Chapter 20: Capital Modelling Methodologies

Question 1

Aim of a capital model

Answer 1

Can be used to help the insurance company determine the LEVEL OF CAPITAL TO HOLD.The model will also enable the company to better understand their risks and inform business decisions.

Question 2

Components of a capital model

Answer 2

1. Future Written Premium income-
   -project premium income separately for each LOB,
   -split by source of business,
   -set assumptions after consulting sales, u/w and senior management
   -when including new business and renewals, allowance will need to be made for expected rates of premium growth (and profitability) in the light of the company’s business plan, the competitive position and the effect of the insurance cycle.
2. Future claims -
   -for each LOB split claims into
   a. Attritional, large, cat claims,
   b. IBNR,
   c. development of notified claims i.e. IBNER,
   d. claims from period of unexpired risk on existing business,
   e. claims from business written in the future
   -Outstanding claims can be estimated using projection methods (eg the chain ladder method).
   -These methods can also be used as the basis for estimating the claim outgo in future periods.
   -As an example of how the elements of insurance risk might be treated, potential claims arising from catastrophes are usually analysed using the output from proprietary catastrophe modelling software, or by using scenario tests if capital modelling software output is not available.
3. Future expenses -
   -commissions are normally a % of WP,
   -rental. staffing cost, it is based on projected business growth,
   -The expenses relating to handling the claims can be allowed for either explicitly or implicitly (ie with the corresponding claims).
   -If the insurer stopped writing any more business, the expenses would just be in respect of the existing business.
4. Ceded Reinsurance -
   Assumptions about ceded reinsurance will need to take into account:
   -any existing outwards reinsurance arrangements
   -any changes that could be made to those arrangements
   -planned new arrangements to cover unexpired exposures or business planned to be written
   -any forecast changes in underlying direct exposures
   -any expected softening or hardening of future reinsurance costs.
   -Allowance should be made for recovery delays and defaults.
   -Alternatively, claims could be projected net of recoveries, with a margin to allow for defaults and an adjustment to the development profiles to allow for recovery delays.
5. Investment returns - Assumptions about future investment return will depend on:
   -the current investment portfolio held
   -investment prospects and expectations around the future economic environment
   -the current and projected future investment policy
   -expectations on premium and claims payment patterns, which impact the run-off of reserves and investment assets.
   -For all types of investment, it is important to make allowance for:
   the expenses of investment,
   the future volatility of capital values and investment income.
6. Environment -
   Economic - A capital model will need to make assumptions about future inflation and future interest rates. These assumptions should be consistent with each other. Allowance may also be made for other features of the economy, for example, the increased moral hazard associated with increased claim frequencies during times of recession.
   The insurance cycle - The model should also take account of the insurance cycle, as should any business plan underlying the model. In particular, it will need to allow for the fact that different classes of business may be at different stages in the cycle.
   Operating environment - The model should also take account of what is happening internally within the company and its potential influence on future cashflows. For example the potential impact of high staff turnover on the ability to meet regulatory deadlines or the loss of an underwriting team on the ability to meet a business plan.
   There should also be some consideration of potential changes in legislation and their impact, for example the potential impact of a change in the Ogden discount rate on future claims payments.
7. Risk measure - A capital model requires a defined risk measure, on which it will be calibrated. This includes the type and confidence, for example, a 99.5% VaR.

Question 3

Explain why we might split premium income by source of business for future written premium income component of a capital model

Answer 3

The premium income from each source of business should be considered separately in order to allow correctly for delays and acquisition costs.

Question 4

The allowance for the cost of running the business once an office has closed should be much more than the normal allowance for claims handling costs. Suggest why this might be the case. (After all, both approaches relate to the cost of settling the existing business…)

Answer 4

As the business runs off, fewer claims will be settled (incurring lower claims handling expenses), so the firm’s fixed expenses will become a larger and larger proportion of overall expenses.Usually, a firm would use new premium income to meet its fixed expenses, but in the case of a run-off business, extra capital must be set aside to finance this.

Question 5

Suggest under what circumstances it would be acceptable to treat all the equity holdings as providing an infinite stream of dividends

Answer 5

Treating equities purely as an income stream assumes that none of the equity portfolio is ever realised. This might be the case if the value of the portfolio was less than the free reserves (ie equities are backing the free reserves not the technical reserves), and the liability outgo would be entirely covered either by income from assets (including equities) or the redemption proceeds from other assets (eg index linked securities). This means that the equities would never need to be realised to meet future liability outgo.

Question 6

Explain what is meant by a 99.5% VaR.

Answer 6

Value at Risk (VaR) generalises the likelihood of underperforming by providing a statistical measure of downside risk. VaR assesses the potential losses on a portfolio over a given future time period with a given confidence level.Consider, for example, a VaR of £10m over the next year with a 99.5% confidence interval. This means that there is only a 0.5% expected probability of the underperformance (relative to a benchmark) being greater than £10m over the next year

The Value at Risk (VaR) is the loss at a predefined confidence level (eg 99.5%), specified over a particular time horizon. Consequently, if an insurer holds capital equal to the VaR, it will remain solvent over a particular time horizon with a probability of the confidence level (eg 99.5%) and be insolvent with probability of one minus the confidence level (eg 0.5%).The use of probabilities and confidence levels in the risk measure seems to imply that we need to use a stochastic model. However, VaR can be used as a risk measure for deterministic models too, but the decision as to what constitutes a 99.5% probability will be very subjective.

Question 7

Stochastic model

Answer 7

One in which we assume some of the variables in the business plan have an underlying probability distribution.

This enables us to describe critical assumptions, and their financial implications, in terms of ranges of possible outcomes.

