Lognormal distribution
Distribution whose logarithm follows a normal law, widely used to model heavy-tailed losses and strictly positive amounts.
Gibrat, 1931, Les Inégalités économiques: a quantity that grows through independent multiplicative shocks ends up lognormal. A random variable X follows a lognormal distribution with parameters μ and σ if ln(X) follows a normal N(μ, σ²). Right-skewed, it fits insurance losses, made of many small claims and a few large ones, rare but possible. Two parameters characterize it, μ (log-scale) and σ (shape), and VaR and TVaR have closed forms: VaR at level α equals e^(μ + σ z_α). It appears in solvency models, notably the Solvency II standard formula for non-life underwriting risk, and in cyber loss modeling. For extreme losses, its tail is more realistic than the normal's.
Textbook case: cyber losses modeled as lognormal μ = 12, σ = 1.2. VaR at 99.5% (Solvency II SCR level) = e^(12 + 1.2 × 2.576) = e^15.09 ≈ 3.58 M EUR.
lognormal distribution, log-normal
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