Chapter 18: Modelling

Question 1

What does the modelling approach depend on?

Answer 1

Approaches to solving actuarial or financial problems

• Approach taken will be driven by purpose of the exercise and nature of the problem
• E.g. far more detailed approach will be required to determine provisions for life insurer’s statutory returns than provide interim update to the provisioning level for internal management purposes during year
• Simple problems can have simple solution that is arrived at by straightforward mathematics

Question 2

Why do we need to develop a model?

Answer 2

The need to develop a model

• Most problems that require actuarial skills involve taking view on uncertain future events
• Possible to take view on various parameters and produce single answer that is appropriate in these best estimate conditions
• If this is done, then the communication of the solution to client needs care as there are uncertainties in underlying assumptions
• I.e. client likely to wish to know the variability of answer provided, should circumstances not be as estimated
• To assess effects of varying assumptions used in producing the answer – necessary to use actuarial model of future events
• The variability of answer might be assessed by carrying out:

 Sensitivity analysis – varying individual assumptions and assessing the impact on results

 Scenario testing – changing many assumptions in combo e.g. to look at the many assumptions that may change if economy were to move into a recession

Question 3

What is a model?

Answer 3

What is a model?

• Cut-down, simplified version of reality that captures the essential features of problem and aids understanding
• Important to be able to communicate results effectively
• Modelling requires balance between realism and simplicity for ease of application, verification and interpretation of results

Question 4

When finding a model what are the various approaches to modelling and what will the merits of each of these approaches depend on?

Answer 4

Finding a model
- Various approaches to modelling when faced with actuarial/financial problem:

 Commercial modelling product could be purchased

 An existing model could be reused – possibly after modification

 A new model could be developed

• Merits of each of these approaches will depend on:

 The level of accuracy required

 The ‘in-house’ expertise available

 The number of times the model is to be used

 The desired flexibility of model

 The cost of each option

Question 5

What are the two types of existing models ?

Answer 5

Existing models

• Existing deterministic or stochastic models can be used

Stochastic models

• Many stochastic asset models in existence – in public and private domains
• Fewer models available for other variables, such as mortality and voluntary discontinuance, but these are starting to be developed

Question 6

What are the key objectives of construction of an actuarial model?

Answer 6

Key objective

• Any model should be fit for purpose for which it is being used
• This is very relevant when model is being purchased from external provider or when existing model is being reused for different purpose (after modification)
• Even with new purpose-built models there is risk of model error – model that replicate past results may still be unreliable in projecting future results

Question 7

What are the operational issues of construction of an actuarial model?

Answer 7

Operational issues
- Model being used should be adequately documented

 So that key assumption and approximations made are understood and so it can be run by other members of staff and improvements introduced over time

• The workings of model should by easy to appreciate and communicate – results should be displayed clearly
• Model should exhibit sensible joint behaviour of model variables

 So model needs to make allowance for variables that are linked to each other: relationship between them need to have been modelled in appropriate way

 Assumption should also be consistent – assumed rate of investment return should be consistent with assumed rate of inflation

• Outputs from model should be capable of independent verification for reasonableness and should be communicable (to whose getting advice)
• Model must not be overly complex = so that results become difficult to interpret and communicate or too expensive and long to run – unless necessary
• Model should be capable of development and refinement
• Model should be capable of being implemented in a range of different ways – to facilitate testing, parameterisation and focus of results
• More frequently the cashflows are calculated, the more reliable the output from the model – but danger of spurious accuracy
• Less frequently the cashflows are calculated, the faster the model can be run and results obtained
• Argument for having shorter time period between cashflows in early years – given that starting points for model should be known with fair degree of certainty and thus early result are most meaningful
• Time period chosen so that it captures key areas of experience
• Decision must be made about how many years into the future the results will be projected

Question 8

How are model points used in the construction of an actuarial model?

Answer 8

The use of model points

• Underlying business being modelled will comprise very wide range of different policies
• These will need to be brought together into manageable number of homogeneous groups
• Groupings need to be made in way that each policy in a group is expected to produce similar results when the model is run
• It is then sufficient enough for a representative single policy in each group to be run through the model and for result to be scaled up to give total set of policies in the group
• Model point – the representative single policy in a group and set of such model points can be used to represent whole of underlying business
• Model point needs to capture most important characteristics of the group of policies it represents

Question 9

How are model points chosen in the construction of an actuarial model?

Answer 9

Choosing model points

• Set of model points will be chosen to represent expected new business under the product
• For an existing product – profile of existing business, modified to allow for expected changes in future, used to obtain model points
• For new product – profile of any similar existing product combined with advice from company’s marketing department would be used
• Number of model points will depend on number of model points that can be handled by the model
• Number of model points used will depend on:

 Computing power available

 Time constraints

 Heterogeneity of the class

 The sensitivity of the results to different choices of model points

 Purpose of the exercise

Question 10

What discount rate is used in the construction of an actuarial model?

