Value-Based-Management in Non-Life Insurance

Foreword The classical way of steering a company by simple performance indicators has become increasingly substituted by value-based-management systems that combine risk and return in a reasonable manner. In this context, the precise determination of the capital requirement as the most relevant factor in producing insurance cover is inevitable. In order to separate the different influences suitably, an adequate performance measurement should be carried out. Internal models in non-life insurance usually focus on the stochastic economic capital after one year in order to determine the capital required. In life insurance, the Market Consistent Embedded Value is believed to define a suitable concept for an economic entity value. In non-life insurance, this concept also works quite well as an alternative fair value approach. This paper has been produced as a result of a long term project that has been initialized and executed by myself together with the other editors with further participation and contribution of Stefan Arens Vanessa Bittner Lars Helmig Laura Hosse Christina Hübner Simon Kaufhold Kristina Klein Sonja Kohl Sebastian Langel Hendrik Meyer Volha Muraskha Anna Naumova Anne Neu Jan Poll Ivana Simic Christoph Wiebe The target of this project was to introduce the topic “Value Based Management in Non-life Insurance” step by step to as many people as possible enabling a profound understanding without requiring too much mathematical knowledge. Therefore, lots of examples have been developed, which are not only as simple as possible, but also as complex as necessary. This project would not have succeeded, if it had not been supported by so many involved parties. Especially, the engagement of the co-editors of this publication has to be pointed out in this context. Furthermore, we would like to thank Mr. Robert G. Price for his careful revision of the original English document. Cologne, March 2013 Maria Heep-Altiner Content 1 INTRODUCTION TO VALUE-BASED MANAGEMENT ....................................1 1.1 B USINESS M ODEL OF I NSURANCE P RODUCTION ...............................................2 1.1.1 Capital Efficiency due to Synergy........................................................4 1.1.2 Risk Mitigation by Reinsurance...........................................................7 1.1.3 Additional Synergy due to Investment.................................................8 1.1.4 Legal Framework.................................................................................9 1.2 V ALUE AND R ISK - BASED M ODELS .................................................................10 1.2.1 Traditional Steering Parameter..........................................................10 1.2.2 Risk-based Steering Parameter ........................................................10 1.2.3 Required Capital versus Available Capital.........................................11 2 REQUIRED CAPITAL......................................................................................14 2.1 E XTERNAL M ODELS .....................................................................................15 2.2 I NTERNAL M ODELS – B ASIC A PPROACH ........................................................19 2.3 I NTERNAL M ODELS – S TOCHASTIC P ROFIT & L OSS A CCOUNT .........................29 2.3.1 Technical Result – Underwriting Risk................................................31 2.3.2 Non-technical Result – Asset Risk ....................................................40 2.3.3 Non-technical Result – Reinsurance Default.....................................43 2.3.4 Non-technical Result – Operational Risk...........................................44 2.3.5 Non-technical Result – Extraordinary Tax Depreciation....................48 2.4 I NTERNAL M ODELS – R EQUIRED C APITAL ......................................................49 2.4.1 Complete Model & Capital Distribution..............................................50 2.4.2 Complete Model & Capital Distribution – Calculation Example .........51 2.4.3 Determination of Required Capital ....................................................54 2.4.4 Capital Allocation...............................................................................57 3 RISK-BASED PERFORMANCE MEASUREMENT.........................................63 3.1 U NDERWRITING P ERFORMANCE M EASUREMENT ............................................63 3.1.1 Traditional Performance Measurement.............................................64 3.1.2 Traditional Performance Measurement – Calculation Example.........66 3.1.3 Risk-based Performance Measurement............................................70 3.1.4 Risk-based Performance Measurement – Calculation Example........71 3.1.5 New Business versus Existing Business – Calculation Example.......78 3.1.6 CoC Requirements and Target Combined Ratios.............................85 3.2 A SSET & U NDERWRITING P ERFORMANCE M EASUREMENT ..............................87 3.2.1 Asset Performance............................................................................87 3.2.2 Asset & Underwriting Performance ...................................................91 3.2.3 Asset & Underwriting Performance – Separate Treatment................95 3.3 A SSET & U NDERWRITING P ERFORMANCE O PTIMIZATION ..............................100 3.3.1 Preliminary Remarks.......................................................................101 3.3.2 Model Approach – Uncorrelated Risks............................................104 3.3.3 Model Approach – General Case....................................................105 3.3.4 Calculation Examples......................................................................107 3.3.5 Conclusion.......................................................................................113 3.4 T REATMENT OF E XTRA D IVIDENDS ..............................................................114 3.4.1 Cost of Capital and Target Premium...............................................114 3.4.2 Extra Dividends According to Underwriting Performance................115 3.4.3 Extra Dividends According to Asset Performance...........................118 3.4.4 Extra Dividends Given Several Accident Years...............................120 4 EMBEDDED VALUE AS FAIR VALUE APPROACH.....................................122 4.1 E MBEDDED V ALUE IN L IFE I NSURANCE ........................................................123 4.2 H ISTORICAL D EVELOPMENT .......................................................................124 4.2.1 Application of Embedded Value ......................................................127 4.2.2 Market Consistent Embedded