How do mixture distributions improve VaR estimates?

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AcadiFi · Accepted Answer

A mixture distribution combines two or more normal (or other) distributions with weights that sum to one. A common two-state model assumes returns are drawn from N(mu1, sigma1) with probability p and N(mu2, sigma2) with probability 1-p. The second component typically has larger sigma and captures stress days, producing fat tails and possibly skew without abandoning tractable math. VaR under a mixture is not a closed-form quantile; you invert the CDF numerically. Example: p = 0.9, mu1 = 0, sigma1 = 0.01, p2 = 0.1, mu2 = -0.01, sigma2 = 0.04. The marginal distribution has volatility higher than a single normal at 0.01 and exhibits negative skew. ```mermaid flowchart LR C[Calm regime
90%] --> M[Mixture return] S[Stress regime
10%] --> M M --> Q[VaR quantile
from inverted CDF] ``` Fitting uses the expectation-maximization algorithm, which alternates between assigning observations to regimes probabilistically (E-step) and updating parameters (M-step). Mixtures are less complex than Markov regime switching because the regime is drawn independently each period rather than transitioning, which trades realism for simplicity. For FRM, remember that mixture VaR requires numerical inversion, that EM fits are sensitive to starting values, and that it is a good middle ground between naive normal VaR and full regime-switching GARCH.

How do mixture distributions improve VaR estimates?

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