Introduction to modelling

Data functions The main uses of data for an actuary are:

• setting premiums and/or contribution rates
• calculating reserves required to meet future liabilities
• preparing statutory returns to demonstrate solvency
• assisting with risk management decisions. e.g. determining appropriate investment, reinsurance and/or bonus distribution strategies
• carrying out experience analysis
  • to assess the suitability of past assumptions
  • to identify unprofitable tranches of business sold in the past

Data qualities

• Poor quality and/or quantity of available data can have a significant adverse effect on the quality of the advice given.
• If data is poor quality, then results may be wrong and, if poor quantity of data, then results will have high degree of uncertainty.
• Problems of data quality and quantity can be a result of:
  • poor management control of data recording and verification
  • poor design of data systems
• If possible, data used for all kinds of purposes should be controlled through one integrated data system. Data used for different purposes will be consistent, reducing possibility of errors.
• However, in some cases, a large quantity of high-quality past data may simply not be readily available.
• New company, new product, new market, new distribution channel, significant changes to terms and conditions of the product, regulatory changes (ban on gender discrimination (can use proxy instead)).

Types Sources Possible issues

• Proposal form:
  • In both life and non-life insurance, the data gathered directly from the policyholder on the proposal form will be crucial.
  • Questions should be clear and unambiguous. Where possible, quantitative (rather than qualitative) data should be requested.
  • Aside from general administrative data (e.g. name, address, bank details), only measurable data likely to affect the claim amount and/or claim timing (or frequency) is required.
  • Data collected not necessarily to price the product at this moment. But they may be used in the future.
• Source uncontrolled by the insurer:
  • There are some circumstances when the actuary will not have full control over the data to be used (e.g. an actuary carrying out a statutory valuation of an occupational pension scheme).
  • In this case, data covering the membership of the scheme and the benefits accrued by each member at the date of the valuation will usually be provided by the employer.
  • However, where the data is provided from an external source, this is particularly important.
  • Need to perform check on such data!
• External source:
  Where poor quality and/or insufficient data is available, the actuary may consider the use of data from an external source (e.g. industry-wide data, reinsurer data, national statistics).
  This can be particularly useful to small insurance companies and companies writing a new (or specialised) class of business, where the quantity of internal data is inadequate to allow credible statistical analysis. For example, the Continuous Mortality Investigation (CMI) Bureau collects and analyses a large quantity of mortality and morbidity data from a range of life insurance companies in the UK.
  However, the main disadvantage of external industry-wide data collection schemes is the possible distortion caused by heterogeneity between different data providers.
  Distortions in collective data schemes can result as not all contributors will be homogeneous with regard to:
  • terms and conditions of policies (there must be difference !)
  • underwriting and claim settlement practices
  • underwriting and claim settlement practices
  • target market
  • nature and/or detail of data requested and stored
• Industry-wide data also tend to be less detailed and flexible than internal data (as data will usually be provided in summarised form, with no access to underlying raw data), and more out-of date (due to time taken to collect, collate and distribute results).
• Life insurance: As mentioned above, the Continuous Mortality Investigation provides mortality data for both assured lives and annuitants (separated by a range of major risk factors) and also morbidity data from critical illness and income protection insurance.
  Demographic data (such as population projections) are produced regularly by the Office of National Statistics.
• Non-life insurance: The Association of British Insurers also has an extensive database covering premium, claim and expense experience for the non-life insurance market as a whole (and subdivided by category covering motor, property, employers’ liability etc), as well as re-insurance data for Marine, Aviation and Transport (MAT) and non-MAT business.

Risk classification

• The main aim of risk classification is to obtain homogeneous classes of data with respect to the factors affecting the risk being analysed. Then, the experience of each class will be more stable and characteristic of the underlying grouping. This allows for more accurate projection of future experience.
• To ensure the changes in the underlying mix of risks will not affect future experience.
• However, separating the data into homogeneous groupings may give insufficient data in some cells (e.g. at very low and/or very high ages). In this case, it may be necessary to combine some groupings and sacrifice some degree of homogeneity for increased credibility

Data verification
Possible checks applied to a given data set include:

• reconciliation with data used at previous valuation
• reconciliation with accounting data
• any inconsistency between shareholdings at start and end of period (adjusted for sales and purchases) may indicate errors in the asset data provided
• checks on any unusual values in the data set
  e.g. very high (or low) sum assured or premium may indicate individual data entry error or systematic problem
• random spot checks on individual data items
  it is particularly important to check data for members (or policyholders) who have significant liabilities
In general, the extent of data verification in any particular situation will depend on the financial significance of any errors made.

