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Why does the arithmetic mean of +50% and -50% yearly returns overstate actual growth?

level: middleimportance: should knowfreq 44%

answer

  1. returns chain, they do not add
  2. work with factors, not percentages
  3. 1.5 times 0.5 is not 1
  4. the nth root of the product
  5. arithmetic is the optimistic bound

basics

~20 s

Growth multiplies rather than adds: +50% then -50% leaves 0.75 of the starting value, yet the arithmetic mean of the returns is 0%. The geometric mean of the growth factors, about 0.866, reproduces the true ending value.

solid answer

~50 s

Returns compound, so the quantity that combines across periods is the growth factor, not the percentage. Here the factors are `1.5` and `0.5`, and `1.5 x 0.5 = 0.75` — a 25% loss over two years. Averaging the percentages gives 0%, which would predict an unchanged balance, so the arithmetic mean is answering the wrong question. The right average is the geometric mean of the factors, `sqrt(1.5 x 0.5) = sqrt(0.75) = 0.866`, roughly -13.4% a year; applying it twice reproduces 0.75 exactly. In general the geometric mean of n positive values is their product raised to the power `1/n`, equivalently `exp` of the mean of their logarithms. For positive numbers not all equal, the arithmetic mean is strictly larger than the geometric mean, so averaging returns arithmetically always overstates compound growth, and the gap widens with volatility.

go deeper

for a junior

Recall that percentage changes chain by multiplying, not adding, so up 50% then down 50% leaves you at 75% of where you started, and know the word for the right average is geometric.

for a middle

Compute it: multiply the growth factors, take the nth root, and show that applying the result n times reproduces the ending value. Be able to state AM-GM and its equality condition.

for a senior

Show reporting judgment — which average a stakeholder actually needs, why a volatile series widens the gap between the advertised average return and the realised compound one, and how you label the figure.

for a principal

Set the convention for how growth is reported across the organisation, including the period, whether it is compound or single-period, and how volatile series are presented so nobody can pick the flattering average.

## Additive versus multiplicative quantities The arithmetic mean is the right average for quantities that **add**: total distance, total spend, total conversions. Divide the total by the count and you get the value that, repeated n times, reproduces the total. Some quantities do not add — they **multiply**. Period-over-period growth is the classic case. A portfolio that grows 50% and then falls 50% does not experience `+50 - 50 = 0`; it experiences `x1.5` then `x0.5`. The right average for a multiplicative quantity is the one that, repeated n times, reproduces the *product*. That is the geometric mean. ## Working the example Start with 100. - After year 1 at +50%: `100 x 1.5 = 150`. - After year 2 at -50%: `150 x 0.5 = 75`. Ending value 75, a 25% loss over two years. Now compare the two candidate averages. - **Arithmetic mean of returns**: `(+50% + -50%) / 2 = 0%`. Applied twice: `100 x 1.0 x 1.0 = 100`. Wrong by 25. - **Geometric mean of factors**: `(1.5 x 0.5)^(1/2) = 0.75^(1/2) = 0.8660...`, i.e. about -13.40% per year. Applied twice: `100 x 0.866 x 0.866 = 75`. Exact. The geometric mean is defined as `(x_1 x x_2 x ... x x_n)^(1/n)` for positive values, and equivalently as `exp((1/n) * sum ln x_i)` — the arithmetic mean carried out in log space and transformed back. That second form is the practical one: logarithms turn multiplication into addition, so the geometric mean is simply the arithmetic mean of the log growth rates. ## Why the arithmetic mean is always the optimistic one The **AM-GM inequality** states that for positive numbers, the arithmetic mean is greater than or equal to the geometric mean, with equality only when all the values are identical. Check it here: arithmetic mean of the factors is `(1.5 + 0.5)/2 = 1.0`, geometric mean is `0.866`, and `1.0 > 0.866`. This is not a coincidence of the example; it is a direction that holds always. Reporting an average return arithmetically therefore **never understates** compound growth, and generally overstates it. The size of the overstatement grows with the dispersion of the returns: a steady sequence of 5%, 5%, 5% has arithmetic and geometric means that coincide, while a volatile 60%, -40%, 30% opens a large gap. That is why the same fund can advertise an impressive average annual return while an investor who held it throughout sees a much smaller compound annual growth rate. ## Where the geometric mean is the correct centre Beyond compounded returns, use it for anything that combines multiplicatively or is naturally a ratio: - **Growth rates of any kind** — user counts, revenue, prices across periods. - **Ratios and index numbers**, where a value of 2 and a value of 1/2 ought to average to 1. The arithmetic mean gives 1.25 and depends on which side of the ratio you put on top; the geometric mean gives exactly 1 and is invariant to flipping the ratio. - **Combining normalised scores on different scales**, where each score is a multiplier rather than an amount. Two constraints to state plainly. The geometric mean is undefined for negative values and collapses to zero if any value is zero — with returns this is handled by averaging *growth factors* (which are positive as long as the asset is not wiped out) rather than the percentages themselves. And it does not reconcile with a total: mean multiplied by n equals the total, but the geometric mean has no such property, so never use it where sums matter. ## The interview shape The question is usually posed as a trap: the interviewer offers +50% then -50% and asks whether you broke even. The complete answer names all three moves. 1. **Reject the framing**: you are down 25%, because returns chain multiplicatively. 2. **Name the right tool**: the geometric mean of the growth factors, about 0.866 per year, or roughly -13.4%. 3. **Generalise**: the arithmetic mean of returns is an upper bound on compound growth by AM-GM, and the gap widens with volatility. A related trap uses percentage changes on the same base: a price that rises 10% then falls 10% ends at `1.1 x 0.9 = 0.99` of its start, down 1%. The asymmetry has the same cause — the second percentage applies to a different base than the first — and it is worth having ready, because a candidate who explains one usually gets asked the other. ## Reporting habit When a stakeholder asks for average growth, ask what they will do with it. If they intend to project a balance forward, they need the compound figure, which is the geometric mean. If they need a single-period expectation for a model, the arithmetic mean of returns is a legitimate answer to a different question. Report which one you used; a growth number without the word arithmetic or compound attached is ambiguous to anyone who knows the difference.

  • What is the geometric mean of n positive values, and how do logarithms make it easy?
    It is the product of the values raised to the power 1/n. Because the logarithm turns products into sums, it equals exp of the arithmetic mean of the logs, which avoids overflow on long series and reduces the computation to an ordinary average in log space. It is defined only for positive values.
  • A price rises 10% then falls 10%. Where does it end up?
    At 0.99 of the starting price, down about 1%. The factors are 1.1 and 0.9 and their product is 0.99, not 1. The fall applies to the larger post-rise base, so equal-looking percentage moves in opposite directions never cancel — the same multiplicative asymmetry as the plus-and-minus-fifty case.
  • When would you still prefer the arithmetic mean of returns?
    When you want a single-period expected return rather than a realised compound path — for instance as an input to a forward-looking model that treats each period independently. It answers what a typical single year looks like, not what a multi-year balance becomes, so label it clearly and never use it to project an ending value.

Adding percentages is like measuring a staircase by summing step labels; multiplying factors is walking it. After up-half and down-half you are standing below where you started.

saying these in an interview costs you the question

  • Says +50% then -50% breaks even
  • Averages percentage returns instead of growth factors
  • Thinks the geometric mean can exceed the arithmetic mean
  • Applies the geometric mean to values that can be zero or negative
  • Uses the geometric mean where a total must reconcile

context

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