Folio Lab/Documentation

Monte Carlo · Return processes

Lognormal GBM Benchmark

Benchmark

gbm_lognormal

The textbook model, included precisely so you can see what it gets wrong. Geometric Brownian motion is the reference point almost every retirement calculator quietly assumes. FolioLab ships it as an explicitly labeled sanity benchmark rather than as its planning model.

The model

Continuous-time GBM,

sampled at a fixed daily step is exactly an IID Gaussian log-return model. There is no approximation and no discretization error to discuss: the daily log return is normal, and successive days are independent. The engine simulates that directly.

Log returns are drawn as through a Cholesky factor of the calibrated covariance, and gross returns follow as , positive by construction. As with the FHS processes, the log mean is normalized so the expected gross return equals the declared daily arithmetic target exactly. Here the innovations really are Gaussian, so the correction is the closed form

rather than the empirical cumulant correction the FHS processes need.

Ledoit-Wolf shrinkage covariance

The covariance is not the raw sample estimate. With assets there are parameters to estimate, and the sample covariance matrix of a realistic portfolio is ill-conditioned: its smallest eigenvalues are dominated by estimation noise, which a Cholesky-based simulator would faithfully reproduce as structure that does not exist.

Ledoit-Wolf shrinkage pulls the sample matrix toward a well-conditioned structured target,

with the intensity chosen analytically to minimize expected squared error. It trades a little bias for a large reduction in variance and guarantees a positive-definite, well-conditioned matrix to factor.

Why it is a benchmark and not an upgrade

GBM is deliberately not a realism improvement over the bootstrap. It is missing the two features that dominate real equity risk, and both omissions push in the same direction: they make the portfolio look safer than it is.

  • No volatility clustering. Every day draws from the same distribution regardless of what just happened, so the model cannot produce a sustained turbulent stretch. Sustained turbulent stretches are what actually break withdrawal plans.
  • No fat tails. Daily equity returns have far more probability mass in the extremes than a normal distribution allows. Under a Gaussian model calibrated to Indian equities, several observed single-day moves are events that should not occur once in the age of the universe. They occurred within one lifetime of market data.
  • No cross-asset tail dependence. A covariance matrix fixes average co-movement, but assets crash together far more often than a Gaussian copula implies. Diversification fails exactly when you need it, and this model cannot express that.

The compensating advantage is that GBM is analytically tractable. Terminal wealth under it is exactly lognormal, so its percentiles can be checked in closed form. That makes it genuinely useful as a reference: if a resampling process produces a median far from the GBM median under the same drift, that difference is attributable to distributional shape rather than to a bug.

When to use it

Use it as a control. Run your plan under GBM and under the block bootstrap and compare. The medians should be close, because both are anchored to the same drift. The lower percentiles, the maximum-drawdown distribution and the depletion probabilities should be visibly worse under the bootstrap. That gap is the cost of the Gaussian assumption, measured on your own portfolio rather than asserted in general.

It is also the right process for reconciling FolioLab against an external tool. Most calculators are GBM underneath, so matching them requires matching the model, not just the inputs.

Do not plan on it. A goal probability computed under GBM is optimistic in a direction you cannot easily quantify without running one of the other processes anyway. It needs only 60 observations to calibrate, which makes it available on almost any panel, and that availability is not an endorsement.

References

  • Samuelson, P. A. (1965). Rational Theory of Warrant Pricing. Industrial Management Review, 6(2), 13–39.
  • Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637–654.
  • Ledoit, O., & Wolf, M. (2004). A Well-Conditioned Estimator for Large-Dimensional Covariance Matrices. Journal of Multivariate Analysis, 88(2), 365–411.
  • Mandelbrot, B. (1963). The Variation of Certain Speculative Prices. Journal of Business, 36(4), 394–419. The original demonstration that returns are not Gaussian.
  • Cont, R. (2001). Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues. Quantitative Finance, 1(2), 223–236.