Four averages appear in return work and they are not interchangeable. Each answers a different question, and picking the wrong one is the commonest quantitative error in published fund material.
How do the arithmetic and geometric means differ?
The arithmetic mean, Σr/n, answers: what is my best single estimate of next period's return, if periods are independent draws from one distribution? It is the right input to a forward-looking expected-return model and the wrong description of a realised path.
The geometric mean, (Π(1+rᵢ))^(1/n) − 1, answers: at what constant rate did the money actually grow? On the +40, −30, +40, −30 series the terminal wealth relative is 1.40 × 0.70 × 1.40 × 0.70 = 0.9604, so the geometric mean is 0.9604^0.25 − 1 = −1.005051%. Ten lakh rupees ends at ₹9,60,400. Report the arithmetic 5.00% as though it described the path and you claim ₹12,15,506, which is ₹2,55,106 of money that never existed.
The gap between the two is roughly half the variance. The population variance of the four returns is 0.1225, so 5.00% − 12.25%/2 = −1.125% against a true −1.005051%, good to 12 basis points on a series with a 35% standard deviation. Volatility is a tax on compounding, and this is the receipt.
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When do you use a weighted or harmonic mean?
The weighted mean, Σwᵢrᵢ, answers: what did the portfolio return, given how much sat in each part? Sixty-two percent of a book in a sleeve returning +11.80% and 38% in a sleeve returning −2.40% gives 0.62(11.80) + 0.38(−2.40) = 6.404%. The unweighted average of the two sleeve returns is 4.70%, wrong by 1.704 points, and wrong in a direction that depends entirely on which sleeve was bigger.
The harmonic mean, n / Σ(1/xᵢ), answers: what did I pay on average, when I spent a fixed sum each time rather than buying a fixed quantity? Worked example 1 shows it reproducing a purchase plan's realised average cost to the paisa, while the arithmetic mean of the same prices overstates that cost by 6.3125%. The other place it is the honest average is a set of valuation multiples, and the relative-valuation module owns that case.
For a set of positive numbers the three sit in a fixed order: harmonic ≤ geometric ≤ arithmetic. Equality holds only when every number is identical. On the purchase plan above: ₹253.97, ₹261.53, ₹270.00. The spread between them is a measure of dispersion in disguise.