Modelling: The Science

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Chapter 7
Asset-Liability Management for Actuaries

• Modelling The Science Art of the Actuary

Shane Whelan, L527
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Introduction

• The work of science is to substitute facts for
  appearances and demonstrations for impressions.
• Ruskin often quoted in publications by SoA
  (US).
• Actuarys role is to assess/measure/evaluate/price
  or reserve for future contingent events and
  assess/measure/evaluate and mitigate the
  associated risk(s).
• To do so, the actuary needs a model
• Building a model is part of the actuarial control
  cycle.
• But that required model (or meta-model) will
  generally build on many smaller models
• e.g. a life table is a model, used as input to,
  say, profit test.
• Your technical training to date has exposed you
  to many simple models we use these as building
  blocks to help model a clients or employers
  problem.

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Actuarial Control Cycle
Context of Business/Economic/Commercial
Environment
Specifying the Purpose of Model
Professional Considerations
Monitoring the experience to update model
Developing a Model
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Examples of Modelling (by Actuaries)

• Assessing the net present value of a capital
  project
• Valuing life assurance liabilities in the context
  on assets held
• For statutory purposes, to help decide bonus
  policy on with profits business, etc.
• Valuing a (defined benefit) pension fund to
  advise on a contribution rate
• Valuing the liabilities of a general insurer in
  the context of assets held
• For statutory purposes, management purposes, or
  otherwise.
• Profit test a new product, and help set premium
  rates.
• Asset-liability Modelling (ALM) to find least
  risk investment portfolio for a given portfolio
  of liabilities.
• To model the future financial trajectory of a
  life office (the model office) to estimate
  future capital needs, future bonus rates,
  investigate impact of new products, etc so to aid
  the management of the company.

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Examples of Modelling (by Actuaries) (Cont.)

• To estimate the appraisal value of a companythis
  is defined as a measure of the present value to
  shareholders of the future stream of
  distributable profits (and residual wind-up value
  if any).
• So the change in appraisal value over time gives
  a measure of value-added.
• Appraisal value is the sum of the following three
  items
• Net worth (of company) the amount that could be
  distributed to shareholders immediately.
• Value of in-force business the present value of
  the future distributable profits to shareholders
  from the existing in-force business.
• Value of new business the present value of the
  future distributable profits to shareholders from
  future new business. Sometimes this component of
  value is called goodwill.
• Embedded value is defined as net worth plus value
  of in-force business.
• A new modelling exercise is liability-driven
  investment (LDI), which attempts to ensure the
  market value of assets exactly increases in line
  with then value of a liability portfolio,
• through swaps (real interest rate, nominal
  interest rate, equity, currency, etc), and
  options (generally OTC) and any other instruments
• This is ALM for the 21st century.

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Modelling

• Model a simple, stylised imitation of a real
  world system or process.
• Used to predict how process might respond to
  given changes enabling results of possible
  actions to be assessed
• or simply to understand how system will evolve in
  the future.
• Other methods being too slow, too risky, or too
  expensive.
• Objective of Model is paramount
• we need to know what is best model and this is
  generally not the most accurate model need to
  balance cost with benefits.
• There is a place for the back of the envelope
  model.
• All models are wrong, but some are useful.
  George Box.
• Constraints in model building are time, budget,
  available data, other resources (your toolkit of
  models) and your know-how (experience).
• e.g., macroeconometric model of economy
• Price of share at each future date
• Model life office

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Modelling

• So model has to be fit-for-purpose
• two models of the same system may be completely
  different if they have different purposes
• Model to value life office for purchase or sale
• Model to determine the solvency position
• Also remember that there are different
  stakeholders, each wanting a different bias in
  the model
• E.g., setting transfer values for early leavers
  of a defined benefit pension scheme.
• Finally remember that professional guidance,
  through guidance notes, can be very prescriptive
• in detailing how the modelling is to be done,
  requirements the model must satisfy (e.g.,
  consistency), or checks that must be made (e.g.,
  data reasonableness and consistent with past
  data). Sensitive testing which is testing the
  sensitivity of the output to parameter inputs
  would generally be necessary.

