The model underlying this Demonstration assumes that catastrophic events, when they occur, create losses that obey a Weibull distribution. This distribution is calibrated such that a materialization of 1 means that the losses equal the value of the property. A high percentage of the time, however, no losses occur at all. For payment of an amount

, the insured may take precautions against loss that reduce the size of any baseline loss

to an amount

, such that the size of the loss is reduced by one half for each marginal unit of precautions. Thus, if

is the deductible and

is the rate of coinsurance, the actual loss suffered by the insured from precautions and any loss

, after consideration of insurer payments, is

. If

is the probability of a catastrophic event occurring, the distribution of losses suffered by the insured may be written as a mixture distribution in which (1) the first component is always zero (no loss); (2) the second component is a transformed Weibull distribution conditioned on the loss's being less than the deductible (the transformation is

); and (3) the third component is a transformed Weibull distribution conditioned on the loss's being greater than the deductible (the transformation is

).

The CDF of this mixture distribution may be written as:

if

;

if

;

if

;

if

;

if

;

if

;

if

;

if

;

if

;

if

;

if

;

if

;

if

;

if

;

if

;

otherwise.

You can then compute the expected quantile of this outcome mixture distribution subject to a weighting function that is nondecreasing in losses and whose domain is the interval

. The spectral measure approach to risk, coupled with an assumption of rational behavior on the part of the insured, permits you to determine the optimal level of precautions by the insured by finding the value of precautions

that minimizes this expectation. It is this relationship between precautions

and spectral measure that is displayed on the graphic above.

It turns out that a closed-form solution to the expected quantile is difficult or impossible to obtain. However, as noted by Oleksandr Pavlyk, an expected quantile is equivalent to the expectation of a random variable

multiplied by a special probability density function, where

is drawn from the outcome distribution. The special probability density function is the PDF of the weighting function evaluated at the CDF of the outcome distribution when in turn evaluated at

. With this method, you can obtain the following closed forms for expected quantile.

By differentiating these spectral measures with respect to

and then bounding the results between 0 and some arbitrarily large value, you can then obtain the optimal level of precautions. The table below shows closed-form values.

Often when you work with spectral measures, you use a weighting function that is nonincreasing rather than one, as here, that is nondecreasing. A nonincreasing weighting function is used here, however, because the underlying outcome is a loss. Risk aversion thus requires that the larger losses be weighted at least as heavily as the smaller losses.

Although the discussion in this Demonstration focuses on insurance against catastrophes, the mathematics underlying it permits its extension to all sorts of insurable losses, provided they are Weibull distributed.

This Demonstration achieves its speed of computation through extensive use of the

*Mathematica* compiler.

Snapshot 1: when the cost of precautions is high, the effect of greater deductibles and coinsurance is small, even when the insured is highly risk averse

Snapshot 2: when the cost of precautions is low, greater deductibles and coinsurance have an enormous effect on expected losses, even when the insured is not particularly risk averse

Snapshot 3: high deductibles and coinsurance result in high levels of optimal precautions

Some literature examining the use of the Weibull distribution in modeling catastrophe losses includes [1–5].

[2] R. Hogg and S. Klugman,

*Loss Distributions*, New York: Wiley, 1984.

[3] C. Kleiber and S. Kotz,

*Statistical Size Distributions in Economic and Actuarial Sciences*, Hoboken, NJ: Wiley, 2003.

[5] M. Legg, L. Nozick, and R. Davidson, "Optimizing the Selection of Hazard-Consistent Probabilistic Scenarios for Long-Term Regional Hurricane Loss Estimation,"

*Structural Safety* **32**(1), 2010 pp. 90–100.

doi:10.1016/j.strusafe.2009.08.002.