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Module 15 · The statistics underneath · lesson 2 of 14 · stage 3 of 3 · advanced

Average return and compound return

8 minadvancedsample lesson
A curve bending upward as it compounds, the geometric rate an account grows at, which sits below the average of its yearly returns.
Objective
Tell an arithmetic average return from the compound rate an account actually grows at, work out both for a short record, and estimate how much a wider spread of returns costs.

An average return and the rate an account grows at are different numbers, and the gap between them is not a rounding error. $10,000 that rises 50% becomes $15,000; a 50% fall from there leaves $7,500. The two returns average zero, and the account is down 25%. A loss needs a larger percentage gain to undo it, so returns that swing both ways compound to less than their average.

$10,000 over two years, every path averaging +10% a year
+10%, then +10%Ends at $12,100: compounds at 10.00% a year
+30%, then −10%Ends at $11,700: compounds at 8.17% a year
+50%, then −30%Ends at $10,500: compounds at 2.47% a year
+70%, then −50%Ends at $8,500: compounds at −7.80% a year

The four paths share one average and end between $8,500 and $12,100. The only thing that separates them is how far each year's return sat from that average.

  • Arithmetic mean return — add the period returns and divide by how many there are. It is what an average of yearly returns reports.
  • Geometric mean return — multiply the growth factors, one plus each return, take the root for the number of periods, and subtract one. It is the rate the money actually compounded at.
  • The gap between the two grows with the spread of the returns — roughly half the variance — so two records with one average can end far apart.
  • The geometric mean is never above the arithmetic mean. The two are equal only when every period's return is the same.
The shortcut: compound rate ≈ average − variance ÷ 2
Average +10%, standard deviation 20%10% − 0.20² ÷ 2 = 8.00%; the +30%, −10% path compounds at 8.17%
Average +10%, standard deviation 40%10% − 0.40² ÷ 2 = 2.00%; the +50%, −30% path compounds at 2.47%
Average +10%, standard deviation 60%10% − 0.60² ÷ 2 = −8.00%; the +70%, −50% path compounds at −7.80%

The shortcut is an approximation, closest when the returns are small, and it shows the shape of the cost: the drag grows with the square of the spread. Doubling the standard deviation from 20% to 40% multiplies the drag by four, from 2 points to 8.

The same paths repeated for twenty years, from $10,000
+10% every year$67,275, compounding at 10.00% a year
+30% and −10% alternating$48,068, compounding at 8.17% a year
+50% and −30% alternating$16,289, compounding at 2.47% a year
+70% and −50% alternating$1,969, compounding at −7.80% a year

A trading record has the same gap. Expectancy, from the proving-it module, is an arithmetic average per trade, while a fixed fraction of a growing account compounds. A position that doubles or loses 60% with equal odds averages +20% a turn, yet one turn each way takes $10,000 to $8,000, a compound rate of −10.56% a turn. The average was positive and the account shrank, which is the arithmetic underneath the warning about betting too large a fraction.

NOTE
A figure quoted as an "average annual return" can be either kind. The compound annual growth rate, also called the annualised return, is the geometric one: the single rate that turns the starting value into the ending value. An arithmetic average of the same years is never below it, so a record that quotes only an average may have compounded at a noticeably lower rate.
RISK
Leverage multiplies the spread of returns, and the drag grows with the square of the spread. Twice the +30%, −10% path is +60%, −20%: the average doubles from 10% to 20%, while the compound rate rises only from 8.17% to 13.14% a year. A fund that resets its leverage daily applies this every day, which is why over months it can fall well short of its multiple of the index.
Key takeaway
An account grows at the geometric mean, not the arithmetic average. At the same average, a wider spread of returns means a lower compound rate — lower by roughly half the variance.
Common mistakes
  • Reading an average of yearly returns as the rate the money grew at.
  • Comparing two records by their average return without looking at how widely each one swung.
  • Expecting a percentage loss and an equal percentage gain to cancel out.
  • Assuming a positive average per trade means a growing account.
Test your knowledge · 5 questions
1. $10,000 rises 50% and then falls 50%. It ends at
2. Two years of +50% and then −30% average +10%. The compound annual rate is about
3. Compared with the arithmetic mean of the same returns, the geometric mean is
4. An arithmetic average of 10% a year with a standard deviation of 20% compounds at roughly
5. The single rate that turns a starting value into the ending value is the
Answer every question to check.
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