Finance

The normix.finance subpackage turns a fitted mixture into portfolio analytics. Because a normal variance-mean mixture is conditionally Gaussian given the latent \(Y\), portfolio quantities — and crucially their gradients and Hessians — are computable by a fast conditional Monte Carlo over \(Y\) alone.

Portfolio projection

Any linear combination \(w^\top X\) of a mixture’s assets is again a univariate member of the same family, with parameters available in closed form. No re-fitting is needed:

from normix.finance import project_portfolio

proj = project_portfolio(model, w)     # a Univariate* distribution of wᵀX
proj.mean(); proj.std()
proj.ppf(0.05)                         # 5% quantile (a VaR level)

This makes it cheap to evaluate many candidate weightings against one fitted model. See A multivariate stock basket.

Tail risk: VaR and CVaR

CVaR(alpha) computes Value-at-Risk and Conditional Value-at-Risk at tail probability alpha:

from normix.finance import CVaR

cvar = CVaR(0.05)                       # 95% level
Y = proj.subordinator.rvs(100_000, seed=0)   # conditional-MC draws
var_95 = cvar.var(proj)                 # deterministic quantile
cvar_95 = cvar.value(proj, Y)           # conditional Monte Carlo over Y

Conditioning on \(Y\) makes the estimator far lower-variance than sampling returns directly.

Differentiable risk

The payoff for the exponential-family structure is analytic risk sensitivities, in both the projected scalar parametrization and the portfolio weights:

cvar.gradient_scalar(proj, Y)   # ∂CVaR/∂(μ̃, γ̃, σ̃)
cvar.hessian_scalar(proj, Y)    # 3×3 Hessian
cvar.gradient_w(model, w, Y)    # ∇_w CVaR  (chain rule through the projection)
cvar.hessian_w(model, w, Y)     # weight-space Hessian

These match finite differences to machine precision and plug straight into gradient- or Newton-based portfolio optimizers. The WeightFunctional helper bundles a risk measure, model, and Y into a callable with .grad and .hess. See Portfolio CVaR and its derivatives for verification and a worked CVaR-reduction loop.

Scaling to many assets

At portfolio scale the factor mixtures replace a dense covariance with \(\Sigma = F F^\top + \operatorname{diag}(D)\), cutting covariance parameters from \(O(d^2)\) to \(O(d r)\) and routing every solve through the Woodbury identity. The GH tail behaviour is retained. See Factor mixtures for a Dow Jones 30 portfolio.

Further reading

The mathematical background — CVaR derivatives, mean–risk optimization, transaction costs, and factor analysis — is developed in the theory notes.