Arithmetic Mean Return

Add up the yearly returns and divide by the number of years. That is the arithmetic mean return — the number most people mean when they say "it averaged 12% a year".

It answers one question well: if next year is drawn at random from the same pattern as the past years, what return should I expect?

It answers a different question badly: what did my money actually compound at? Because gains and losses of the same percentage are not symmetrical — a 50% fall needs a 100% rise to recover — the simple average of volatile returns always sits above the true growth rate.

The workbook's illustration uses two choices for ₹1,00,000 over two years:

Choice 1 "averages" 25% a year and leaves the investor exactly where they started. Choice 2 averages only 10% and actually grows the money by ₹21,000.

The workbook's rules:

• For a one-year holding period, arithmetic and geometric returns are identical.
• Over longer periods the geometric return is always less than the arithmetic return, except when every yearly return is exactly the same.
• The arithmetic average is the best estimate of a future year's return, statistically, based on a random distribution of prior years.

Arithmetic mean return = (R₁ + R₂ + … + Rₙ) ÷ n

The workbook's Chapter 20 caselet: a PMS investment earns 15%, 12%, −10%, −8% and 25% over five years, net of all charges.

Arithmetic mean = (15 + 12 − 10 − 8 + 25) ÷ 5 = 34 ÷ 5 = 6.80%

Now check it against what ₹1 actually became:

1.15 × 1.12 × 0.90 × 0.92 × 1.25 = 1.3331
Geometric mean (TWRR) = 1.3331^(1/5) − 1 = 5.92%

Illustrative extension: on a ₹1 crore investment held throughout, compounding at the geometric 5.92% gives the true ending value of about ₹1.333 crore. Compounding at the arithmetic 6.80% would predict about ₹1.389 crore — an overstatement of roughly ₹5.6 lakh. A marketing brochure quoting the 6.80% "average" is not lying about the arithmetic; it is answering the wrong question.

Chapter 20 (Performance Measurement and Evaluation of Portfolio Managers), section 20.2.3, sets arithmetic against geometric mean return with the −50%/+100% example, and a Chapter 20 caselet asks for the simple arithmetic average (6.80%) alongside TWRR (5.92%) and MWRR (6.09%) on the same data. Expect a question on which average is always higher, and when the two are equal.

• Arithmetic mean ≥ geometric mean, always. Equal only when every period's return is identical (or there is only one period).
• The arithmetic mean is for forecasting a single future period; the geometric mean is for describing long-run compounded growth.
• Include the sign. −10% and −8% are subtracted, not added — the caselet's explanation says "add all the returns with the sign".
• A manager can raise the arithmetic average by adding risk without improving the ending value — the workbook's point about the ₹100 → ₹50 → ₹100 manager.
• Do not confuse the arithmetic mean of returns with the weighted average return of a portfolio's holdings (section 20.2.10) — that is a different calculation across securities, not across time.