For pension: main data required of each individual member

• name
• date of birth
• gender
• date of retirement (particularly if there is any minimum guaranteed payment term)
• current level of pension
• class of member (as this may affect entitlement to spouse’s pension and/or future pension increases)
• marital status and age of spouse
• scheme member or spouse/dependent of scheme member

scheme-wide data entries

• scheme trust deed and rules (to ensure the correct benefits are valued)
• the valuation report from the previous valuation (for reconciliation purposes)
• details of all the assets held by the scheme at the current valuation date (to enable valuation of these assets)
• any contributions made by the sponsoring employer since the previous valuation

For life products: rating factors (rating factors are often measurable proxies for risk factors)

• age
• genders
• smoking status
• in-force duration – because of high level of underwriting
• distribution channel – different distribution channel will attract different customers
• sum assured – the higher the sum assured, the more likely the anti – selection will be

When analysing claims data from a portfolio of travel insurance business, what are likely to be the main risk factors?

• age – more complications for older people, destination (US expensive health care expense)
• duration of stay (The longer you stay, the higher the insurance will cost)
• type of cover (e.g. are dangerous sports covered?)
• occupation
• reason for trip (business or pleasure)
• gender is probably not a major factor here – There is not enormous different claim inception rate between the genders

The key requirements of an actuarial model

• The model being used must be valid, rigorous enough for its purpose and adequately documented
• The model chosen should be capable of adequately reflecting the risk profile of the financial products, schemes, contract or transactions being modelled.
• So at the planning stage, the requirements of all stakeholders should be brought into account and the budget/timescales/etc should be established
• At the model design stage, the methods or other models available to test the model should be considered, so that the model built can be adequately tested.
• The parameters used must allow for all those features of the business being modelled that could significantly affect the advice being given.
• The inputs to the parameter values should be appropriate to the business being modelled and take into account any special features of the provider and the economic and business environment in which it is operating.
• The workings of the model should be easy to appreciate and communicate. This is both the structure of the model and how the parameterisation has been determined. The model should exhibit sensible joint behaviour of model variables.
• The outputs for the model should be capable of independent verification for reasonableness. The results should be displayed clearly and should be communicable to those to whom advice will be given.
• The model 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

Deterministic or stochastic?

• Features
  • In a deterministic model, the parameter values are fixed at the outset and the model output will be a single realisation of the future experience.
  • Sensitivity analysis is then used to assess the variability of the model output as a result of changing one or more parameter values.
  • In a stochastic model, one or more of the parameters is represented by a specified probability distribution.
  • Then, the model is run a large number of times and the value of these parameters is simulated from the chosen distribution each time. Thus, the model output will be a range of different possible future scenarios, giving an understanding of the distribution of outcomes.
• Advantages / Disadvantages of using a stochastic model
  • Advantages
    • allows uncertainties in future outcomes (i.e. risks faced) to measured objectively,
    • allows pay-offs from options and guarantees to be modelled.
    • Stochastic models are particularly useful in assessing the impact of financial guarantees. because such guarantees will lead to a pay-out in some future circumstances and not in others ... which cannot be replicated easily in a deterministic model (where pay-out will definitely occur or definitely not occur, depending on assumptions used.
  • Disadvantages
    • more difficult (and, thus, expensive) to design, construct and explain. --> greater model risk
    • Harder to parameterise -> greater parameter risk
  • In practice, a combination of deterministic and stochastic models may be used. Stochastic models are often used for more volatile parameters (e.g. investment return), whereas deterministic models are usually used for more stable parameters (e.g. mortality).
  • fluctuations in more volatile variables such as investment returns will tend to have a more significant impact on volatility of profits (and, hence, financial health of company) than fluctuations in more stable variable such as mortality. but, not also true (e.g. profits from term assurance business relatively insensitive to investment experience).

