How to Calculate Average Return
Two different averages get called "average return" and they give different answers. The arithmetic mean sums the period returns and divides by the count. The geometric mean (also known as the Compound Annual Growth Rate or CAGR) compounds them and takes the nth root. For any investment that varies year to year, the geometric mean is the one that matches what actually happened to your money.
Arithmetic mean formula:
Arithmetic mean = (r₁ + r₂ + ... + rₙ) / n
Geometric mean (CAGR) formula:
CAGR = (EV / BV)^(1/n) − 1
Equivalently, given a list of period returns r₁..rₙ, the geometric mean equals the nth root of the product of (1 + each return), minus 1. This is the only average that compounds back to the right ending value.
Arithmetic vs Geometric: Why They Disagree
Suppose an investment returns +50% in year one and −50% in year two. The arithmetic mean is (50 + −50) / 2 = 0%. That sounds like you broke even. But $100 grows to $150 then falls by half to $75. The geometric mean is √(1.50 × 0.50) − 1 = −13.4%. The arithmetic mean overstates the realized return by 13.4 percentage points a year. This gap is called volatility drag or variance drain, and it grows with the variance of the returns.
The rough approximation: geometric mean ≈ arithmetic mean − σ²/2, where σ is the standard deviation of returns. For a portfolio with a 10% arithmetic mean and 20% volatility, the realized CAGR is roughly 10% − 0.20²/2 = 8%. That two-percentage-point haircut is real money: over 30 years, $10,000 compounding at 8% is $100,627, while $10,000 compounding at 10% is $174,494. The volatility drag costs $73,867 of ending wealth.
Key Concepts & Definitions
Return %
The percentage change in value over one period. A return of 10% means the value at the end of the period is 10% higher than at the start. Negative returns are losses. When you enter returns in this calculator, use percentage points (10 for 10%, not 0.10).
Total return
The cumulative return over the entire period, expressed as a percent. If $10,000 grows to $18,000, the total return is 80%. Total return ignores time; CAGR is the annualized version.
Volatility (standard deviation)
The dispersion of returns around the mean. The calculator uses the sample standard deviation (denominator n−1) of the period returns. Higher volatility means a wider range of outcomes, which mechanically reduces the geometric mean relative to the arithmetic mean.
Multiplier
The growth factor: ending value divided by starting value. A 3.5x multiplier means the investment more than tripled. The relationship to CAGR is multiplier = (1 + CAGR)^n.
Comparison Table: Asset Class Historical Averages
These are long-run historical averages, not tied to any specific year and not a prediction. Individual decades have varied dramatically. Use these as anchors, not promises.
| Asset class | Nominal annual return | Volatility (annual σ) | Worst single year (approx) |
|---|---|---|---|
| US large-cap stocks | ~10% | ~16% | −37% |
| US small-cap stocks | ~12% | ~20% | −40% |
| International developed stocks | ~8% | ~17% | −43% |
| Intermediate Treasury bonds | ~5% | ~6% | −13% |
| Real estate (REITs) | ~8% | ~18% | −38% |
| Gold | ~6.5% | ~16% | −33% |
| Cash / T-bills | ~3% | ~1% | 0% |
Sequence-of-Returns Risk
For a portfolio you are not withdrawing from, the order of returns does not change the ending value — only the geometric mean matters. But once you start drawing income, order suddenly matters enormously. Two retirees with the same CAGR can end up with vastly different outcomes if one experiences a big loss early in retirement and the other experiences it late, because withdrawals during the down years lock in the losses.
This is why retirement planning emphasises a bond and cash buffer, a lower-than-average withdrawal rate in the first few years, and stress-testing against bad-sequence scenarios — not just average return. Run the same average return number through the retirement calculator with different sequences and you will see the difference.
Average Return vs Total Return vs CAGR
People use the phrase "average return" loosely. Disambiguate before quoting a number. Total return is cumulative and not annualized — useful for short-horizon comparisons. Arithmetic average is the simple mean of period returns — useful only as an estimate of next-period expected return for an independent, identically distributed process. CAGR is the compounded annualized return — the right number for comparing investments over different time horizons.
