Arithmetic, geometric and harmonic mean returns
The arithmetic mean is the simple average and estimates one period. The geometric mean is the constant rate that reproduces actual ending wealth: compound growth. The harmonic mean is the average cost per unit when you invest a fixed amount each period. Always: harmonic ≤ geometric ≤ arithmetic.
Before this: Holding period return
On the exam
- Calculate and interpret different approaches to return measurement over time and describe their appropriate uses
Calculate and interpret arithmetic, geometric and harmonic means, and describe when to use each. Expect a geometric mean calculation, a "which mean fits" question, and a fixed-amount purchase where the harmonic mean is the average cost.
What is the arithmetic mean return?
It is the simple average of the period returns. It is an unbiased estimate of what a single future period will return, and it is the version statistics uses.
The arithmetic mean return is the right tool for a one-period-ahead forecast. It is the wrong tool for describing what happened over several periods, as the next section shows.
What is the geometric mean return?
It is the constant annual rate that would have produced the same ending wealth. It answers the question “how fast did my money actually grow?”
The geometric mean return works on growth factors (1 + R), never on the raw returns.
- Actual wealth
- Growing at the arithmetic mean
- Growing at the geometric mean
USD 1 ends at 1.056. At the arithmetic mean you would expect 1.331; at the geometric mean, 1.056.
Show working
Arithmetic: (60.0% + -40.0% + 10.0%) ÷ 3 = 10.00%
Geometric: (1.600 × 0.600 × 1.100)1/3 − 1 = 1.05601/3 − 1 = 1.83%
Harmonic (on 1 + R): 3 ÷ (1/1.600 + 1/0.600 + 1/1.100) − 1 = -6.27%
Order check: harmonic ≤ geometric ≤ arithmetic.
The geometric mean is never above the arithmetic mean. They are equal only when every return is identical, and the gap widens as returns become more volatile.
What is the harmonic mean?
It is the right average when a fixed amount of money buys a varying number of units. A fixed amount buys more units when the price is low, so the average cost is pulled toward the lower prices.
That pattern of buying a fixed amount at regular intervals is cost averaging , and the harmonic mean of the prices is what it costs you per unit.
Which mean does the question want?
Match the wording of the question to the mean.
| The question asks for… | Use |
|---|---|
| Average annual return over past years, compound growth, “annualized return over the period” | Geometric |
| Expected return for the next single period, sample mean for statistics | Arithmetic |
| Average cost per share with equal amounts invested | Harmonic |
Exam tip When a question says “compound”, “annualized” or “over the period”, it wants the geometric mean.
Go deeper Why volatility drags the geometric mean down
A close approximation links the two means: the geometric mean is about the arithmetic mean minus half the variance, . Take returns that alternate +20% and −20%. The arithmetic mean is 0%. After one up year and one down year, 1.20 × 0.80 = 0.96, so you have lost 4% over two years, a geometric mean of about −2.02% a year. The approximation gives 0% − (0.20)²/2 = −2%. The more the returns swing, the larger the penalty, which is why steadier portfolios can beat more volatile ones with the same average return.
Go deeper The harmonic mean of price-to-earnings ratios
Suppose you hold equal amounts in two shares with P/E ratios of 10 and 40. The arithmetic mean of 25 sounds right but is misleading. Each unit of money in the first share buys 0.10 of earnings, and in the second 0.025. With equal money in each, the average earnings per unit of money is (0.10 + 0.025) ÷ 2 = 0.0625, which is a P/E of 16. That equals the harmonic mean, 2 ÷ (1/10 + 1/40) = 16. The harmonic mean works because it averages the earnings yields, which are the quantities that actually add up across equal investments, and then flips back to a ratio.
Go deeper Trimmed and winsorized means
When a data set has extreme outliers, a plain average can mislead. A trimmed mean discards a fixed percentage of the largest and smallest observations before averaging. A winsorized mean keeps them but replaces the extremes with the nearest remaining values. Both reduce the influence of outliers. They belong to the statistics module that follows this one, where they sit alongside the median and the mode. For now, the point is only that “mean” is a family of tools and each has a job.
Exam traps
- Reporting the arithmetic mean when the question asks for compound or annualized performance.
- Multiplying raw returns instead of (1 + R). A negative return breaks the root.
- Taking the n-th root of the sum of growth factors instead of their product.
Checkpoint
3 exam-style questions. About 90 seconds each.
1A portfolio returns 20%, −10% and 5% over three years. Its geometric mean annual return is closest to:
2To describe how fast a fund's value actually grew over the past five years, an analyst should most appropriately use the:
3An investor invests USD 600 each month in a fund whose unit price is USD 10, USD 12 and USD 15 in the three months. The average cost per unit is closest to:
Key takeaways
- Arithmetic: the simple average, the best guess for one period ahead.
- Geometric: compound growth, the correct way to report past multi-period performance.
- Harmonic: average cost with a fixed amount invested each period.
- Harmonic ≤ geometric ≤ arithmetic, and the gap grows with volatility.
Where this comes back
- Portfolio Risk and Return: Part I Portfolio Management
Questions candidates ask
Why is the geometric mean lower than the arithmetic mean?
Losses hurt compound growth more than equal-sized gains help, because each return applies to a changed balance. A 50% gain followed by a 50% loss leaves you down 25%. The more returns vary, the wider the gap. They are equal only when every return is identical.
When should you use the harmonic mean in finance?
When you invest a fixed amount each period, the harmonic mean of the prices is your average cost per unit. It is also used to average ratios such as price-to-earnings across holdings, where the arithmetic mean would be pulled up by a few large values.
Which mean return should I use for CFA Level 1 questions?
Geometric for past performance over several periods, such as an annualized return. Arithmetic for the expected return of a single future period, or as a sample statistic. Harmonic for average cost with equal amounts invested. If a question says compound or annualized, it wants geometric.
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