Chapter 22: Capital Modelling - Allowance for diversification

Question 1

State why diversification effects arise.

Answer 1

Diversification effects arise because the various risks from a company’s operations are not perfectly correlated.

Since risks are not perfectly correlated, the sum of the standalone capital requirements for each risk will usually be greater than the capital requirement for all risks combined. In other words, an adverse event (eg a catastrophic claim event) should not affect diversified policies simultaneously (by definition), and so some credit can be given for the level of diversification. This will result in a lower overall capital requirement.

Question 2

State why correlation effects arise.

Answer 2

Correlations may arise because specific types of risks affect different segments of the company’s overall portfolio. Correlations may also arise between the different types of risks.

Correlation can occur between:

• risks in the same class – this is often called risk clash
• risks in different classes – this may be called class correlation.

Question 3

Capital modelling investigations

Mnemonic: When to allow for diversification (SPAR SPA)

Answer 3

Capital modelling investigations include:

1. assessing solvency capital requirements
2. allocating the capital held between classes, products or individual policies for
   
   • performance measurement,
   • pricing,
   • business planning and
   • strategy setting
3. reinsurance purchasing
4. asset allocation studies
5. studies of enterprise level risks such as credit risk and operational risk.

SPAR SPA:

1. Solvency assessment
2. Performance measurement
3. Asset allocation
4. RI purchasing
5. Strategy setting
6. Pricing
7. Assessing Enterprise level risks

Question 4

Allocating the total capital held between classes, products or individual policies may be necessary for (3)

Answer 4

• performance measurement (to accurately assess the profit as a percentage of capital required)
• business planning and strategy setting
• pricing (Premiums include a capital / profit loading reflecting the cost of capital)

Question 5

Capital Modelling investigations - assessing solvency capital requirements.

Answer 5

1. The assessment of solvency capital requirements is a ‘ground-up’ exercise and could be at:

• • product level
• • class of business level, or
• • whole company portfolio level.

2. The assessment would also be net of reinsurance, with reinsurance being modelled either explicitly or implicitly.
3. We assess the sum of the capital requirements for the various risks from each portfolio segment separately
4. This sum is typically more than the capital requirement from all risks analyzed together for the whole portfolio because of diversification.
5. However, there can be negative diversification or accumulation of risk because of operations risks associated with group structures and policies. There risks are dealt with by the enterprise risk management processes and they can be a source of negative diversification benefit leading to an additional capital requirement.
6. In quantifying diversification effects, we will need to make assumptions about the extent of applicable dependencies (including correlation coefficients).
7. To do this, we need to understand the various assumptions underlying the modelling of the risks in each portfolio and how they could interact when combined. This is necessary before we can assess the capital requirements of the overall portfolio.
8. Dependencies can occur between: - risks in the same class – this is often called risk clash and risks in different classes – this may be called class dependencies.
9. The allowance for dependencies within the stochastic model can either be implicit within the modelling process or explicit through the use of dependency assumptions (including correlation coefficient matrices).
10. An implicit allowance might involve reducing the variability assumption when fitting a distribution.

Question 6

Explain how operational risks may affect the diversification adjustment in a capital model.

Answer 6

A company may be exposed to operational risks associated with its corporate group structures and policies. These risks, which are likely to be targeted by the group’s enterprise risk management process, may be a source of negative diversification effects, that is, require additional capital to be held.

Question 7

How can dependencies in a capital model be addressed? State the two methods by which such dependencies / correlations might be allowed for in a capital model.

Answer 7

In quantifying diversification effects, we will need to make assumptions about the extent of applicable dependencies (including correlation coefficients). To do this, we need to understand the various assumptions underlying the modelling of the risks in each portfolio and how they could interact when combined. This is necessary before we can assess the capital requirements of the overall portfolio. Dependencies can occur between:   risks in the same class – this is often called risk clash risks in different classes – this may be called class dependencies.

The allowance for correlation within the stochastic model can either be implicit within the modelling process or explicit through the use of correlation matrices.

An implicit allowance might involve reducing the variability assumption when fitting a distribution.