A stochastic model can be very complex and its results difficult to interpret.

It is worth remembering that the output from a stochastic model is only as useful as the underlying data input allows. As such, we should start the model process by gaining a thorough knowledge of the underlying data. Similar data issues apply equally to deterministic processes.

Question 8

9 steps in building a stochastic model

Answer 8

• specify the purpose of the investigation
• set the risk measure eg. VaR
• select an appropriate model structure
• decide which variables to include, and their interrelationships.
• determine the types of scenarios to develop and model. eg. interest rates, competitive environment, etc.
• collect group and modify the data,
• choose a suitable density function for each of the stochastic variables
• estimate the parameters that should be used for each variable.
• test and validate the reasonableness of the assumptions and their interactions. If the goodness of fit is not acceptable, then attempt to fit a different density function(s).
• ascribe values to the deterministic variables
• construct a model based on the chosen density functions.

Question 9

Running a stochastic model (3)

Answer 9

Once the model has been built,

• run the model many times, each time using a random sample from the chosen density function(s). The constructed model then calculates the net profit based on the values simulated from each pdf in the model.
• produce a summary of results that shows the distribution of the modelled results after many simulations have been run.
• run the model using different distributions / parameters to check sensitivity.

Question 10

3 Advantages of stochastic model

Answer 10

• test a WIDER RANGE OF SCENARIOS
• we can derive a PROBABILITY DISTRIBUTION OF OUTCOMES
• A stochastic approach explores all possible combination of stresses and can rank these against the chosen risk measure.

Question 11

9 Advantages of deterministic models
Exam style question - April 2015, Q3

A general insurance company is considering building a computer model to determine its capital requirements. Outline the advantages of building and using a deterministic rather than a stochastic model. [8]

Answer 11

• the model is usually easier to design and quicker to run
• we can introduce more detail and ensure use an intelligent selection of scenarios
• we can often make the results more comprehensible (understandable)
• Deterministic models are more readily explicable to a non-technical audience
• By developing stresses and scenarios we can help link the capital model with the risk register, helping to integrate capital and risk management
• It is clearer which economic scenarios have been tested.
• It is important to consider potential cause and effect relationship between risks.
• Even where we have used a stochastic model, stress tests using a deterministic model are useful to check / validate the model for reasonableness and to calibrate assumptions.
  ____________________________________________
  It is important to consider potential cause-and -effect relationships between risks.
  We may model such relationships better using deterministic relationships rather than relying on statistical dependence structures.
  It is more straightforward and, therefore, quicker and cheaper to build a deterministic model than a stochastic one.
  Deterministic model is easier to sense check…
  and easier to flex.
  It does not require the same level of expert resource as a stochastic model..
  and gives rise to less risk of model error. There is less danger of parameter error…
  and less danger of spurious accuracy, particularly in the tail.
  A deterministic approach may be appropriate where there is less data.
  By reducing the computational power necessary to generate many thousands of simulations, we can introduce more detail in other dimensions, such as detailed descriptions of reinsurance programmes or treatment of underlying risks.
  This may aid the intelligent selection of a limited number of scenarios.
  It could be more efficient than a stochastic model where we hope that the important scenarios appear amongst a larger number of randomly generated outcomes.
  We can integrate the capital model more closely with risk management, by extending the scenario modelling to scenario planning and “what-if” analysis.
  We commonly use stress and scenario tests for those risks that cannot easily be modelled quantitatively and where more subjective judgment is required.
  This allows us to concentrate more on the more important areas of the distribution of outcomes for the key risks when a full specification of the distributions is impossible.
  By developing deterministic stresses and scenarios, we can help to link the capital model with the risk register, helping to integrate capital and risk management.
  It can be easier to communicate the results of stress and scenario tests to senior management, and to give them comfort as to the reasonableness of the overall capital value.
  It is important that users of the output understand the results from the model as well as methods and assumptions.
  By showing the effect of a limited range of stresses and scenarios – some of which may have been developed in consultation with those users – we can often make the results more comprehensible to them.

Question 12

5 Features of a good model

Mnemonic - VAN FUR UP CAVE

Answer 12

• model should be valid, complete and adequately documented
• adequately reflects the risk profile of the classes of business being modelled.
• parameter values used should be appropriate for the classes of business, and investments being modelled.
• The outputs from the model and the degree of uncertainty surrounding them should be capable of independent verification for reasonableness and should be readily communicable to whom advice will be given.
• The model should be sufficiently detailed to deal adequately with the key risk areas and capture homogeneous classes of business, but not excessively complex so that the results become difficult to interpret and communicate or the model becomes too long or expensive to run.
• The model should be sufficiently flexible. The model should be capable of development and refinement
• In addition, a range of methods of implementation should be available to facilitate testing, parameterisation and focus of results.
• have all parameters clearly identified and justified
• be structured and documented so that it can be understood by senior management and board members who do not have actuarial expertise.

VAN FUR UP CAVE:

Valid
Adequately documented
Not overly complex

Flexible
Understandable by managers
Reflect risk profile

Uncertainty should be verifiable
Parameters identified and justified

Complete
Appropriate parameters
Verifiable
Easy to communicate results

Question 13

5 Additional features of a good STOCHASTIC model

Answer 13

A good stochastic model should:

• be rigorous i.e. strictly applying to constraints and principles and self-consistent
• be capable of being run with changed parameters for sensitivity testing.
• use a large number of simulations to avoid simulation error
• have a robust software platform.

Question 14

Deterministic models

Answer 14

A deterministic approach is one in which we assign fixed values to the variables or parameters (interest rate, inflation rates, claims rates and so on). Under a deterministic approach, we produce a single run for each set of fixed values. The standard formula under the Solvency II regime is an example of a deterministic model.

