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CFA Level I Exam · Portfolio Risk and Return: Part I

Holding Period Return and Return Measures for CFA Level 1

Updated 7 October 2026 · Fact-checked

Holding period return (HPR) is the total gain on an investment over one period: (end value − start value + income) ÷ start value. Other measures build on it. Chain HPRs for geometric and time-weighted returns, solve for IRR for money-weighted returns, and convert nominal to real returns by dividing growth factors.

Understand Holding Period Return and Return Measures

A holding period return measures what you earned on an investment over a single stretch of time. It includes price change and any income, such as dividends or interest, and it is shown as a percentage of what you paid at the start.

One period is easy. Several periods need an average. The arithmetic mean adds the period returns and divides by n. It is the best estimate of the return in a single future period. The geometric mean compounds the returns and takes the nth root. It gives the true average growth rate over the whole span. Geometric is never larger than arithmetic. They are equal only if all returns are the same. The bigger the volatility, the bigger the gap.

When you add or withdraw cash during the period, two views exist. The money-weighted return is the IRR of all cash flows. It reflects the timing and size of your deposits, so it measures your own result. The time-weighted return chains the sub-period HPRs and ignores flows. It measures the manager's skill, because the manager does not control when clients add money.

Returns are also quoted in different forms. Gross return is before management fees and other costs (and often before taxes). Net return is after fees and expenses. Nominal return includes inflation. Real return strips inflation out, so it shows the gain in purchasing power.

Annualizing lets you compare returns over different lengths. Compound a short-period return up, or take a root of a long-period return down. Always keep returns as decimals inside formulas.

Key formulas to remember

Holding period return
HPR = (P₁ − P₀ + D) ÷ P₀ = (P₁ + D) ÷ P₀ − 1
D is income received during the period. Reinvested income needs care.
Multi-period HPR
HPR = (1 + R₁)(1 + R₂)…(1 + Rₙ) − 1
This is also the time-weighted cumulative return when R values are sub-period returns.
Arithmetic mean return
R̄ = (R₁ + R₂ + … + Rₙ) ÷ n
Best single-period forecast. Overstates compound growth when returns vary.
Geometric mean return
R_G = [(1 + R₁)(1 + R₂)…(1 + Rₙ)]^(1/n) − 1
Use for compound growth over past periods. Always ≤ arithmetic mean.
Annualizing a return
Annual return = (1 + HPR)^(365 ÷ days) − 1
For a period shorter than a year, this compounds up. For longer periods, the exponent is below 1.
Money-weighted return
Σ CFₜ ÷ (1 + IRR)ᵗ = 0
Treat deposits as outflows and ending value as an inflow. Solve with the calculator.
Real return (exact)
(1 + real) = (1 + nominal) ÷ (1 + inflation)
Approximation: real ≈ nominal − inflation. Use exact when options differ closely.
Net return
Net return ≈ gross return − fees and expenses
Fees are usually a percentage of assets. Net is what the investor keeps.

How to solve Holding Period Return and Return Measures questions

Start by deciding what the question really asks, then pick the matching measure.

  1. 1Read the stem for the key word: holding period, average, annualized, money-weighted, time-weighted, real, or net.
  2. 2List the data: prices, income, cash flows with dates, inflation, fees. Convert percentages to decimals.
  3. 3Check whether there are cash flows in or out during the period. If so, the question probably contrasts money-weighted and time-weighted.
  4. 4For time-weighted, find each sub-period HPR first, using the value just before and after each flow. Then chain them.
  5. 5For money-weighted, write the cash flow timeline and solve for IRR on the calculator.
  6. 6For averages over several periods, use arithmetic for a one-period expectation and geometric for realized compound growth.
  7. 7Annualize if the periods differ from a year, or if the options are annual figures.
  8. 8Test the three options for reasonableness. Geometric below arithmetic and real below nominal are quick filters.

Quickest way: Growth-factor shortcut

When to use it: Use for any chained, averaged or inflation-adjusted return question.

  1. Turn every return into a growth factor: 1 + R.
  2. Multiply factors to chain periods. Divide by the inflation factor for real return.
  3. For geometric mean, take the nth root of the product. On the BA II Plus, enter the product, then press yˣ, then enter 1 ÷ n, then =.
  4. Subtract 1 at the end and read the percentage.
  5. Eliminate options: a geometric mean larger than the arithmetic mean is wrong, and a real return above the nominal return (with positive inflation) is wrong.

