Simulating stock market returns using Monte Carlo › 4. Computing Final Portfolio Values

Michael BrenndoerferJuly 19, 20252 min read
Simulating stock market returns using Monte Carlo › 4. Computing Final Portfolio ValuesOne section · Sections in this article5 / 8 ↓
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The Compound Growth Formula

The fundamental principle of compound growth is applied to calculate how the investment grows over time. The mathematical formula is:

Future Value=Initial Investment×∏t=1T(1+rt)\text{Future Value} = \text{Initial Investment} \times \prod_{t=1}^{T}(1 + r_t)

Where:

  • ∏\prod (capital Pi) means "product of" - multiply all terms together
  • rtr_t is the return in year tt
  • (1+rt)(1 + r_t) converts a return percentage to a growth factor

Why This Works

  • A 10% return means money grows by a factor of 1.10
  • A -5% loss means money is multiplied by 0.95
  • Over multiple years, all these factors are multiplied together

Example Calculation

If returns of [8%, -2%, 15%] occur over 3 years:

  • Growth factors: [1.08, 0.98, 1.15]
  • Total growth: 1.08 × 0.98 × 1.15 = 1.217
  • 100becomes:100 becomes: 100 × 1.217 = $121.70
In[4]:
Code
# Converting annual returns to growth factors (1 + return)
growth_factors_matrix = 1 + annual_returns

# Calculating cumulative growth by multiplying across years (axis=1)
# This gives the total growth factor for each 10-year scenario
cumulative_growth_factors = growth_factors_matrix.prod(axis=1)

# Applying compound growth to get final portfolio values
final_portfolio_values = INITIAL_INVESTMENT * cumulative_growth_factors

print("Example Compound Growth Calculations:")
print(f"   Scenario 1 returns: {annual_returns[0, :].round(3)}")
print(f"   Growth factors: {growth_factors_matrix[0, :].round(3)}")
print(f"   Cumulative growth: {cumulative_growth_factors[0]:.3f}")
print(f"   Final value: ${final_portfolio_values[0]:.2f}")
print(f"\nSummary Statistics:")
print(f"   Number of scenarios calculated: {len(final_portfolio_values):,}")
print(f"   Minimum final value: ${final_portfolio_values.min():.2f}")
print(f"   Maximum final value: ${final_portfolio_values.max():.2f}")
print(f"   Average final value: ${final_portfolio_values.mean():.2f}")
Out[4]:
Console
Example Compound Growth Calculations:
   Scenario 1 returns: [ 0.126 -0.076  0.193  0.221 -0.213 -0.115  0.099  0.033  0.077 -0.048]
   Growth factors: [1.126 0.924 1.193 1.221 0.787 0.885 1.099 1.033 1.077 0.952]
   Cumulative growth: 1.228
   Final value: $122.84

Summary Statistics:
   Number of scenarios calculated: 10,000
   Minimum final value: $33.48
   Maximum final value: $1165.34
   Average final value: $214.67
In[5]:
Code
# Creating comprehensive statistical summary
portfolio_summary = (pd.Series(final_portfolio_values, name='Final Portfolio Value ($)')
                     .describe(percentiles=[.05, .25, .5, .75, .95])
                     .round(2))

# Reordering for better presentation and adding meaningful labels
risk_metrics = portfolio_summary.loc[['min', '5%', '25%', '50%', 'mean', '75%', '95%', 'max']]
risk_metrics.index = ['Worst Case', '5th Percentile (VaR)', '25th Percentile', 
                     'Median', 'Expected Value', '75th Percentile', 
                     '95th Percentile', 'Best Case']

# Calculating additional useful metrics
prob_loss = (final_portfolio_values < INITIAL_INVESTMENT).mean() * 100
prob_double = (final_portfolio_values >= 2 * INITIAL_INVESTMENT).mean() * 100
expected_return_annualized = (risk_metrics['Expected Value'] / INITIAL_INVESTMENT) ** (1/YEARS) - 1

print("Portfolio Value Distribution After 10 Years")
print("=" * 50)
print(risk_metrics.to_string())
print(f"\nRisk Analysis:")
print(f"   Probability of losing money: {prob_loss:.1f}%")
print(f"   Probability of doubling investment: {prob_double:.1f}%")
print(f"   Expected annualized return: {expected_return_annualized:.1%}")
print(f"   Range of outcomes: ${risk_metrics['Worst Case']:.2f} - ${risk_metrics['Best Case']:.2f}")
Out[5]:
Console
Portfolio Value Distribution After 10 Years
==================================================
Worst Case                33.48
5th Percentile (VaR)      91.25
25th Percentile          144.07
Median                   195.04
Expected Value           214.67
75th Percentile          264.35
95th Percentile          404.81
Best Case               1165.34

Risk Analysis:
   Probability of losing money: 7.6%
   Probability of doubling investment: 48.0%
   Expected annualized return: 7.9%
   Range of outcomes: $33.48 - $1165.34