Answer 10

Rate for discounting cashflows

• For each model point – cashflows projected, allowing for reserving and solvency margin requirements, on the basis of set of base values for parameters in model
• Net projected cashflows will then be discounted at rate of interest = risk discount rate
• Could be rate that allows for:

 Return required by company

 Level of statistical risk attaching to cashflows under particular contract i.e. variation about the mean as represented by the cashflows

• Statistical risk is intended to encompass all types of risk – comprises the model risk, parameter risk and random fluctuation risk
• The level of statistical risk could be assessed:

 Analytically – by considering the variances of individual parameter values used

 By using sensitivity analysis – with deterministically assessed variations in parameter values

 By using stochastic models for some, or all, of the parameter values and simulation

 By comparison with any available market data

• Stochastic modelling approach achieved by:

 varying the important parameter values in model according to their assumed probability function

 and recalculating the rate of return for each new scenario

• by running many simulations, a good idea of variance of the rate of return can be found
• OR, stochastic discount rate could be used
• IN THEORY, separate risk discount rate should be applied to each separate component of cashflows – as statistical risk of each component will differ
• IN PRACTICE, single risk discount rate is commonly used, bearing in mind the average risk of the product
• This keeps it simple
• And difficult (time and data requirements) to analyse accurately the variability of different cashflow components

Question 11

A deterministic or stochastic model?

Answer 11

A deterministic or stochastic model?

• Deterministic model – parameter values are fixed at the outset of running the model and result is a single outcome
• Sensitivity analysis and scenario testing can then be carried out to assess variability of results
• Stochastic model – estimates at least one of the parameters by assigning it a probability distribution
• Run a large number of times and values of stochastic parameters selected from distributions on each run
• Outcome is a range of values giving understanding of likely distribution of outcomes

Merits of a deterministic model

Advantages:

• More readily explicable to a non-technical audience
• Clearer what economic scenarios have been tested
• Cheaper and easier to design
• Quicker to run

Disadvantages:

• Requires thought as to range of economic scenarios that should be tested

 Limited economic scenarios testes = danger that certain scenarios are not identified

• Users can get blinded by science by complex models – assuming they must work but without verifying or testing this

Merits of stochastic model

• Tests wider range of economic scenarios
• Programming is more complex and run time longer but benefit is quality of the result
• Depends on parameters used in any standard investment model
• Actuary must decide if increase amount of info provided by model justifies significant additional computations
• Other NB considerations:

 degree of spurious accuracy introduced,

 increased difficulty in interpreting and communicating results

 and questionable accuracy of distribution functions that are replacing the deterministic values

• stochastic models are important in assessing impact of financial guarantees or to allow for investment mismatching risks – because good at allowing for uncertainty involved

Question 12

How can a combination of deterministic and stochastic modelling be used?

Answer 12

A combination of deterministic and stochastic modelling

• In many cases problem solved by combining the two
• Variables whose performance is unknown and risk associated with them is high might be modelled stochastically
• While other variables can be modelled deterministically
• Stochastic approach usually limited to economic assumptions
• Demographic assumptions modelled deterministically

Question 13

Discuss the dynamism in the model.

Answer 13

Dynamism of the model

• In all cases dynamism of model is vital
• Means that asset and liability parts of the model and all assumptions are programmed to interact as the would in real life
• E.g. inflation and interest rates are consistent
• Rules need to be determined as to how various features would interact in different circumstances
• Actuarial judgement may be required in choosing and using model and in setting parameters and interactions between different features
• Interactions important when assets and liabilities are being modelled together

Question 14

How is a deterministic model developed?

Answer 14

Developing a deterministic model

• Specify the purpose of investigation
• Collect, group and modify data
• Choose form of the model – identifying its parameters or variables
• Ascribe values to parameters using past experience and appropriate estimation techniques
• Construct model based on expected cashflows
• Test model in order to identify any build errors and correct if necessary
• Check that goodness of fit is acceptable – attempt to fit another model if first choice doesn’t fit well
• Run model using estimates of values of variables in the future
• Run the model several times to assess the sensitivity of the results to different parameter values
• Model might be run under different scenarios to test the robustness of the results to many parameters changing at the same time

Question 15

How is a stochastic model developed?

Answer 15

Developing a stochastic model

• Stochastic modelling would involve same process, with additional or alternative steps:
• Choose suitable density function for each of the variables to be modelled
• Specify correlation between variables
• Run the model many times – each time using random sample from chosen density function
• Produce summary of results that shows distribution of the modelled results after many simulations

Question 16

What should the sensitivity analysis look at?

