Geometric vs Arithmetic vs Time-Weighted Return: Three Means, Three Answers

The 50% loss / 100% gain asymmetry

The starting intuition. A portfolio loses 50% in year 1, then gains 50% in year 2. What was its average annual return?

• Arithmetic mean: (−50% + 50%) / 2 = 0%. Looks fine.
• Geometric mean (CAGR): √(0.5 × 1.5) − 1 = −13.4%. You lost money.

The geometric mean is right. If you start with $100, lose 50% (down to $50), then gain 50% (up to $75) — you ended at $75, a 25% loss, which over 2 years is −13.4% CAGR. The arithmetic mean ignores the compounding asymmetry: it takes more than a 50% gain to recover from a 50% loss, because the gain compounds off a smaller base.

This isn't a contrived example. For any volatile return series, arithmetic mean ≥ geometric mean (AM-GM inequality). For investments with meaningful vol, the gap is real money over time. The gap has a name: volatility drag.

Volatility drag: the σ²/2 rule

A precise approximation: for a return series with arithmetic mean μ and standard deviation σ, the geometric mean is approximately:

geometric ≈ arithmetic − σ²/2

(in continuous compounding; for discrete returns it's a tiny variation but the intuition is identical). So:

• 10% return, 10% vol: drag = 0.5%, geometric ≈ 9.5%. Modest.
• 10% return, 20% vol: drag = 2%, geometric ≈ 8%. Noticeable.
• 15% return, 40% vol: drag = 8%, geometric ≈ 7%. The strategy looks great on arithmetic mean and mediocre on geometric.
• 20% return, 70% vol (crypto-style): drag = 24.5%, geometric ≈ −4.5%. The arithmetic mean is positive, the compounded reality is negative.

This is why high-vol strategies often underperform their backtest expectations. Backtests often display arithmetic means; real compounded performance follows geometric.

What is arithmetic mean used for?

Three things specifically:

1. Expected single-period return

"What return should I expect next month?" The arithmetic mean is the correct answer in expectation. If past monthly returns averaged 1.2% arithmetically, your best unbiased estimate for next month is 1.2%.

2. Sharpe ratio numerator

The standard Sharpe ratio formula uses arithmetic excess return: Sharpe = (arithmetic mean excess return) / (stdev of excess return). Some practitioners report "geometric Sharpe" (using CAGR in the numerator), but it's nonstandard and not directly comparable to published Sharpes.

3. Markowitz mean-variance optimization

Markowitz portfolio theory derives its optimal weights from arithmetic means of asset returns. Substituting geometric means produces sub-optimal portfolios — the math explicitly requires arithmetic inputs. (This is a subtle gotcha that has produced lots of academically wrong but practically deployed portfolios.)

What is geometric mean used for?

Two things:

1. CAGR — actual realized growth

CAGR = (end_value / start_value)^(1/years) − 1

This is the constant annual growth rate that would have produced the actual end value from the actual start value. It's what your portfolio actually grew at. It equals the geometric mean of annualized returns. Use the CAGR Calculator to compute directly.

2. Long-term wealth projection

If you're projecting wealth N years out, use geometric mean (or run a Monte Carlo simulation with median path — which approximates geometric). Arithmetic mean systematically overshoots long-term wealth by σ²T/2 over horizon T. For a 20-year projection at 20% vol, that's 4% — non-trivial.

The QuantOracle Monte Carlo Simulation Calculator shows both the mean and median terminal value across simulated paths; the median is closer to the geometric expected outcome and is the more useful planning figure for risk-aware retirees and traders.

What is time-weighted return?

Time-weighted return (TWR) is the geometric return computed in a way that strips out the impact of intermediate cash flows. For portfolios with no cash flows, TWR = geometric return = CAGR. For portfolios with deposits/withdrawals, they diverge.

The mechanism: TWR chains the geometric returns of sub-periods between cash flows, weighting each sub-period equally regardless of how much capital was deployed at that time. This isolates manager skill from cash-flow timing decisions (which are usually outside the manager's control).

Contrast with dollar-weighted return (also called IRR or money-weighted return), which weights periods by capital level. IRR captures the actual investor experience but conflates manager skill with the timing of when money was added or withdrawn.

