Geometric Mean — Formula, Applications in Finance and Growth

Geometric Mean

The geometric mean is the nth root of the product of n positive values. It is the appropriate average for ratios, rates of change, and multiplicative processes.

$G = \left(\prod_{i=1}^n x_i\right)^{1/n} = \exp\left(\frac{1}{n}\sum_{i=1}^n \ln x_i\right)$

Why Geometric Mean for Growth Rates?

Suppose a stock grows: Year 1: +100%, Year 2: −50%.

• Arithmetic mean: (100% − 50%) / 2 = +25% per year → sounds great
• Actual result: $100 →$200 → $100 → back where you started (0% per year!)
• Geometric mean: √(2.00 × 0.50) − 1 = √(1.0) − 1 = 0% → correct!

import numpy as np
from scipy.stats import gmean
import pandas as pd
import matplotlib.pyplot as plt

# ==========================================
# Example 1: Investment Returns
# ==========================================
annual_returns = [0.25, -0.10, 0.40, -0.20, 0.15]  # +25%, -10%, etc.
multipliers = [1 + r for r in annual_returns]

# Geometric mean of multipliers
geo_mean_multiplier = gmean(multipliers)
arith_mean_multiplier = np.mean(multipliers)

geo_cagr = geo_mean_multiplier - 1
arith_mean_return = np.mean(annual_returns)

print("Annual returns:", [f"{r:.0%}" for r in annual_returns])
print(f"Arithmetic mean return: {arith_mean_return:.2%} ← WRONG for compounding")
print(f"Geometric mean (CAGR):  {geo_cagr:.2%} ← CORRECT for compounding")

# Verify: compound at geometric mean
initial = 1000
final_actual = initial * np.prod(multipliers)
final_geometric = initial * (1 + geo_cagr)**5
print(f"\nActual final value: ${final_actual:.2f}")
print(f"Projected via CAGR: ${final_geometric:.2f}")

# ==========================================
# Example 2: Population Growth
# ==========================================
years = [2018, 2019, 2020, 2021, 2022, 2023]
population = [50000, 53000, 56180, 59551, 63124, 66911]

growth_rates = [population[i]/population[i-1] for i in range(1, len(population))]
geo_mean_growth = gmean(growth_rates)
print(f"\nAnnual growth rates: {[f'{r:.4f}' for r in growth_rates]}")
print(f"Geometric mean growth rate: {geo_mean_growth:.4f}")
print(f"Expected CAGR: {geo_mean_growth - 1:.2%}")

# Verify
n_years = len(population) - 1
cagr_from_endpoints = (population[-1]/population[0])**(1/n_years)
print(f"CAGR from endpoints: {cagr_from_endpoints - 1:.2%}")

# ==========================================
# Example 3: Geometric vs Arithmetic Mean
# ==========================================
print("\n=== AM-GM Inequality ===")
print("Arithmetic mean is ALWAYS ≥ Geometric mean (for positive values)")
for trial in range(5):
    x = np.random.uniform(1, 100, 10)
    am = np.mean(x)
    gm = gmean(x)
    print(f"  AM = {am:.3f}, GM = {gm:.3f}, AM ≥ GM: {am >= gm}")

Log-transform Shortcut

The geometric mean of X equals exp(arithmetic mean of log X):

# Same result, numerically more stable for many values
data = np.random.uniform(1, 100, 1000)

gm_direct = gmean(data)
gm_log = np.exp(np.log(data).mean())

print(f"Direct gmean: {gm_direct:.6f}")
print(f"Via log:      {gm_log:.6f}")

When to Use Each Mean

Key Takeaways

1. Use geometric mean for growth rates and ratios — arithmetic mean overstates
2. AM ≥ GM always (AM-GM inequality), with equality only when all values are identical
3. CAGR (compound annual growth rate) is the geometric mean of annual multipliers
4. Log transformation converts geometric mean to arithmetic mean of log-transformed data
5. The geometric mean is undefined if any value is zero or negative
6. Investment returns, population growth, inflation rates — all geometric means in finance