Monte Carlo · Return processes

Multivariate Gaussian Process

Comparison

gaussian

The thin-tailed reference case. It is included so the cost of assuming normality can be measured on your own portfolio rather than argued about in the abstract, and it is never the default.

Construction

Daily simple returns are drawn directly from a multivariate normal calibrated to the historical daily mean vector and covariance matrix:

where is the Cholesky factor of the sample covariance with a small diagonal jitter for numerical stability. Days are IID. There is nothing else to the model.

Note the difference from the lognormal GBM benchmark, which is Gaussian in logreturns. That distinction is not cosmetic. GBM has positive gross returns by construction; this process works in simple-return space, so its support is unbounded below and it depends on the engine's disclosed lower gross-return clip to keep wealth non-negative.

Why it materially understates Indian-market tails

This is not a marginal calibration quibble. Daily equity returns have excess kurtosis in the range where Gaussian tail probabilities are wrong by orders of magnitude, not percentages. Moves that a fitted normal treats as effectively impossible occur repeatedly in any reasonably long sample of NSE history.

Three specific failures, all of which bias results toward looking safe:

Thin tails. Value at Risk and CVaR at the 99% level come out systematically too small, and the maximum-drawdown distribution is too tight.
No skew. The normal is symmetric. Equity returns are negatively skewed, so the model gives large gains and large losses equal weight when reality does not.
Gaussian dependence. Correlations are matched on average, but the normal copula has asymptotically independent tails: it cannot produce the everything-falls-together days that dominate real portfolio drawdowns.

Because it is IID, it also has no volatility clustering, so it cannot generate a sustained bad stretch. For a withdrawal plan, that omission alone can move the depletion probability substantially.

When to use it

Use it as the null case in a comparison. Its value is entirely diagnostic: the distance between the Gaussian result and the block bootstrap result is the non-normality of your portfolio, quantified in the units you care about, whether that is goal probability, drawdown or depletion.

It is also useful for validating intuition against classical portfolio theory, which is built on exactly this assumption. Mean-variance optimization is optimal under a Gaussian world; simulating that world explicitly shows what the optimizer was implicitly promising.

Never plan on it. If you want a parametric process with realistic tail behaviour, the Student-t is strictly closer to the data, and the resampling processes are closer still.

References

Mandelbrot, B. (1963). The Variation of Certain Speculative Prices. Journal of Business, 36(4), 394–419.
Fama, E. F. (1965). The Behavior of Stock-Market Prices. Journal of Business, 38(1), 34–105.
Cont, R. (2001). Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues. Quantitative Finance, 1(2), 223–236.
Embrechts, P., McNeil, A. J., & Straumann, D. (2002). Correlation and Dependence in Risk Management: Properties and Pitfalls. In Risk Management: Value at Risk and Beyond, 176–223.

Related: Kurtosis and Skewness quantify, on your actual holdings, the departures from normality this process ignores.