Question 15

Explain what the main difference is between stress and scenario tests

Explain what the main difference is between stress+scenario tests and sensitivity analysis

Answer 15

In a stress test, a single parameter is varied. Stress tests therefore analyse the impact of individual risks in isolation. In a scenario test, a combination of parameters is varied. Scenario tests therefore analyse the combined impact of a number of risks.

So stress testing and scenario testing are used to gain an insight into the uncertainty of our results.
Sensitivity testing is used to gain an insight into which parameters have the greatest effect on the model result.

Question 16

Steps in running a deterministic model

Answer 16

A deterministic model involves the following steps:

• specify the purpose of the investigation
• set the risk measure, eg Value at Risk (VaR)

-collect data, group and modify data
By data, we mean information relating to the policies (ie the mix of business and claims history) being modelled. It may be too time consuming to run each individual policyholder through the model. Therefore it is common to group together policyholders with similar characteristics into ‘model points’.
-choose the form of the model, identifying its parameters and variables
Parameters are any factors which would affect the decisions we make as a result of running the model. For example, if investment returns influence the premium charged then investment return is a parameter. Variables are the factors we are trying to test when running the model. For example, if we are setting the premiums for a product, then the premium is the variable.
-ascribe values to the parameters using past experience and appropriate estimation techniques, taking into account the risk measure being used and any correlations between parameters
The value assigned to any parameter is usually referred to as the assumption for that parameter. The full set of assumptions is referred to as the basis of our model.
- construct a model based on the expected cashflows
- check that the goodness of fit is acceptable and, if not, attempt to fit a different model
This can be done by running a past year and comparing the model with the actual results. run the model using the selected variables run the model using different parameters to check sensitivity. The model may also be run under different scenarios, ie testing the robustness of the results to many parameters changing at the same time, rather than changing single parameters in isolation

Question 17

Choosing a suitable density function for claim severity when using a stochastic model

Answer 17

-One way to choose a suitable distribution for claim severity would be to plot each of the observed claims in a bar chart (or equivalent), by size of claim:
-Now convert the left-hand scale so that the total area under the curve is 1, ie by dividing through by the total number of observed claims.
-Then select a function, y = f(x) which has a similar shape to our plotted data. This is the probability density function (PDF).
The following loss distributions are often used:
-Frequency: Poisson, negative binomial
-Severity: log-normal, Weibull, Pareto.
-We would then select a method of fitting to find parameter values for our chosen distribution. eg method of moments or method of maximum likelihood and then select the one that gives the best fit to our data.
-The method and parameters that have been fitted would be scrutinised using a number of statistical tests to determine how well the observed claims fit the modelled claims.
-Particular attention must be paid to the ‘important’ part of the distribution. For example, some classes of insurance have very skew claim amount distributions. Care should be taken that the fitted distribution has a sufficiently long tail.
-In these cases, a distribution such as the Pareto which has a relatively long, thick tail should be used. With excess of loss reinsurance care should be taken to ensure there is a good fit close to the excess point.
-If the goodness of fit of the model is not adequate, the distribution or the parameters should be altered until the fit is good enough

Question 18

Give an example of a financial guarantee that might arise in general insurance

Answer 18

A motor policy, which promises to refund the first year’s premium if the policyholder makes no claims in a five-year period.

Question 19

Advantages of deterministic models

Answer 19

There are a number of benefits of a deterministic approach:

• The model is usually easier to design and quicker to run.
• It is important to consider potential cause and effect relationships between risks.
• We may model such relationships better using deterministic relationships rather than relying on statistical dependence structures.
• It is more straightforward and, therefore, quicker to build a deterministic model than a stochastic one.
• By reducing the computational power necessary to generate many thousands of simulations, we can introduce more detail in other dimensions, such as detailed descriptions of reinsurance programmes or treatment of underlying risks.
• This may aid the intelligent selection of a limited number of scenarios.
• It could be more efficient than a stochastic model where we hope that the important scenarios appear amongst a larger number of randomly generated outcomes.
• We can integrate the capital model more closely with risk management, by extending the scenario modelling to scenario planning and ‘what-if’ analysis.
• It is clearer what economic scenarios have been tested. As discussed above, the disadvantage of this point is that it requires thought as to the range of economic scenarios that should be tested. Since only a limited number of economic scenarios will be tested, there is a danger that certain scenarios, which could be particularly detrimental to the company, are not identified.
• We commonly use stress and scenario tests for those risks that cannot easily be modelled quantitatively and where more subjective judgment is required. This allows us to concentrate more on the important areas of the distribution of outcomes for the key risks when a full specification of the distributions is subject to substantial potential error.
• By developing deterministic stresses and scenarios, we can help to link the capital model with the risk register, helping to integrate capital and risk management (this would also apply in a stochastic environment by considering each individual simulation as a scenario).
• It can be easier to communicate the results of stress and scenario tests to senior management, and to give them comfort as to the reasonableness of the overall capital value.
• A deterministic model is more readily explicable to a non-technical audience (eg users of results of the model and senior management), since the concept of variables as probability distributions is not easy to understand.
• It is important that users of the output understand the results from the model. By showing the effect of a limited range of stresses and scenarios – some of which may have been developed in consultation with those users – we can often make the results more comprehensible to them.
• Deterministic models are good for checking / validating results of a stochastic model.
• Deterministic model stress tests can be used in conjunction with the results generated from a stochastic model. This provides additional context to the stochastic results as well as providing either independent validation or appropriate challenge.

