Financial · Live
Average Return Calculator,
arithmetic vs. geometric.
Enter a series of annual returns, positive or negative, and instantly see the arithmetic mean, geometric mean (CAGR), volatility drag, and a year-by-year balance chart that shows why the two averages always diverge.
Inputs
Your return series
Annual returns · 7 years
Enter each year's return as a percentage — positive or negative. The geometric mean (CAGR) is always ≤ the arithmetic mean; the gap is the volatility drag on your portfolio.
Geometric mean (CAGR)
7 years
Arithmetic average +6.57%, volatility drag −0.69%
- Arith. mean
- +6.57%
- Total return
- +49.2%
- Ending balance
- $14,915.24
- Std deviation
- 12.09%
Growth chart
Actual balance vs arithmetic projection
Schedule
Year-by-year breakdown
| Year | Return | Balance | Cumulative |
|---|---|---|---|
| Y1 | +12.5% | $11,250.00 | +12.5% |
| Y2 | -5.2% | $10,665.00 | +6.7% |
| Y3 | +18.3% | $12,616.70 | +26.2% |
| Y4 | -2.1% | $12,351.75 | +23.5% |
| Y5 | +9.7% | $13,549.87 | +35.5% |
| Y6 | +24.1% | $16,815.39 | +68.2% |
| Y7 | -11.3% | $14,915.25 | +49.2% |
Complete guide
What is an average return calculator?
An average return calculator takes a series of investment returns, one per year typically, and produces the two most important averages: the arithmetic mean and the geometric mean. Understanding the difference between them is one of the most important concepts in long-term investing.
Most people instinctively reach for the arithmetic average: add up all the returns and divide by the number of years. That is correct for a single-year average. But for multi-year compounding, which is how every real investment actually works, the geometric mean is the right number. It is the single annual rate that, applied every year, produces the same ending balance as the actual return series. This is also called the Compound Annual Growth Rate (CAGR).
The two formulas
Geometric mean = (∏(1+r₁)(1+r₂)…(1+rₙ))^(1/n) − 1
where r₁…rₙ are the period returns as decimals (e.g., 0.125 for 12.5%)
The arithmetic mean treats each year independently. The geometric mean accounts for compounding; each year's return is applied to the previous year's ending balance, not the original principal. Because of this, the geometric mean is always less than or equal to the arithmetic mean. The two are equal only when every period return is identical.
The +50% / −50% trap
Here is the clearest demonstration of why arithmetic averages mislead investors. Suppose an investment earns +50% in year 1 and −50% in year 2:
$10,000 × 1.50 = $15,000 after year 1
$15,000 × 0.50 = $7,500 after year 2
Total return = −25%
Geometric mean = (1.50 × 0.50)^(1/2) − 1 = −13.4% per year
The arithmetic average says 0%; you "broke even" on average. But you lost a quarter of your money. The geometric mean of −13.4% per year tells the true story: the sequence of +50% and −50% destroys value, even though the simple average is zero.
This effect, where volatility causes the geometric mean to lag the arithmetic mean, has a name: volatility drag or variance drain.
Volatility drag: the formula
The gap between the two averages is not random. For most investment return series, it approximates:
where σ is the population standard deviation of period returns
This approximation is exact for log-normally distributed returns (the standard model for equity prices) and a useful estimate for real-world data. A portfolio with 15% annual standard deviation loses roughly 0.15² / 2 = 1.1% per year to volatility drag relative to its arithmetic average. A more volatile portfolio with 25% standard deviation loses 0.25² / 2 = 3.1% per year.
This is why portfolio diversification and risk management matter even when you are not trying to reduce headline risk: lower volatility compounds better, even at the same arithmetic average.
CAGR: the investor's north star
The Compound Annual Growth Rate (CAGR) is mathematically identical to the geometric mean when calculated from period returns. You can also compute it directly from starting and ending values:
Example: $10,000 → $18,500 in 7 years
CAGR = (18,500 / 10,000)^(1/7) − 1 = 9.21% per year
CAGR is the standard metric for comparing investment performance across different time periods and asset classes because it normalizes for compounding. A fund that grew 150% over five years has a different CAGR than one that grew 150% over ten years. Without CAGR, the comparison is meaningless.