Value...............................................128 4.3 E MBEDDED V ALUE IN N ON - LIFE I NSURANCE ................................................132 4.3.1 Differences between Life and Non-life Insurance............................132 4.3.2 MCEV Principles for Non-life Insurance ..........................................133 4.3.3 General Approach...........................................................................139 4.4 E MBEDDED V ALUE IN N ON - LIFE I NSURANCE – C ALCULATION E XAMPLE ..........141 4.4.1 Example Company..........................................................................141 4.4.2 Net Asset Value...............................................................................144 4.4.3 Value of In-Force Business .............................................................147 4.4.4 Market Consistent Embedded Value...............................................150 4.4.5 MCEV versus Economic Capital......................................................152 4.5 C ONCLUSION ............................................................................................156 GLOSSARY..........................................................................................................158 BIBLIOGRAPHY...................................................................................................162 LIST OF FIGURES ...............................................................................................164 LIST OF ABBREVIATIONS ..................................................................................169 - 1 - 1 Introduction to Value-based Management The concept of value-based management (VBM) arose from the fact, that at the end of the last century companies were becoming more and more complex. Managers started to consider the capital used for investments as well as the capital costs. Alfred Rappaport is regarded as one of the co-founders of value-based management. His classic book “Creating Shareholder Value”, which set corporate strategy in relation to the shareholder value, was published in 1986. Nowadays, this management approach is defined as follows: “Value-based management is an approach to management whereby the company’s overall aspirations, analytical techniques, and management processes are aligned to help the company maximize its value by focusing management decision-making on the key drivers of shareholder value.” 1 The focus of VBM is on the shareholder value. We have to bear in mind that this is a one-sided approach, which does not consider the perspective of other stakeholders. Important for a value-based management approach is the rate of return, demanded by the shareholders from the insurance company. Due to several risks which influence the shareholder value, an effective value-based management always includes a comparison of risk and the generated value. The answers to the following questions provide the basis for a successful valuebased approach: • How can risks and values be defined, measured and compared? • Which parameters and techniques increase the shareholder value? • Do the shareholders get a risk adjusted rate of return for their capital? • Does the rate of return exceed the capital costs? On the basic of those aspects the management of an insurance company has to decide, how to steer the business according to a value-based approach: • Regarding new business  premium calculation (premium risk). • Regarding existing business  reserve setting (reserve risk). 1 Scarlett 2001, Value Based Management, p. 2. - 2 - Before focusing on the different aspects of value-based management in the following chapters, it is necessary to understand the insurance business model. In the basic construction of insurance business we find the reasons not only for the profitability of the business model but also for the difficulties with respect to other business models. 1.1 Business Model of Insurance Production For a good understanding of the business model of insurance the differences between insurance and a typical consumer good (e.g. cars) should be outlined. In the following figure the differences regarding production and sale in connection with the allocation of risks are shown. Production Sale Consumer Good Pre Sale Transparency of production, costs are almost certain.    relatively small risk Post Production Volume of sales is ambiguous.    relatively high risk Insurance Cover Post Sale No transparency of production, production costs are ambiguous (amount and date of payment).    relatively high risk Pre Production Volume of sales is known, possibly minimal volume needed.    relatively small risk Figure 1: Insurance Cover versus a Typical Consumer Good If you consider a typical material good (e.g. a car), the production has to be finished before it can be purchased. The costs of production are covered by the producer of the good and have already been paid before the car goes on sale. So there is a risk of sale, which implies the possibility that the producer cannot sell the cars he has produced for the price that covers the production costs. In contrast an insurance product is an immaterial good, which is produced after the contract has been signed. If during an agreed period of insurance, a specified uncertain event occurs, the insured person will be indemnified by the insurer for the financial loss. 2 Because 2 Carter; Lucas; Ralph 2000, Reinsurance, p. 3. - 3 - there is no visible production of insurance and the only physical item the consumer receives is the policy, the achievements of insurance cover very often seem quite nontransparent. Those aspects especially imply that neither the policyholder nor the insurer knows • whether the insured event will occur, • when it will occur and • how (and to which extent) it will occur. As a result of this uncertainty it is difficult to calculate the claims payments and to determine the premium. Due to the risk transfer between the insured person and the insurance company the insurer needs an estimation of the expected claims and their distribution over time. Because claims often occur after a time lag, the insurer has to establish a reserve. The accounting is performed on an accrual basis. The premium in non-life insurance is normally paid for a period of one year and is charged directly or within a short period after the insurance contract has been signed or renewed. All in all the premiums have to cover the costs of the insurance company, which can be divided into three types: • Acquisition costs (at the beginning of the contract), • administration costs (during the contract) and • claims payments (after a time lag - sometimes of several periods). So the underwriting of insurance includes the risk that the premium could be calculated too low to cover the costs and claims payments over time. This uncertainty is called underwriting risk. To cover the underwriting risk, the insurer needs financial supply: on the one hand by the premiums of the insured, on the other hand by additional capital supply. To summarize: The amount and point in time of future claims payments are uncertain and have to be secured by capital. So capital is the most important production factor of insurance production. - 10 - 1.2 Value and Risk-based Models To implement a value-based management, special models are needed to measure risk and value. As a consequence, the management of an insurance company can derive decisions from the current value and risk situation. 