What this tool does
Paste a series of period returns (monthly, quarterly, annual — whatever you have) and the calculator returns five numbers that together describe an investment's historical performance: arithmetic mean, geometric mean (CAGR), total cumulative return, volatility, and Sharpe ratio. You also get a bar chart of the per-period returns and a growth curve showing how $10,000 would have evolved over the same window.
Arithmetic vs geometric mean — which to use?
The arithmetic mean averages period returns additively. The geometric mean multiplies them. For any non-zero variance, the geometric mean is always less than the arithmetic mean, and the gap is roughly σ² / 2 — the so-called volatility drag.
Use the geometric mean (CAGR) for:
- Stating past performance of a fund, stock, or portfolio.
- Projecting how a lump sum would grow under a constant return assumption.
- Comparing the historical performance of competing investments.
Use the arithmetic mean for:
- Feeding into CAPM or similar single-period expected-return models.
- Unbiased estimate of the next period's return under i.i.d. assumptions (rarely realistic for real markets).
The formulas
Arithmetic mean = (r₁ + r₂ + … + rₙ) / n
Geometric mean = [(1+r₁)(1+r₂)…(1+rₙ)]^(1/n) − 1
CAGR = (End / Start)^(1/years) − 1 (same thing, if returns are annual)
Cumulative return = (1+r₁)(1+r₂)…(1+rₙ) − 1
Volatility (σ) = √( Σ(rᵢ − r̄)² / (n − 1) ) (sample std dev)
Sharpe ratio = (annualised geom mean − rf) / annualised σ
Annualised σ = σ_period × √k (k = periods per year)
Annualised geom = (1 + geom_period)^k − 1Time-weighted return (TWR)
For mutual fund and manager performance reporting the industry standard is TWR — the geometric mean of periodic returns, ignoring external cash flows (deposits and withdrawals). If you enter a fund's published monthly or quarterly returns, the geometric mean output of this tool is the TWR. Money-weighted return (IRR) is a different animal — it accounts for when cash was added or withdrawn, and is what this tool's sibling IRR Calculator computes.
Volatility drag — the real reason geometric is lower
An investment that alternates +50% and −50% returns has arithmetic mean 0% but geometric mean roughly −13%. The variance in returns eats compound growth. The approximation geometric ≈ arithmetic − σ²/2 holds well for small returns and makes the cost of volatility concrete: a portfolio with 20% arithmetic mean but 40% volatility gives up ~8 percentage points to drag, ending with a ~12% compound rate.
Sharpe ratio cheat-sheet
| Sharpe | Verdict |
|---|---|
| < 0 | Underperforming the risk-free rate — not worth the risk. |
| 0 – 1 | Subpar. Most vanilla long-only portfolios land here. |
| 1 – 2 | Good. Well-diversified equity portfolios. |
| 2 – 3 | Very good. Top quartile hedge funds. |
| > 3 | Excellent — or your data is wrong. Double-check. |
Tips & gotchas
- Units matter. If you enter
10, the tool reads it as 10% (i.e., 0.10). If you mean 10 basis points, enter0.1. - Period consistency. Don't mix monthly and annual returns in the same list. Pick one period.
- Set the correct “Period =” dropdown so annualisation is right. 12 monthly returns → select Month (k=12). 10 yearly returns → Year (k=1, no-op).
- Sharpe uses geometric. We use the annualised geometric mean (not arithmetic) as the numerator, which is the more conservative and realistic convention.
- Sample vs population σ. This tool uses sample standard deviation (divide by n−1), the standard choice for historical return analysis.
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