The Effects of Coinsurance and Deductibles on Optimal Precautions for Weibull-Distributed Loss

This Demonstration examines the effects of coinsurance and deductibles on optimal precautions for Weibull-distributed losses, such as those frequently used to model hurricanes, earthquakes, and other catastrophes. The issue is important because regulation of deductibles and coinsurance mediates two dueling goals. One goal is to provide insureds and those who depend on them with fairly complete insurance. A second goal, however, is to protect the public, who often backstop catastrophe insurance schemes, from precaution-decreasing moral hazard likely to result when insurance is very complete. Thus, in assessing the virtue of regulations requiring heightened deductibles and coinsurance, it is important to consider the magnitude of loss reductions likely to result: the greater the effect, the stronger the case for regulation. This Demonstration estimates such magnitudes based on a theoretical model of insured behavior.
To interact with this Demonstration, you select a level of risk aversion. Higher levels of risk aversion result in worse outcomes being weighted progressively more heavily than better ones. You select the price of taking one unit of precautions in a system calibrated such that each marginal unit of precaution reduces the expected loss from accidents by half. You select the probability that a catastrophic event such as a windstorm or an earthquake occurs at all; during many policy periods it is quite likely that no catastrophic event will occur. You then select two parameters to the Weibull distribution of baseline losses if an event occurs. Baseline losses are calibrated such that a materialization of 1 means that the loss is equal to the value of the property insured. The top graphic of the Demonstration then produces the PDF of the resulting Weibull distribution. The blue zone (often very thin) represents losses less than the deductible; the green zone represents losses in excess of the deductible. A dotted line shows the mean value of the Weibull distribution. Finally, you select the deductible and coinsurance levels of the policy. The deductible is measured as a percentage of the value of the property insured. Coinsurance is measured as a percentage of loss in excess of the deductible.
The Demonstration responds to these control selections with two other panels of information. A bottom graphic shows the "spectral measure" (a kind of weighted average) of the insured's net losses as a function of the level of precautions taken by the insured. A rational insured should be expected to select the level of precautions that minimizes this spectral measure. A black point on the lower graphic shows this optimal precaution level. A grid on the right shows a number of statistics created by the controls and various computations. Statistics in the top gold zone largely recapitulate the controls selected by the user. Statistics in the blue zone show optimal precautions, losses as a fraction of what would occur if no precautions were taken, and the spectral measure of losses for the optimal level of precautions. Statistics in the red zone show the effect of changing insurance policy parameters. Each "loss " represents the reduction in loss that would be caused by the change described for the row. Thus a row that reads "loss from 20% higher coinsurance 0.393" means that increasing the coinsurance level by 20 percentage points would change optimal precautions such that losses would be reduced by 39.3% from the level induced by optimal precautions with the current insurance policy parameters. The higher these "loss s," the more effective policy changes would be in reducing the losses caused by catastrophes.
The Demonstration shows that the effect of coinsurance changes depends critically on such factors as the cost of the precautions. When the cost of precautions is high, changes in coinsurance rates from low levels may have little or no effect on the optimal level of precautions. Other regulations, such as mandatory conditioning of indemnity obligations on maintenance of certain precautions, may be necessary to reduce the risk of loss. When the cost of precautions is low, however, changes in coinsurance rates and deductibles may have dramatic effects. Reductions of 50% or more from increases in coinsurance rates by 20 percentage points are not uncommon, nor are reductions of 10% or more from increases in deductibles by 2 percentage points (such as going from 2% of the property value to 4% of the property value).


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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 ;
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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].
[1] C. Watson and M. Johnson, "Hurricane Loss Estimation Models: Opportunities for Improving the State of the Art," Bulletin of the American Meteoreorological Society, 85(11), 2004 pp. 1713–1726. journals.ametsoc.org/doi/pdf/10.1175/BAMS-85-11-1713.
[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.
[4] W. K. Härdle and B. L. Cabrera, "Calibrating CAT Bonds for Mexican Earthquakes," Journal of Risk and Insurance, 77(3), 2010 pp. 625–650. doi:10.1111/j.1539-6975.2010.01355.x.
[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.
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