Coefficient of variation (CV): concise explainer
What it is
– The coefficient of variation (CV) is a unitless measure that compares the amount of dispersion (variability) in a data set to its mean (average). In finance, CV is commonly used to compare volatility (risk) relative to expected return across assets or portfolios.
– Why unitless matters: because CV is a ratio, you can compare returns expressed in different units or scales (for example, monthly vs. annual returns, or dollar vs. percentage changes).
Key definitions
– Mean (μ or x̄): the arithmetic average of a data series (e.g., average annual return).
– Standard deviation (σ for population, s for sample): a measure of how far observations typically are from the mean; it quantifies dispersion or volatility.
– Coefficient of variation (CV): CV = standard deviation ÷ mean. Often shown as a decimal or multiplied by 100 to give a percentage.
Formulas
– Population CV: CV = σ / μ
– Sample CV (common in empirical finance): CV = s / x̄
– Optional percent form: CV% = (s / x̄) × 100
How to calculate — step‑by‑step
1. Choose the return series and sample period (e.g., monthly returns for 5 years).
2. Compute the mean of the series (x̄).
3. Compute the sample standard deviation (s) of the series.
4. Divide the standard deviation by the mean: CV = s / x̄.
5. Optionally multiply by 100 for a percent value.
Excel quick method
– Use STDEV.S(range) for a sample standard deviation and AVERAGE(range) for the mean.
– Example formula: =STDEV.S(A1:A60)/AVERAGE(A1:A60)
– For percent: =STDEV.S(A1:A60)/AVERAGE(A1:A60)*100
Checklist for computing and interpreting CV
– [ ] Confirm whether you need sample (s) or population (σ) standard deviation.
– [ ] Ensure returns are measured on the same frequency (both annual or both monthly).
– [ ] Compute mean and standard deviation.
– [ ] Check mean is not zero or negative (see pitfalls).
– [ ] Calculate CV and compare across assets or strategies.
– [ ] Interpret lower CV as a better historical volatility-to-return tradeoff for a risk-averse viewpoint; interpret carefully for other objectives.
Worked numeric example
Suppose two funds with long-run annual statistics:
– Fund A: mean return x̄ = 8.0% (0.08), sample standard deviation s = 12.0% (0.12).
– Fund B: mean return x̄ = 10.0% (0.10), sample standard deviation s = 18.0% (0.18).
Compute CVs:
– CV_A = 0.12 / 0.08 = 1.50 → CV% = 150%
– CV_B = 0.18 / 0.10 = 1.80 → CV% = 180%
Interpretation: Fund A has a lower CV, meaning its volatility per unit of average return has been lower historically. A risk‑averse investor might prefer Fund A on that basis; a risk-seeking investor could accept the higher CV of Fund B for the chance of higher mean return. This is an historical comparison, not a forecast.
What CV tells you (and what it does not)
– Tells you: how large dispersion is relative to the mean. Useful for comparing risk per unit of return.
– Does not tell you: directional bias (positive vs. negative returns) beyond the mean, nor does it substitute for more complete risk analysis (drawdowns, skewness, correlation, tail risk).
Advantages
– Unitless—enables comparisons across assets with different scales or units.
– Simple to compute and easy to interpret as a relative risk metric.
– Useful when screening many assets for risk-adjusted stability.
Disadvantages and pitfalls
– Mean near zero: if the mean is close to zero, CV becomes unstable and can be misleading.
– Negative mean: a negative expected return makes the ratio hard to interpret.
– Ignores distribution shape: CV uses mean and standard deviation only; it ignores skewness and fat tails.
– Sensitive to sampling choices: different time windows or return frequencies change both mean and SD, altering CV.
Practical notes
– Use consistent data frequency (e.g., all annualized) when comparing CVs.
– Consider complementary metrics (Sharpe ratio, Sortino ratio, maximum drawdown) to capture other aspects of risk-adjusted performance.
– Remember CV is retrospective—based on historical mean and volatility—and does not guarantee future behavior.
Sources for further reading
– Investopedia — Coefficient of Variation (CV): https://www.investopedia.com/terms/c/coefficientofvariation.asp
– Wikipedia — Coefficient of variation: https://en.wikipedia.org/wiki/Coefficient_of_variation
– Microsoft Support — STDEV.S function (Excel): https://support.microsoft.com/en-us/office/stdev-s-function-4d6a2f2f-0113-4497-b5d5-01c6d12b7f32
– Khan Academy — Standard deviation and variance (educational): https://www.khanacademy.org/math
https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data
Worked numeric example (step‑by‑step)
– Data (monthly returns for one asset, expressed as decimals): r = [0.01, 0.02, −0.01, 0.03, 0.00].
– Step 1 — Arithmetic mean (monthly):
mean = (0.01 + 0.02 − 0.01 + 0.03 + 0.00) / 5 = 0.01 (1.00% per month).
– Step 2 — Sample standard deviation (monthly). Using the sample SD (divide by n−1):
deviations = [0, 0.01, −0.02, 0.02, −0.01]
squared sum = 0^2 + 0.01^2 + (−0.02)^2 + 0.02^2 + (−0.01)^2 = 0.001
sample variance = 0.001 / (5 − 1) = 0.00025
SD = sqrt(0.00025) = 0.015811 ≈ 1.5811% per month.
– Step 3 — Coefficient of variation (monthly):
CV_monthly = SD / mean = 0.015811 / 0.01 = 1.5811 → 158.11%.
Interpretation: volatility is about 1.58 times the average monthly return.
– Step 4 — Annualize (optional, but keep frequency consistent):
– Annualized arithmetic mean ≈ mean_monthly × 12 = 0.01 × 12 = 0.12 (12% per year).
(Note: arithmetic mean annualization assumes simple summation; for compounded returns use geometric mean.)
– Annualized SD ≈ SD_monthly × sqrt(12) = 0.015811 × 3.4641 = 0.05480 ≈ 5.48% per year.
– CV_annual = 0.05480 / 0.12 = 0.4567 → 45.67%.
Key assumptions in the example
– Sample SD (n−1) was used, which is standard for historical-return estimates.
– Annualization assumes independent, identically distributed (i.i.d.) returns (SD scales with sqrt(time)). If returns are serially correlated, the sqrt rule may be invalid.
– Arithmetic mean was used for simple annualization; for long horizons the geometric mean (compounded return) is often more appropriate.
Practical checklist before you compute CV
1. Choose your return measure: arithmetic vs geometric (decide based on your question).
2. Use consistent frequency: convert all inputs to monthly, quarterly, or annual before comparing.
3. Pick SD estimator: sample SD (STDEV.S) for historical samples; population SD (STDEV.P) if appropriate.
4. Inspect the mean: if mean ≈ 0 or negative, CV is unstable or misleading—consider alternatives.
5. Check for serial correlation, skewness and fat tails—CV ignores these features.
6. Use complementary metrics: Sharpe ratio (uses risk‑free rate), Sortino ratio (downside risk), max drawdown.
How to compute in spreadsheets and Python
– Excel / Google Sheets:
– Mean: =AVERAGE(range