Explanation

When estimating the average return of an investment over multiple consecutive periods (time-weighted returns), the geometric mean is most appropriate because it accounts for the compounding effect of returns over time.

Why Geometric Mean is Correct:

1. Compounding Effect: Investment returns compound over time. If you have returns of +20% in year 1 and -10% in year 2, the arithmetic mean would be 5%, but the actual cumulative return would be:

   • Year 1: $100 × 1.20 = $120
   • Year 2: $120 × 0.90 = $108
   • Total return = 8% over 2 years, not 10% (5% × 2 years)

2. Geometric Mean Calculation:

   • Geometric mean = [(1 + R₁) × (1 + R₂) × ... × (1 + Rₙ)]^(1/n) - 1
   • For returns +20% and -10%: [(1.20) × (0.90)]^(1/2) - 1 = (1.08)^(0.5) - 1 ≈ 0.0392 or 3.92%
   • This accurately reflects the annualized return

Why Other Means are Incorrect:

• Arithmetic Mean: Simply averages the returns without considering compounding, which overstates the actual return when volatility is present.
• Harmonic Mean: Used for averaging rates when the denominator varies (like average speed over equal distances), not appropriate for investment returns.

Key Takeaway:

For multi-period investment returns, always use the geometric mean to calculate the compound annual growth rate (CAGR) or time-weighted rate of return.