Average Return
Compute arithmetic and geometric average returns on investments to measure portfolio performance over multiple periods
Formula
Geometric Mean = [(1+R₁) × (1+R₂) × ... × (1+Rₙ)]^(1/n) - 1
Geometric mean calculates true compound return by multiplying growth factors (1 + return rate) for each period, taking the nth root, and subtracting 1. This accounts for compounding and is always more accurate than arithmetic mean for multi-period investment returns.
Worked Examples
Example 1: Why Geometric Mean Matters
Problem:Investment returns: Year 1: +50%, Year 2: -50%. What's the real average return?
Solution:Arithmetic mean:\n(50 + (-50)) ÷ 2 = 0%\n\nGeometric mean:\n√[(1+0.50) × (1-0.50)] - 1\n= √[1.5 × 0.5] - 1\n= √0.75 - 1\n= 0.866 - 1 = -13.4%\n\nActual result: \n$100 → $150 (after year 1)\n$150 → $75 (after year 2)\n\nYou lost $25 despite 0% arithmetic average!\n\nGeometric mean correctly shows -13.4% loss.
Result:Geometric: -13.4% (correct) | Arithmetic: 0% (misleading)
Example 2: Real Portfolio Example
Problem:5-year investment returns: +10%, -5%, +12%, +8%, +15%. What's the true annual growth rate?
Solution:Arithmetic mean:\n(10 - 5 + 12 + 8 + 15) ÷ 5 = 40 ÷ 5 = 8.0%\n\nGeometric mean (CAGR):\n[(1.10) × (0.95) × (1.12) × (1.08) × (1.15)]^(1/5) - 1\n= [1.4536]^0.2 - 1\n= 1.0777 - 1 \n= 0.0777 = 7.77%\n\nGeometric mean stays below arithmetic mean because compounding and volatility reduce the true multi-year growth rate.\n\nActual portfolio growth:\n$10,000 → $14,536 over 5 years\nCAGR: 7.77%
Result:CAGR: 7.77% (true compound growth)
Example 3: Volatile Portfolio
Problem:Volatile stock returns: +30%, -20%, +25%, -15%, +35%, -10%. Average return?
Solution:Arithmetic:\n(30 - 20 + 25 - 15 + 35 - 10) ÷ 6 = 45 ÷ 6 = 7.5%\n\nGeometric:\n[(1.30)(0.80)(1.25)(0.85)(1.35)(0.90)]^(1/6) - 1\n= [1.3426]^(1/6) - 1\n= 1.0503 - 1 = 5.03%\n\nDifference: 7.5% - 5.03% = 2.47%\n\nHigh volatility creates a large gap between means. Geometric mean reflects actual compounded returns. Your money grew at 5.03%/year, not 7.5%.
Result:Arithmetic: 7.5% | Geometric: 5.03% (2.47% gap)