Question 20

Combining deterministic and stochastic approaches

Answer 20

In many cases a 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 these reasons, the stochastic approach is often limited to the economic assumptions, with demographic assumptions being modelled deterministically. It may be appropriate to use a blend of approaches:
-stochastic models for some risk categories
-stress and scenario tests for other risk categories
-ad-hoc methods for yet other categories.
In fact, there are various ways in which stochastic and deterministic approaches can be combined in a single model. For example, when modelling claim frequency and average claim size separately, we could:
-Determine the number of claims stochastically and associate this with a deterministic mean claim cost. Ideally the claim numbers would be divided into various homogeneous groups in terms of claim size.
-Determine the claim amounts stochastically for the (deterministically chosen) expected number of claims.
-Determine both claim amounts and numbers stochastically, using a collective risk model.

Question 21

3 Key objectives of any capital requirement regime

Answer 21

To ensure that:
Senior management focus on risk management - a risk management framework should be central to this process.
There is a link between risk and capital setting - in making an assessment of capital adequacy, a firm should:- identify the significant risks facing the business- assess their impact (both prior to and post having controls in place)- quantify how much capital is required
The capital model is being used within the decision making process - we demonstrate this through clear documentation of all prudential risks, processes and controls.
The overarching objective is PH protection and for ensuring solvency of insurers

Question 22

two broad approaches available to firms when producing a capital model,

Answer 22

1. stress and scenario tests
   - Stress tests consider different factors in isolation. Scenario tests consider several factors at a time.
   - This approach is a deterministic approach, where the user chooses which stresses and scenarios to test.
2. economic capital models (also known as stochastic models or dynamic financial analysis (DFA) models).
   - An economic capital model is a more integrated, holistic approach.
   - It systematically models the effects of many interrelated risk factors using simulation techniques.
   - Although these are significantly different in application, they are not in principle different, as a stochastic model is based on stress and scenarios weighted by probabilities.
   - In a DFA model, stress tests are generated automatically and often cannot be ‘seen’.

Question 23

Uncertainty in a capital model - sources(i)List the main assumptions that will be needed.(ii) Suggest other possible sources of error.

Answer 23

There are many potential sources of uncertainty in the parameters used to assess the capital required, but the most common is lack of credible relevant data.
The aim is for the data to represent a best estimate of current claims experience. So it is necessary to try to:
-group the data into broadly homogeneous groups
-make sure that the estimates for outstanding claims are best estimates, rather than deliberately cautious
-consider whether the emergence of any missing IBNR claims would change the shape of the distribution of claims
-convert into constant money terms and try to identify the sources of inflation
-if data is from more than one year adjust for trends in claim frequency, changes in policy conditions, large claims, catastrophes, etc.
-Where appropriate, we supplement the firm’s data by using external data.
-We should do additional sensitivity tests on the more uncertain assumptions.
(i)Assumptions:
-premiums in respect of business already written, and assumed premium income for future business
-claim frequencies
-claim amounts
-claims inflation
-claim payment profiles
-expenses (including fixed and variable expenses, and in respect of new and existing business)
-investment returns
-lapse rates
-business volumes and mix
-reinsurance costs, recoveries and payment profiles
(ii)Sources of error
-using an inappropriate model
-assuming distributions that are not appropriate to the business being modelled
-using the wrong parameters
-not correctly allowing for interdependencies between variables
-misinterpreting the output
-not checking the output for reasonableness
-placing too much reliance on the output

Question 24

Validation and testing in a capital model (4)

Answer 24

1. Stress testing
2. Scenario testing
3. Sensitivity testing
4. Back testing

Question 25

Validation and testing in a capital model 1. Stress Testing

Answer 25

Stress testing quantifies the effect of varying a single parameter, and is useful for understanding the potential impact of individual risks in isolation.Stress tests are a necessary tool in assisting an insurer to manage its risks and maintain adequate financial resources to deal with the risks to which it is exposed. We can use stress tests to identify and quantify the impact of different stress scenarios on an insurer’s expected financial position. Stress tests may be deterministic but we often develop them with probability distributions in mind.The stress tests should be calibrated to the appropriate risk measure (eg 99.5% VaR). We expect that prudent, well-managed insurers would undertake stress testing as a matter of good corporate governance, even if they also use more sophisticated models.

Question 26

List three reasons why stress tests may be needed when using a stochastic model.

Answer 26

Stress tests are needed:
to validate the model output for reasonableness
to help with calibrating assumptions
to test the impact on the results of uncertain assumptions.

Question 27

Validation and testing in a capital model 2. Scenario Testing

Answer 27

The next step, following simple stress testing, is scenario testing. This quantifies the effect of a change in a combination of parameters. It is useful for considering the combined effect of a number of risks and the cumulative effect of several different mitigating actions occurring at the same time.Scenario testing should look at the impact of unlikely, but not impossible, adverse scenarios.Actuaries should carefully consider which risks the firm is exposed to when designing the scenarios to be used as part of a capital assessment. They must also consider the relationships between different types of risks. They may actually model such relationships better using deterministic models rather than stochastic ones.We can also use stress and scenario testing to test the output from a stochastic model.

Question 28

Validation and testing in a capital model 3. Sensitivity Testing

Answer 28

Sensitivity testing is the process of testing the extent to which the results of a capital model change as a result of making a small change to an assumption in the model.
As a minimum standard and part of the validation and sign off process, all capital models must be subject to some level of sensitivity analysis.

Question 29

Purpose of sensitivity testing

Answer 29

The purpose of sensitivity testing is to identify the more sensitive assumptions in the capital model; that is, which assumptions, if changed, would have the greater impact on the results of the capital model.
We can then pay attention to the more sensitive assumptions.
We can pay greater attention to justifying and documenting the rationale for selecting values of these assumptions, because they are more critical to the results of the capital model.
So stress testing and scenario testing are used to gain an insight into the uncertainty of our results.
Sensitivity testing is used to gain an insight into which parameters have the greatest effect on the model result.