1.2.1 Traditional Steering Parameter Traditional steering parameters are often based on the balance sheet information of an insurance company that is publicly available. Examples for steering indices are (net) profit and its relation to volume. Others are premium volume, cost ratio and combined ratio. But all these traditional steering parameters normally disregard the underlying risk of an insurance undertaking. Because of the fact, that the consideration of risk is basic for a value-based approach, new risk based steering parameters must be implied. 1.2.2 Risk-based Steering Parameter The focus of value-based management is a sufficient risk analysis, as it helps to assess the solvency of an insurance company and to increase the shareholder value. In the following, modern and well known metrics for a risk-based performance evaluation are introduced. Return on Risk Adjusted Capital (RORAC) As an alternative to the traditional Return on Equity (ROE), RORAC is a risk adjusted steering parameter where the following relation holds: RORAC = Return / Required Capital. The RORAC shows the relation between return and the required capital and is important for risk adjusted performance measuring. To consider the risk, the economic return is favored in comparison to the return based on book values (e.g. the German GAAP return). If the insurer increases risk without changing the profit situation, the RORAC decreases because more capital is required. - 11 - Economic Value Added (EVA 7 ) Another risk adjusted steering parameter is the Economic Value Added where the following relation holds: EVA = Return – Cost of Capital = Return – Required Capital · CoC Ratio. The EVA as an absolute number shows the return of a business line minus costs of capital. A positive number implies an added value whereas a negative EVA shows a value destruction. The costs of capital are the product of capital required and the cost of capital ratio. The cost of capital ratio is the extra dividend ratio the investor demands and is influenced by external and internal effects. If the risk increases without changing the profit situation, more capital will be needed and the EVA will decrease. If the return is smaller than the required capital costs, the EVA will become negative and the business unprofitable. This is somewhat crucial if the CoC ratio is chosen artificially high. Risk Adjusted Return on Capital (RAROC) A further risk adjusted parameter is the Risk Adjusted Return on Capital where the following relation holds: RAROC = EVA / Available Capital. To calculate the RAROC the required capital and a model of capital costs are needed. The RAROC is an index without dimension whereas the EVA is an absolute number. 1.2.3 Required Capital versus Available Capital The task of value-based management is to compare the available and required capital. In a sufficient situation the available capital is equal to the required capital or even higher. 7 Stern Stewart & Co. has trademarked the abbreviation EVA. - 12 - Figure 7: Value Management versus Risk Management On the one hand, an evaluation of the available capital is needed to analyze the actual value. Thus valuation models like the security principle (e.g. German GAAP), best estimate (e.g. US GAAP and partly IFRS actual status) or fair value (e.g. IFRS final status) are used to determine the available capital. Those are quite traditional valuation approaches which do not consider any risks. On the other hand, the required capital specifies the amount of capital which is needed to cover the risks taken by the insurance company. Within Solvency II context, the required capital is defined as the capital needed to protect the company at 99.5% security level. Risk models to determine the required capital are described in the next chapter. If the amount of available capital is lower than required capital, the underwriting of new business will not be possible in the same way as before. In such a case, the insurer can undertake the following options to continue business: Reduction of risk The risk of a gross portfolio can be reduced by different techniques which are described below: Decrease of Volume The risk volume can be decreased through cancellation of contracts or products, through risk exclusion or through limit (sum insured) decrease. But less volume may lead to less profit. Value Management Traditional Approach Available Capital Valuation Models - Security Principle - Best Estimate - Fair Value VBM Risk Management Advanced Approach Required Capital Risk Models - External Models - Internal Models - 13 - Increase of Premiums If possible, increasing the premium is the best solution to increasing the available capital. Although the volume remains the same, the capital and profit situation is improved. But market competition has to be taken into account. Change of Risk Structure In order to improve the risk structure, the insurer can check its risk portfolio and make changes to the underwriting. In the example mentioned above, the standard deviation was 10,000 but if it is possible to reduce this to 8,000 by risk-adjusted underwriting, the portfolio size remains the same but becomes less of a risk for the insurer. Purchase of reinsurance Reinsurance reduces the risk, but it affects the profit-situation. This will be analyzed in more detail in the next chapter. Increase of capital The last option is injection of new capital from the shareholders but this reduces the profit situation. The first insurer has to develop and improve methods for measuring risk and capital, which is necessary to secure the risk correctly. The return should be adequate to a special risk structure. For the evaluation of available capital the balance sheet capital (e.g. German GAAP or IFRS) or the (virtual) economic capital (Embedded Value in life insurance, shareholder’s net asset value in non-life insurance) can be used. - 14 - 2 Required Capital The profit and loss situation for insurance companies