Simulating stock market returns using Monte Carlo › 3. Generating Random Returns

Michael BrenndoerferJuly 19, 20251 min read
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The Heart of Monte Carlo Simulation

Thousands of possible future scenarios will be generated by randomly sampling investment returns from a normal distribution. This is the core of Monte Carlo simulation:

Key Insight: Annual returns are assumed to follow a normal distribution with:

Understanding the Output Structure

  • Rows: Each row represents one possible future scenario (simulation path)
  • Columns: Each column represents a year in the 10-year horizon
  • Values: Each value is a randomly sampled annual return for that year and scenario
In[3]:
Code
# Setting up reproducible random number generation for consistent results
rng = np.random.default_rng(seed=42)

# Generating random annual returns for all simulation paths
# Shape: (N_ITER, YEARS) = (10,000 scenarios × 10 years)
annual_returns = rng.normal(loc=MU, scale=SIGMA, size=(N_ITER, YEARS))

print(f"Generated Returns Matrix:")
print(f"   Shape: {annual_returns.shape}")
print(f"   Each row = one possible 10-year future")
print(f"   Each column = returns for a specific year across all scenarios")
print(f"\nSample of first 5 scenarios (first 5 years):")
print(pd.DataFrame(annual_returns[:5, :5], 
                  columns=[f'Year {i+1}' for i in range(5)],
                  index=[f'Scenario {i+1}' for i in range(5)]).round(3))
Out[3]:
Console
Generated Returns Matrix:
   Shape: (10000, 10)
   Each row = one possible 10-year future
   Each column = returns for a specific year across all scenarios

Sample of first 5 scenarios (first 5 years):
            Year 1  Year 2  Year 3  Year 4  Year 5
Scenario 1   0.126  -0.076   0.193   0.221  -0.213
Scenario 2   0.212   0.197   0.090   0.249   0.150
Scenario 3   0.052  -0.022   0.263   0.057   0.016
Scenario 4   0.401   0.019   0.003  -0.042   0.172
Scenario 5   0.191   0.161  -0.020   0.115   0.098