Question 8

Capital allocation for performance measurement is a ‘top-down’ exercise. Explain what is meant by a ‘top-down’ exercise in a capital model and why it would be appropriate here

Answer 8

A ‘top-down’ exercise is the opposite of a ground-up exercise. It involves starting with an overall position and subsequently breaking it down into individual components. In this context, the overall capital required would be modelled and the results then broken down (ie allocated between) different classes of business or products. If a ground-up approach was used, then capital would be calculated for each policy (or group of policies), but since there would be no allowance for diversification benefits, capital would be overstated for each policy (or group of policies) and so profits understated

Question 9

Methods of allocating capital

Answer 9

• a marginal capital method
• the Shapley method

Question 10

Capital allocation:
Percentile method

Answer 10

This method is applied to the output of a stochastic model.
We would take the simulation which determines the capital requirement and assess how the loss in that simulation was made up.
E.g., if we ran 10 000 simulations, the simulation giving the 50th worst results (1 in 200 loss) would be analysed to assess what the underwriting result was in each class, what the investment return was, etc.

In practice the insurer may look at a number of simulations on either side of this (to reduce randomness).

Question 11

Explain why a lower percentile risk measure may be used when capital is alocated to different classes/products?

Answer 11

When we allocate the capital, we may use a different risk measure from that used in assessing the capital requirement. For example, a group’s solvency capital requirement may be based on a target percentile in the tail of the underlying aggregate loss distribution. But we may allocate the diversified capital down to individual classes of business or products for a company in the group with reference to a lower percentile, to prevent over-allocation to catastrophe-type business.

Question 12

Marginal capital method

Answer 12

We allocate capital with reference to the marginal capital requirements of each segments.
I.e. we consider the additional capital that would need to be held if the element was to be added to the business.

Question 13

Shapley method

Answer 13

An extension to the marginal capital method, based on game theory.

We allocate the capital with reference to an average of the marginal capital requirements, assuming that the class under consideration is added to the overall portfolio first, second, third, …

I.e. for any one class of business, calculate each capital amount that would be required if we add the class to the overall portfolio first, second, third, etc. 
We then take the ACTUAL CAPITAL REQUIREMENT AS THE AVERAGE of these amounts 

The advantage is that the allocated capital is NOT DEPENDENT ON THE ORDER in which it was allocated to each class.

Question 14

Proportions Method

Answer 14

May also allocate economic capital to each class of business in proportion to its contribution to the risk metric on a standalone basis.

We would first calculate the required capital separately for each line of business, and work out what proportion of each of these is of the total of the individual capital requirements.

We then calculate the aggregate capital requirement, allowing for any diversification benefits, and allocate this to each class, according to the proportions calculated.

Question 15

Outline the factors to consider when choosing the capital allocation method.

Answer 15

In general, when we select the method for the allocation of capital across lines of business, we should have regard to the use to which the results will be put and consider desirable properties of the results, such as stability over time. There is not necessarily one method that is best suited in all cases. Typically, we would compare the results from several methods of allocation. We would use our judgement when recommending or setting the final allocation.

Question 16

Suggest reasons why a company may hold more capital than its capital model would suggest.

Answer 16

1. to increase business volumes by providing a high degree of confidence that it can meet its obligations to policyholders
2. to benefit from cheaper financing terms
3. to improve its standing in the market in the eyes of investment analysts
4. to reduce the risk of having to call on shareholders or members for further finance if losses are greater than expected
5. to smooth dividends to shareholders, who prefer less volatile returns
6. to meet the requirements of other stakeholders, eg debt providers, whose interests may be subordinated to those of policyholders
7. to enable it to develop the business, eg launching new products or for financing the takeover of a rival insurer to enable it
8. to undertake a more aggressive investment (and hence pricing) strategy
9. a company may hold more capital than its capital model may suggest is needed. Amongst other reasons, the company may require the excess capital to support its credit rating or choose to hold surplus capital so that it can take advantage of business opportunities in the future. With the publication of SCR ratios, there is now also market pressure to be seen to have ratios within a certain range. _

