Savings Goal Monte Carlo Simulator Calculator

Every trial here is a discretised path of geometric Brownian motion with a deposit stream layered on top. In each month the balance is multiplied by the exponential of (mu minus sigma squared over two) divided by twelve, plus sigma over the square root of twelve times a standard normal draw, and the monthly deposit is then added at the end of the month as an annuity-immediate. The subtraction of half the variance is the Ito correction, often called volatility drag: because compounding is multiplicative, the median growth rate sits below the arithmetic mean, and including the term keeps the simulated distribution consistent with the expected return that was entered. Normal deviates come from the Box-Muller transform driven by a seeded thirty-two-bit generator, so identical inputs always reproduce identical output rather than flickering on every re-render. The deterministic comparison shown alongside the percentiles is not a rival approximation but the exact analytic mean of the simulated distribution. Since the expected value of the monthly growth factor equals the exponential of mu over twelve, and the deposits are fixed and independent of the returns that follow them, the projection built from a constant factor of exp(mu/12) reproduces E[B(N)] exactly; in the worked examples above the sampled mean lands within 0.03 and 0.28 percent of it. That identity is what makes the skew visible. For a lump sum with no deposits the median is the mean multiplied by exp(minus sigma squared T over two), and the probability of beating the plain average-return projection is exactly one minus the standard normal cumulative distribution evaluated at sigma root T over two. At sixteen percent volatility over fifteen years that is 37.8 percent, with the median at 82.5 percent of the mean. A deterministic calculator can therefore report a comfortable surplus for a plan that fails more often than it succeeds. Two further outputs deserve attention. The probability that the balance ever touched the goal is always at least the probability that it finished above the goal, and the gap between them counts the trials that hit the target and then surrendered it, which is the cleanest in-model illustration of sequence-of-returns risk. And because the success probability is a sample proportion, it carries a standard error equal to the square root of p times one minus p over n: roughly 1.6 percentage points at a thousand trials near a fifty percent reading, 0.7 at five thousand and 0.5 at ten thousand. Sensible inputs come from long-run history, with broad US large-cap equity near ten percent nominal at fifteen to twenty percent annual standard deviation, investment-grade bonds around four to six percent at mid single-digit volatility, and a sixty-forty blend about seven to eight percent at nine to eleven percent. The assumptions are strong ones. Independent, identically distributed lognormal returns understate fat tails and ignore volatility clustering and mean reversion; mu and sigma are held constant for the whole horizon; deposits are never missed and nothing is ever withdrawn; and taxes, fund fees and inflation are all outside the model.

Monte Carlo methods were invented at Los Alamos in the years just after the Second World War. Stanislaw Ulam described conceiving the idea in 1946 while convalescing and playing solitaire, wondering whether repeated random trials might settle a combinatorial question faster than exact enumeration; John von Neumann saw its application to neutron diffusion, Nicholas Metropolis supplied the code name after the casino in Monaco, and the two published The Monte Carlo Method in the Journal of the American Statistical Association in 1949, with early runs executed on ENIAC. The normal deviates used by this simulator come from a transform George Box and Mervin Muller published in the Annals of Mathematical Statistics in 1958, which converts a pair of uniform random numbers into a pair of independent standard normal ones. The financial model arrived along a separate track. Louis Bachelier's 1900 doctoral thesis Theorie de la speculation treated price movements as Brownian motion decades before economists took the idea seriously. Paul Samuelson revived it in geometric, log-normal form in the mid 1960s, which rules out the negative prices ordinary Brownian motion permits and underpins the Black-Scholes and Merton option pricing papers of 1973. Phelim Boyle then showed in a 1977 Journal of Financial Economics article how options could be valued by simulation instead of closed-form solution, establishing Monte Carlo as a mainstream tool of quantitative finance. Applying all of this to ordinary household savings goals is much more recent. Through the 1980s retail financial projections were almost entirely deterministic, quoting a single assumed rate. William Bengen's October 1994 article in the Journal of Financial Planning, which tested withdrawal rates against actual historical return sequences rather than against an average, made the profession confront sequence-of-returns risk directly, and the 1998 Trinity study by Philip Cooley, Carl Hubbard and Daniel Walz reinforced the point. Financial Engines, founded in 1996 by the Nobel laureate William F. Sharpe with Joseph Grundfest and Craig Johnson, put simulation-based forecasts in front of ordinary retirement plan participants for the first time. Over the following decade probability of success displaced the single projected balance as the headline number in mainstream planning software, and US regulators responded in kind: FINRA's rule governing investment analysis tools obliges firms presenting simulated outcomes to disclose the assumptions behind them and to make clear that the results are hypothetical rather than predictive.