Where does the log-logistic distribution show up in finance?

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AcadiFi · Accepted Answer

Log-logistic has PDF f(x) = (beta/alpha)*(x/alpha)^(beta-1) / (1 + (x/alpha)^beta)^2. Its hazard is non-monotonic: rising then falling (for beta > 1). This makes it suitable for modeling phenomena where risk initially grows then wanes, like startup bankruptcy (risk peaks around year 2-3 then declines), or loan prepayment (peak around seasoning).

Brookhaven Credit Analytics fits log-logistic to time-to-default for venture-backed SaaS firms with alpha = 3.2 years, beta = 2.1. The hazard peaks around year 2.8 then declines as surviving firms demonstrate viability. Median survival = alpha = 3.2 years. Survival function is S(t) = 1 / (1 + (t/alpha)^beta), easy to compute without numerical integration unlike lognormal.

The distribution's tail is heavier than lognormal but lighter than Pareto. In finance, fit log-logistic when observed hazard plots show a clear hump rather than monotonic behavior. Goodness of fit via Anderson-Darling; closed-form quantiles make VaR calculations efficient. Relation to logistic: log(X) follows a logistic distribution, giving the 'log-' prefix. Not heavy-tailed enough for extreme operational losses, so typically combined with a Pareto tail.

Where does the log-logistic distribution show up in finance?

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