Question 17

3 Types of capital to allocate

Answer 17

• Total capital
• Economic capital
• Excess capital

Note: Total capital = economic capital + excess capital

Question 18

Capital Modelling investigation - cost of allocated capital for pricing

Answer 18

1. the total capital needs to be allocated across each individual policy in order to determine the theoretical (or ‘technical’) price that should be charged for each policy.
2. We should consider how the allocated capital compares to that needed on an underwriting year basis, which may be subject to different diversification effects. Capital modelled on an underwriting year basis may be subject to different diversification effects because claims in any one underwriting year cohort can arise from incidents that occur in different accident years. They can therefore be subject to different (diversifying) claims environments.
3. For expanding or contracting portfolios, the capital needed to support the reserves that would be held until all claims from the specific underwriting year are fully paid may differ from the reserve risk component of the allocated capital. This would be based on the size of the total reserves brought forward. So, the capital needed to support claims from a specific underwriting year will form just a part of the total reserves. The total reserves will be the sum of the reserves for each underwriting year. For a stable portfolio, each underwriting year reserve would form a similar portion of the total reserve. However, for an expanding (contracting) portfolio, the most recent underwriting year reserves will form a relatively large (small) portion.
4. Similar differences may arise with the underwriting risk component. Where material differences arise, we should discuss with the underwriter and the management of the company the implications of setting pricing loads with reference to the allocated capital.
5. The effect from diversification in pricing would be partly offset by the cost of any excess capital.
6. Starting from a well-diversified position, in order to diversify further, we may need to sell more business. This will have offsetting effects on the capital requirements:  higher business volumes will increase the amount of capital required, eg to meet new business strain; typically, capital gets more and more expensive to raise, so the cost of capital may increase  higher business volumes should improve diversification, which will reduce the capital required.
7. In determining a ‘technical price’ benchmark, we should consider if the relevant load for the cost of capital should be based on the diversified or non-diversified capital requirement. Allowing for diversification will reduce the capital required and so should reduce the premiums, which will improve competitiveness. However, we cannot necessarily guarantee that the business sold will be diverse, so it may not be prudent to allow fully (or at all) for this.

Question 19

Capital Modelling investigation - business strategy.

Answer 19

As a company will be looking to optimise the use of its capital, in any modelling investigations, we should allow for:

1. expected changes in the mix of business by class
2. the marginal effects of adding new classes or products, and
3. the effectiveness of the company’s reinsurance protection programme in reducing the volatility of the retained risks.

In cases where new classes of business or products are being considered, we will investigate the relevant additional solvency capital requirement and the corresponding diversification effects on the overall company’s portfolio. We should consider carefully the assumptions made for the new class. For a new class, we will need external data or prior experience to parameterise the modelling assumptions.

Question 20

Capital modelling investigations for reinsurance purchasing

Answer 20

1. We often investigate the capital requirement of a particular class of business or product alongside the relevant reinsurance protection requirement. Holding capital and purchasing reinsurance are, to some extent, alternatives. An insurer with high levels of free capital will be able to cope better with volatile claims experience, single large claims or accumulations / catastrophes, and so will have less need for reinsurance.
2. When we advise a company on optimising its reinsurance protection programme, we must consider the effects of different reinsurance strategies on the company’s overall solvency and economic capital requirements
3. The modelling required will depend on the reinsurance that we are considering:
   
   1. If the reinsurance covers just single classes, we will need to choose the assumptions within the modelling of the gross losses for that class so that we can evaluate the reinsurance recoveries.
   2. If the reinsurance covers multiple classes, we will also need to consider how to allocate the recoveries back to the individual contributing classes.
   3. For whole account reinsurance protections, it is again important to allocate the recovery back to individual classes.
4. In addition to allocating the recoveries from these policies, we can also use the stochastic model as a basis for allocating the reinsurance premium at a class level. Depending on the type and amount of reinsurance purchased, this will change the diversification of the retained (net) business. Drivers of such change will include:
   
   1. tail diversification (insurance risk)
   2. mix of risk by type (for example, less insurance risk but more credit and liquidity risks).
5. By purchasing reinsurance, insurance risk (of the retained business) should be reduced, in particular in the tails, ie reinsurance is likely to cover extreme risks. However, with reinsurance, the insurer will be exposed to reinsurer default, hence credit risk may increase.

Question 21

Capital Modelling investigation - Asset allocation studies

Answer 21

The ESG used within the stochastic model will enable us to analyse the impact of different asset allocations. The ESG will ensure the consistency of assumptions throughout the modelling process.