fluctuates from year to year on the basis of risks caused by random fluctuations, errors and changes. If those risks occur significantly – considerably higher claims payments than expected, errors in the calculation of premiums, changes in external influences (e.g. judicial decisions, price levels) – then the previously collected premiums are insufficient and a considerable loss arises. If the claims (and administration) payments are higher than the premiums in such years, the exaggeration of loss must be covered by the insurer’s capital. The greater a company’s capital, the lower the danger of insolvency and therefore the higher the probability of a lasting guarantee of given performance promises. This is economically desirable, because insolvency of an insurance company has an impact on the whole economy. As the supply of capital requires costs, insurers try to determine the (minimal) amount of capital which is appropriate according to the risk. Thus – as already mentioned in the previous chapter – the required capital is very important for the value based management of an insurer. To determine the required capital, there are several approaches. These approaches can be divided into • external models and • internal models. In some countries, such as Switzerland and the United States, internal models are not authorized by the government. In the EU both models are permitted within the framework of Solvency II regulations. The following will explain the differences between external and internal models for determining the required capital. The external models are only outlined briefly, whereas focus will be placed upon the internal models as these have at least the same requirements as the external models. Furthermore, internal models are more adequate for corporate management. The section about the internal models will be divided into the three sub-chapters: • Basic approach, • stochastic profit & loss account and • required capital - 15 - After explaining the basic approach for the internal models, the individual components of a stochastic profit & loss account will be clarified. On this basis, the method of determining the required capital will be illustrated. 2.1 External Models As mentioned above, external models provide a more simplistic view of the risk situation than internal models, because they are characterised by closed formulas and simplified bottom-up-approaches are usually factor models. Bottom-Up-Approach in this context implies that the capital requirement must be determined separately first for each category of risk. Afterwards, the capital requirements from the individual risk categories are aggregated to an overall capital requirement. Based on a specific time horizon, factor models compare the available capital with the required capital resulting from the insurance company’s risk position. The disadvantage of a factor model is that it does not describe any qualitative connections. Thus no statements can be made about the insurance company’s actual position in relation to risk. External models can be divided into: • solvency models (as the Solvency II model) and • rating models (as the Standard & Poor model – S&P model) Rating models for instance are relatively similar to the solvency models in their calculation, but may not be used to determine the required capital. They only serve for rating purposes. Solvency Models Solvency models indicate solvability rules for capital adequacy of insurance companies. Besides the already mentioned Solvency II model, which applies to the insurance companies in the EU, there are other solvency models worldwide, such as the Swiss (Swiss Solvency Test) and the US-American (RBC standards) solvency model. To regulate capital resources, Solvency II, for example, uses a two-stage approach which consists of a Solvency Capital Requirement (stage 1) and a Minimum Capital Requirement (stage 2). The Solvency Capital Requirement (SCR) corresponds with the capital which the insurance company should have at its disposal in order to have a high probability (at least 99.5%) of not being technically ruined by the losses occurring during the - 16 - following period of one year. The Minimum Capital Requirement (MCR) reflects the provision of a minimal level of the insurance company's own funds and corresponds with the amount of capital, below which the continuance of the insurance business can be endangered. A breach of the Minimum Capital Requirement leads to serious measures, which can culminate in a withdrawal of the business license. The amount ordinarily required to be maintained by the insurance company corresponds to the Solvency Capital Requirement (target capital). According to Solvency II, this may be determined either by a uniform so called “Standard Formula“ as an external solvency model or by an internal model which will be explained in later sections. 8 For a sufficient capitalization an insurance company must have at its disposal available capital of at least the same amount as required capital, i.e. Available Capital / Required Capital ≥ 100 %. Concerning the model structure, solvency models have changed over time but there are four main risk categories that determine the general model framework. Asset risks Default risks Asset default Market risk Asset default Reinsurance default Currency risk Interest rate risk Underwriting risks Operational risks Premium risk IT failure Reserve risk Management error etc. Figure 8: Risk Categories for Solvency Models 9 8 Heep-Altiner a.o. (2010), p. 8-11; Heep-Altiner a.o. (2011), p. 5-9. 9 Heep-Altiner a.o. (2010), p. 12. - 17 - The table above illustrates the four main risk categories covered by solvency models (and that must be covered by all other models at least) which will be explained in the following. Asset Risks Asset risks are subdivided into special subcategories, e.g. asset default, market risk, (foreign) currency risk and interest rate risk. The asset default can also be categorized as default risk. Default Risks In principle those risks include the default of assets as well as the default of reinsurance (regarded as an asset), but within the framework of solvency models the asset default risk has been classified as an asset risk. In order to reduce the risk of reinsurance default, a minimum rating should be required with respect to the reinsurer chosen. Underwriting Risks Underwriting risks are divided into premium risk and reserve risk. The premium risk is limited exclusively to incorrectly calculated premiums or unusually high losses from new business. The reserve risk is the risk that the reserves for outstanding claims of the existing business are too low. An underwriting loss therefore arises, if