Actuarial Applications of Multifractal Modeling

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

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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)

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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

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Example multifractal field
Population Density in the Northeast USA
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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
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Universality (CLT for multifractals)
It is thought that a 2-parameter family of
generating distributions make up the central
limit of random multifractals.
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Generate multiplicative cascade
2.0
1.4
Building block distribution has average density
1
0.6
0.0
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Application portfolio allocation
Poisson Model
Actual Portfolio
Multifractal Model
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Hail Swath Simulation
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Multiple Realizations of Hail Swath
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• Actuarial Applications of Multifractal Modeling
• 1
• Introduction and Spatial Applications
• John A. Major, ASA, MAAA