Using the stochastic model, we can assess the impact of the company’s asset portfolio on the overall capital requirement. The model will help us to consider questions such as:

1. Do we have enough liquidity in our portfolio?
2. Should we invest in bonds that match longer-tail liabilities by duration?
3. What should the security profile of bonds be, given the liability profile and the current investment environment? For example, how much government-backed, how much can be AAA, AA and so on?
4. Should we consider equities / property for the longest liabilities or the excess capital?

ESG models can incorporate correlations between different variable, and can therefore allow for any diversification benefits of the asset allocation. 1

Question 22

Capital Modelling investigation - enterprise level risks

Answer 22

In addition to insurance risk, the company faces a range of other risks that we need to consider within the stochastic model. We should consider risks such as liquidity, market, credit and operational risks.

We should model each of these, either implicitly or, more likely, explicitly. We can then assess the impact of each risk on the overall capital requirement. We then need to consider how to allocate any change in the economic capital assessment as a result.

Correlation assumptions in a stochastic model should allow for the diversification effects of these risks, for example, by an adjustment to the variability assumptions or by an explicit assumption to the correlation matrix. Alternatively, a deterministic capital model should investigate a range of scenarios in order to assess the impact on capital of diversification between these risks.

Question 23

Setting diversification assumptions in a capital model

Answer 23

In capital modelling we model the diversification effect between the modelled components of the business (eg between the loss ratios of classes on new business). Diversification occurs where one class experiences higher than expected losses when the other class experiences lower. When modelling diversification effects we need to make assumptions about the dependencies between the modelled components in question. Generally, the greater the diversification effects between modelled components of the business, the weaker the dependency between those components. Dependency assumptions can be:

1. implicit (ie where the modelling of the components of the business are driven by a common risk factor – eg the modelling of loss ratios for new motor and liability business both use a common claims inflation factor)
2. explicit (ie where components of the business are modelled such that the dependency between the modelled components exhibit explicit dependency assumptions selected by the modeller).

When modelling diversification effects both implicit and explicit dependency assumptions can be used. In most capital modelling work an explicit dependency assumption has two parts. The first part is a dependency structure, and the second part is the parameters used in that structure. However, there can be cases where an explicit dependency assumption has no dependency structure part.

The parameters used in the dependency structure are assumptions of, in most cases, the correlation between the modelled components (ie an average measure of the strength of association between the components). In this case the parameters are either the actual correlation coefficients assumed or a parameter that drives the correlation.

Question 24

Explain how a lack of data can increase the difficulty of modelling diversification effects.

Answer 24

Ideally when we estimate the effect of diversification, we would begin by gathering appropriate historical data and applying statistical tests to measure correlation. However, for many risks there may be insufficient historical data for analysis. Or where such data exists, it may not be sufficient to assess correlation around the tails of the distributions.

Question 25

Describe how correlations can be assessed in a capital model

or how to set diversification assumptions?

Answer 25

(1) Ideally when we estimate the effect of diversification, we would begin by gathering appropriate historical data and applying statistical tests to measure correlation. (2)for us to consider causes and effects to assess the key drivers of correlation (3) to consider the likely scenarios in which the particular risks will occur at the same time.

Question 26

Implementing diversifications in a capital model

Answer 26

There are a number of ways in which we can allow for diversification effects in capital modelling work. These include:

1. linking assumptions (also known as a ‘driver’ dependency)
   
   1. If two assumptions within a capital model are explicitly linked through formulae, this will introduce an implicit dependency between them. An example would be to express a claims inflation assumption as a margin above price inflation. This is often referred to as a ‘driver’ dependency. Another example, within motor insurance, would be to link the inflation assumption for bodily injury claims to an index of average earnings.
2. incorporating an explicit dependency assumption between marginal distributions in a stochastic model, eg:
   
   1. correlation factors or matrices
   2. copulas
3. using the standard approach for summing variances in a deterministic model.

Question 27

Outline why explicit correlation assumptions might be preferable to implicit ones

Answer 27

For some assumptions, it may be appropriate to apply an explicit correlation factor (or a correlation matrix for multiple parameters) within a capital model. Simple correlation factors give rise to symmetric dependency structures. For some risks, the correlation between them may vary in a more complex way (for example, having a greater level of dependency in the tail of the distributions than around the mean). _

Question 28

Explain what is meant by a copula

Answer 28

A copula is a mathematical relationship between the individual distributions of random variables and the joint distribution of those variables. Copulas allow us greater flexibility when we model multiple dependencies than single correlation factors allow us. I

Question 29

Explain the impact of the choice of copula when modelling the relationship between variables.

