Parametric and Nonparametric Models of Joint Returns: Density, Dependence, and Portfolio Tail Risk

We compare parametric, empirical and neural models of joint asset returns to determine whether greater flexibility improves density, dependence and portfolio risk. Gaussian, shrinkage-Gaussian, Student and meta-t laws provide parametric benchmarks; meta-t joins asset-specific Student margins with a Student copula. Filtered historical simulation (FHS) supplies empirical innovations. Normalizing flows learn invertible density transformations, variational autoencoders learn latent-variable distributions, and Wasserstein GANs learn scenario generators. Separate volatility and correlation controls distinguish conditioning from innovation modeling. Ten estimation datasets per Student, meta-t and Heston synthetic law, and 2,514 CRSP test dates in five-basket and 100-stock panels, support paired comparisons with multiplicity control. Meta-t achieves the best observed exact density fit in both CRSP panels. Under common one-percent validation selection, no tested neural model establishes a risk-score improvement over FHS; alternative checkpoint rules change rankings. Student's 100-stock advantage depends on the comparison family. Variance filtering produces the largest point-score gains, with inconclusive adjusted tests. Synthetic results distinguish correct specification from architectural flexibility and separate likelihood quality from expected-shortfall accuracy. A Wishart–CIR extension relaxes a restricted covariance model's dispersion ceiling but fails the conditional numerical-stability gate. The evidence makes conditioning, specification and validation design central to assessing distributional flexibility.