Finance & Economics · Portfolio Management · Portfolio Risk
Monte Carlo Simulation Calculator
Estimates the probability distribution of portfolio outcomes over time using Monte Carlo simulation with user-defined return and volatility assumptions.
Calculator
Formula
W_t is the portfolio value at time t. W_0 is the initial portfolio value. \mu is the expected annual return (as a decimal). \sigma is the annual volatility (standard deviation of returns, as a decimal). t is the time horizon in years. Z is a standard normal random variable (Z ~ N(0,1)), sampled independently for each simulation path. The term (\mu - \sigma^2/2) is the drift-adjusted expected log return, accounting for the Ito correction under geometric Brownian motion.
Source: Hull, J.C. — Options, Futures, and Other Derivatives (10th ed.). Geometric Brownian Motion framework, Chapter 15.
How it works
Monte Carlo simulation is a computational technique that models uncertainty by running thousands of hypothetical scenarios based on random sampling. In finance, it is most commonly applied to portfolio growth problems where future returns are unknown but can be characterized by a probability distribution. The fundamental insight is that real-world portfolio values do not follow a simple straight-line trajectory — they fluctuate according to market randomness, and understanding the full range of plausible outcomes is essential for sound financial planning.
This calculator uses the Geometric Brownian Motion (GBM) framework, the same mathematical model that underpins the Black-Scholes options pricing formula. Under GBM, the portfolio value at time t follows a log-normal distribution. The key formula is W_t = W_0 · exp[(μ − σ²/2)t + σ√t · Z], where μ is the expected annual return, σ is the annual volatility (standard deviation of returns), t is the time horizon in years, and Z is a standard normal random variable. The term (μ − σ²/2) is the Itô-corrected drift, which accounts for the mathematical difference between arithmetic and geometric averages under log-normal dynamics. Because log-normal distributions are positively skewed, the median outcome is always less than the mean (expected value) — an important and often overlooked distinction in financial planning.
The outputs of this calculator span the full distribution: the mean (expected value), the median (50th percentile), the 90th and 10th percentiles representing optimistic and pessimistic scenarios, and the 5th percentile worst case. The probability of loss quantifies the likelihood that the portfolio finishes below its starting value. These statistics allow investors to answer critical planning questions: What is the chance I run out of money in retirement? How much upside do I realistically have? What is my downside in a bad decade? Annual contributions are also supported, modeled as a continuous stream of capital added to the portfolio each year.
Worked example
Consider an investor with a $100,000 initial portfolio, contributing $6,000 per year, targeting a 20-year horizon. The portfolio is invested in a diversified equity index fund with an expected annual return of 8% and annual volatility of 15%.
Step 1 — Compute the log-normal drift: The Itô-corrected drift is (0.08 − 0.5 × 0.15²) × 20 = (0.08 − 0.01125) × 20 = 0.06875 × 20 = 1.375.
Step 2 — Median outcome (Z = 0): Portfolio growth = $100,000 × e^1.375 ≈ $100,000 × 3.955 = $395,507. The contribution future value at median uses a mid-range approximation, adding roughly $55,000–$70,000 for annual contributions. Median total ≈ $450,000–$470,000.
Step 3 — Expected (mean) outcome: Under GBM, the mean uses the full arithmetic drift e^(μt). Mean portfolio growth = $100,000 × e^(0.08 × 20) = $100,000 × e^1.6 ≈ $100,000 × 4.953 = $495,303. Adding contribution FV = $6,000 × (e^1.6 − 1) / 0.08 ≈ $6,000 × 49.15 ≈ $294,900. Expected total ≈ $790,203.
Step 4 — Pessimistic scenario (10th percentile, Z = −1.28): Portfolio growth = $100,000 × e^(1.375 + 0.15 × √20 × (−1.28)) ≈ $100,000 × e^(1.375 − 0.859) = $100,000 × e^0.516 ≈ $167,516. With pessimistic contribution growth, total ≈ $195,000–$215,000.
Step 5 — Probability of loss: d = (0.08 − 0.01125) × 20 / (0.15 × √20) = 1.375 / 0.6708 ≈ 2.05. P(loss) = Φ(−2.05) ≈ 2.02% — quite low over a 20-year equity horizon, illustrating why long time horizons dramatically reduce loss probability.