Question 30

Sensitivity testing - how to apply judgement and what is the extent of sensitivity testing to be done.

Answer 30

Applying judgement
When selecting assumptions, we must often make subjective judgements. (It is rare that we can choose assumptions on the basis of data analysis alone.) There might be several possible selections for an assumption, each appearing to be equally reasonable, yet we must decide which to select. In such a situation, it is informative to carry out a sensitivity test to find out the change to the capital model result from making a different (but equally reasonable) selection for the assumption under consideration.

Extent of sensitivity testing:

Sensitivity testing should be as comprehensive as possible. However, in complex models, particularly where there are a lot of classes of business being modelled, it may not be practical to sensitivity test each and every assumption. For example, it may not be practical to sensitivity test the assumed loss ratio standard deviation for each class individually. In this case, we could test the sensitivity of the capital model results to changing the loss ratio standard deviation of all classes by a fixed multiple.
When it is not practical to sensitivity test an assumption for each class individually, we could sensitivity test the assumption for the two or three largest classes

Question 31

Communicating the results of sensitivity analyses

Answer 31

The results of sensitivity analyses should be communicated to the senior management to ensure that they understand the uncertainty associated with the setting of parameters.

Question 32

Validation and testing in a capital model 4. Back Testing

Answer 32

Back testing is the process of comparing actual experience with model output.
It tests how well the model predicts the range of outcomes that actually occur.
Purpose of back testing:
Back testing is essential in order to ensure that the model is a sufficiently accurate reflection of the real world.
As discussed above, one of the key requirements for a good model is that it should adequately reflect the risk profile of the classes of business being modelled, and back testing is a key tool in the assessment as to whether this requirement is being met.
It may indicate shortcomings in the model that are not detected by other tests.
In the development of a model, many assumptions are based on an analysis of historical data.
There is therefore a presumption that past performance is a good indicator of future performance.
Back testing can be used to assess the validity of this assumption.
Analysing the results of back testing:
Any significant deviations between actual and predicted values that are identified by back testing will need to be analysed in order to understand the reason or reasons behind them.
An assessment will need to be made as to whether the deviation is simply a consequence of the expected random variation, a more systematic effect such as a permanent change in the environment, or an erroneous assumption or parameter.
If the deviation is deemed to be a systematic change, then changes may be made to the model or parameters.

Question 33

Documentation / Audit trail in a capital model

Answer 33

There should be a clear audit trail from the impact of any financial calculations to the relevant risk capital allocation in the capital model, whatever modelling approach is adopted.
We should justify and document the rationale for selecting assumptions, especially the more critical assumptions.
We should also document the known limitations associated with the selected assumptions, and any testing that has been carried out to assess the materiality of the assumptions.
Documentation should also justify the methodology selected and mention the alternatives considered and why they were rejected.
In addition, where we have considered particular risk issues, we should document the issues considered, how we considered them and the reasons behind the conclusions and findings.

Checks and documentation:
It is important that the data used to build a model of the business is as close to that in the actual business as possible.
Therefore we should reconcile the above data items to the firm’s financial statements and business plans, and document explanations of differences.
This will be possible where the model is being run at a date for which financial statements exist. However, this will not always be the case.

Question 34

2 Uses of data in a capital model

Answer 34

• to create the model of the business as at the run date of the capital model
• as inputs to selecting assumptions used to simulate the firms results and capital over the period covered by the capital model

Question 35

14 Main items of data needed to create the model of the business

Mnemonic - CRUDE CALF PUP
Claims payment profiles
RI programmes
Unpaid gross claims
Details of operational risks
Expenses

Credit exposures
Asset values
Large losses
Future reinsurance costs

Planned premiums (gross and net)
Unexpired premiums (gross and net)
Planned RI programmes

Answer 35

• Gross and net of reinsurance unexpired premiums at the as at date of the capital model, by class of business
• Gross and net of reinsurance premium planned to be written over the new business period to be covered by the model, by class of business.
• Gross unpaid claims at the as at date of the capital model, by class of business.
• Claims payment profiles (that is, sizes, frequencies and settlement patterns)
• Policy limits, and the likelihood of claims reaching such limits
• The costs of future reinsurance
• The reinsurance programmes to which gross unpaid claims are subject, each reinsurer’s participation on the programme and the extent to which claims paid have used up coverage available on these programmes.- The total reinsurers’ share of unpaid claims with, to the extent that it can be ascertained, each reinsurer’s share of the total.- The reinsurance programme to which claims arising from unexpired business is subject. If the actual programme at the time of carrying out the modelling is not known, a planned programme is needed.- The planned reinsurance programme to which claims arising from unexpired business is subject. If the actual programme at the time of carrying out the modelling is not known, a planned programme is needed.- The planned reinsurance programme to which claims covered by the model are subject.- The expenses of the firm.- The value of the assets by asset category- Credit exposures; e.g. broker balances- Details of operational risks - normally identified in a risk map.

