Chapter 17-Modelling

Question 1

1. Describe why the solution to many actuarial problems involves building a model.

Answer 1

There are various approaches that can be taken to produce the solution to an actuarial or financial problem. Simple problems can have a simple solution that is arrived at by some straightforward mathematics, for example calculating the yield on a fixed-interest asset, or the present value of a series of known cashflows.
However, most problems that require actuarial skills involve taking a view on uncertain future events. It is possible to take a view on various parameters, such as future economic conditions, future mortality rates, or the amount of business that a provider might write in future, and produce a single answer that is appropriate in these best estimate conditions. If this is done then the communication of the solution to the client needs care, because of the uncertainties in the underlying assumptions.
In these circumstances the client is likely to wish to know the variability of the answer provided, should circumstances not be as estimated. To assess the effects of varying the assumptions used in producing the answer, it is normally necessary to use an actuarial model of future events.

Question 2

2. What is a model?

Answer 2

A model can be defined as ‘a cut-down, simplified version of reality that captures the essential features of a problem and aids understanding’. The final phrase in this definition recognises the importance of being able to communicate the results effectively.
Modelling requires a balance to be struck between realism (and hence complexity) and simplicity (for ease of application, verification and interpretation of results).

Question 3

3. List three potential approaches to obtaining a model and list five criteria that should be considered in deciding between these approaches.

Answer 3

When faced with an actuarial or financial problem, there are various approaches to modelling:
* a commercial modelling product could be purchased
* an existing model could be reused, possibly after modification
* a new model could be developed.
The merits of each of these approaches will depend on the following:
* the level of accuracy required
* the ‘in-house’ expertise available
* the number of times the model is to be used
* the desired flexibility of the model
* the cost of each option
* fitness for purposes

Question 4

4. State a type of model of which there are now many in existence. For what variables are there fewer models available?

Answer 4

There are now many stochastic asset models in existence, in both the public and private domains. There are fewer models available for other variables, such as mortality and voluntary discontinuance, but these are starting to be developed.

Question 5

5. Outline the key objective when constructing an actuarial model, including when this objective is particularly relevant.

Answer 5

Any model should be fit for the purpose for which it is being used. This is particularly relevant when a model is being purchased from an external provider or when an existing model is being reused for a different purpose, possibly after modification. Even with new purpose built models, the potential for model error remains - a model that replicates past results may still prove unreliable in projecting future results.

Question 6

6. Describe nine operational issues that need to be considered in relation to model design and construction.

Answer 6

Operational issues
* The model being used should be adequately documented.
* The workings of the model should be easy to appreciate and communicate. The results should be displayed clearly.
* The model should exhibit sensible joint behaviour of model variables.
* The outputs from the model should be capable of independent verification for reasonableness and should be communicable to those to whom advice will be given.
* The model, however, must not be overly complex so that either the results become difficult to interpret and communicate or the model becomes too long or expensive to run, unless this is required by the purpose of the model. It is important to avoid the impression that everything can be modelled.
* The model should be capable of development and refinement - nothing complex can be successfully designed and built in a single attempt.
* A range of methods of implementation should be available to facilitate testing, parameterisation and focus of results.
* The more frequently the cashflows are calculated the more reliable the output from the model, although there is a danger of spurious accuracy.
* The less frequently the cashflows are calculated the faster the model can be run and results obtained.

Question 7

7. Discuss the merits of deterministic and stochastic models.

Answer 7

Deterministic models
The advantages:
* A deterministic model is more readily explicable to a non-technical audience, since the concept of variables as probability distributions is not easy to understand.
* It is clearer what economic scenarios have been tested.
* The model is usually cheaper and easier to design and quicker to run.
The disadvantages:
* it requires thought as to the range of economic scenarios that should be tested.
* Also, users can get ‘blinded by science’ by complex models, assuming they must be working correctly, but without verifying or testing this.