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Classifying Models

• Deterministic Model Unique output for given set
  of inputs. The output or inputs are not random
  variables.
• Stochastic Model Output is a random variable.
  Perhaps some inputs are also random variables.
• So models not just the expected output but our
  uncertainty of that output. Hence models risk
  too.
• A deterministic model can be seen as a special
  case of a stochastic model.
• Revision See my class-pages on Introduction to
  Models Stochastic Model and printout text of
  Chapter 1.

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Contrasting Deterministic Stochastic Models
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Further Terminology in Modelling

• Scenario modelling using different sets of
  economic conditions to forecast deterministically
  the model predictions
• E.g., best estimate, prudent, pessimistic and
  optimistic bases
• This outlines the range of possible outcomes, in
  a simpler and quicker manner than stochastic
  modelling
• Can use percentile values of parameters, if
  distribution known, to approximate a stochastic
  model
• Scenario testing this is scenario modelling
  above, but used to ensure compliance with a
  minimum standard set by the regulator. Generally
  each of the scenario tested is adverse relative
  to current conditions.
• Resilience testing is an example of this for life
  offices, where the appointed actuary must certify
  that assets will cover liabilities even if long
  term interest rates rise or fall by 3 and equity
  values fall by 25 on the valuation date. The
  extra amount of reserves to comply with this is
  known as the resilience test reserve.

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Further Terminology in Modelling

• Sensitivity Testing running the model with
  different assumptions (e.g. mean value of
  parameters, distribution of parameters, etc) to
  assess which assumptions the output is most
  sensitive to.
• This is crucial in validating the model for use
• It highlights key dependencies of model which
  must be communicated
• It is used to obtain a deeper understanding of
  the process
• E.g., profit testing, where better understanding
  might prompt change in product design.
• Models can be divided into
• Demographic models used to model numbers of
  individuals in different categories
• e,.g., decrement model of life table or
  multi-state model for sickness or disability
  insurance
• Economic models used to model relationship
  between economic, financial and investment
  drivers
• E.g., inflation, earnings, interest rates, equity
  yields and returns. Particular attention must be
  paid to the relationship between them.
• Other models defined by what they model.

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Further Terminology in Modelling

• Models can also be divided by the (mathematical
  or financial) properties of the model
• Such as the generic form of model - market
  consistent models, no-arbitrage models, diffusion
  processes, mean reversion process, ARCH, Markov,
  Levy process,.
• Fully dynamic model is a stochastic model which
  incorporates decision making rules, dependent on
  future output.
• E.g., if used for model office it would change
  bonus declarations on with profits business
  depending on prevailing interest rates and
  forecast asset shares, etc.
• Dynamic modelling is used in dynamic financial
  analysis (DFA))
• Customer lifetime value (CLV) models the value
  of customers rather than contracts, as customer
  loyalty and propensity to purchase more products
  is an intangible asset.
• Can answer questions
• such as the discount or special terms banks might
  give to students to start an account with them
• Or the goodwill to be paid for a business with
  customers that overlap yours

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Attributes of a Good Model

• Model must be relevant to exercise at hand,
  produce outputs that are credible, and be
  adequately documented.
• This is the minimum that can be expected.
• The model should shed light on the risk profile
  of the process modelled (e.g., financial product,
  scheme or contract design)
• All factors that could significantly affect the
  advice being given are incorporated in the model
  or modelling exercise.
• Any financial drivers risk discount rate,
  statutory reserves, etc, reasonable variation in
  parameters.
• The estimated parameter values of the model
  should reflect the business being modelled and
  the economic and business environment.
• This means the pecularities of the product size
  of premium, early lapse rate, presence of options
  or guaranteed, etc.
• The parameters in the model should be
  self-consistent.
• So inflation, return on assets, risk discount
  rate, lapse rate, escalation of expenses, should
  all be mutually consistent.

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Attributes of a Good Model (Cont.)