Question 36

14 Possible assumptions used in capital modelling

Mnemonic - REDUCE DOG CRITIC

Answer 36

• Gross written premium
• Ceded premiums, including premium to reinstatement reinsurance and purchases of any reinsurance on a “losses occurring during” basis needed to cover claims occurring after the capital model date arising from business written prior to the capital model date
• Ultimate gross claims by class of business, split by attritional, large and catastrophe claims
• Claims payment profiles
• Gross reserve movements, by class of business
• Reinsurers’ share of gross ultimate claims
• The proportion of reinsurers’ share of gross claims that the firm is unable to recover, by reinsurer
• Reinsurance exhaustion
• Reinsurer downgrade assumptions (possible changes in default risk)
• Expenses, acquisition and administration
• Inflation and investment returns by asset class
• Operational losses
• Tax and dividends
• Relationships / correlations between different components of the model

REDUCE DOG CRITIC
RI share of ultimate claims and RI bad debt
Exhaustion of reinsurance and reinsurer
Downgrade assumptions
Ultimate gorss claims Ceded premiums
Expenses
Dividends
Operational losses
GWP
Catastrophe claims
REserve movements (gross) by COB
Inflation 
TAx
Investment returns split by asset class
Claims payment profiles

Question 37

Classification of business in a capital model

Answer 37

The classes of business to which premiums and unpaid claims are allocated in a capital model should be as close as possible to the classification of business used by the firm for internal management reporting, reserving and business planning.
This is not necessarily easy to achieve, as a firm may use different classifications of business for different internal reporting purposes.
We should aim in the capital model to use a business classification:
with which others in the firm will be able to identify
which is not at too high a level, such that the accuracy of the results from the capital model would be unacceptable (that is, too few groups so that some groups are too heterogeneous)
which is not so detailed that the model becomes overly complex, has overly long run times, or parameter error for many of the classes becomes unacceptably high because there is paucity of data in many of the classes.
It is important to communicate and document what business classification has been used in the model and why.

Question 38

Specific assumptions in a capital model for Gross written premium

Answer 38

Assumptions:Future rate changes and planned exposures, or gross written premium: If the capital model allows for more than one year of new business, it may be appropriate to simulate premium rates and exposures.If we allow for just one year of new business, it may be fit for purpose to assume written premiums will be as per the business plan (but with stress-testing to estimate the impact of variation from this).If we assume changes to rates and exposures, we should allow for the potential for these being interdependent; that is, an increase in the former could cause a reduction in the latter.New business rates and business retention rates.Data Needed:Historical rate changes. Underwriters’ views.The firm’s business plan. Historical retention rates.Historical new business rates.

Question 39

Specific assumptions in a capital model for Ceded reinsurance premiums

Answer 39

Assumptions needed:Reinsurance rates in respect of contracts covering unexpired and planned future written business or ceded premiums: If the capital model allows for more than one year of new business, it may be appropriate to simulate reinsurance rates and apply them to simulated gross written premium or to the amount of layers of reinsurance planned to be purchased.If we allow for just one year of new business, it may be fit for purpose to assume ceded premiums will be as per the business plan (again stress-testing may be employed).Changes in conditions from existing to future reinsurance arrangements.We should allow for the cost of reinstating reinsurance if gross claims exceed retention levels. (Reinstatement costs will be a term in the reinsurance arrangement.)The potential for market forces, say following a major loss event, increasing the costs of reinsurance on a ‘losses occurring during’ basis in respect of unexpired business at the capital model date by more than usually expected.Data Needed:Terms of reinsurance contracts entered into to which unpaid claims and claims from unexpired business are subject.Planned reinsurance programmes to which claims from business planned to be written will be subject.Firm’s business plan.Historical changes in reinsurance rates.

Question 40

Specific assumptions in a capital model for Gross claims (excluding catastrophes) by class of business, split by attritional and large claims

Answer 40

Assumptions needed:If simulated from loss ratios (applied to premium) – a distribution, mean and standard deviation of loss ratios for each class.If simulated from claim frequency (applied to an exposure measure) and severity – distribution, mean and standard deviation of frequency and severity for each class.Depending on the class and available data, it may be appropriate to simulate large claims from frequency and severity information and to simulate other (attritional) claims from loss ratios.Assumptions of loss ratios and claim severity should allow for future claims inflation.Data needed:Historical loss ratios, frequencies and severities.We should revalue these to the date of the capital model by adjusting for inflation, rate changes and changes to terms and conditions.Historical inflation rates.Input from other areas of the business (mostly underwriters and claims handlers), particularly their views on expected and extreme underwriting losses.

Question 41

Specific assumptions in a capital model for Catastrophe claims

Answer 41

Assumptions needed:Frequency and severity, or a distribution, of the firm’s exposure to each catastrophe event.For natural catastrophes, the assumed frequency should allow for events occurring in clusters due to climatic forces.The assumed severity should allow for potential inadequate resources being available to mitigate or repair damage.Data needed:For modelling catastrophe claims firms typically license catastrophe models provided by specialist catastrophe modellers (because they have more expertise and data than available to the firm). If so the data needed is exposures to catastrophe events related to unexpired business and new business covered by the model.Frequency and severity of historical catastrophes might be of use for validating results of the catastrophe modelling.

Question 42

Specific assumptions in a capital model for Volatility of gross outstanding claims

Answer 42

Assumptions needed:If the ultimate cost of outstanding claims are simulated at aggregate class level – a distribution, mean (usually best estimate gross reserve) and standard deviation of best estimate gross reserves for each class.If simulated at individual claim level – claim frequency for each past year (to be applied to exposure for that year) and claim severity for each past year. These assumptions should be affected by:  numbers of claims reported claims settledcase reserves plus part payments on non-settled claims, that is, a Bayesian approach.Claims inflation on unpaid claims in respect of business earned to the date of the capital model.Data needed:  Best estimate reserve.Carried reserve (if different). Historical reserve movements.Input from business (mostly claims, reserving and underwriting staff), particularly their views on extreme reserve movements.Number of reported and estimated IBNR claims.Case reserves.Past settled claims (revalued to the date of capital model). Historical inflation rates.

