Average Return Calculator

Q: Why is the geometric mean different from the arithmetic mean?

The arithmetic mean simply averages the percentages. The geometric mean accounts for compounding — a 50% gain followed by a 33% loss returns to the starting point (0% CAGR), but the arithmetic average shows +8.5%. The geometric mean correctly reflects actual investor outcomes because money compounds period to period.

Q: What is CAGR and when should I use it?

CAGR (Compound Annual Growth Rate) is the steady annual rate that would grow an investment from its starting value to its ending value over the holding period. Use CAGR to compare investments held for different periods (both normalized to an annual basis) and to communicate long-term investment performance accurately.

Q: What is standard deviation in investment returns?

Standard deviation measures the variability (volatility) of annual returns around the average. A portfolio with 8% average return and 15% standard deviation typically delivers returns between -7% and +23% in two-thirds of years. Higher standard deviation means higher volatility and risk.

Q: What is the historical average return of the S&P 500?

The S&P 500 has returned approximately 10% annually nominal (before inflation) and 7% real (after inflation) since 1926. The geometric mean is typically 1% to 2% lower than the arithmetic mean due to volatility drag. Individual years vary from -37% (2008) to +38% (1995), with the average realized only over very long periods.

Q: How does volatility reduce actual compounded returns?

Volatility drag means that high year-to-year volatility reduces the geometric mean below the arithmetic mean. A portfolio that returns +40% then -28.6% has an arithmetic mean of +5.7% but a geometric mean of 0% — the two years cancel out. Smoother returns with the same arithmetic average produce higher real wealth.

Q: What is a realistic expected CAGR for different asset classes?

Broad US equities (S&P 500): 10% nominal, 7% real. International developed: 7% to 8% nominal. Emerging markets: 8% to 10% nominal but higher volatility. Bonds: 4% to 5% nominal. Balanced 60/40 portfolio: 7% to 8% nominal. Real estate: 8% to 12% total return including appreciation. Cash/savings: 2% to 5%.

Calculate the true compound annual growth rate (CAGR) from investment start and end values, or analyze a series of annual returns to find the geometric mean (true compounded return), arithmetic mean, and standard deviation. The geometric mean is always lower than the arithmetic average and is the only accurate measure of actual compounded performance.

Average Return Based on Cash Flow

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Average and Cumulative Return

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Guides & Reference

How It Works

CAGR FormulaAnnualized return from start/end values

CAGR calculates the constant annual growth rate that produces the same result as the actual investment performance over the holding period. It is the most accurate single number to communicate investment returns over multiple years.

CAGR = (Ending Value / Beginning Value)^(1/Years) − 1

$10K → $25K in 10yr: (2.5)^(0.1) - 1 = 9.6% CAGR

Geometric MeanTrue compounded return from annual data

For a series of annual returns, the geometric mean accurately reflects the actual compounded investor experience. It is always lower than the arithmetic mean when returns vary — the greater the variability, the larger the gap.

Geometric Mean = [(1+R1)(1+R2)...(1+Rn)]^(1/n) − 1

Returns: 20%, -10%, 15%: geom = (1.2×0.9×1.15)^(1/3)-1 = 7.86%

Arithmetic MeanSimple average of percentages

The arithmetic mean adds all return percentages and divides by the number of periods. It is a useful summary statistic but overstates actual compounded returns because it ignores the sequential compounding effect of gains and losses.

Arithmetic Mean = (R1 + R2 + ... + Rn) / n

Returns: 20%, -10%, 15%: (20-10+15)/3 = 8.33% arithmetic mean

Volatility DragWhy geometric < arithmetic

The gap between geometric and arithmetic mean is called volatility drag. Higher variability creates a larger gap. A portfolio returning 30% and -30% alternately has an arithmetic mean of 0% but a geometric mean of -5.4% — the portfolio shrinks over time despite averaging 0%.

Geometric ≈ Arithmetic − (Variance / 2)

30% and -30%: arithmetic 0%, geometric -5.4% (money lost)

Standard DeviationInvestment risk measurement

Standard deviation measures how much annual returns deviate from the average. The S&P 500 has a standard deviation of about 15% to 16%. In a normal distribution, about 68% of years fall within 1 standard deviation, and 95% within 2 standard deviations of the mean.

σ = √[Σ(Ri − Mean)² / n]

Mean 9%, σ=15%: 68% of years between -6% and +24%

Sharpe Ratio ConceptRisk-adjusted return evaluation

The Sharpe ratio divides excess return (CAGR minus risk-free rate) by standard deviation. It measures return per unit of risk. A Sharpe ratio above 1.0 is good; above 2.0 is excellent. Compare funds on Sharpe ratio to identify which gives the best risk-adjusted performance.

Sharpe = (CAGR − Risk-free Rate) / Standard Deviation

9% CAGR, 5% risk-free, 15% σ: Sharpe = (9-5)/15 = 0.27

Quick Reference

Common examples — verify instantly above.