Answer 29

There are many different copula structures based around various probability distributions. They differ primarily in the part of the distribution in which they apply the strongest relationship between the two variables.

Question 30

Gumbel copula - when is it used

Answer 30

Question 31

Explain how a deterministic allowance for diversification could be made.

Answer 31

Question 32

Explain the importance of using actuarial judgement when choosing correlation assumptions

Answer 32

Given the lack of historical data on correlations between risks under stressed situations, we will have to apply our judgement when we set correlation assumptions. We should have regard to:  the purpose of the capital modelling work  the available data (if any), and  the sensitivity of the results to the level of diversification chosen. In cases where the advice given may vary significantly depending upon the correlation assumptions underlying the diversification adjustment, it is important that we communicate the relevant uncertainty. It may be helpful that we provide results on alternative assumptions, to assist the users of the capital modelling results to understand the financial implications of deviations from the assumptions made. _

Question 33

S2013, Q7:

Answer 33

The amounts of capital required are significant
..which indicates that the syndicate is probably large…
..or writes business that is particularly risky
In particular reserving risk is very high..
..possibly due to writing a significant amount of long tailed business
..and/or reflecting that the syndicate has been in operation for some time
Underwriting risk is also material
…so the syndicate intends to continue to write a significant amount of business
..perhaps increasing proportion written in capital intensive areas such as
catastrophe reinsurance
..though given size of reserving risk could have been larger in the past and
shrunk
The aggregate insurance risk of 550 is not possible if other figures are correct
(and assuming that insurance risk comprises only reserving and underwriting
risk)
..even if risks were fully dependent would give maximum capital requirement
of 525.
..indicating that the correlations in the model are incorrect or some other error.
Or that there are other contributing components not shown in the table (e.g.
catastrophe risk)
Market risk is relatively low…
..as would typically be the case for a Lloyd’s syndicate.
..indicating investments are probably low risk such as gilts rather than equities

Counterparty default risk is unusually high
..suggesting that significant reinsurance
..from lower grade reinsurers may be utilised
..and/or significant funds remain with third parties (brokers/loss funds at
MGAs)
..and/or investment counterparties may be weak.
There is some overlap between market risk and counterparty risk so relativities
may or may not be odd.
Operational risk appears to be low relative to other risks
..possibly due to inadequacies in modelling for this risk
As expected, liquidity risk is fairly low as secure investments…
…and future premium income from new business.
The total syndicate capital requirement seems quite low compared to its
constituent parts
..again suggesting issues with correlation assumptions or application
..or alternatively that the total figure is discounted whilst components are not.
Calculation of fully diversified syndicate capital requirement (£433m) (square
root of sum of squares of each assumed independent risk component)
…compared to assuming complete dependence (£685m)
Note: other assumptions could be used and hence different answers obtained
and should be given marks if reasonable explanation (e.g. using the £550

aggregate risk figure given if assumption that includes other components).
No group risk indicates that syndicate is stand-alone (unless omitted, given
other problems with the figures).
It would be useful to know on what basis the figures have been calculated
..and the mix of business written

Question 34

Q5, A2015:

Answer 34

Comment that the figures may be pre/post diversification credit.
Property cat has a relatively high capital allocation relative to premium written..
..may reflect concentration risk by location
..reasonable as exposure to hurricanes/earthquakes means significant catastrophe risk.
..and therefore results can be very volatile thus capital for underwriting risk..
..although unlikely to have large reserves so reserving risk will be small.
Allowance for demand surge or liquidity risk
Professional indemnity also has quite a high capital requirement relative to premium
..likely to be dominated by reserving risk as long tailed business
..may also be allocated more market risk as benefits from investment income.
GL also long tail but only written for one year so unlikely to have high reserves
..however as new class of business may be allocated more capital for new business risks
i.e. uncertainty around pricing, lack of historic data etc. (underwriting risk).
Pet insurance meanwhile has been written for a number of years..
..and normally has low reserves and fairly stable loss ratios..
..all which contribute to its overall low capital requirement.
Motor class written for a number of years so likely to have significant reserves
..possible risk of legal changes increasing cost of settling claims..
..may be considering aggregation with GL as both exposed to bodily injury claims.
Motor class is written direct so lower credit risk than brokers holding balances.
No information on past premium and mix - historical premium may be different to expected future premium
Perceived political risk