Question 43

Specific assumptions in a capital model for Reinsurers’ share of gross ultimate claims occurring post model date and gross reserve movements, by reinsurer

Answer 43

Assumptions needed:If it is necessary to estimate this component, (ie because it is not practical to apply reinsurance programmes to simulated gross claims and gross reserve movements), then the assumption needed is ratios of net to gross claims for each class.Data needed:Historical ratios of net to gross claims by accident / underwriting year for each class.Reinsurance programmes.Input from business (mostly underwriters) on their views of ratios of net to gross claims in respect of future earned and written business.Data

Question 44

Specific assumptions in a capital model for Proportion of reinsurers’ share of claims not recovered

Answer 44

Assumptions needed:Probability of default of the reinsurer and the expected loss to the insurer if the reinsurer does default.These assumptions should take into account the potential of major industry wide losses causing increased reinsurer defaults and amount of each resultant loss.Data needed:Historical reinsurer failures. Credit ratings for current reinsurers.Views of the firm’s reinsurance department and brokers. Data from rating agencies.

Question 45

Specific assumptions in a capital model for Expenses

Answer 45

Assumptions needed:Proportion of acquisition costs to gross written premium, by class of business.Administration costs. Inflation.Data needed:The firm’s business plan. Historical inflation.

Question 46

Specific assumptions in a capital model for Inflation, expected returns, volatilities and dependencies between the modelled economic series

Answer 46

Assumptions needed:Assumptions needed depend on the method used to simulate these and are outside the scope of this Core ReadingData needed:Depends on methods used.Data usually needed includes: historical inflation, government bond yields (by duration and territory), corporate bond yields (by duration and territory), equity returns and exchange rates.

Question 47

Specific assumptions in a capital model for Operational losses

Answer 47

Assumptions needed:Risks that have the potential to give rise to operational losses.The likelihood of each risk materialising. Expected costs if the risks materialise. The variability around these costs. A distribution of losses arising from each risk.The extent to which multiple risks might materialise during the period covered by the capital model.Data needed:The firm’s risk register.Input from the business (mostly risk and operations management areas of the business), particularly their views on the likelihood of operational losses occurring and the magnitude of extreme operational losses

Question 48

Specific assumptions in a capital model for tax

Answer 48

Assumptions needed:The tax rate on profit retained in a financial year. The tax offset from carried forward losses.Data needed:Rules from the local taxation authority.The

Question 49

Aggregation methodologies for bring together all risks in a capital model

Answer 49

It is likely that we will use more than one approach, model or stress test in considering all risks that the company faces. We should bring together the results of these different elements and produce a single result: one that can be compared both to available capital and other metrics.In carrying out this aggregation, we should consider both the method used to combine the results of different elements and the level of correlations that might exist between these elements.In a stochastic model, we typically aggregate risks by determining joint probabilities using Monte Carlo simulation techniques. The method should capture the correlation between variables in the more extreme stresses that are likely to be of interest to the regulator.Correlations are by no means constant across the whole of a joint distribution. It is likely that results are more highly correlated under extreme scenarios, ie in the tails of the distribution. The capital model should aim to capture these effects.We can achieve this using an appropriately heavy-tailed distribution together with a correlation matrix, or using a copula (discussed in Chapter 22).In some cases, modelling underlying drivers (for example, loss severity directly linked to economic factors) may be applied to capture correlations.If we have not fitted statistical distributions, or we cannot determine a joint distribution, then we should use more approximate methods of combination. Across different risks, it may be suitable to use a variance / covariance approach to aggregate stand-alone risk measures, and apply realistic disaster scenarios that assess the reasonableness of the calculated overall capital charge.In either case, we should state and clearly justify the dependency assumptions between various categories such as the following:  underwriting classes of business (for example, between motor and household business)risk types (for example, between underwriting risk and reserving risk) years of accountlegal entities within a group, including between entities in different territories.Correlation assumptions will typically be subjective and based on a high level of judgement. It is good practice, when justifying selected correlations, to consider historical events, the range of correlations that could be deemed as equally valid, and the consequent impact on the capital models of such correlations.

Question 50

Data needed in capital model - CASHFLOWS

Answer 50

“Data will be needed in order to predict all cashflows in terms of both timing and amount. These cashflows include:
     premiums
claims – this data will often need to be split into unexpired risks, IBNR, etc reinsurance premiums and recoveries expenses
investment income.
Data will also be needed on risks, as these may affect the likelihood of cashflows occurring, or lead to additional cashflows.
Information will be needed on both existing business and future business. The main items of data needed to create the model of the business are: 

Gross and net of reinsurance unexpired premiums at the as at date of the capital model run, by class of business.
Gross and net of reinsurance premium planned to be written over the new business period to be covered by the model, by class of business.
(A theoretically preferable alternative to gross written premium is planned written exposures and gross premium per unit of exposure. However, it may not be practical for the business to have this level of data available.)

Estimated future gross claims relating to claims that have occurred at the as at date of the capital model run, by class of business.
The Details of large individual reported but not settled claims at the as at date of the capital model run and whether they are likely to reach limits, and the associated limits.
Claims payment profiles (that is, sizes, frequencies and settlement patterns).
Exposures by location and peril to assist in catastrophe modelling (eg exposures to European windstorm, or UK flood, or Californian earthquake). Examples of measures of exposures include: sum insured, maximum possible loss, probable maximum loss, estimated maximum loss. These exposures should relate to unexpired business and to the new business period.

The reinsurance programmes to which gross outstanding claims are subject. Each reinsurer’s participation on the programme, and the extent to which claims paid have used up coverage available on these programmes. (In practice some of this data may not be available – see next bullet point below.)