Stochastic models
The advantages:
* A stochastic model tests a wider range of economic scenarios, including those that may not be thought of under a deterministic model
* It allows better for the random nature of variables and the correlations between them
* It is more useful for assessing the impact of financial guarantees and options or to allow for investment mismatching risks
* It is potentially more objective in incorporating allowance for uncertainty in risks
* It can provide more insights into the variability of the results as it provides a distribution of results
The disadvantages:
* The programming is more complex and the run time longer

Question 8

8. Explain, with an example relating to pricing an investment guarantee, how a combination of a stochastic and deterministic model could be used.

Answer 8

In many cases the problem can be solved by a combination of stochastic and deterministic modelling. Variables whose performance is unknown and where the risk associated with them is high might be modelled stochastically, while other variables can sensibly be modelled deterministically.
For example, a model for pricing an investment guarantee attached to a life insurance policy might use a stochastic investment model, but would be unlikely to model fluctuations in mortality rates other than deterministically. This is because it is normally self-evident which direction of movement in mortality rates would give rise to financial difficulties.

Question 9

9. Outline with examples how a model can be made ‘dynamic’.

Answer 9

In all cases the dynamism of the model is vital. Rules need to be determined as to how the various features would interact in different circumstances. For example, how life assurance bonus rates would vary with fixed-interest yields, how policy lapse rates would vary with economic conditions, or how unemployment rates would vary with economic conditions. These interactions are usually much more important than the type of model.
Considerable actuarial judgement may be required in choosing and using the model and in setting the parameters and interactions between the different features.

Question 10

10. Give an example of a key use of a model in an insurance company.

Answer 10

A model could be developed to determine a premium or charging structure for a new or existing product that will meet an insurance company’s profit requirement.

Question 11

11. Explain the use of ‘model points’ in a model and how suitable model points can be obtained for use in product pricing.

Answer 11

A model point is a set of data representing a single policy or a group of policies. It captures the most important characteristics of the policies that it represents.
The insurer may have very many policies and it may not be practical to run all the individual policies through the model.
Instead these policies are arranged into groups of those that would give very similar results. Each group is represented by a single model point, thus reducing the number of data points that need to be run through the model.

Question 12

12. Outline what ‘running a pricing model’ with model points actually entails.

Answer 12

For each model point, cashflows would be projected, allowing for reserving and solvency margin requirements, on the basis of a set of base values for the parameters in the model. The net projected cashflows will then be discounted at a rate of interest, the risk discount rate.

Question 13

13. Outline how the risk discount rate for the model can be determined.

Answer 13

This could be a rate that allows for:
* the return required by the company; and
* the level of statistical risk attaching to the cashflows under the particular contract, ie their variation about the mean as represented by the cashflows themselves.

The level of statistical risk could be assessed:
* in some situations, analytically - by considering the variances of the individual parameter values used
* by using sensitivity analysis, as described below, with deterministically assessed variations in the parameter values
* by using stochastic models for some or all of the parameter values and simulation
* by comparison with any available market data.
Alternatively a stochastic discount rate could be used.
In theory, a separate risk discount rate should be applied to each separate component of the cashflows, as the statistical risk associated with each component will be different. In practice a single risk discount rate is commonly used, bearing in mind the ‘average’ risk of the product.

Question 14

14. In a model used to set premiums or charges, how will the key objective of the model be expressed?

Answer 14

The premium or charges for the model point can then be set so as to produce the profit required by the company.

Question 15

15. State five things that might need to be reconsidered in the process of pricing a product, as a result of the premiums or charges produced have been considered for marketability.

Answer 15

The premiums, or charges, produced need to be considered for marketability. This might lead to a reconsideration of:
* the design of the product, so as either to remove features that increase the risk within the net cashflows, or to include features that will differentiate the product from those of competing companies
* the distribution channel to be used, if that would permit either a revision of the assumptions to be used in the model, or a higher premium or charges to be used without loss of marketability
* the company’s profit requirement
* the size of the market
* whether to proceed with marketing the product.

Question 16

16. Discuss how and why a model of the business of the whole company will be used in the process of pricing a new product.

