Balance & Equity Over Time
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Payment Composition
Key Milestones
Cost Breakdown
Cumulative Interest
Equity Build-Up (%)
Amortization Schedule
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See the full range of outcomes — not just one number. Compare loan scenarios side by side. Model real market volatility on your investments. Free, private, runs entirely in your browser.
See how extra payments, different terms, and frequencies change your total cost. Save and compare up to 3 scenarios side by side.
See the best-case, worst-case, and most-likely outcomes for any investment — from CDs to crypto — using real volatility modeling.
Add contributions, withdrawals, and one-time events. See how they interact with growth and volatility over any time horizon.
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This page documents the mathematical models, formulas, and assumptions that power both the Loan Analytics and Asset Simulator modules. Every calculation runs client-side in your browser — no data is sent anywhere.
The periodic payment for a fixed-rate fully-amortizing loan is computed using the standard annuity formula:
Each payment is split into an interest component and a principal component. The interest portion for period k is the outstanding balance multiplied by the periodic rate. The principal portion is the remainder:
Extra payments are added directly to the principal portion each period, which reduces the outstanding balance faster and shortens the loan term. The principal-interest crossover occurs at the period where the principal portion first exceeds the interest portion — a milestone that depends on the rate, term, and extra payment amount.
The Asset Simulator supports six compounding modes. The future value after time t depends on the compounding frequency:
Continuous compounding represents the mathematical limit as n approaches infinity. In practice, the difference between daily and continuous compounding is negligible for typical APYs, but the continuous formula is analytically cleaner and is the default because it connects naturally to the GBM stochastic model.
When volatility is non-zero, the Asset Simulator models price dynamics using Geometric Brownian Motion — the same framework underlying the Black-Scholes option pricing model. The stochastic differential equation is:
For simulation, we discretize this into monthly time steps using the exact solution:
The drift-adjustment term is the Itô correction. It ensures that the expected value of the GBM process equals the deterministic growth path. Without this correction, volatility would systematically inflate the expected outcome due to the asymmetry of log-normal distributions.
The simulator generates multiple independent GBM paths (configurable, default 500). Each path is a sequence of monthly values obtained by repeatedly applying the discretized GBM step. Cash flows (contributions, withdrawals, one-time events) are applied after the growth step each month.
Random numbers are generated using the Box-Muller transform, which converts pairs of uniform random variables into standard normal random variables:
At each monthly time step, the values from all simulation paths are sorted and percentiles are extracted using order statistics. For N paths and percentile q, the value is taken at index . The displayed bands are the 5th, 10th, 25th, 50th (median), 75th, 90th, and 95th percentiles.
The final-value distribution histogram bins all terminal values across paths. The probability analysis computes the empirical probability of exceeding specific return thresholds by counting the fraction of paths above each target.
Constant volatility: GBM assumes volatility (σ) is constant over time. Real markets exhibit volatility clustering — periods of high volatility tend to follow other high-volatility periods (GARCH effects). GBM underestimates the frequency of extreme events.
Log-normal returns: GBM produces log-normally distributed returns. Real financial returns have fatter tails than the log-normal distribution predicts. This means crashes of the magnitude seen in 2008 or 2020 are more likely than the model suggests.
No mean reversion: The model does not incorporate mean reversion — the tendency of valuations to return to historical averages over long periods. This is most relevant for very long simulation horizons (20+ years).
Fixed rates: Loan calculations assume a fixed interest rate for the entire term. In practice, adjustable-rate mortgages (ARMs) and refinancing opportunities can significantly alter outcomes.
No taxes or fees: The simulator does not account for capital gains taxes, dividend taxes, management fees, inflation, or transaction costs, all of which reduce real returns. A 10% nominal return might be 6-7% after taxes and fees.
Monthly granularity: All simulations use monthly time steps (Δt = 1/12). For very high volatility assets, finer granularity could yield different percentile estimates, though the expected values remain unbiased due to the Itô correction.
Cash flow timing: Contributions and withdrawals are applied at the end of each monthly step (post-growth). Withdrawals are capped at the available balance to prevent negative values.