the calculated premium or the accrued reserves are lower than needed. Operational Risks Operational risks are not originally insurance-specific risks. They include all operating risks which can cause losses in a business. For example, management errors or the failure of administrative systems belong to this category. For each group of risks considered, a separate capital requirement is calculated. The individual capital requirements are aggregated to obtain the total capital requirement where different correlations are taken into account. The aggregation of the individual risks can be distinguished conceptually between two assumptions: 1. The risks R 1, …, R k with the capital requirements C 1, …, C k are assumed to be fully dependent on each other as well as on the residual risk. 2. The risks R k+1, …, R n with the capital requirements C k+1, …, C n are assumed to be correlated with the correlations ρ ij . - 18 - In summary the following general aggregation formula can be established: C total = C 1 + ... + C k + (∑ i>k C i2 + ∑ i,j>k ρ ij · C i · C j ) 1/2 This aggregation rule shall be illustrated in the following example for an insurance company with the following values: Operational Risk (OR) 100.0, Underwriting Risk (UW) 400.0, Asset Risk (A) 300.0. The capital requirements due to underwriting risk and asset risk are considered to be totally independent where the following assumptions hold with respect to the capital requirements due to operational risk: Totally Independent OR: C total = [C UW ² + C A ² + C OR ²] 1/2 = [400.0² + 300.0² + 100.0²] 1/2 = 509.9 Fully Dependent OR: C total = [C UW ² + C A ²] 1/2 + C OR = [400.0² + 300.0²] 1/2 + 100.0 = 600.0 This example shows how much influence the dependence structure has upon the determination of the capital requirement. The capital requirements of the subgroups are identical in both variants but their dependency is different. Thus different total capital requirements result. 10 Rating Models In the following, the rating models will be explained on the basis of the S&P model. The S&P model (like solvency models) is a factor model and also results from a bottom-up approach. Concerning the underwriting risk, this model represents a simple approach consisting of an entity factor, a premium factor and a reserve factor for the determination of the capital requirement. It is used for ratings purposes. 10 Heep-Altiner a.o. (2010), p. 11-14; Heep-Altiner a.o. (2011), p. 7-11. - 19 - The entity factor depends on the level of security targeted at the individual company level. The following diagram shows the entity factors for the relevant rating categories according to S&P. 11 Rating Class Entity Factor Financial Security AAA Above 175 % Outstanding AA 150 % - 174 % Excellent A 125 % - 149 % Very good BBB 100 % - 124 % Good To obtain a stable S&P rating, a company should orient towards the higher limit of a range in the calculation of its capital resources. Thus, possible negative events can be absorbed without being downgraded to a lower rating. Accordingly, an entity factor of 125 % would indicate a stable BBB rating rather than an A rating. The premium and reserve factors depend on the risk structure of a segment. These factors are provided as fixed values by S&P. The following describes the capital allocation system according to S&P: RC(1) = Entity Factor · Premium Factor · Premium, RC(2) = Entity Factor · Reserve Factor · Reserve at the Begin of Period 2, … RC(t) = Entity Factor · Reserve Factor · Reserve at the Begin of Period t. At the beginning of the first period, the capital requirement is calculated by the multiplication of the entity factor with the premium factor for each segment and the premium (as volume measure). In the following periods, the multiplication takes place with the reserve factor and the residual reserve at the beginning of the new period (as volume measure) instead of the premium factor and the premium. 12 2.2 Internal Models – Basic Approach Within the Solvency II framework insurers can also establish their own internal models instead of using external models to determine their capital requirements in order to reflect their business risks which have been described in the previous 11 Heep-Altiner a.o. (2010), p. 56. 12 Heep-Altiner a.o. (2010), p. 55-57. - 26 - Stochastic FV at t = 1 Nominal Value of the Zero Bond: 1,000.00 Duration of the Zero Bond: 4 risk-free Rate (stochastic  ESG) 6.0% Risk Spread (stochastic  ESG) 3.0% FV 1 = 1,000 / (1 + 0.060 + 0.030) 4 = 708.43 Change in FV at t = 1 ∆FV 1 = FV 1 – FV 0 = 708.33 – 712.99 = - 4.56 In the example outlined, the stochastic fair value of the zero bond after the expiration of the period clearly results from the simulated risk-free interest rate and the simulated interest rate spread. 16 Monte-Carlo Simulations By the means of Monte-Carlo simulations, an empirical capital distribution at the end of the period can be generated by the simulated stochastic profit and loss account and the deterministic capital at the beginning. It is necessary that as many simulations as possible will be carried out in order to show extremely rare events. Exceptional circumstances, such as operational risks, which occur very rarely and whose consideration is extremely important due to solvency reasons, can only be shown accurately by a multitude of simulations. The following figure illustrates possible developments towards the stochastic capital after one period (based on a deterministic capital of 500 at beginning) for five simulated paths. 16 Heep-Altiner a.o. (2010), p. 66-68. - 27 - 0 500 1.000 0 1 Path 1 Path 2 Path 3 Path 4 Path 5 Figure 9: Capital after one Year for given Monte Carlo Simulations 17 For approximating the distribution of the capital after one year as many Monte Carlo simulations of the stochastic profit and loss account have to be carried out as possible. Only then, can very rare events with a highly negative influence on the profit and loss account be considered. After a sufficient large number of simulations, the majority of the simulations is distributed around the mean value. It can be observed that the distribution of the capital after one year is limited at the positive tail, because the maximal profit after one year is limited. On the other hand, relatively high claims payments may occur so that the distribution is relatively unlimited at the negative tail. As a consequence the distribution is normally left skewed. From a solvency perspective, the focus lies on the negative tail of the distribution, where the cases are illustrated in which the capital approaches nil and the insurance company is threatened with insolvency. The scenarios in which the capital after one year is above the mean value are less diversified than the scenarios in which the capital is below the mean value. 