Additional Info
Information on level of reinsurance purchased for each unit would be needed
..to comment on respective levels of credit risk
Growth in class of business over time

Relative expense levels by class of business
Other capital allocation methods may give very different answers…
…as may a different actuary using different judgements.

Question 35

A2016, Q2

Answer 35

Question 36

(i) Define diversification in the context of capital modelling. (2)
(ii) Discuss applications of capital modelling where diversification needs to be considered. (10)
(iii) List dependencies that should be considered in a typical capital model, giving an example of the drivers of dependency in each case.(6)

Answer 36

(i) Diversification

1. Diversification arises because risks are not perfectly correlated.
2. Therefore the poor experience of one risk is offset by better experience on another risk.
3. In the context of capital modelling, diversification benefits arise because the capital required for the company as a whole will be less than the sum of the capital requirements for each risk / portfolio segment separately.

(ii) When diversification should be considered in capital modelling
* Assessing solvency requirements

The solvency capital requirement would be assessed on a ground-up basis, allowing for correlations within / between classes. A correlation matrix may be used to consider negative diversification, eg capital to cover group risk.

• Capital allocation for performance measurement purposes

Capital may be allocated between classes, products, or individual policies. When capital is allocated to each segment, an implicit diversification benefit is made. A different risk measure may be used to allocate capital than the one used to assess the capital requirement. This would help prevent over / under-allocation to catastrophe-type business. Judgement should be used to decide which method of allocation to use. Consideration should be given to how the results will be used. We should also consider the stability of the results over time. Economic capital should be allocated in proportion to each class’ contribution to the risk metric. Excess capital should probably be allocated in the same proportion as the risk-based capital.

• Capital allocation for pricing purposes

Capital should be allocated to each individual product so that the technical price reflects its individual risk profile. Capital should be allocated on an underwriting year basis because the diversification effects may be different for each. The portion of capital allocated to reserve risk may be different to the capital required until all reserves are fully paid. Consult with underwriters / management to understand the impact of loading prices with allocated capital.

The company should consider whether the cost of capital loading should allow for diversification effects. Diversification effects will be offset in part by the cost of excess capital.

• Capital allocation for business planning

We should allow for the diversification effects of writing products / changing the mix of business. Allowance should be made for different types of correlations

• Reinsurance purchasing

The level of diversification will depend on the type / amount of reinsurance. Reinsurance will increase tail diversification, since it is likely to cover extreme risks. Reinsurance will increase credit risk. Asset allocation studies Asset allocations can be analysed with an ESG, which can allow for correlations between variables. This will help ensure consistency and optimal asset allocation.

• Studies of enterprise level risks

These can be modelled implicitly / explicitly. The capital model should allow for the joint behaviour of these risks and should consider how to allocate its capital to each risk.

(iii) Dependencies in a capital model  dependencies between risk types, eg economic downturn could lead to poor investment returns and higher theft rates  dependencies within risk types, eg asset values within any one class of asset often tend to move together  dependencies between classes of business, eg the property and motor portfolios may both be affected by severe weather  dependencies within a class of business, eg a fire in a commercial property could result in a property damage claim and a business interruption claim  geographic dependencies, eg insurance risks within any one area will be affected by floods

correlations in the tail of the distribution, eg an earthquake could give rise to large workers’ compensation claims and large property claims  dependencies between origin years, eg due to correlations in underwriting standards, etc  dependencies between underwriting year and accident year, eg where it is unclear which origin year a claim should be allocated to

dependencies within the different sources of operational / group risk, eg poor administration systems or poor governance can both give rise to reputation damage  dependencies between asset classes and between individual assets, eg a recession is likely to cause a fall in the value of both equities and bonds  dependencies between counterparty credit risks, eg catastrophes could cause the default of several reinsurers simultaneously