Example Models

@RISK Models

Minimum edition: @RISK 6.0 Industrial. This model assumes that an insurance company is offering a local organization an insurance policy that will guard the organization against large heating oil costs from excessively cold winters. There are three sources of uncertainty: the price of heating oil, the weather, and the amount of heating oil required. The first two of these are modeled with @RISK's Time Series Fit tool, based on real historical data.

This model uses @RISK to run a discrete-event simulation of insurance claims through time. It assumes that a company starts with an initial number of customers and an initial amount of cash. There are three types of events, and the times between each are assumed to be exponentially distributed. A type 1 event is when a customer makes a claim for a random amount, a type 2 event is when a current customer leaves the company, and a type 3 event is when a new customer joins the company. All of the company's customers pay a premium at a given daily rate.

This model enables an insurance company to analyze the possibility of being reinsured. Without reinsurance, the company pays all claims, net of deductibles, for its policy holders. With reinsurance, it pays a fixed premium to another insurance company, the reinsurer. There is a reinsurance deductible. If the company's liability for all claims, net of deductibles, is less than this deductible, the company is liable for all of it. However, if this liability is greater than the reinsurance deductible, the company is liable only for the deductible; the reinsurer pays the rest.

@RISK’s RiskCompound function uses two distributions to create a single new input distribution, streamlining insurance models that must account for frequency and severity of claims. This function takes two arguments, each typically an @RISK distribution function. In a given iteration, the value of the first argument specifies the number of samples which will be drawn from the distribution entered in the second argument. Those samples from the second distribution are then summed to give the value returned by the RiskCompound function.

For example, the function:

RiskCompound(RiskPoisson(5),RiskLognorm(10000,10000))

would be used where the frequency or number of claims is described by RiskPoisson(5) and the severity of each claim is given by RiskLognorm(10000,10000). Here the sample value returned by RiskCompound is the total claim amount for the iteration, as given by a number claims sampled from RiskPoisson(5), each with an amount sampled from RiskLognorm(10000,10000).

RiskCompound can eliminate hundreds or thousands of distribution functions from existing @RISK models by encapsulating them in a single function. The result is models that are much simpler to use, and run much faster. Two examples illustrate claims modeling using RiskCompound.

This model contains a portfolio of potential claims of different types. Each claim has different parameters for the distributions of frequency, severity, and duration. The model illustrates a powerful feature of the RISKCOMPOUND function: the argument that corresponds to the severity can be a reference to a cell containing a formula, rather than just a single distribution function.

This example shows how you might model the uncertainty involved in payment of insurance claims. To model this properly, you must account for the uncertainty in both the total number of claims and the dollar amounts of each claim made. This is done using the RiskCompound function.

Suppose that the company is required by law to have enough money on hand to pay all the claims with the probability of 95%, and that it can only set aside \$2000 for the purposes of this particular insurance product. On the other hand, a simulation of the model shows that the 95th percentile of the Total Payment Amount is around \$2700. Assume further that the company can purchase from a larger company an insurance policy against the number of claims being in the top decile. The policy under consideration specifies that if the number of claims falls within the top decile, the larger company will satisfy all the claims. The smaller company can model the situation with the policy in place by using Stress Analysis to stress the distribution for total number of claims from the 0th to 90th percentile. With the modified distribution the 95th percentile of the Total Payment Amount is reduced to around \$1650. If the policy costs up to \$350, the smaller company can purchase it and keep \$1650 on hand to comply with the law.

Would the larger company be willing to sell the policy for under \$350? There is a 10% probability that it will be required to make payments under the policy. The payments can be analyzed using the same model and stressing the distribution for total number of claims from the 90th to 100th percentile. This analysis shows the mean payment to be around \$2800. Since there is only a 10% probability that claims will need to be satisfied, the mean cost to the larger company is around \$280. Hence, it does not seem unreasonable for the larger company to sell the policy for \$350.

In many circumstances one wishes to calculate the aggregate impact of many possible yes/no type events. For example, it is often important to answer questions such as "What is the loss amount that will not be exceeded in 95% of cases?" Simulation is usually required to answer such questions. In this model, the "yes/no" events are modeled using Binomial distributions. The results profile shows a multi-peaked distribution, which is typical when there are discrete-type inputs. It can be seen that a provision level of around \$700,000 is necessary to cover 95% of the cases.

Possible generalizations to this model that could be made (and which are explored in more detail on Palisade training courses) include:

a) Assessing the impact of changing the loss resulting from each event into a distribution, rather than assuming a fixed amount.

b) Assessing the impact if mitigating actions could be developed for certain events, so that, e.g., the amount of loss were reduced if these events occur (or the probabilities of events are reduced or both).

c) Creating dependencies or correlations between the occurrence (and/or magnitude) of some of the events.

d) Replacing the Binomial distribution with a Poisson distribution so that each event could occur more than once per period.

This example models different types of insurance claims from different lines of business and sums them in order to calculate an estimated total claims paid out for the next year. It incorporates @RISK’s distribution fitting to define distribution functions for claim amounts, and illustrates the use of correlations to describe relationships between different types of claims. The RiskCompound function is used to combine frequency and severity of claims, simplifying the model.

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