Let's start with a practical example of a large corporation with three basic insurance risks: earthquake exposure to buildings, workers' compensation insurance, and general liability insurance.

Earthquake risk is defined as the potential for loss to buildings and property from a large earthquake as well as business interruption following the event. For our sample company, management has chosen to insure earthquake risk with a policy that covers $25 million in business and personal property with a 5 percent per occurrence retention. Earthquake sprinkler leakage is not covered.

For workers' compensation, management has chosen to buy a retention policy with a $1 million per occurrence retention, with no upper limitation, as it is a statutorily unlimited coverage.

The general liability coverage is represented by a $25 million per occurrence limit and a $250,000 per occurrence retention.

Now that we have the insurance coverage, we can assume the risk of loss for each of the three lines of coverage follows basic loss distributions as follows:

1. Earthquake (EQ). Loss frequency has a Poisson distribution with mean , and severity has a Pareto distribution with parameter ,

Exhibit 25.4 Mean Retained Losses by Line

Retention

Limit

Current

EQ

5%

$25,000,000

$2,500,501

WC

$1,000,000

Statutory

$3,163,992

GL

Portfolio

$250,000

$25,000,000

$1,597,373

$7,261,866

2. Workers' compensation (WC). Loss frequency has a Poisson distribution with mean λ = 50, and severity has a lognormal distribution with parameters μ = 10, σ = 1.5.

3. General liability (GL). Loss frequency has a Poisson distribution with mean λ = 10, and severity has a lognormal distribution with parameters μ = 12, σ= 1.0.

Notice that because the retentions are rather large, we are more focused on the tail portion of the loss distributions. We have decided not to use correlations for this example, to allow the reader to more easily follow and replicate the figures. In reality, correlations would be a key input into the model and would help determine the optimal risk transfer structures.

Exhibit 25.4 is a brief summary of the expected losses for the insurance policy and to the corporation below retentions and above insurance limits. The intent of this exhibit is to show the risk profile of the corporation using the assumed distributions listed earlier.

Note that there are many methods for fitting proper distributions and selecting the parameters to ensure good fits of historical data. Curve fitting is well beyond the scope of this chapter, and we will let the reader peruse other sources for details on loss distribution fitting.

With the knowledge of the current risk profile, we can now seek to optimize the portfolio and the insurance purchase by selecting different insurance options for our portfolio. By "options" we mean to choose different risk transfer contracts that can be used to modify the risk profile of the corporation. This can be done by taking a mathematical approach (using increments off of the current program) or by selecting common insurance contract terms known in the insurance marketplace. Exhibit 25.5 lists the options using the two different methods.

As one can see, there is almost an unlimited amount of options in the mathematical approach. The possibilities are only limited by your computing power.

Exhibit 25.5 Portfolio Options under the Mathematical Approach

Option #1

Option #2

Option #3

Option #4

Option #5

EQ

5% retention

5% retention

5% retention

5% retention

10% retention

$20M limit

$30M limit

$40M limit

$50M limit

$25M limit

WC

$250K retention

$500K retention

$2M retention

$3M retention

$4M retention

Statutory limit

Statutory limit

Statutory limit

Statutory limit

Statutory limit

GL

$500K retention

$1M retention

$2M retention

$3M retention

$500K retention

$25M limit

$25M limit

$25M limit

$25M limit

$30M limit

Exhibit 25.6 Portfolio Options under the Coverage Availability Approach

Option #1

Option #2

Option #3

Option #4

Option #5

EQ

5% retention

5% retention

5% retention

5% retention

10% retention

$20M limit

$50M limit

$75M limit

$100M limit

$25M limit

WC

$250K retention

$500K retention

$2M retention

$5M retention

$10M retention

Statutory limit

Statutory limit

Statutory limit

Statutory limit

Statutory limit

GL

$500K retention

$2M retention

$5M retention

$10M retention

$500K retention

$25M limit

$25M limit

$25M limit

$25M limit

$30M limit

It should also be noted that the selections for the different options are based on simple increments from the current values. These options may not be available in the insurance marketplace. This is somewhat intentional, as the goal is to find the optimal mathematical solution and then find the insurance option that gets closest to that optimal solution. The coverage availability approach is shown in Exhibit 25.6.

You will notice a subtle change in Exhibit 25.6, as indicated by the bolded options. The difference here is that we have selected options that can be knowingly purchased in the insurance marketplace. For more historical reasons than anything else, insurance risk transfer has been based around round numbers for retentions and limits. By using these options, we are guaranteeing (assuming the entity is insurable) viable options for the corporation.

Now the mathematicians can begin their number crunching. Using the options for Exhibit 25.5, we can determine the expected risk spend (expected losses to the corporation, which are the losses below the retention and above the limits) and the tail value at risk (TVaR) for each option, and then plot them on a graph. We have done this for each line described earlier and combined all the lines in a portfolio. We have assumed no correlations in the portfolio, to keep the mathematics and logic easier for the reader to follow.

To obtain Exhibits 25.7 to 25.10, we have run a simulation model using a Monte Carlo simulator. There are various software programs that provide the capability to simulate losses by using different distributions. Readers may wish to try the parameters within their own software to follow along.

Exhibit 25.10 provides the assumed insurance premiums for each of the mathematical options. In reality, we would work with insurance brokers to obtain insurance quotes for each of the options to arrive at a true market price for each option. The option exists to use an actuarial estimate of premium, which is not preferred. The reason an actuarial estimate of premium is not preferred is that the market does not always follow actuarial estimates and can often fall to other vagaries of market pricing (underwriting judgment, capital constraints, class restrictions, premium goals, etc.). Therefore, we recommend using different quotes provided by insurance brokers for each option. Given insurance premiums are presented in Exhibit 25.11.

Now with the options plotted (using our modeled losses, TVaR, and insurance premiums), we have created an efficient frontier and can determine the best option for a given level of risk. Ideally, we would select more than five options, and the options would be more complex. The beauty of the process is that it can be as simple

Exhibit 25.7 Earthquake Modeled Options

or complex as one desires. The process is flexible so as to handle different risk measures (not just TVaR) and can optimize different costs of risk (losses, insurance spend, internal costs, etc.).

It is also important to have an enterprise understanding of our risk appetite and tolerance. By having a formal statement of risk appetite, we can use that knowledge in the proper selection of the options in our efficient frontier.