Historical average returns: S&P 500 as a reference
The U.S. equity market provides the most-studied long-run return data. Understanding these benchmarks helps you evaluate whether your own return series is strong, weak, or in line with the market.
| Period | Arith. mean | CAGR (geometric) | Std deviation |
|---|---|---|---|
| S&P 500 (1928–2024) | ~11.7% | ~10.2% | ~19.5% |
| S&P 500 (2000–2024) | ~10.8% | ~7.9% | ~17.6% |
| S&P 500 (2010–2024) | ~13.9% | ~13.0% | ~14.0% |
| Global equities | ~10–12% | ~8–10% | ~16–20% |
| 10-yr US Treasury | ~5–6% | ~4–5% | ~8–10% |
| Cash / T-bills | ~3–4% | ~3–4% | ~3–4% |
Approximate figures. Returns include dividends reinvested; inflation not adjusted.
Notice the consistent gap between arithmetic and geometric means for equities (~1.5–2%) vs. near-zero gap for cash (~0%). This perfectly reflects the volatility-drag formula: equity standard deviation of ~18% implies drag of 0.18² / 2 ≈ 1.6% per year, while cash has near-zero standard deviation.
Why the chart shows two lines
The calculator draws two lines on the growth chart:
- Actual balance (amber area): the portfolio value after applying each year's real return. This is what actually happened.
- Arithmetic projection (dashed line): what the portfolio would be worth if it had grown at the arithmetic mean every year. This is an optimistic fiction.
The gap between the two lines is the accumulated volatility drag. It grows over time and with return volatility. Because the geometric mean is always ≤ the arithmetic mean (Jensen's inequality), the actual balance is always at or below the arithmetic projection. The longer and more volatile the return series, the more the two lines diverge.
How to use this calculator for portfolio analysis
- Personal portfolio review. Enter your actual year-by-year investment returns to see your true CAGR vs. what the simple average would suggest. The difference shows your portfolio's volatility cost.
- Fund comparison. Enter a fund's annual returns to compute its CAGR, then compare to the fund's advertised arithmetic average to understand how much the marketing number overstates actual performance.
- Rebalancing impact. A rebalanced portfolio typically has lower volatility than a drifting one, which means lower volatility drag and a higher geometric mean, even if the arithmetic average is unchanged.
- Sequence-of-returns risk. Enter the same set of returns in reverse order to see how the sequence affects the ending balance. The arithmetic mean is sequence-independent; the geometric mean is not.
Which average should you report?
Always use the geometric mean (CAGR) to represent investment performance. The arithmetic mean is only appropriate for estimating expected one-period returns in a forward-looking model. For any backward-looking evaluation of what an investment actually delivered, CAGR is the correct metric.
The SEC and CFA Institute both require geometric returns for multi-year performance reporting. When a mutual fund advertisement shows "average annual return," it is required by law to state the geometric figure. If you ever see an arithmetic average presented as the fund's "average return," that is either an error or misleading marketing.
Worked example: the sample data
The calculator's default data uses seven years of returns: +12.5%, −5.2%, +18.3%, −2.1%, +9.7%, +24.1%, −11.3%.
= 46.0 / 7 ≈ 6.57%
Product = 1.125 × 0.948 × 1.183 × 0.979 × 1.097 × 1.241 × 0.887
≈ 1.4914
Geometric mean = 1.4914^(1/7) − 1 ≈ 5.87%
Volatility drag = 6.57% − 5.87% = 0.70%
$10,000 starting balance → $14,914 ending balance
The arithmetic average of 6.57% would imply a final balance of $15,572, an overstatement of $658 or 4.4%. Over a 30-year horizon, the same level of volatility drag would overstate the ending balance by tens of thousands of dollars.
Disclaimer
Past returns do not predict future performance. The arithmetic mean, geometric mean, and standard deviation are descriptive statistics about a historical data series. They do not imply that any future return will match these figures. All investing involves risk, and the value of investments can fall as well as rise.