• The outputs of the model should appear reasonable
• Reproduce historic episodes
• Capable of independent verification/peer review
• Possible to communicate key results to client
• Subject to all the above, the simplest model is
  the best
• As is cheaper to develop and run
• Easier to interpret and communicate
• Everything should be made as simple as possible,
  but not simpler." Albert Einstein

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Deterministic or Stochastic Model?

• A deterministic model is, in general, simpler
  than a stochastic model
• And, in particular, its results are easier to
  communicate (try taking of gamma distributions
  and Levy processes to Trustees of a pension
  fund!)
• It is easier to develop, to interpret and quicker
  to run.
• It is also clearer what scenarios have been
  testedbut these are not implicit in the model,
  but made external to it as part of the wider
  modelling process.
• However, only a limited number of scenarios are
  run..
• The modelling exercise may have missed one that
  is particularly detrimental this is important
  when contract has embedded options (e.g., to
  extend life cover without underwriting) or
  guaranteed (e.g., surrender value not lower than
  premiums paid). So make scenario testing implicit
  in model a stochastic model.
• We might need to model explicitly the probability
  of each outcome (e.g. to price embedded option).
  Hence we require a stochastic model.

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Deterministic or Stochastic Model? (Cont.)

• Sometimes model must allow for dynamic feedback
  that is the future evolution of system depends on
  what happens in the future this requires a
  stochastic model top trace the different possible
  paths and their likelihood.
• E.g., bonus declarations or policy depends on
  performance of assets
• E.g., discretionary rises to pensions in payment
  given level of inflation and past service surplus
  at the future time.
• Stochastic models are more complex so need to be
  satisfied that
• extra output (and time needed to develop and run
  and interpret and communicate) is justified
• Do we know underlying distributions of the
  parameter(s) we sufficient accuracy (or are we
  just introducing spurious accuracy?)
• Considerable judgment required to factor in
  variability of parameters, relationship between
  parameters (correlation, coppulas), and dynamic
  decision feedback.
• Often a combination of stochastic and
  deterministic models are used
• Economic models (where the output has a high
  dependency on inputs) are often modelled
  stochastically.
• Demographic models are often modelled
  deterministically (as variability is less
  material to output).

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Quick Question

• Indicate whether a deterministic or stochastic
  model is appropriate to
• Price a guarantee on the lowest interest rate
  that an annuity will be sold at in the future.
• To set a contribution rate on a defined benefit
  pension scheme
• How much capital a company should maintain to
  that the probability of insolvency within a year
  is less than x.
• For statutory valuation of a life office
• What reinsurance arrangements (excess of loss,
  stop loss, etc) gives best value for money when
  claims variability is set to prescribed limit.
• The net present value of a project
• The asset portfolio that best matches
  salary-related benefits.

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Building a Deterministic or Stochastic Model 10
Helpful Steps

• Set well-defined objectives for modelthe purpose
  of the model/investigation
• Plan how model is to be validated
• i.e., the diagnostic tests to ensure it meets
  objectives
• Define the essence of the structural model the
  1st order approximation. Refinement and details
  can come later.
• This involves specifying the form of the model,
  identifying the parameters, input and output
  variables.
• Involve experts on the real world system to get
  feedback on conceptual model
• Collect analyse data for model (and any other
  parameters)
• Ascribe values (or specified distributions) to
  the parameters using past experience, appropriate
  estimated techniques, and (properly documented)
  professional judgement.
• NB If stochastic model, specify correlation (or
  other relationship) between variables.

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Building a Model 10 Helpful Steps

• Test the reasonableness of the output from the
  model and otherwise analyse output.
• Check that goodness-of-fit is acceptable by
  reproducing past episodes.
• Estimate distribution of the error term in model
  (from not modelling certain factors that have an
  affect on output)
• Start again if fit not acceptable maybe moving
  to 2nd order model.
• Test sensitivity of output to input parameters
• i.e., ensure small change to inputs has small
  affect on output. We do not want a chaotic
  system in actuarial applications if so,
  redesign product, reinsure, or take other action
  to make financial output tractable.
• If stochastic model ensure result reasonably
  robust to assumed distribution of input
  parameters where unknown
• Perform scenario modelling best estimate,
  cautious, optimistic, etc.
• This involves changing the parameter inputs (the
  set of assumptions underlying the parameterised
  model is often termed the basis. Typically there
  is an economic basis and a demographic basis.)
• NB If stochastic model, run model many times
  using random sample from the input random
  variables, producing an empirical distribution of
  the output random variable(s).