When a fund advertises a "10-year average return", they almost always mean the 10-year CAGR. When a one-page fact sheet shows year-by-year returns and then an "average", that average is sometimes arithmetic — misleading. Check the methodology.
Dollar-Cost Averaging and Average Return
Dollar-cost averaging (DCA) does not change the asset’s average return, but it does change your realized return on dollars invested. Because you are buying more shares when prices are low, your money-weighted return often beats a lump-sum return into a volatile asset. The CAGR of the underlying asset is the same; the path you take matters. The investment calculator can model the path including DCA contributions.
Using Average Return to Project Future Wealth
If you have a defensible long-run CAGR estimate, projecting forward is straightforward: FV = PV × (1 + CAGR)^n. But pick the geometric mean, not the arithmetic mean, otherwise you overstate the expected ending value. And remember the average is itself uncertain — the actual outcome will be a distribution, not a point. The compound interest calculator handles the FV math; this calculator gives you the CAGR to feed in.
Edge Cases & Advanced Scenarios
1. Extreme outliers
A single outlier period — a 100% gain or a 60% loss — dominates the arithmetic mean. The geometric mean is somewhat more robust because it works through multiplication, but a single −100% return drives the geometric mean to −100% no matter what came before. If your data includes obvious anomalies, consider whether they should be in the average at all.
2. Mixed positive and negative returns
If any single period return is −100% or worse, the geometric mean is undefined (or −100%) because the product becomes zero or negative. In practice this represents total ruin, and an average return is no longer a meaningful summary — the investment is gone.
3. Negative or zero starting value (mode 2)
CAGR is undefined when the starting value is zero or negative because the ratio EV/BV is meaningless. If you are tracking a position that started from nothing (e.g. a savings plan that began at $0), use the annual-returns table mode instead, anchored to the first non-zero balance.
4. One-period investment
With only one period of return data, the arithmetic and geometric means are equal and the standard deviation is undefined. That single number is the holding-period return; calling it an "average" is technically correct but contains no new information.
5. Sub-annual periods
If your returns are monthly or quarterly, the CAGR formula gives the per-period compounded rate. To annualize a monthly mean of m, compute (1 + m)^12 − 1. Doing the wrong conversion is one of the most common mistakes in performance reporting.
What To Do With Your Result
- If your geometric mean is much lower than the arithmetic mean, look at the volatility number. Wide gaps indicate a bumpy ride — consider whether the risk profile matches your goals.
- Compare the CAGR to a relevant benchmark (S&P 500, total bond index, or a target-date fund). Beating a benchmark by 1% over a decade is real; under-performing by 1% costs serious money long term.
- Plug the CAGR into the compound interest calculator to project how the same return rate compounds over your remaining horizon.
- Use the volatility number to stress-test your retirement plan — replay a sequence with the worst years first to see how a real bad-sequence retirement would have played out.
- If you are evaluating an investment proposal that quotes only an arithmetic average, ask for the CAGR and the year-by-year returns. The difference between the two is the marketer’s thumb on the scale.
Two Worked Examples
Example A: Stock fund over 6 years
Returns: +12%, −8%, +25%, +5%, −15%, +20%. Arithmetic mean = (12 − 8 + 25 + 5 − 15 + 20)/6 = 6.5%. Multiplier = 1.12 × 0.92 × 1.25 × 1.05 × 0.85 × 1.20 = 1.380, so total return is 38.0% and CAGR = 1.380^(1/6) − 1 ≈ 5.55%. The 0.95 percentage point gap between arithmetic and geometric is the volatility drag.
Example B: $10,000 to $18,000 over 6 years
Multiplier = 1.80. CAGR = 1.80^(1/6) − 1 = 0.1029 = 10.29% per year. Even though the total return is 80%, the annual rate that compounds to that is just over 10%. This is mode 2 of the calculator.
Cross-links
Pair this calculator with the ROI calculator for single-period analysis, the investment calculator to model contributions, and compound interest for future-value projections.