18 The structure of a capital distribution at the end of the period is illustrated in the following figure. This distribution of capital is typically left skewed. This means that the distribution is limited at the positive tail but runs out at a negative tail. 17 Heep-Altiner a.o. (2010), p. 63. 18 Heep-Altiner a.o. (2010), p. 62-64. - 28 - 0,00 1,25 -500 -250 0 250 500 750 1000 1250 Figure 10: Distribution of the Capital after one Year 19 The costs of modelling a capital distribution by an internal model are hardly justifiable on the basis of the solvency requirements. It is therefore recommendable to use the results for steering purposes, as the simulated distribution provides the following controlling information: • Technical Ruin  defines the SCR at t = 0, • Minimum Capital Required  defines the MCR at t = 1, • Solvency Capital  defines the SCR at t = 1, • Rating Capital  defines the RCR at t = 1. Technical ruin occurs, if the capital at the end of the period falls below zero. Due to the Solvency II requirements the capital at the beginning of the period must be high enough such that technical ruin occurs only once in 200 years. The minimum capital required at the end of the period defines the next steering level. If the capital at the end of the period fells below this level, this would imply ruin for the shareholder. Even if business activities were not prohibited, the supervisory authorities would take over the management of the business in this case. This would be the equivalent to an “expropriation” of the owner. The next level is the solvency capital at the end of the period. If this level is not reached at the end of the period the supervisory authorities would contact the in19 Heep-Altiner a.o. (2010), p. 64. - 29 - surance company and demand adequate actions to solve the problem by the end of the following period. Another important focus is on securing the rating capital at the end of the period. The downgrading of a company’s rating due to a decrease in capital at the end of the year can have the result that in the following year less business can be written and that the investment returns demanded by the shareholders cannot be not produced. If (on the base of a simulation model) the probability of falling below an intended level is too high, the company should undertake appropriate management measures. 20 Because the minimum capital required, the solvency level, and the rating level have to be evaluated at t = 1 it is necessary to simulate the distribution also at t = 2 (or to proceed some type of approximation) in order to establish whether the level can be reached in the next period. By checking the distribution above, it is obvious that the company does not comply with the solvency requirements at t = 0. This example will be discussed more intensively in the following sections. 2.3 Internal Models – Stochastic Profit & Loss Account 21 In this section, we will focus in more detail on modeling different risks within a stochastic profit & loss (P&L) account. A short overview of the risk categories has already been given in the previous section. Concerning stochastic P&L account, there is a split between • Technical Result (Underwriting Risk) and • Non-technical Result with underlying o Asset Risk, o Reinsurance Default Risk, o Operational Risk and o Other Risks such as Extraordinary Tax Depreciation. The stochastic P&L is necessary for the determination of the required capital by stochastic simulations. The most important component of the stochastic P & L is the 20 Heep-Altiner a.o. (2010), p. 64-66. 21 This chapter is a short summary of the chapters 3 to 5 from “Interne Modelle nach Solvency II”, Heep-Altiner, Kaya, Krenzlin, Welter. - 30 - ordinary P&L due to the yearly business budget. An overview of how to model the underwriting and asset risks is shown in the figure below. Figure 11: Underwriting and Asset Risk 22 Concerning underwriting and asset risks, we can differentiate between existing and new business. The existing business is reflected in the existing reserves and the assets covering those reserves. The reserve risk reflects the possible volatility of the existing business, which occurs due to the change of the reserves. The real claims amount may differ significantly from the estimated value. The new business is reflected in the incoming premium and outgoing claims. It is important to calculate the premium risk-appropriately in order to cover the claims. The premium risk reflects the risk that the premium – even if calculated appropriately – is insufficient to pay an extraordinary claims experience. Within the context of the stochastic modeling, management rules play an important role in any case. Those rules cover aspects like the Strategic Asset-Allocation or the coverage of the solvability 23 . Apart from the underwriting and asset risks there are also other stochastic influences, which affect the P&L result from the non-technical side. The most important of them are reinsurance default and operational risk together with extraordinary tax depreciation. Those aspects should be considered in an internal model. 22 Heep-Altiner, Maria 2011, Internes Holdingmodell nach Solvency II, p. 109 23 Nikolic, Hrabovszki 2012, Interpretation von Modellergebnissen, p. 3 Underwriting & Asset Risk Existing Business New Business Asset Model Reserve Model Claims Model Portfolio & Asset Model Technical Result + Non-technical Result – Asset = P & L before other Risks - 31 - 2.3.1 Technical Result – Underwriting Risk As already could be seen in the figure above and will be illustrated afterwards, the underwriting risk can be split into • reserve risk for the existing business and • premium risk for the new business. Among the variety of models for reserve evaluation, we can choose for instance a chain ladder model. With the help of a stochastic model, we can see possible developments of our reserves. As a consequence, the chain ladder model is stochastic; thus we obtain stochastic best estimates together with a distribution of these values. In a one-period-model, only the stochastic of the next diagonal is relevant so that the full volatility is not realized. With a stochastic reserve model, we can measure the reserve risk as well as the run-off risk. The reserve risk reflects the possible deviations from a given best estimate and the run-off risk reflects the possible volatility of the payment pattern. A claims model is necessary to determine the premium risk and the run-off risk of the new business. The premium risk reflects the possible insufficiency of the premium to pay the