$10K → $25K, 10yr

CAGR

9.6% CAGR

$50K → $100K, 8yr

CAGR

9.05% CAGR

Returns: 20%,-10%,15%

Geometric mean

7.86% (vs 8.33% arithmetic)

Returns: 30%,-30%

Geometric mean

-5.4% (loses money!)

S&P 500 historical

Geometric CAGR

~10% nominal, ~7% real

S&P 500 standard deviation

Annual variability

~15-16%

Volatility drag

σ=20%, arithmetic=10%

Geometric ≈ 8% (drag = 2%)

Rule of 72

At 9.6% CAGR

Doubles in ~7.5 years

Tips & Shortcuts

✓

Always report investment performance using CAGR (or geometric mean), never simple arithmetic average. The arithmetic average overstates actual investor outcomes and is misleading for comparing funds.

✓

Volatility drag means that reducing portfolio volatility can increase actual wealth even with the same arithmetic return. This is why diversification across uncorrelated assets adds value beyond simple return averaging.

✓

When evaluating a fund manager's track record, require at least 10 to 15 years of performance data to distinguish skill from luck. Short-term performance (3 to 5 years) has very high random variation.

✓

Use standard deviation to compare the risk of two similar funds. If Fund A has 8% CAGR and 20% standard deviation and Fund B has 7.5% CAGR and 12% standard deviation, Fund B offers better risk-adjusted return.

✓

The geometric mean from a long series of annual returns is mathematically identical to CAGR from start and end values. Use whichever data you have available — both give the same answer.

✓

Factor in your personal tax situation when evaluating investment returns. The after-tax CAGR is what matters, and it varies depending on whether gains are short-term (ordinary income rate) or long-term (lower capital gains rate).

Common Mistakes to Avoid

Reporting arithmetic average as investment performance

Mutual fund companies and advisors sometimes cite arithmetic averages to make performance look better. Always ask for and report the geometric mean (CAGR), which reflects actual compounded investor wealth.

Ignoring the volatility drag when comparing investments

Two portfolios with the same arithmetic average return produce very different wealth if one is more volatile. The more volatile portfolio has a lower geometric mean and lower actual returns. Prefer smooth, consistent returns.

Judging a fund manager on 3 to 5 years of performance

Short-term outperformance has enormous random variation. Even a below-average manager will appear to outperform for 3 to 5 years by chance. Require 10 to 15 years of data and compare against the appropriate benchmark, not the broader market.

Comparing returns from different periods without annualizing

A 50% total return means nothing without knowing the time period. Always express returns as CAGR for meaningful comparison across investments with different holding periods.

Not separating return into CAGR and volatility components

Two funds with 8% CAGR can have very different risk profiles: one might have 10% standard deviation and another 25%. Without knowing the standard deviation, you cannot evaluate whether the return justifies the risk.

Using the geometric mean when arithmetic mean is appropriate

For projecting the expected value of a single future outcome (not the median), the arithmetic mean is technically correct due to Jensen's inequality. Use geometric mean for evaluating past compounded performance; use arithmetic for single-period expected value projections.

Frequently Asked Questions

The arithmetic mean simply averages the percentages. The geometric mean accounts for compounding — a 50% gain followed by a 33% loss returns to the starting point (0% CAGR), but the arithmetic average shows +8.5%. The geometric mean correctly reflects actual investor outcomes because money compounds period to period.

CAGR (Compound Annual Growth Rate) is the steady annual rate that would grow an investment from its starting value to its ending value over the holding period. Use CAGR to compare investments held for different periods (both normalized to an annual basis) and to communicate long-term investment performance accurately.

Standard deviation measures the variability (volatility) of annual returns around the average. A portfolio with 8% average return and 15% standard deviation typically delivers returns between -7% and +23% in two-thirds of years. Higher standard deviation means higher volatility and risk.

The S&P 500 has returned approximately 10% annually nominal (before inflation) and 7% real (after inflation) since 1926. The geometric mean is typically 1% to 2% lower than the arithmetic mean due to volatility drag. Individual years vary from -37% (2008) to +38% (1995), with the average realized only over very long periods.

Volatility drag means that high year-to-year volatility reduces the geometric mean below the arithmetic mean. A portfolio that returns +40% then -28.6% has an arithmetic mean of +5.7% but a geometric mean of 0% — the two years cancel out. Smoother returns with the same arithmetic average produce higher real wealth.

Broad US equities (S&P 500): 10% nominal, 7% real. International developed: 7% to 8% nominal. Emerging markets: 8% to 10% nominal but higher volatility. Bonds: 4% to 5% nominal. Balanced 60/40 portfolio: 7% to 8% nominal. Real estate: 8% to 12% total return including appreciation. Cash/savings: 2% to 5%.

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