Monte Carlo Simulation
Monte Carlo simulation is a brute-force forecasting method: instead of using formulas, you run thousands of random scenarios and observe the distribution of outcomes. The process: (1) Estimate return distributions for each asset (mean, volatility, correlations); (2) Generate random draws from these distributions; (3) Simulate portfolio evolution over time (daily/monthly steps); (4) Repeat 10,000+ times; (5) Analyze the distribution of final wealth. For example, running 10,000 simulations of a 60/40 portfolio over 30 years shows the range of outcomes — median $2.5M, 10th percentile $1.2M, 90th percentile $4.8M. Monte Carlo captures non-linearity, path dependency, and rebalancing effects that formulas miss. Used for retirement planning, VaR estimation, and stress testing.
| S_t | Asset price at time t |
| \mu | Expected return (drift) |
| \sigma | Volatility |
| \Delta t | Time step (1 month) |
| Z | Standard normal random variable |
- Returns are log-normally distributed
- Volatility and correlations are constant
- No jumps or regime changes
- Monthly time steps for computational efficiency
Value at Risk (VaR)
The maximum expected loss at a given confidence level — but doesn't tell you how bad the tail is.
CVaR / Expected Shortfall
The average loss in the worst-case scenarios beyond VaR.
Portfolio Volatility
Standard deviation of portfolio returns — total risk including diversification effects.
Efficient Frontier
The set of portfolios offering the highest return for each level of risk.
Sharpe Ratio
Risk-adjusted return: excess return divided by volatility.
Covariance Matrix
Captures how asset returns move together — the foundation of diversification.