Actuarial Applications of Multifractal Modeling 1 Introduction and

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

Background n n A new paradigm l Fractals -- early ‘ 80 s. l Multifractals -- late ‘ 80 s to early ‘ 90 s. 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 l n Exhibits “scale-invariance” l n e. g. , dollars exposed per square mile e. g. , number of hailstones per square foot Statistically self-similar at various scales How to simulate l multiplicative cascade

Example multifractal field Population Density in the Northeast USA

The K(q) curve summarizes moment scaling q = 1. 4 K = 0. 273 “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.

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 2. 0 1. 4 0. 6 0. 0 “Building block” distribution has average density = 1

Application: portfolio allocation Poisson Model Actual Portfolio Multifractal Model

Hail Swath Simulation

Multiple Realizations of Hail Swath

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