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Actuarial Applications of Multifractal Modeling 1 Introduction and Spatial Applications John A. Major, ASA, MAAA Guy Carpenter & Co., Inc. Background. A new paradigm Fractals -- early ‘80s. Multifractals -- late ‘80s to early ‘90s. Applies to: financial time series (Mandelbrot)
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Actuarial Applications of Multifractal Modeling 1 Introduction and Spatial Applications John A. Major, ASA, MAAA Guy Carpenter & Co., Inc.
Background • A new paradigm • Fractals -- early ‘80s. • Multifractals -- late ‘80s to early ‘90s. • Applies to: • financial time series (Mandelbrot) • rain, clouds, etc. (Schertzer & Lovejoy) • population density (Appleby) • insured property values (Lantsman, Major & Mangano)
What is a multifractal? • A type of intensity field • e.g., dollars exposed per square mile • e.g., number of hailstones per square foot • Exhibits “scale-invariance” • Statistically self-similar at various scales • How to simulate • 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 2.0 1.4 “Building block” distribution has average density = 1 0.6 0.0
Application: portfolio allocation Poisson Model Actual Portfolio Multifractal Model
Actuarial Applications of Multifractal Modeling 1 Introduction and Spatial Applications John A. Major, ASA, MAAA