Answer 16

The net cashflows in respect of the model points, appropriately scaled up for the expected new business under the product, will be incorporated into a model of the business of the whole company. It is possible for the desired level of profitability to be reached in aggregate, without requiring every single model point to be profitable in its own right. If certain model points are unprofitable, the aggregate profitability of the business is then exposed to changes in mix and volume of the contracts sold.
The actuary can assess the impact on capital management of writing the product, by observing the modelled amount and timing of cashflows. If capital is a problem, this may lead to a reconsideration of the design of the product to reduce or amend the timing of its financing requirement.

Question 17

17. State what has to be done once appropriate prices have been determined for the model points that were chosen for the pricing process.

Answer 17

Once acceptable premiums or charges have been determined for the model points, premiums or charges for all contract variations can be determined.

Question 18

18. Outline how the capital requirement and return on capital would be assessed for a new product.

Answer 18

The net cashflows for the model points described in the section on pricing above can be grossed up for the expected new business and used to assess the amount of capital that will be required to write the product either on a regulatory or an economic basis.
Any one-off development costs can be added, to the extent that they are not amortised and included in the cashflows used. This gives the total capital requirement and can be compared with the profits expected to emerge from the product to determine the expected return on that capital.

Question 19

19. For a benefit scheme, the equivalent to determining the price for a product is to set the future financing strategy. Describe how a model would be used to set a future financing strategy.

Answer 19

For a benefit scheme, the equivalent to determining the price for a product is setting the future financing strategy, and similar modelling techniques can be used.
The existing membership can be divided into categories and represented by a set of model points. Similar potential new members can be represented, perhaps by a single model point at the average entry age and salary.
A potential financing strategy is determined, in terms of both the amount and timing of future contributions. The cashflows from the existing assets and future contributions can be modelled, as can the liability cashflows, taking all the possible decrements into account.
Unlike an insurance company, a benefit scheme can show a deficit at a point in time (ie the value of accumulated assets does not exceed the value of accrued liabilities), provided that there is a sponsor with a good enough covenant to make good the shortfall. However, the scheme does need to be solvent to the extent that it has sufficient assets to meet benefit outgo as it falls due. A well-designed model will check this feature as well as determining the discounted value of asset and liability cashflows.
Considerations such as the choice of risk discount rate, and the need to test sensitivities to changes in conditions are all similar to those in product pricing.

Question 20

20. Describe the use of models for risk management.

Answer 20

Cashflow models are used in risk management to determine the amount of capital that it is necessary to hold to support the risks retained by a financial institution.
As well as the full corporate model to assess capital requirements, models of specific risks can be used to determine the extent of a risk event that will occur at a given probability, even if a full stochastic model is too slow, too complex, or otherwise not used.
For example, a company that is targeting being able to withstand a 0.1% probability of ruin needs to know what equity market fall to test in a deterministic scenario.
A standard equity market stochastic model can be used and calibrated to historical performance of the market being considered. By running the model several thousand times and ranking the results, the equity fall that gives the one in a thousand worst result can be found.

Question 21

21. Explain why the use of models is limited in relation to valuing the liabilities of a life insurance company or pension scheme.

Answer 21

The normal procedure for determining life assurance or pension scheme liabilities is to value the benefits for each policy or scheme member individually. In many territories this may be required by legislation or regulation.
Consequently for published results there is little scope for using model points. However, before finalising a published basis, many ‘what if’ questions might be asked. These could be answered by running a model of the business. For smaller schemes or sections of a company’s business it might be just as quick to run the whole data file to answer the question and eliminate the model risk, given the current speed of computers.

Question 22

22. Outline how a stochastic model can be used when determining provisions for existing commitments.

Answer 22

As part of assessing a realistic provision it is necessary to consider the effect of changes in economic scenarios. For example, using a stochastic model of possible asset movements, a provision that would be adequate in all but a small proportion of scenarios can be determined.

