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CFA Level I Exam · Returns of Financial Assets and Instruments

Holding Period Return, Arithmetic and Geometric Mean

Updated 5 October 2026 · Fact-checked

Holding period return (HPR) is the total return over one period: (ending value − beginning value + income) ÷ beginning value. To average several periods, use the arithmetic mean for a typical single-period return, the geometric mean for compound growth over time, and the harmonic mean for average cost of equal-money purchases.

Understand Return Measures: HPR, Arithmetic and Geometric Mean

A holding period return measures what you earned on an investment over one stretch of time. It adds price change and income (dividends or interest), then divides by the starting value. It can cover a day, a month or ten years.

Once you have returns for several periods, you often want one number to summarise them. There are three common averages. Each answers a different question, so the exam tests whether you pick the right one.

The arithmetic mean adds the returns and divides by the count. It is the best estimate of the return you expect in a single future period. The geometric mean compounds the returns and takes the nth root. It gives the constant per-period return that would produce the same ending wealth. It describes past compound growth.

The geometric mean is never larger than the arithmetic mean. They are equal only when every return is the same. The more the returns vary, the wider the gap. This is why volatility drags down compound growth.

The harmonic mean is for ratios such as price per share when you invest a fixed amount each period. It gives the average price paid per share. Order of size for positive, unequal values: harmonic < geometric < arithmetic.

Key formulas to remember

Holding period return
HPR = (P1 − P0 + D1) ÷ P0 = (P1 + D1) ÷ P0 − 1
P0 is the starting price, P1 the ending price, D1 income received in the period.
Multi-period HPR
HPR = (1 + R1) × (1 + R2) × … × (1 + Rn) − 1
Link returns by multiplying growth factors. Never add them.
Arithmetic mean return
Mean = (R1 + R2 + … + Rn) ÷ n
Use for the expected return of a single period.
Geometric mean return
RG = [(1 + R1)(1 + R2)…(1 + Rn)]^(1/n) − 1
Use for compound growth over past periods. Each 1 + R must be positive.
Harmonic mean
XH = n ÷ Σ(1 ÷ Xi)
Use for average price when a fixed money amount buys each time. All values must be positive.
Ordering of means
Harmonic ≤ Geometric ≤ Arithmetic
Holds for positive values. All three are equal only if all observations are equal.

How to solve Return Measures: HPR, Arithmetic and Geometric Mean questions

Use this routine for any question on return measures.

  1. 1Read what is asked: a single-period return, a compound or past average, an expected next-period return, or an average purchase price.
  2. 2Write each period's return as a decimal and convert to growth factors (1 + R).
  3. 3For one period, compute (ending price + income − beginning price) ÷ beginning price.
  4. 4For compound growth, multiply the growth factors, take the nth root, and subtract 1.
  5. 5For a typical single-period expectation, add the returns and divide by n.
  6. 6For average price with equal money invested each time, use the harmonic mean of the prices.
  7. 7Check the result: geometric should not exceed arithmetic. Then pick the option that matches, watching rounding.

Quickest way: Pick the mean first, then calculate

When to use it: Use when time is short, at about 90 seconds per question.

  1. Decide the mean from the wording: 'compound' or 'over the period' means geometric; 'expected' or 'single period' means arithmetic; 'average price' means harmonic.
  2. Compute the arithmetic mean first. It gives an upper bound, so you can discard any geometric option above it.
  3. On the calculator, multiply growth factors, then use yx with 1/n. BA II Plus: enter product, press yx, enter 1 ÷ n, press =, then subtract 1. HP 12C: product, ENTER, n, 1/x, yx.
  4. Pick the remaining option and confirm it lies below the arithmetic mean.

Common mistakes in Return Measures: HPR, Arithmetic and Geometric Mean

  • Averaging returns arithmetically to describe compound growth.

    The arithmetic mean is quick and familiar.

    Fix: If the question asks for past growth of wealth over time, use the geometric mean.

  • Forgetting to add 1 before multiplying returns.

    Returns look like ready-made multipliers.

    Fix: Always convert to growth factors, 1 + R, then subtract 1 at the end.

  • Leaving out income when computing HPR.

    Focus falls on price change.

    Fix: Include dividends or interest received in the period in the numerator.

  • Choosing the arithmetic mean for the average price of fixed-money purchases.

    Students treat prices like any other data.

    Fix: If equal money is invested each period, the average price paid is the harmonic mean.

  • Picking a geometric answer larger than the arithmetic mean.

    Calculation slips, such as forgetting the root.

    Fix: Use the ordering as a check. A geometric mean above the arithmetic mean signals an error.

Worked examples

Example 1

A share is bought at $40. It pays a $2 dividend and is sold a year later for $42. What is the holding period return?

Show the solution
  1. HPR = (P1 − P0 + D1) ÷ P0.
  2. P1 − P0 = 42 − 40 = 2.
  3. Add dividend: 2 + 2 = 4.
  4. Divide by 40: 4 ÷ 40 = 0.10.

Answer: 10.0%

Example 2

A fund returns +20%, −10% and +15% in three consecutive years. What are the arithmetic mean and the geometric mean annual returns, to one decimal place?

Show the solution
  1. Arithmetic mean = (20 − 10 + 15) ÷ 3 = 25 ÷ 3 = 8.33%, or 8.3%.
  2. Growth factors: 1.20 × 0.90 × 1.15.
  3. 1.20 × 0.90 = 1.08; 1.08 × 1.15 = 1.242.
  4. Cube root of 1.242: 1.075 cubed is 1.2423, so root ≈ 1.0749.
  5. Geometric mean ≈ 7.5%.
  6. Check: 7.5% is below 8.3%, as expected.

Answer: Arithmetic mean 8.3%; geometric mean 7.5%

Exam tips

  • Questions are three-option MCQs. The wrong options are often the arithmetic mean when geometric is asked, so work out both and match the wording.
  • Use the ordering harmonic ≤ geometric ≤ arithmetic to eliminate options without full calculation.
  • Watch for the word 'income' or 'dividend' in HPR questions. It must go in the numerator.
  • Practise the yx key on your approved calculator until the root step takes under 20 seconds.
  • Conceptual questions may ask which mean suits a purpose. Match: expected return to arithmetic, past compound growth to geometric, average cost to harmonic.

Practice questions from Returns of Financial Assets and Instruments

Return Measures: HPR, Arithmetic and Geometric Mean in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Return Measures: HPR, Arithmetic and Geometric Mean: frequently asked questions

What is the difference between arithmetic and geometric mean return?

The arithmetic mean is the simple average of period returns. The geometric mean is the constant compound rate that gives the same ending wealth. The geometric mean is lower whenever returns vary, because losses reduce the base that later gains compound on.

How do I calculate geometric mean return on the CFA calculator?

Add 1 to each return, multiply the factors, raise the product to the power 1/n using the yx key, then subtract 1. For three periods, the exponent is 1 ÷ 3. Convert the result to a percentage.

When do I use the harmonic mean?

Use it for the average price paid when you invest the same money amount at each purchase, as in cost averaging. It weights low prices more heavily because the same money buys more units. It is always at or below the arithmetic mean for positive values.

Can the geometric mean be larger than the arithmetic mean?

No, not for positive growth factors. They are equal only when every return is identical. If your geometric answer is higher, recheck the calculation.