The total reinsurers’ share of gross outstanding claims with, to the extent that it can be ascertained, each reinsurer’s share of the total.
(This is a theoretically ‘second best’ alternative to using the actual reinsurance programmes to which gross outstanding claims are subject. However, there may be situations where this is the only option, or only practical option, available. For example, records of reinsurance programmes covering very old years may have been lost. Also it may be that even if the actual reinsurance programmes are fully available, using them in the model may make the model overly complex or cause it to have overly long computer run times.)
  
The reinsurance programme to which claims arising from unexpired business is subject. If the actual programme at the time of carrying out the modelling is not known, a planned programme is needed.
The planned reinsurance programme to which claims relating to the new business period are subject, and the cost of that reinsurance.
The extent to which occurrence-based programmes may overlap with exposure periods that would not otherwise be included in the model. It may therefore be necessary to include allowance for such exposure to ensure the application of the reinsurance on modelled exposures is accurate.
 
 
The expenses of the firm.
The value and details of assets by asset category (other than the reinsurers’ share of technical provisions, as this is covered above).
Credit exposures, for example, broker balances.
Details of operational risks – often identified in a risk register. “

Question 51

Sources of error in a capital model

Answer 51

“using an inappropriate model
assuming distributions that are not appropriate to the business being modelled using the wrong parameters
not correctly allowing for interdependencies between variables misinterpreting the output
not checking the output for reasonableness placing too much reliance on the output”

Question 52

Differences between stochastic and deterministic - Advantages and disadvantages

Answer 52

“Merits of deterministic models:
1. Deterministic models are easy to develop and implement
2. Less costly and relatively lesser effort.
3. It is less time consuming to build or use.
4. Can be understood easily
5. Communication of results to non-technical audience is also easier
6. Using intelligent selection of stresses and scenarios, critical combinations can be evaluated
7. Where judgement is required, in a deterministic model, it is flexible to implement such modifications
8. It is easy to link the model to the risk register and hence use it for decision making. I.e. risk management and capital planning integration is easier
9. It is used in validating the reasonableness of results of a stochastic model.
10. Uses lesser computational power
11. Modifications to the model is not overly complex.
Merits of stochastic models:
1. Evaluates a wider range of scenarios some of which could not have been easy for deterministic models to consider or come up with.
2. Produces a probability distribution of outcomes/results
3. We can rank the various combination of outcomes and the results against a risk measure to evaluate capital requirements
4. We can use various risk measures on the same results
5. May be required by regulators as a satisfactory capital model methodology.
6. Useful to calculate confidence intervals
7. It is useful to consider knock on consequences of severe economic or insurance risks”

Question 53

Stress Testing and its use

Answer 53

“Stress testing quantifies the effect of varying a single parameter. It is a preliminary step in performing a scenario test
Stress tests may be deterministic but they are often developed with probability distributions in mind.
[
Stress tests are a necessary tool in assisting an insurer in managing its risks and maintaining adequate financial resources to deal with the risks to which it is exposed. They can be used to identify and quantify the impact of different stresses on an insurer’s expected financial position.
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Stress tests are often used
for modelling future catastrophe claims
for analysing the impact of unlikely (but not impossible) adverse stresses of one parameter
to perform additional tests on uncertain assumptions
as a matter of good corporate governance, even if more sophisticated models are also used
to improve an insurer’s understanding of the potential impact of individual risks in isolation
to identify and investigate the risks incurred by extreme movements in parameters. “

Question 54

Scenario testing and its use

Answer 54

“Scenario testing quantifies the effect of a change in multiple parameters. It is a natural extension to a stress test.
to consider the combined effect of a number of risks and the cumulative effect of several different mitigating actions occurring at the same time
to allow for interactions between variables that would not be considered under stress tests
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within stochastic models to explore combinations of randomly-generated values for the key parameters by exploring scenarios that you might otherwise not have imagined to be important
  
to help to understand the relationships between different types of risks as a communication tool, to make the range of results more comprehensible to help with scenario planning and ‘what-if’ analysis
Stress and scenario tests can also be used: 
–
as a broad approach when using a capital model: –
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stress and scenario tests are used explicitly with deterministic models
stress and scenario tests are used implicitly with stochastic models – they are effectively based on stresses / scenarios weighted by probabilities …
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… in a DFA model, stresses and scenarios are generated automatically and often cannot be ‘seen’
to validate the output from stochastic models for reasonableness to help calibrate assumptions in stochastic models
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to provide a transparent link between the capital model and the insurer’s risk register [1⁄2]
to provide a useful mechanism for entering into a dialogue with senior management and give them comfort as to the reasonableness of the overall capital value
for those risks that cannot easily be modelled quantitatively and where more subjective judgement is required …
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… this allows a better degree of focus on the more important areas of the distribution of outcomes for the key risks, when a full specification of the distributions is impossible”

Question 55

“Sensitivity testing and its uses

Practice Q.20.4”

Answer 55

“Sensitivity testing is the process of testing the extent to which the results of a capital model change as a result of making a small change to an assumption in the model.
Sensitivity testing should be used: 
to identify the more sensitive assumptions in the capital model, ie which assumptions, if changed, would have the greater impact on the results of the capital model …
… attention can then be focused on the more sensitive assumptions, and greater attention can be given to the justification and documentation of the rationale for selecting values of these assumptions
  [1⁄2] [1⁄2]
as part of any stochastic modelling process (to run the model using different distributions and parameters to check sensitivity)
to help select assumptions – where there are several possible selections for an assumption that each appear equally reasonable, a sensitivity test may be carried out to find out the effect of making different selections for the assumption under consideration
  as a minimum standard and part of the validation and sign off process
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to help understand the variability and uncertainty in parameter values so that these can be communicated to the board and senior management.”