Building a cash flow model
The key steps involved in developing a deterministic model are:

• 1) specify the problem that needs to be solved
  • determine the premium to be charged for new business, or
  • calculate the current contribution rate for a defined benefit pension scheme, or
  • calculate the reserve required to be held in respect of existing liabilities, or
  • assess the expected profits arising from existing business
• 2) collect and verify the required data
  • e.g. data relating to existing policyholders or scheme members and past experience
• 3) group the data into broadly homogeneous sub-groups
  • in practice, it may be very time-consuming to include each individual policyholder and/or scheme member
  • thus, it is common to group together individuals with similar characteristics into “model points”
• 4) identify the key parameters affecting the amount and/or timing of the required future cash flows
  • will depend on the nature of business being modelled
  • e.g. investment return is likely to be crucial in modelling future cash flows from long-term life and pensions business, but much less important for short-term nonlife business
• 5) If deterministic models are used: assign values to these key parameters using past experience and appropriate statistical estimation techniques
  • Subject CT4 looked at fitting a model to past mortality experience and similar techniques can be used for other demographic factors (e.g. withdrawals and retirements)
  • Subject CT6 looked at fitting distributions to claim frequencies and amounts for non-life insurance business
  • Subject CP1(1) looks at techniques for analysing past investment experience ─ if necessary, an expense analysis can be used to allocate past expenses appropriately to individual policies
  • If stochastic models are used: assign probability distribution to parameters.
• 6) project the expected cash flows (both in and out) in each future time period using the chosen set of parameter values (or “valuation basis”)
• 7) verify the goodness of fit of the chosen model ─ this can be done by projecting forward the data from a previous time period and comparing the model output with the actual experience
• 8) make adjustments as required to the structure of the model (in particular, the interactions between parameters) until an appropriate goodness-of-fit is obtained
• 9) run the model using the current data and estimates of the likely future values of the key parameters ─ these estimates may differ from the past experience and may also include margins on grounds of prudence (if appropriate)
  If stochastic models are used: allow for correlation between the variables and run model many times to estimate distribution of key output variables empirically.
• 10) use the model output to develop an initial solution to the problem specified
• 11) repeat several times using different combinations of parameter values to assess sensitivity of results obtained ─ this will also allow the factors that have the most significant effect on the future outcomes to be identified

Data Sourse + Proper adjustments + Other important notices Economic variables generally affect the amount of the future cash flows (e.g. investment return, price inflation), whereas as statistical variables affect the timing of the future cash flows.
The set of assumptions used in any particular case is known as the valuation basis.
The key considerations when setting the valuation basis are

• the use to which the assumptions will be put
  • e.g. to price a new contract or to estimate the reserve required to meet the current liabilities
• the actuary should take particular care over the choice of the assumptions that will have most financial significance
  • e.g. for a final-salary pension scheme, why is the rate of investment return likely to be a much more important assumption than the rate of early retirement?
• consistency between the various assumptions in the basis
• any legislative or regulatory constraints