When to use which:

• TWR for comparing manager skill (industry standard, GIPS-compliant)
• IRR for measuring actual investor experience (useful for individual performance reporting)

A concrete example: same returns, different stories

A portfolio with three years of returns: +30%, −20%, +30%. What was the average?

The 2.3 percentage-point gap is the volatility drag. Both numbers are correct — they answer different questions. The arithmetic 13.3% is the right expected single-period return going forward. The geometric 11.0% is the right answer to "what did the portfolio actually grow at over those 3 years?"

If you started with $100,000, after the three years you had $135,200. (CAGR check: 100,000 × 1.11³ = 135,205 ✓.) Using arithmetic mean would project 100,000 × 1.133³ = 145,500 — overstating reality by $10,300.

The decision rule

1. Forward-looking wealth projection over N years → geometric mean. Or equivalently, use the Monte Carlo Calculator and read the median terminal value.
2. Reporting realized performance → CAGR (= annualized geometric mean). Use the CAGR Calculator.
3. Sharpe ratio, Sortino, Calmar, any risk-adjusted return ratio → arithmetic mean in the numerator. The Sharpe Ratio Calculator uses arithmetic by default.
4. Markowitz mean-variance optimization → arithmetic means as inputs. Geometric inputs produce wrong portfolios.
5. Comparing managers / strategies fairly → time-weighted return (TWR). For portfolios without cash flows, this is the same as CAGR.
6. Individual investor performance reporting → either TWR or IRR depending on whether cash-flow timing was the investor's decision. If yes, IRR. If no, TWR.

Common mistakes to avoid

Reporting arithmetic mean as "average return"

Most performance tearsheets unfortunately do this. If you see a fund advertising "15% average return" with 25% volatility, the geometric reality is around 11.9% — the realized growth investors actually experienced. Always check whether the number cited is arithmetic or geometric; with high-vol strategies the gap is meaningful.

Using arithmetic mean in long-horizon Monte Carlo

GBM Monte Carlo simulations use log-normal returns parameterized by μ (drift) and σ (vol). Plugging in the arithmetic mean of historical returns as μ overshoots the terminal wealth distribution because the actual realized drift is closer to (arithmetic − σ²/2). For wealth projections, plug in the geometric mean directly OR use the arithmetic mean with σ correction.

Comparing Sharpe ratios computed with different conventions

If one manager reports Sharpe using arithmetic excess return (standard) and another uses geometric (nonstandard), their numbers aren't directly comparable. Geometric-Sharpe is always lower. Verify both use the same convention before drawing conclusions.

Confusing CAGR with arithmetic mean of returns

A portfolio with 10% annual returns for 5 years has CAGR = 10% (because the geometric mean of constant returns equals each return). But a portfolio averaging 10% with vol will have CAGR < 10%. The two are equal only when returns are constant — which they aren't in any real strategy.

Related calculators

• CAGR Calculator — geometric/time-weighted annualized return from start/end values
• Sharpe Ratio Calculator — uses arithmetic mean by standard convention
• Monte Carlo Simulation Calculator — shows both mean and median terminal values (arithmetic vs geometric expected)
• Value at Risk Calculator — uses arithmetic mean returns + stdev (the parametric VaR model)

Related comparisons

• Sharpe vs Sortino vs Calmar — all three use arithmetic mean in their numerator
• VaR vs CVaR vs Max Drawdown — risk metrics that depend on the arithmetic/geometric distinction
• Kelly vs Fixed Fractional vs Optimal-f — Kelly maximizes geometric growth specifically (not arithmetic)

References

• Kelly Jr., J. L. (1956). "A new interpretation of information rate." Bell System Technical Journal 35(4), 917-926. Kelly criterion explicitly maximizes the geometric growth rate, not arithmetic.
• Markowitz, H. (1952). "Portfolio Selection." Journal of Finance 7(1), 77-91. Portfolio theory uses arithmetic means.
• Sharpe, W. F. (1966). "Mutual fund performance." Journal of Business 39(1, Part 2), 119-138. Sharpe ratio uses arithmetic excess return.
• Jorion, P. (1997). "Value at Risk: The New Benchmark for Managing Financial Risk." — standard reference for VaR/risk metric mathematics including mean conventions.
• Bodie, Kane, & Marcus, "Investments" (any recent edition) — Chapter 5 covers arithmetic vs geometric mean in detail with worked examples.