This example demonstrates the wide dispersion of outcomes that Monte Carlo simulation captures: an investor might end up with $200,000 in a bad scenario or over $1,000,000 in an exceptional one, even with identical starting conditions.
Limitations & notes
Constant parameters assumption: This model assumes μ and σ remain fixed throughout the simulation horizon. In reality, market regimes shift — volatility clusters and expected returns vary with economic conditions. Fat-tailed distributions (leptokurtosis) in real asset returns mean extreme events are more common than GBM predicts.
Log-normal return distribution: GBM models returns as log-normally distributed. This excludes jump processes (sudden crashes, flash events) and cannot capture correlation breakdowns during market stress, when diversification benefits typically collapse precisely when they are most needed.
Annual contribution simplification: Contributions are modeled as a continuous or end-of-year annuity approximation. In practice, dollar-cost averaging on a monthly or weekly schedule has a slightly different mathematical profile, and this calculator does not account for the sequence-of-returns risk that affects withdrawal strategies in retirement.
No inflation adjustment: All figures are in nominal terms. Real purchasing power of future outcomes may be substantially lower depending on prevailing inflation rates over the horizon.
Tax and fee drag excluded: Management fees, transaction costs, capital gains taxes, and dividend reinvestment tax implications are not incorporated. These can meaningfully reduce net returns, particularly over long horizons.
Single-asset approximation: This calculator treats the portfolio as a single asset with aggregate return and volatility characteristics. Multi-asset portfolios with low correlations will have different risk-adjusted outcomes that require full covariance matrix simulation.
Frequently asked questions
Why is the median outcome lower than the expected (mean) outcome in Monte Carlo simulation?
Under geometric Brownian motion, portfolio values follow a log-normal distribution, which is positively skewed. The mean is pulled upward by the possibility of very large gains in a small number of favorable scenarios, while the median represents the midpoint where half of all simulated paths finish above and half below. The difference is captured by the Itô correction: the mean uses the full arithmetic return μ, while the median uses the drift-adjusted log return (μ − σ²/2). Higher volatility widens this gap significantly.
How many simulations does a robust Monte Carlo analysis require?
In practice, professional-grade Monte Carlo simulations run 10,000 to 100,000 paths to achieve stable percentile estimates. This calculator uses analytical percentile formulas derived from the log-normal distribution, which are mathematically equivalent to running an infinite number of simulations. This approach eliminates sampling error and produces exact percentile values given the input assumptions, making it more reliable than a small finite simulation for a quick reference tool.
What expected return and volatility should I use for a diversified stock portfolio?
The US stock market (S&P 500) has historically delivered nominal arithmetic returns of approximately 10–11% per year with annual volatility around 15–20%. A globally diversified equity portfolio might assume 7–9% expected return and 12–16% volatility. Bonds and balanced portfolios have lower return and volatility assumptions. Always stress-test with both your base-case assumptions and more conservative inputs — financial plans should be robust to pessimistic scenarios.
What does the probability of loss output mean, and how should I interpret it?
The probability of loss is the statistical likelihood that the portfolio's final value falls below the initial investment, given the specified return, volatility, and time horizon. It is computed using the log-normal CDF. A very low probability of loss (e.g., 2%) over 20 years for a typical equity portfolio does not mean losses are impossible — it means that in approximately 1 out of 50 equivalent scenarios, the final balance would be negative. The worst-case 5th percentile output complements this by showing the dollar value at that tail scenario.
Can I use this Monte Carlo calculator for retirement planning?
Yes — this calculator is well-suited for retirement accumulation planning. Enter your current portfolio balance, expected annual contribution (401k, IRA, taxable savings combined), the time until retirement, and your asset allocation's historical return and volatility parameters. The percentile outputs will show a range of plausible retirement nest eggs. For the withdrawal (decumulation) phase, note that sequence-of-returns risk is critically important and is not captured here — consider using a dedicated retirement drawdown simulator for that phase of planning.
Last updated: 2025-01-15 · Formula verified against primary sources.