claims. The run-off risk reflects the volatility of the payment patterns of the new business. Premium and Reserve Risk Due to solvency requirements, underwriting risk must be split into premium risk and reserve risk. The premium risk reflects the risk of the premium in the current year being insufficient to cover the losses. The reserve risk reflects the risk of the reserve for the existing business at the beginning of the year being insufficient at the end of the year. Consequently the non-technical result net can be structured as follows: Net Premiums – Net Costs – Net Claims Payments – New Business – Allocation to Net Reserves – New Business – Net Claims Payments – Existing Business + Change in Net Reserves – Existing Business _____________________________________ = Net Non-technical Result Premium Risk Reserve Risk - 32 - Concerning the reserve risk, the expected value of the non-technical result net should be zero. This means that on average the payments should be equal to the changes of reserve for the existing business. Thus, the reserve risk reflects the variability of the existing business result due to payments and change of reserves in the existing business. Moreover, the premium and reserve risk could be decreased significantly by reinsurance. For solvency requirements, it is very important to measure how the reinsurance decreases the risk and thus the required capital. The decrease of risk by reinsurance depends on the type of the reinsurance because the risks can be ceded proportionally or non-proportionally. In many cases, it may not be sufficient to buy only proportional reinsurance due to possible big claims amounts in the tail of a claims distribution. New Business Model As already mentioned we need a claims model for the new business to analyze the structure of the claims distribution. Moreover, the claims model is necessary in order to see how reinsurance affects the required capital. It may be quite important to model more than the total claims amount in order to analyze the real impact of a reinsurance solution. There should therefore be at least a split into • Base Claims, • Nat Cat Claims and • Major Claims. This split enables us to check the efficiency of the reinsurance solution. Moreover, different reinsurance treaties should be used to secure those different claims types. Base claims have a high frequency with low claims amount. Therefore, they need not be reinsured at all or only on a proportional basis. This type of claims can be estimated by a global distribution of aggregate losses using, for example, the panjer recursion. Nat Cat claims arise from one event and relate to many policy holders. Those events are caused by natural hazards, which are modeled using an event model with event tables from external providers or individually depending on the company’s own portfolio structure. The Nat Cat claims are usually reinsured on an XL per occurrence basis. - 33 - Major claims have a low frequency, but a high claims amount. They are so very volatile that a stochastic simulation is quite important. Because of the low frequency and the high amounts, major claims are reinsured on an XL per risk basis 24 . In order model this type of claims adequately, they must be split into a claims number and a claims size model. Therefore one needs • a frequency model for the claims number and • a severity model for the claims size. For a better understanding of the impact of the reinsurance on the major claims we will outline those two model types in more detail with an example. Frequency Model To model the frequency we apply a Poisson model. The Poisson distribution is one of the simplest discrete distributions suitable for a frequency model. This distribution depends only on the Poisson parameter λ. If N is the number of claims, than P[X = N] = (λ N / N!) · e -λ E[X] = λ Var[X] = λ The Poisson parameter λ defines the expected value as well as the variance of this distribution. Being an average, λ does not need to be integral. Usually the expected value is not equal to the variance. Therefore, it has to be checked whether the observed parameter fits in the hypotheses “expected value = variance” or not. The probabilities as well as the accumulated probabilities of a Poisson distribution with parameter λ = 4.32 are shown in the figure below. 24 Heep-Altiner, Maria 2010, Internes Modell nach Solvency II, p. 19. - 34 - 0% 20% 40% 60% 80% 100% 120% 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Claims Number P[X = N] P[X ≤ N] Figure 12: Poisson Distribution Given this Poisson distribution, a randomly drawn quantile of 62.80% results in an expected claims number of five. Severity Model To model the claims amount we apply a Pareto model. It is a continuous distribution, which depends on the parameters K (threshold) and α (Pareto parameter). The Pareto parameter α must be positive; it determines how fast the distribution function trends to 100%. The variance and the expected value depend on the parameters α and K. Moreover, this distribution is very special because if α is less than 1 we do not have an expected value; if α is less than 2 we do not have a variance etc. Especially the following relations hold: P[X < x] = 1 – (x / K) - α E[X] = (α · K) / (α – 1) Var [ X ] = (α · K 2 ) / [(α – 1) 2 · (α – 2)] The Pareto distribution is suitable for modelling the major claims amount, because it starts at a threshold K (corresponding to an excess point of an XL treaty). More- - 35 - over, the Pareto distribution has a relatively heavy tail, which can be seen in the figure below: 0% 50% 100% 150% 0 5 10 15 20 25 30 Claims Size Density Function Distribution Function Figure 13: Pareto Distribution The density function and the distribution function of the Pareto distribution shown in the figure above are based on the following parameter: Pareto Parameter α 3.57 Threshold K 5.00 E[X] = (3.57 · 5.00) / (3.57 – 1) = 6.94. Var[X] = (3.57 · 5.00 2 ) / [(3.57 – 1.00) 2 · (3.57 – 2.00)] = 8.57. So far, we have modelled the expected claims number and the expected claims size. Assuming independency between claims number and claims size we obtain E[S] = E[N] · E[X] E[S] = 4.32 · 6.94 = 29.98 - 42 - In the following, we would like to demonstrate how the asset risk can influence the FV. We therefore focus on the interest rate risk and the spread risk. Calculation Example – Interest Rate Risk In order to demonstrate an example for the interest rate risk we consider a risk-free zero bond given the following parameter: Deterministic FV at t = 0 Nominal Value of the Zero Bond: 1,000.00 Duration of the Zero Bond: 5 Risk-free Rate (deterministic) 4.0% FV 0 = 1,000 / (1 + 0.040) 5 = 821.93 Stochastic FV at t = 1 Nominal