Arithmetic return is the simple average of returns. Geometric (compound) return reflects what you actually earn after compounding — it's always lower when returns vary. Monte Carlo simulations must use arithmetic returns as inputs because compounding happens naturally across iterations.
Arithmetic and geometric returns are two different ways to measure investment performance, and confusing them is one of the most common errors in retirement planning. The arithmetic return is the simple average of periodic returns. The geometric (compound) return accounts for the sequential compounding of returns and reflects actual wealth accumulation.
How It Works
Arithmetic return: (R₁ + R₂ + ... + Rₙ) / n
Geometric return: (∏(1 + Rᵢ))^(1/n) - 1
A concrete example:
| Year | Return | Portfolio (starting $100,000) |
|---|---|---|
| 1 | +50% | $150,000 |
| 2 | -50% | $75,000 |
- Arithmetic average: (50% + -50%) / 2 = 0%
- Geometric return: √(1.50 × 0.50) - 1 = -13.4%
- Actual result: you lost $25,000
The arithmetic return suggests you broke even. The geometric return tells you the truth: you lost 13.4% annually in compound terms. The gap is caused by variance drain.
Why It Matters for Retirement Planning
The distinction is critical for two reasons:
-
Simulation inputs: Monte Carlo simulations must use arithmetic returns. The compounding happens naturally as returns are applied year by year. Using geometric returns as inputs would understate growth because the volatility penalty would be applied twice.
-
Setting expectations: when financial advisors quote "average stock returns of 10%," they usually mean arithmetic. The geometric (what you actually earn) is closer to 7-8% for a diversified equity portfolio. Over a 30-year retirement, this 2-3% gap compounds to an enormous difference in terminal portfolio value — making it essential to use the correct return type for each context.
Frequently Asked Questions
- Which return should I use when planning for retirement?
- For Monte Carlo simulations, always input arithmetic (average) returns. The simulation naturally compounds them across iterations, producing realistic geometric growth including variance drain. If you input geometric returns, the simulation will understate expected growth because it would double-count the volatility drag effect.
- Why is the geometric return always lower than the arithmetic return?
- Because of variance drain (volatility drag). When returns vary, compounding penalizes volatility. A +20% gain followed by -20% loss gives an arithmetic average of 0%, but a geometric return of -2% (you end with 96% of your starting value). The more volatile the returns, the larger the gap between arithmetic and geometric averages.