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Data underlying model

• Communicate and document results and the model.
• Arrange for emerging actual experience to be
  monitored in a suitable way.
• i.e., put in place a system to collect data so
  that actual experience can be compared with
  expected experience
• Review model and update in the light of new data
  and other changes. This is part of the on-going
  monitoring.

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Case Study Models for Pricing

• We want to develop a model to help determine the
  charging structure (e.g. premium) for a new
  product that meets the companys profit
  requirement.
• This, as so often in actuarial modelling, will
  have at its heart a cashflow model
• The different income (premiums) and outgo from
  office (expenses, claims, etc) at each future
  time
• Two primary questions
• How frequently do I model the cashflows?
• Annually, monthly, etc.
• Perhaps monthly for first few years and
  thereafter annually. Variable time period that
  exploits key sensitivities of the profit.
• What premium is assumed, what age, what term of
  policy, what gender of policholder ? Or, in
  general, what any other rating factor or metric
  for size or term of policy?
• Here the choice is infinite. Use the concept of
  model point

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Model Point

• A model point is a representative single points
  to use to model key characteristic of a larger
  group.
• Each within the larger group acts in all
  important respects, like the model point.
• So, a judicious selection of model points enables
  us to model the entire system.
• We just need to scale up the model point
  multiply the result at the model point by the
  expected number in the subgroup it represents.
• Normally one does not have to use model points in
  valuing existing business/doing valuations
• Because regulation requires valuation
  policy-by-policy
• Because it is simpler to do all pension fund
  valuations.
• But can use model points for what if
  investigations.
• In the pricing example, the model point is a
  policy.
• Some experimentation might be needed to establish
  the model points.
• Then choose the number so that, when scaled up,
  the expected new business is satisfactory
  modelled.

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Case Study Models for Pricing (Cont.)

• For each model point the cashflows are projected
• Allowing for future reserves on the statutory
  basis
• Allowing for solvency margin requirements
• The net projected cashflows are discounted at the
  risk discount rated
• Reflecting the return required by the company
• With due allowance for the risk (variability)
  inherent in the cashflow this requires some
  judgement
• Variability in model (if stochastic)
• Parameter risk
• Model risk
• Could use a stochastic discount rate.
• Should a different discount rate be applied to
  each type of cashflow?

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Case Study Models for Pricing (Cont.)

• The premium or charging structure for each model
  point can now be set so as to produce the
  required profit.
• So scale up.
• So, adopt common scale so average profit over
  business is satisfactory
• Beware of cross-subsidy
• Interpolate for policies between model points.
• The most important part of pricing is to set
  competitive premiums (subject to profit margin)
• If not, try to differentiate product from
  competitors by adding features.
• Or remove features than are expensive (options)
  or add to the riskiness (guarantees)
• Reconsider distribution channel (e.g., so as to
  increase average premium size)
• Reconsider companys target profit
• Reconsider whether to go ahead with the product
• Consider capital requirements of the policy, and
  their timing.
• Maybe redesign if too capital intensive.
• What about one-off development costs not
  amortised in cashflowmust add in.
• The pricing model can be developed to model the
  future cashflows of the entire businessthe model
  office

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Example

• A unit-linked policy guarantees to pay a maturity
  value of the greater of the bid value of units or
  the sum of premiums paid. On termination prior to
  maturity, the surrender value is the bid value of
  the units.
• Outline the steps involved in pricing the
  guarantee, the type of modelling involved, and
  the key assumptions determining the price.

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Completes Chapter 7
Asset-Liability Management for Actuaries

• Modelling The Science Art of the Actuary

Shane Whelan, L527