The arithmetic mean of a return series is generally treated as a fair summary of it. It is always at least as high as what an investor actually experienced, and the gap widens with volatility rather than narrowing.

What follows concerns why the two measures diverge, the exact relationship governing the size of the gap, and the November 2023 rule that removed the choice between them from Indian fund advertising.

Zero Percent Average, Seventy-Six Percent Loss

The canonical case: a portfolio returns +50% one year and -50% the next. The arithmetic mean is 0%. What happens to a rupee is that it becomes 1.50, then 0.75 - a 25% capital loss, a CAGR near -13.4%.

Repeat that cycle for a decade and the rupee becomes roughly 24 paise, a 76% cumulative loss, while the average return across all ten years remains exactly 0%.

Value of ₹1 compounding through an alternating +50%/-50% return cycle for ten years, against what the value would be if the 0% arithmetic average held every year.

The Gap Has a Closed Form

This is not an artefact of an extreme example. Compound growth is approximately the arithmetic mean minus half the variance of returns - G ≈ μ - ½σ² - a relationship that falls directly out of the mathematics of compounding and has been documented as “variance drain” since a 1995 paper by Tom Messmore.

The same correction term appears in continuous time, when geometric Brownian motion is solved through Itô’s lemma. Two independent derivations reaching the same adjustment is what makes this a property of compounding rather than a quirk of discrete annual measurement.

The practical reading is that every unit of volatility costs compounded return even when the advertised average has not moved. Two funds quoting an identical mean return, one of them twice as volatile, did not deliver the same outcome to anyone who held them.

Why AMFI Removed the Choice

SEBI had flagged that return illustrations in fund advertising could lead investors to read a volatile product as something close to a fixed return. AMFI’s Best Practices Guidelines Circular No. 109/2023-24, dated 1 November 2023, restricted advertisements to CAGR and ruled out a simple average of yearly returns as the headline figure.

The same circular fixed the maximum past return a scheme may quote: 12.64% for the Sensex and 12.93% for the Nifty, derived from the mean of 10-year rolling CAGRs between 1 June 2013 and 30 May 2023. That second provision matters as much as the first - it closes the route of raising the advertised number by switching measures.

The Old Figures Were Not Dishonest

A simple average is a real if incomplete description of a return series, and pre-2023 figures computed that way were not fabricated. The two numbers legitimately differ for any volatile investment.

What the rule recognises is that the difference is not neutral in expectation. It always favours the arithmetic figure, and the advertiser always chooses which to print - so the choice itself was the problem, not any individual calculation.

What This Means for Allocators

Convert any quoted average to a compounded figure before comparing across funds, particularly for anything holding volatile or non-linear exposures.

Use the variance term as a quick sanity check. Half the variance is the approximate haircut, and it is computable from the same factsheet that quotes the mean.

And treat the 12.64% and 12.93% ceilings as what they are - a regulatory cap on advertised history, not a forecast, and not a number any scheme may exceed by changing its method.

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