Answer 16

Reliability of the results

Understanding potential variability of experience

Model error

Parameter error

Alternative ways of allowing for risk

Question 17

What should the sensitivity analysis look at? - Reliability of results and model error

Answer 17

Reliability of the results

• Results from model depend on model itself and values assigned to the parameters
• Output should not be assumed correct

Model error

• If model developed is not appropriate for the financial products, schemes, contracts or transactions being modelled – possibility of model error
• Checks of goodness of fit needed to assess the suitability of the model

Question 18

What should the sensitivity analysis look at? - Understanding the potential variability of experience

Answer 18

Understanding potential variability of experience

• Stochastic model tries illustrating the potential variability of the experience – but results that it produces are dependent on accuracy of model and its parameter values
• For deterministic model potential for uncertainty is greater since fewer scenarios are tested
• Re-running with different but feasible parameter values will produce alternative results and help illustrate potential deviations
• Re-running with series of different sets of parameter values (chosen from PDF for such values) will help illustrate range in which actual experience may lie or even creating probability distribution for this experience
• Extent to which this can be achieved will depend upon:

 confidence with which probabilities can be assigned to different sets of parameter values used

 time it takes to run the model

 the associated costs

• running the model on different sets of assumptions is useful in highlighting errors in a model

Question 19

What should the sensitivity analysis look at? - Parameter error

Answer 19

Parameter error

• Effect of mis-estimation of parameter values can be checked by carrying out a sensitivity analysis
• Involves assessing effect on output of model of varying each of the parameter values
• Any correlation between different parameters should be allowed for
• When model used for pricing, results from sensitivity analysis will help assess the margins that need to be incorporated into parameter values
• E.g. consider steps actuary takes if they identify that product is unduly sensitive to withdrawal rates and mortality rates
• If product profitability is sensitive to any factor the results may indicate need to redesign the product OR increase margins in assumptions
• If product is too sensitive to increasing withdrawal rates, then reduction in surrender values should be considered
• If too sensitive to mortality, then reinsurance programme could be revised
• OR, additional margins could be included in pricing basis to reflect increased risk
• When models used to assess return on capital and profitability of existing business – results enable actuary to quantify effect of departures from chosen parameter values when presenting results

Question 20

What should the sensitivity analysis look at? - alternative ways of allowing for risk

Answer 20

Alternative ways of allowing for risk

• Statistical risk associated with the parameter values – can be allowed through the risk element of the risk discount rate
• Alternative would be to use predetermined discount rate and then assess the effect on results of the model of statistical risk
• Predetermined discount rates – means discount rate arbitrarily set by shareholders (perhaps based on return on risk-free assets)
• We discount cashflows at risk-free rate and used more pessimistic parameter values instead of best estimate parameter values
• Degree of pessimism should correspond to risk margin introduce in first approach
• Where a PDF can be assigned to a parameter – may be possible to derive variance of profit or return on capital analytically
• BUT, difficult to objectively assign a PDF to value of any parameter
• A sensitivity test can be carried out
• They will both help in assessing margins or in quantifying the effect of departures from chosen parameter values
• Using sensitivity analysis is a pragmatic, transparent and informative way of understanding the parameter risk without difficulty of deciding upon some arbitrary PDF to represent the uncertainty

Question 21

What are some of the common applications of actuarial models?

Answer 21

Using models in pricing

Setting future financial strategies

Risk management models

Valuing liabilities

Valuing options and guarantees

Question 22

What are some of the common applications of actuarial models? - Using models in pricing

Answer 22

Using models in pricing

• Model could be developed to determine a premium or charging structure for new or existing product that will meet insurance company’s profit requirement
• A model will be needed to set the premium
• For pricing purposes model points are used to represent profile of business expected under a product
• Company will use model to monitor the appropriateness of premium rates at regular intervals, to:

 Check that business is profitable

 Check that rates are appropriate for all groups

 Ensure the rates remain competitive

Question 23

Using models in pricing - Meeting profit requirements

Answer 23

Meeting profit requirements

• Premium or charges for model points can be set so as to produce profit required by company
• Difficult to achieve for all model points – in particular for small policies
• Because element of expenses incurred in relation to each policy will be fixed – allowing for fixed costs makes it difficult to achieve adequate profitability
• May instead focus on total profitability resulting from group of model points
• Group chosen to represent the expected new business mix and volume for product – there will be a degree of cross-subsidy within the group of model points
• Introducing cross-subsidies generates risk that actual business mix is not as assumed (insurer must be aware)