Information sources

• Historical data
  Historical data is likely to be the main source used in determining assumptions about future experience. There is a wealth of past data available to the actuary covering:
  • investment returns on the each of the main asset classes, split between income and capital gain
  • price and salary inflation • mortality (and other decrements – e.g. withdrawal, morbidity)
  • claim frequencies and claim amounts (for non-life insurance)
• current forecasts of future experience
  However, this should be combined with an analysis of recent trends in the data and current forecasts of future experience.
  Examples of current forecasts of future experience include:
  • government policy statements (e.g. with regard to future inflation and/or economic growth targets)
  • the difference between current yields on fixed-interest and index-linked bonds gives an approximation of the markets’ expectation of future price inflation
  • discussions with company directors and/or pension scheme sponsors
    can be used in setting salary growth and salary scale assumptions. may also be suitable when deciding on new entrant assumption (although unlikely that this will be needed in most DB schemes at the moment ... as many are closed to new entrants).
  • For statutory valuations of DB pension schemes, the key assumptions to be used will usually be defined by legislation. To prevent the results for different schemes being influenced by the choice of the assumptions ... e.g. could artificially increase security of accrued benefits by using very optimistic assumptions.
• Relevance
  The relevance of the past data must be considered when projecting future experience.
  Factors that may reduce in the relevance of historic data include:
  • changes in underlying economic and social conditions
  • changes in market structure
  • changes in terms and conditions and/or design of product
  The relevance of past data must be balanced against the need for sufficient data to make for a credible analysis.
  Adding more past data will reduce the standard error of any statistical estimates, but is data from many years ago really relevant when estimating future experience?
  The actuary must consider a number of other key issues including:
  • changes in the experience over time
    • e.g. for mortality data, it is common to identify any trends in the data, and to allow for these in future projections
    • this may also be useful when projecting claim frequencies and amounts in non-life insurance
    • what issues are important to consider here? is it really a trend and not just due to random fluctuations in the data? if so, can the trend be expected to continue in future?
  • effect of abnormal fluctuations
    • e.g. when projecting future claim amounts for non-life business, it may be appropriate to remove any large one-off claims from the data before identifying any trends
  • the past data should be split into homogeneous groups with regard to the factors affecting future experience
    • e.g. age, sex and type of employment are likely to affect future rates of salary growth and mortality
    • however, small data sets can lead to credibility problems
  • where heterogeneity remains, the actuary should be careful to avoid reaching spurious conclusions as a result of changes in the mix of the underlying population
  • consideration must be given to any changes in the way the past data has been recorded over time
• Other factors to consider
  • (1) Purpose of the valuation
    • This will be very important in determining the level of prudence included in the assumptions regarding future experience.
    • A valuation basis used for product pricing is likely to include little or no margins on grounds of prudence (often referred to as a best estimate basis).
      competition will constrain the extent of margins included in the premium basis (as, if premium is too high, business volumes may be too small to be viable)
    • However, a valuation basis for the purpose of calculating reserves (particularly statutory reserves used for demonstrating solvency) will often include significant margins in the assumptions used.
    • This increases the security of the promised benefits, but does not materially affect the cost of the benefits (other than the opportunity cost of the capital set aside).
    • because any excess funds held (as a result of overly prudent margins) will be released when the benefit is paid (or when cover otherwise ends).
  • (2) Significance of assumptions
    The actuary should be very clear on which assumptions will have a significant financial impact on the results. Then, any errors in estimating these parameters may have a serious effect on future profitability.
  • (3) Consistency of assumptions
    In most cases, the actuary will be interested in projecting both asset and liability cash flows (as asset cash flows will be used to meet liability cash flows as and when they arise). Thus, the relative value of many of the key parameters is much more important that the absolute values.
• Assumptions for pricing
  Assumptions will be required for all factors affecting the amount and/or timing of the future liability outgo.
  We now consider the process of setting the appropriate assumptions, with regard to:
  • the expected future experience
  • the extent to which margins against future adverse experience are appropriate
  Different assumptions
  • (1) Demographic Assumptions
    • The main demographic assumption required to price a life insurance contract is with regard to the future mortality experience.
    • The mortality rates used should reflect the expected future mortality experience of the potential policyholders.
    • In this case, three factors are crucial to consider:
      • the target market for the contract (which will depend largely on the type of product and the methods by which the product is sold )
      • the level of underwriting applied
      • allowance for improvements in future mortality experience
      • Even for a very large company, the mortality assumptions used are likely to be based on a suitable standard mortality table.
      • The relevance of the company's own past data must be considered, particularly where there has been a change in underwriting procedures and/or sales methods used.
      • An appropriate allowance for improvement in future mortality is particularly important for annuity contracts.
      • The effect of selective withdrawals on past and expected future mortality must also be considered. However, this will depend on the level of surrender value payable.
  • (2) Investment return
    The importance of the assumption regarding future investment return will depend crucially on:
    • the size of the reserve built up under the contract
      • type of contract is crucial here
      • e.g. investment return assumption is less important for term assurance contracts, as the reserve built up is small
    • the extent of any inherent investment guarantees
      • nature of contract is crucial here
      • less risky assets will usually be held to back non-profit liabilities, as the investment return is guaranteed in full
    • The assumed rate of future investment return will depend on the likely mix of the assets held to back the contract in future.
    • The level of free assets will determine the extent to which the company can depart from a matched investment position in the hope of achieving higher returns.
    • The actuary needs to consider the expected long-term return in future on each of the main asset classes.
    • Allowance should be made for the effects of any expected future changes in the taxation treatment of investments and/or regulation relating to the type of assets to be held.
    • Setting an assumption for the future investment return will also be crucial in a pension fund valuation, but is likely to be much less significant in pricing for non-life insurance.
    • because most business is short-tail → period of investment of premiums is limited → investment return achieved is less significant.
  • (3) Expenses and commission
    Allowance for expenses should include marginal costs such as:
    • initial acquisition costs (e.g. commission)
    • underwriting
    • administration (both initial and renewal)
    • renewal commission (if appropriate)
    • investment
    • claim/maturity administration
    In addition, a contribution to the fixed costs of the business (e.g. rent, salaries) should also be included.
    An expense analysis will usually be conducted using the past experience to determine an appropriate contribution to the overall expenses for each policy.
    Or, for simplicity, we could assume that each in-force policy makes the same contribution to the total fixed expenses of the company.
    some types of product will require more administration -> appropriate they contribute more to fixed costs
  • (4) Withdrawals
    • The assumed rates of future withdrawals should also reflect the expected experience of the potential policyholders.
    • Again, where possible, the company's own past data for the particular contract should be used.
    • However, if this is not available or is insufficient, then experience from similar contracts and/or industry-wide data may be used.
    • What factors are likely to affect future rates of withdrawal: external factors – recession, changes in legislation, actions of competitors
    • internal factors – customer service, failing to meet PRE, level of bonus rates, surrender terms, sales procedures, solvency position, launch of new products
  • (5) Pension schemes and non-life insurance
  • (6) Margins
    An additional cushion for adverse (i.e. worse than expected) future experience can be made by:
    • incorporating explicit margins into each of the assumptions in the valuation basis and calculating the premium using the equivalence principle
    • allowing for the risk through use of an explicit risk discount rate and an appropriate cash flow model