Question 23

23. Outline the key modelling issues in relation to valuing options and guarantees.

Answer 23

In most cases the options and guarantees that give a provider of benefits on future financial events cause for concern are those that are dependent on future investment returns, or an investment value (yield or capital value) at some future point in time. Because of the uncertainty, a stochastic investment model should be used to assess the provisions necessary for such guarantees.
If future returns exceed a certain level, or if a value or index is above (or below) a fixed value at some future point, there will be no cost to the company. But if they are below (or above) that level, there will be a cost, which increases as returns reduce. Hence a range of future investment scenarios should be tested.

Question 24

24. Describe how the variability of modelling results can be illustrated.

Answer 24

The results from the models depend on the model itself and the values assigned to the parameters in the model. Models should not be treated as black boxes, the output of which is assumed to be correct.
The use of a stochastic model goes some way to illustrating the potential variability of the experience, but the results that it produces are still dependent on the accuracy of the model and its parameter values. In the case of a deterministic model, the potential uncertainty of the results is greater, because fewer scenarios are tested.
The re-running of a model (deterministic or stochastic) with different, but feasible, parameter values will produce alternative results and hence help to illustrate the potential deviations. The re-running with a series of different sets of parameter values, perhaps chosen from a probability distribution for such values, will help to illustrate the likely range in which actual experience may lie, perhaps as far as creating a probability distribution for this experience.
For example, consideration of the effect of a change in the membership profile of a funded pension scheme may be needed to illustrate the extent of potential variability in future contributions if the model used is based on a stable membership profile.

Question 25

25. Outline two types of errors that can arise in a model and how they can be investigated.

Answer 25

Model error means that a model is developed that is not appropriate for the financial products, schemes, contracts or transactions being modelled. Model error can be assessed by checking the goodeness of fit of the model output against actual data, taking account of expected changes in experience into the future.
Parameter error means incorrectly setting parameter values used when the model is run. It can involve individual parameters and/or correlation between parameters. Parameter error can be assessed by carrying out sensitivity analysis to consider the effect of varying each of the parameter values. When such sensitivity testing is carried out allowance must be made for any correlation between parameters.

Question 26

26. Explain the limitations of actuarial models and suggest ways in which these limitations can be minimised.

Answer 26

Limitations of models
* The results of a modelling exercise are only as good as the underlying model, ie prone to model error.
* The results of the exercise will depend upon the data used, ie there is a risk of data error if proper records have not been maintained.
* The results depend upon the suitability of the assumptions used, ie prone to parameter error.
* The level and timing of cashflows is uncertain and so actual experience will differ from the model result (ie prone to random error).

Minimising the limitations
* Model error: consider lots of potential models; employ suitable expertise to identify the most appropriate model.
* Data error: ensure data is regularly updated and perform data checks.
* Parameter error: carry out sensitivity tests to identify the key assumptions and pay careful attention to the setting of these important assumptions
* Random error: a stochastic model may be deemed appropriate in order to illustrate a wide range of potential outcomes,

Question 27

27. Describe how the variability of modelling results can be used to help in:
    
    • pricing
    • assessing the return on capital or profitability of business.

Answer 27

In the case of a model used for pricing, the results from the sensitivity analysis will help to assess the margins that need to be incorporated into the parameter values.
In the case of models used to assess return on capital and profitability of existing business, the results will enable the actuary to quantify the effect of departures from the chosen parameter values when presenting the results of the model to the company.

Question 28

28. Outline four ways of allowing for statistical risk in a model.

Answer 28

The statistical risk associated with the parameter values can be allowed for through the risk element of the risk discount rate.
An alternative would be to use a predetermined discount rate and then assess the effect on the results of the models of statistical risk.
Where a probability distribution can be assigned to a parameter, it may be possible to derive the variance of the profit or return on capital analytically.
More generally, a sensitivity analysis, as described above, can be carried out. Whichever of these two is used, they will again help in assessing margins or in quantifying the effect of departures from the chosen parameter values when presenting the results of the model.