Common mistakes in Holding Period Return and Return Measures

  • Averaging returns with the arithmetic mean when asked for compound growth.

    The arithmetic mean is simpler and familiar from other subjects.

    Fix: If the question asks for the average annual growth of an investment over past years, use the geometric mean.

  • Calculating real return as nominal minus inflation when the options are close.

    The approximation is quick and widely used.

    Fix: Use (1 + nominal) ÷ (1 + inflation) − 1 whenever the answer choices are within a few basis points.

  • Mixing up when to use money-weighted and time-weighted returns.

    Both use cash flows, and the names sound alike.

    Fix: Time-weighted judges the manager, who does not control flows. Money-weighted reflects the investor's own experience and depends on timing.

  • Leaving out income in the HPR or forgetting to divide by the starting value.

    Candidates focus on price change only.

    Fix: Always add dividends or interest to the numerator and divide by the beginning price.

  • Using the wrong exponent when annualizing.

    Confusion between days and years, or between compounding up and down.

    Fix: Exponent = periods in a year ÷ length of your period. A 90-day return uses 365 ÷ 90.

  • Entering returns as whole numbers in the geometric formula, for example using 8 instead of 0.08.

    Percentages are typed straight into the calculator.

    Fix: Divide by 100 first. Then 1 + R is a growth factor such as 1.08.

Worked examples

Example 1

A fund returned +20%, −10% and +15% in three consecutive years. What is the geometric mean annual return? A) 7.5% B) 8.3% C) 8.7%

Show the solution
  1. Convert to growth factors: 1.20, 0.90 and 1.15.
  2. Multiply: 1.20 × 0.90 = 1.08. Then 1.08 × 1.15 = 1.242.
  3. Take the cube root: 1.242^(1/3). Since 1.075³ ≈ 1.2423, the root is about 1.0750.
  4. Subtract 1: about 7.5%.
  5. Check: the arithmetic mean is (20 − 10 + 15) ÷ 3 = 8.33%. The geometric mean must be lower, which confirms the answer and rules out B and C.

Answer: A) 7.5%

Example 2

A bond investor earns a nominal return of 6.0% in a year when inflation is 2.5%. What is the real return, to the nearest 0.1%? A) 3.4% B) 3.5% C) 3.8%

Show the solution
  1. Use the exact relationship: 1 + real = 1.06 ÷ 1.025.
  2. Divide: 1.06 ÷ 1.025 = 1.034146.
  3. Subtract 1: 0.034146, which is 3.4%.
  4. The approximation 6.0% − 2.5% = 3.5% is close but is not the exact answer. Option C is too high for either method.

Answer: A) 3.4%

Exam tips

  • Questions are three-option and standalone, so first apply quick sanity checks: geometric ≤ arithmetic, real < nominal when inflation is positive.
  • When flows occur mid-period, expect a time-weighted calculation that needs sub-period returns. Compute the value just before each flow.
  • If two options differ by a few basis points, use the exact real return formula, not the subtraction shortcut.
  • Learn the story: if a large inflow arrives just before strong returns, the money-weighted return exceeds the time-weighted return. If the inflow arrives just before weak returns, the money-weighted return falls below it. Time-weighted return is unaffected by the timing of flows. Conceptual questions often test this.
  • Budget about 90 seconds per question. Chain growth factors on the calculator in one pass rather than computing each period separately.

Practice questions from Portfolio Risk and Return: Part I

Holding Period Return and Return Measures in other exams

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

Holding Period Return and Return Measures: frequently asked questions

What is the holding period return formula for CFA Level I?

HPR = (ending price − beginning price + income) ÷ beginning price. It covers one period of any length. For several sub-periods, chain the growth factors and subtract 1.

What is the difference between arithmetic and geometric mean return?

The arithmetic mean is the simple average of period returns and estimates a typical single-period return. The geometric mean is the compound average growth rate over several periods. The geometric mean is lower unless all returns are equal.

What is the difference between money-weighted and time-weighted return?

Money-weighted return is the IRR of the cash flows and depends on when you add or withdraw money. Time-weighted return chains sub-period returns and removes the effect of flows. Use time-weighted to compare managers.

How do I calculate real return from nominal return?

Divide one plus the nominal return by one plus the inflation rate, then subtract 1. For example, 6% nominal and 2.5% inflation gives about 3.4% real. Nominal minus inflation is only an approximation.