Terminal Portfolio Value Distribution to Machine Precision

We present a deterministic, semi-analytical framework for computing the complete distribution of a portfolio's terminal value at horizon \(T\) for correlated lognormal assets. Unlike traditional approaches, this method requires no Monte Carlo simulation. It directly yields the full probability density, the cumulative distribution, all moments, and exact quantile metrics such as loss Value-at-Risk (VaR) and loss Expected Shortfall (ES). The method is not closed-form in the classical sense (no finite elementary formula exists for the sum-of-lognormals distribution), but it achieves machine precision through a rapidly converging spectral scheme with controllable error.

The method builds on the Hermite-COS representation developed in The Spectral Lognormal Distribution (Nagy, 2026): the terminal portfolio value distribution is expressed as a finite trigonometric polynomial whose coefficients are computed from Gauss-Hermite quadrature over the correlation eigenstructure. The CDF is then a deterministic sine series, the PDF is a cosine series, the lower-tail terminal-value quantile is obtained by root-finding on the CDF, and the corresponding loss VaR and loss ES follow immediately from \(L_T = S_0 - S_T\). Both the quadrature and the spectral expansion converge exponentially, so with standard parameters (\(Q = 40\), \(N = 128\)) the results match Monte Carlo to 4--5 significant figures — deterministically, in under one second.

We demonstrate and validate the framework on three \(\\)100{,}000\( portfolios spanning the practitioner landscape:

1. Bitcoin-heavy crypto portfolio (\)\sigma_{\max} = 90\%\(, skewness = 3.6): extreme volatility where delta-normal underestimates the 1% loss by \)\\(128{,}000\) — more than the portfolio itself. The Hermite-COS method matches Monte Carlo (5M paths) to within \(\\)53\( on VaR.

2. Long-short equity hedge portfolio (net +40%, gross 200%): positions that can go negative. The method matches MC VaR 1% to within \)\\(10\), natively handling short positions.

3. Diversified 60/40 multi-asset portfolio: the standard institutional setup. Matches MC VaR 1% to within \(\\)27\(, revealing \)\\(4{,}687\) of hidden tail risk that the Gaussian assumption misses.

All results are validated against Monte Carlo with \(5 \times 10^6\) paths. Every risk measure agrees to 4--5 significant figures. The Hermite-COS computation completes in 0.4--0.8 seconds per portfolio and is perfectly deterministic: running the same portfolio twice always gives the same answer, unlike Monte Carlo where the VaR 1% fluctuates by \(\\)19\(--\)\\(21\) between seeds.

Keywords: terminal portfolio value distribution, loss Value-at-Risk, loss Expected Shortfall, lognormal sum, semi-analytical, Fourier-cosine method, Gauss-Hermite quadrature, deterministic spectral risk.