Monte Carlo Risk Simulator Calculator

Each simulated year multiplies the portfolio by the exponential of mu minus half sigma squared plus sigma times a standard normal draw, then adds the annual contribution. That middle term is the Ito correction, commonly called volatility drag: because compounding is multiplicative, a portfolio with 15 percent annual volatility loses roughly 1.1 percentage points of compound growth relative to its arithmetic mean return. Including it keeps the median of the simulated distribution consistent with the expected return you entered. Thousands of independent paths are generated, the ending values sorted, and percentiles read directly off the sorted array. Value at Risk here is the shortfall of the 5th percentile against total capital contributed, and the Sharpe ratio uses a fixed 3 percent risk-free rate. Sensible inputs come from long-run history: US large-cap equities have returned roughly 10 percent nominal with annual standard deviation near 15 to 20 percent, investment-grade bonds around 4 to 6 percent with similar single-digit volatility, and a 60/40 blend about 7 to 8 percent with 9 to 11 percent volatility. The model's weaknesses are well known. Normally distributed log returns badly understate tail events; the October 1987 crash was effectively impossible under a lognormal assumption. Constant mu and sigma ignore volatility clustering, regime shifts and any mean reversion. Taxes, fees, inflation and behaviour are absent. Because the sampling is unseeded, results shift slightly between runs, with the extreme percentiles noisiest at 1,000 trials.

The Monte Carlo method was created at Los Alamos in the immediate postwar years. Stanislaw Ulam later recounted conceiving it in 1946 while convalescing and playing solitaire, wondering whether repeated random trials might answer a combinatorial question faster than exact analysis. He raised it with John von Neumann, who saw its application to neutron diffusion, and Nicholas Metropolis suggested the code name after the casino in Monaco frequented by a relative of Ulam. Metropolis and Ulam published The Monte Carlo Method in the Journal of the American Statistical Association in 1949, and early runs were carried out on ENIAC. The finance side has a separate lineage. Louis Bachelier's 1900 doctoral thesis Theorie de la speculation modelled price movements as Brownian motion, decades before the idea took hold. Paul Samuelson revived it in the 1960s in geometric, log-normal form, which prevents the negative prices ordinary Brownian motion allows and underpins the Black-Scholes-Merton option pricing work of 1973. Phelim Boyle then showed in a 1977 Journal of Financial Economics paper how options could be priced by simulation rather than closed-form solution, bringing Monte Carlo squarely into quantitative finance. William F. Sharpe had introduced the reward-to-variability ratio in 1966, later renamed after him, and Value at Risk became an industry standard after J.P. Morgan published the RiskMetrics methodology in 1994.