Simulation path
One possible future in a Monte Carlo run: a full sequence of monthly returns, and optionally inflation, from today to the end of your plan.
Sampling error of a success rate ≈ √(p × (1 − p) ÷ n)
Explanation
A Monte Carlo simulation is made of many paths. Each path is one complete possible future: a month-by-month sequence of investment returns (and, if you choose, inflation) running from the start of your plan to its horizon. The app replays your entire plan through that sequence: contributions, withdrawals, taxes, loan interest and rebalancing.
Some paths start with a boom, some with a crash, and some are uneventful. Together they show the range of outcomes your plan could face. Your success rate is the share of paths that never ran out of money, and the percentile bands come from sorting the paths at each point in time.
How many paths? More paths give a more stable estimate, but each one takes time to compute. The precision improves with the square root of the number of paths, so you need four times as many paths to halve the error. For a 90% success rate, one standard error is about:
- 100 paths: ±3.0 percentage points
- 1,000 paths: ±0.95 points, roughly ±1
- 10,000 paths: ±0.3 points
About 95% of the time, the true figure for your assumptions lies within two standard errors of the estimate. Percentiles far out in the tails, such as the 5th, need more paths to settle than the median does.
Ways to build paths:
- Random (parametric): returns are drawn from a distribution using your expected returns, volatilities and correlations.
- Historical: each path starts in a different historical year and replays the actual returns that followed, in order. This keeps real booms, crashes and streaks, but history offers only a limited number of starting years.
- Bootstrap: each year of a path is drawn at random from history. This uses real yearly returns but makes many more combinations than history alone provides.
Example
You run 1,000 paths of a 30-year retirement, and 900 of them never run out of money: a 90% success rate.
- Standard error: √(0.9 × 0.1 ÷ 1,000) = √0.00009 ≈ 0.0095, or about ±1 percentage point
- A rough 95% range: about 88% to 92%
With 10,000 paths and the same 90% result, √(0.9 × 0.1 ÷ 10,000) = 0.003, or ±0.3 points, so the range narrows to about 89.4% to 90.6%. A difference of 1 point between two plans means little at 1,000 paths but is meaningful at 10,000.