Simulating stock market returns using Monte Carlo › 5. Statistical Analysis and Risk Assessment

The beauty of Monte Carlo simulation lies in its ability to quantify uncertainty. Instead of a single "expected" outcome, a full distribution of possibilities is obtained. The key statistics can be analyzed as follows:

Key Percentiles Explained:

• 5th percentile: Only 5% of scenarios do worse than this (downside risk)
• 25th percentile: First quartile - represents poor but not catastrophic outcomes
• 50th percentile (median): Half of scenarios do better, half do worse
• 75th percentile: Third quartile - represents good outcomes
• 95th percentile: Only 5% of scenarios do better than this (upside potential)

These risk metrics are calculated below:

Key Insight: Notice that the median ($195) is lower than the expected value ($215). This is not an error - it's a fundamental characteristic of investment returns.

Why This Happens:

1. Skewed Distribution: Investment returns exhibit positive skewness - there are occasional very large gains that pull the average upward, but losses are bounded (you can't lose more than 100%).

2. Arithmetic vs Geometric: The expected value uses arithmetic averaging of outcomes, while actual compound growth follows geometric progression. A few extremely successful scenarios significantly raise the arithmetic mean.

3. Practical Implication: The median represents the "typical" outcome - half of all scenarios do better, half do worse. The expected value is mathematically correct but influenced by extreme positive outcomes.

Real-World Meaning: If you ran this investment 100 times, you'd be more likely to end up near the median ($195) than the expected value ($215). The expected value includes the impact of those rare scenarios where your portfolio might grow to $500+ or even $1000+.