Question 24

Using models in pricing - Competitive premiums

Answer 24

Competitive premiums

• Premium or charges need to be considered for marketability
• Might lead to reconsideration of:

 Design of product – either remove features that increase the risks within net cashflows or include features that differentiate products from competitors

 Distribution channel used – revision of assumptions used in model or higher premium or charges used without loss of marketability

 Company’s profit requirements

 Size of the market

 Whether to proceed with marketing the product

Question 25

Using models in pricing -

Business strategy

Answer 25

Business strategy

• Net cashflows in respect of the model points (appropriately scaled up for expected new business) will be incorporate into a model of the business of the whole company
• Scaled up means multiplies up in order than individual model point results represent the expected volume of new business
• Possible for desired level of profitability to be reached in aggregate without requiring every model point to be profitable
• If certain model points are unprofitable the aggregate profitability if business is exposed to changes in mix and volume of contracts sold
• Actuary can assess the impact on capital management of writing the product by observing modelled amount and timing of cashflows
• If capital is a problem – can lead to reconsideration of design of product to reduce or amend timing of its financing requirements
• Once premium or charges determined for model points – then premiums or charges for all contract variations can be determined
• May be achieved by interpolating between premium for various model points
• As long as model points satisfy profit criterion then interpolated values should also satisfy the profit criterion

Question 26

Using models in pricing - Assessing the capital requirements and the return on capital

Answer 26

Assessing the capital
requirements and the return on capital

• Net cashflows for model points can be grossed up for expected new business and used to assess amount of capital required to write product
• Any one-off development costs can be added –not amortised (spread over a period) and included in cashflows used
• Gives total capital requirement and compared with profits expected from product to determine the expected return on that capital
• Design of product or profit requirement may need to be amended to ensure require return on capital is achieved

Question 27

What are some of the common applications of actuarial models? - Setting future financial strategies

Answer 27

Setting future financial strategies

• For benefit scheme the equivalent to determining price is setting the future financing strategy
• Existing membership can be divided into categories and represented by set of model points
• Potential new members can be represented by single model point at average entry age and salary
• E.g. existing members divided into active, deferred and current pensioners
• Potential financing strategy is determined – in terms of amount and timing future contributions
• Benefit scheme can show deficit a point in time provided that there is sponsor with good enough covenant to make good the shortfall
• Scheme does need to be solvent to extent that has sufficient assets to meet benefit outgo as it falls due
• Well-designed model will check this feature and determine the discounted value of asset and liability cashflows
• Choice of risk discount rate and need to test sensitivities to changes in conditions – similar to those in product pricing

Question 28

What are some of the common applications of actuarial models? - Risk management models

Answer 28

Risk management models

• Cashflow models used in risk management to determine amount of capital that it is necessary to hold to support risks retained by financial institution
• Models of specific risks can be used to determine extent of risk event that will occur at given probability – even if stochastic model too slow, complex or not used
• Full corporate model to assess capital requirements can be used
• Standard equity market stochastic model can be used – calibrated to historical performance of market considered
• By running model many times and ranking results – the equity fall that gives one in many worst results can be found

Question 29

What are some of the common applications of actuarial models? - Valuing liabilities

Answer 29

Valuing liabilities

• Normal procedure for determining life assurance or pension scheme liabilities = value benefits for each policy or scheme member individually
• May be required by legislation or regulation
• Regulators was assurance that company has sufficient provisions to meet claims of all actual policyholders and risk covered
• For published results there is little scope for using model points
• Many ‘what if’ questions asked (answered by running a model of the business) before finalising published basis:

 Impact on ability to meet regulatory requirement of fall in investment returns

 Impact on finances of changing mortality experience

 Implications of failure of reinsurer or of insurer

• For small schemes or sections of company’s business – might be as quick to run whole data file to answer the question and eliminate model risk
• Necessary to consider effect of changes in economic scenarios when assessing a realistic provision

Question 30

What are some of the common applications of actuarial models? - Valuing options and guarantees

Answer 30

Valuing options and guarantees

• Options and guarantees that give provider cause for concern are those that are dependent on future investment returns or investment value at some future point in time
• Stochastic model used to assess the provision necessary – because of uncertainty
• Stochastic model can provide info on likelihood of option or guarantee applying together with the associated costs
• If future returns exceed certain level or value or index is above a fixed value – there will be no cost to the company
• BUT if future returns are below that level – cost increases as returns reduce
• If value or index is below fixed value at the future point – there could be a cost
• Hence a range of future investment scenarios should be tested

Other options

• Insurance products can also include options that are not dependent on investment outcomes but relate to death or sickness cover

 E.g. product might include option to take out new term assurance without providing further evidence of health

 When modelling such products the potential cashflows if option exercised and take-up rate of option both need to be allowed for