Actuarial Applications of Multifractal Modeling 1 Introduction and Spatial Applications John A. Major, ASA, MAAA Guy Carpenter & Co., Inc.

Background n A new paradigm l Fractals -- early ‘80s. l Multifractals -- late ‘80s to early ‘90s. n Applies to: l financial time series (Mandelbrot) l rain, clouds, etc. (Schertzer & Lovejoy) l population density (Appleby) l insured property values (Lantsman, Major & Mangano)

What is a multifractal? n A type of intensity field l e.g., dollars exposed per square mile l e.g., number of hailstones per square foot n Exhibits “scale-invariance” l Statistically self-similar at various scales n How to simulate l multiplicative cascade

Example multifractal field Population Density in the Northeast USA

The K(q) curve summarizes moment scaling “Trace Moment Graph” shows how the qth-moment scales with resolution. Each q yields a different slope K for the relation. The relation between q and K reveals the multifractal nature of the field. q = 1.4 K = 0.273

Universality (CLT for multifractals) It is thought that a 2- parameter family of generating distributions make up the “central limit” of random multifractals.

Generate: multiplicative cascade “Building block” distribution has average density =

Application: portfolio allocation Poisson ModelActual PortfolioMultifractal Model

Hail Swath Simulation

Multiple Realizations of Hail Swath

Actuarial Applications of Multifractal Modeling 1 Introduction and Spatial Applications John A. Major, ASA, MAAA