@RISK is widely used in insurance and reinsurance for premium pricing and loss reserves modeling. A 2006 survey identified @RISK as the third most widely-used software by actuaries, after Microsoft Office and in-house actuarial tools. Download and install a free trial version of @RISK to view the models in full.

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Insurance Claims with the RiskCompound Function
@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.

Example model: Claims.xls
Example model: RiskCompound.xls

Insurance Claims with RiskCompound Cell Referencing
The RiskCompound function allows for the sampling of frequency-severity distributions. This is often required in insurance modelling, as well as in some operations management situations. For example, to determine the total insurance claims payout, one must account for the uncertainty in both the total number of claims (frequency) and the dollar amount of each claim made (severity).

A powerful feature of the function is that the argument that corresponds to the severity may be a reference to a cell containing a formula (rather than just a single distribution function).

For example, one could use the function in the form RiskCompound(RiskPoisson(5), A10).  The Poisson distribution would describe the frequency (occurrence) of events (e.g. an individual sample may determine that three events occur), and cell A10 would contain a formula that is separately evaluated for each of these three events (before returning the sum of these three as the sampled value of RiskCompound).

A simple example could be A10 = RiskLognorm(10000,1000)/(1.1^RiskWeibull(2,1)), with the Weibull distribution representing the time to settlement of an insurance claim, which is used to discount the basic claim value sampled from the Lognormal distribution of severity. For example, once a claim is filed for a nominal amount, the actual payment may be delayed due to court actions or disputes, which may reduce the cost of the claim to the insurer.

In more complex models, one may need to remember that the entered formula for the severity needs to be less than 256 characters (the use of a user-defined function in the formula can often help to achieve this). Also, it is important that all @RISK distributions that are required in the severity sample need to be entered in the cell’s formula (i.e. in the formula for cell A10 in this example), and not referenced in other cells.

In this model, we have a portfolio of potential claims of different types. Each claim has different parameters for the distributions of frequency, severity, and duration.

Example model: RiskCompoundCellReferencing.xlsx

Stress Testing Insurance Claims
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.

Example model: ClaimsStress.xls

Event and Operational Risks
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.

Example model: EventandOperationalRisks.xls

Claims Payouts with Correlations,
Fitting, and RiskCompound
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.

Example model: ClaimsPayout.xls