geometric average, not an arithmetic one, and that distinction is the reason it is correct. Averaging annual percentages arithmetically overstates the result whenever those percentages vary — which they always do.

Why the geometric version is the right one

Take an investment that rises 50% then falls 40%.

Arithmetic average: (50 − 40) / 2 = +5% a year. This is wrong, and it is wrong in a specific direction.

What actually happened: ₹100 → ₹150 → ₹90. You have lost 10% over two years.

CAGR: $(90/100)^{1/2} - 1 = -5.1\%$ a year.

The arithmetic mean is not an approximation of the truth here; it has the sign wrong. The reason is that percentage changes compound rather than add — a 40% fall applies to the larger amount the 50% rise created. Any time you see an average of annual returns, check which average it is, because the arithmetic one flatters a volatile series automatically.

This is the single most common arithmetic error in investment writing, and it is the reason regulators mandate CAGR rather than leaving the choice of average open.

What CAGR requires

Three assumptions, and they are exactly the limits of the measure:

One amount in at the start, nothing added or taken out. CAGR is computed from two values and a time span. If you added money in month seven, it is not the measure you need — chapter 3 is.

A whole number of periods, or a fractional exponent. Nine months is $n = 0.75$, and chapter 1 explained why annualising under a year is a projection rather than a measurement.

Both end values known and comparable. For a fund this means NAV to NAV, which brings in a subtlety: SEBI's performance-calculation note requires that "all payouts during the period have been reinvested in the units of the scheme at the then prevailing NAV." So a published CAGR is a total return figure that assumes you reinvested every distribution. If you took the payouts as income and spent them, your own outcome differs from the published one — not because the figure is wrong but because it describes a different investor.

What CAGR hides

The path. This is deliberate and it is worth being explicit about, because the hiding is the feature that makes comparison possible and the defect that makes CAGR insufficient alone.

Two investments with identical CAGR:

	Year 1	Year 2	Year 3	CAGR
A	+10%	+10%	+10%	10%
B	+60%	−30%	+18%	10%

Identical by this measure, and not remotely the same experience. B had a year in which a third of the money disappeared. Anyone who needed to withdraw during year two in B received a very different outcome, and anyone with ordinary human responses may not have held on — which is the behaviour gap the Risk subject documents.

So CAGR answers "how fast did this grow" and is silent on "how unpleasant was it". Chapter 6 adds the missing dimension.

The start and end dates. A CAGR depends entirely on two chosen dates. SEBI's requirement that figures be computed from the last day of the month-end preceding the advertisement exists to stop the end date being selected, and the standard periods — 1, 3, 5 and 10 years and since inception, against the benchmark's Total Return Index — exist to stop the start date being selected. Rolling returns, covered in the Mutual funds subject, are the stronger answer to the same problem.

Working the problem

+50% in year one, −40% in year two.

End value: ₹100 × 1.50 × 0.60 = ₹90.

CAGR: $(90/100)^{1/2} - 1 = 0.9487 - 1 = -5.13\%$ a year.

So the investment lost about 5.1% a year over two years, against an arithmetic average of +5% that would have suggested a gain.

What a reader seeing only the CAGR would fail to learn:

That the swings were enormous. −5.1% a year is consistent with two mild losses. It is also consistent with what actually happened: a 50% gain followed by a 40% collapse. The measure cannot distinguish them.

When the loss arrived. Someone who needed the money at the end of year one was up 50%; someone who needed it at the end of year two was down 10%. Same investment, opposite outcomes, one CAGR.

Whether the investor stayed. A 40% fall in year two is where most people sell. The CAGR describes the investment's path, not any particular person's.

What risk was taken to get it. Losing 5.1% a year on something stable and on something wildly volatile are different failures, and only the second tells you the next two years could be worse.

The general lesson: CAGR is the correct answer to the question it asks, and quoting it alone invites a reader to assume a smoothness that was never there. It should travel with a volatility figure or a worst-window figure, which is why chapter 6 exists.

The point

CAGR is the constant annual rate that connects a start value to an end value, and being a geometric average it is correct where an arithmetic average of annual returns is not — a 50% gain followed by a 40% fall averages to +5% arithmetically and is actually −5.1% a year. It assumes a single amount invested at the start with nothing added, and for funds it assumes every payout was reinvested at the prevailing NAV, so it describes a particular investor who may not be you. Its deliberate blindness to the path is what makes it comparable across periods and what makes it insufficient on its own.