Timesheet analysis is used whenever staff is discussed as a source of expense. Timesheet analysis of the relevant staff’s use of time may be helpful in this analysis
Reasons

• determining a suitable expense loading for pricing and reserving
• identify inefficiencies and cost overruns (which will reduce profitability).
• analyse profitability of individual classes of existing business (to help plan new business strategy)
• identify sources of surplus for profit distribution (e.g. to shareholders, with-profit policyholders)

process

• Expense categorization
  expense analysis: the total expenses incurred during a given period are (notionally) allocated across the in-force business to give an appropriate contribution to the expenses for each individual contract.
  These expenses can then be classified as:
  • fixed
  • variable ─ i.e. vary according to the volume of business handled (in terms of number of contracts written, premium income received or claim amount payable)
  • In addition, expenses are further classified as direct (i.e. belonging directly to a particular class of business) or indirect
  • Direct expenses may include underwriting costs, commission, administration and claim settlement costs. Indirect expenses (or overheads) often relate to support functions such IT and human resources, as well as rental costs.
  • variable expenses can usually be classed as direct, but fixed expenses can be either direct or indirect.
• Expense allocation
  • direct expense: In some cases, the entire expense can be allocated to a single tranche of business (or product line). However, in most cases, it will be allocated between different tranches of business on the basis of timesheets collected over the period.
  • Indirect expenses
    • In general, they tend to be allocated to each of the sources of direct expense as pragmatically as possible (before these direct expenses are then allocated to individual tranches of business).
    • rental costs might be allocated to each department (both direct and indirect) on the basis of floor space, then
    • indirect costs from the HR department (including the contribution to rental costs) could be allocated to each direct department on the basis of number of staff, and
    • indirect costs from the IT department (including the contribution to rental costs) could be allocated to each direct department on the basis of hours of CPU usage
  In addition to allocating the expenses to each tranche of business, it is appropriate to further sub-divide these expenses as either initial, renewal or claim expenses.
  In practice, this may require further timesheet analysis noting whether the work undertaken was for new business, in-force business or claims.

Expense loadings for premium rating
expenses can be included as

• a fixed amount per contract ─ e.g. administration expenses (both initial and renewal)
• a percentage of the premium charged ─ e.g. commission
• a percentage of the benefit payable under the contract ─ e.g. underwriting costs, claim settlement costs

Investment expenses will often be a key component. in the premium calculation: often by using a lower assumed rate of investment return (i.e. net of expenses associated with buying and selling).
Taxation: will depend on how taxation is applied. unusual to be levied on an associated with buying and selling). individual policy basis. could use net (of tax) investment return or work gross of tax (and allow for in required profit margin).
In practice, the insurer may often decide to cross-subsidise between different groups of products.
to increase competitiveness of different tranches of business (e.g. high/low premiums, different distribution channels), to increase market share (e.g. loss leaders), perceived fairness
In general insurance, the lower administration expenses incurred on renewals can be used to cross-subsidise with new business or to offer lower premium rates to existing policyholders.
Also, when using past expense data to determine an appropriate expense loading for future new business, the actuary must allow for future expense inflation.
For pension schemes, the situation is somewhat different. In practice, much of the work (e.g. investment management, actuarial advice) will be assigned to external providers who will charge a fee for their services. Thus, an expense analysis is not usually required.