Value of the Zero Bond: 1,000.00 Duration of the Zero Bond: 4 Risk-free Rate (stochastic  ESG) 6.0% FV 1 = 1,000 / (1 + 0.060) 4 = 792.09 Change in FV at t = 1 ∆FV 1 = FV 1 – FV 0 = 792.09 – 821.93 = - 29.83 The increase of the risk-free interest rate from 4.0% to 6.0% produces a loss of 29.83. Calculation Example – Spread Risk In the second example we would like to show the impact of a change of the risk spread given the following situation: - 43 - Deterministic FV at t = 0 Nominal Value of the Zero Bond: 1,000.00 Duration of the Zero Bond: 5 Risk-free Rate (deterministic) 4.0% Risk Spread (deterministic) 3.0% FV 0 = 1,000 / (1 + 0.040 + 0.030) 5 = 712.99 Stochastic FV at t = 1 Nominal Value of the Zero Bond: 1,000.00 Duration of the Zero Bond: 4 Risk-free Rate (stochastic  ESG) 4.0% Risk Spread (stochastic  ESG) 5.0% FV 1 = 1,000 / (1 + 0.040 + 0.050) 4 = 708.43 Change in FV at t = 1 ∆FV 1 = FV 1 – FV 0 = 708.33 – 712.99 = - 4.56 The increase of the risk spread from 3.0% to 5.0% produces a loss of 4.56. 2.3.3 Non-technical Result – Reinsurance Default Any loss arising from reinsurance default basically depends on the probability and the size of such default as well as the volume of reinsurance written. The probability of RI default can be determined by the credit worthiness of reinsurers involved in business relationship with primary insurer. In its turn, the credit worthiness of a particular reinsurer can be classified according to his credit rating. It is possible that the discharge of CoC given an efficient reinsurance solution can decrease significantly, if the selected reinsurer has a poor credit worthiness. The ceded part of capital costs represents a discharge in capital costs. The risk margin, on the contrary, is a burden on capital costs; it depends on the reinsurer’s credit worthiness or rating. In other words, the risk margin is a price for the potential RI default. The following table illustrates an example of a possible insurer's RI structure according to the ratings of the reinsurers according to the default probabilities (e.g. provided by rating agencies): - 44 - Figure 20: Structure of a Reinsurance Portfolio As it can be seen, most reinsurance contracts in this example are concluded with reinsurers having ratings from AAA to BBB. The average default probability is 1.02%, which reflects an average BB reinsurance structure. This, however, may be crucial for an industrial insurer. The overall default probability as well as the RI exposure change after a period of one year and should be modeled stochastically, e.g. in the following way: Determin. RI default at t = 0 RI Exposure: 1,000.0 Average Default Probability: 1.02% Expected Default: 10.2 Stoch. RI Default at t = 1 RI Exposure: 1,050.0 Average Default Probability: 5.00% Expected Default: 57.5% The expected default has changed significantly due to a very high stochastic realization of the average RI default probability. 2.3.4 Non-technical Result – Operational Risk Operational risks additionally affect the P&L result. They arise from business risks and are not insurance specific (e.g. IT-defaults, management mistakes and wrong process organization). Any operational risk affects the balance sheet negatively either as cash flow in the current period or in form of a bad debt reserve at the end of the period. By law, German insurers for example have to provide information about their operational risks in the appendices to the annual reports – usually in form of a so called - 45 - “risk map”. In order to create such a map the insurers have to identify, evaluate, and control their own risks. However, due to the lack of statistical data this can be done based solely on the systematic self-assessment. An example of a quantitative risk map is shown in the figure below: Risk No. Amount Probab. Exp. Value STD with Corr. of 0% 1 10 0.1% 0.01 0.32 2 50 0.1% 0.05 1.58 3 100 0.1% 0.10 3.16 4 500 0.1% 0.50 15.80 5 10 1.0% 0.10 0.99 6 50 1.0% 0.50 4.97 7 100 1.0% 1.00 9.95 8 500 1.0% 5.00 49.75 9 10 10.0% 1.00 3.00 10 50 10.0% 5.00 15.00 13.26 55.64 Total Figure 21: Quantitative Risk Map The insurer estimates, according to a self-assessment, the amount A and probability P of a risk occurrence. The expected value EV and the standard deviation STD of a single risk can be calculated as follows: EV = A · P STD = A · (P · (1 - P)) 1/2 The expected value is linear; therefore the total value can be calculated by just adding the individual values. The standard deviation does not behave in a linear fashion. However, taking into account an assumed average correlation of 0% the total value can be calculated as follows: STD(X 1 + … + X n ) = (VAR(X 1 ) + … + VAR(X n )) 1/2 In the given example the total expected value of operational risk equals 13.26 and the total standard deviation equals to 55.64. It is very much evident that operational risks have a very high coefficient of variation (defined as CV = STD / EV), which - 46 - equals 419.9% in this case. For normal P&L risks, the CV is typically below 100%. Because of this, the density function is highly right skewed. In the figure below an approximation of the distribution by a lognormal distribution is illustrated. 0.0% 20.0% 40.0% 60.0% 80.0% 100.0% 120.0% 0.0 50.0 100.0 Figure 22: Distribution Function of Operational Risks The distribution function converges slowly against 100% because of the high coefficient of variation. This distribution reflects the fact that the expected losses due to operational risks are quite low. On the other hand, there are very high realizations having a big impact on the risk situation of an entity. In the following chart the risk map according to the given example is illustrated. Amount high medium low Probability low medium high 1 2 3 4 5 6 7 8 9 10 Figure 23: Qualitative Risk Map - 47 - The amount of possible operational risks is shown on the vertical axis and the probability of their occurrence on the horizontal axis. Both values are divided into three classes: low, medium and high. In total there is a classification in three different risk areas: • high risks  red area, • medium risks  yellow area, • low risks  green area. The red area represents the highest risks. Any risk located in this area occurs with a medium or high probability and causes a middle or high loss. Insurers should take appropriate measures in order to reduce or eliminate the number of such risks or to reduce the amount of loss. The yellow area describes medium risks. These risks have either high probability of occurrence combined with small amount of loss or low probability of occurrence with high level of damage. The insurers should constantly monitor these risks and prevent any movement from the medium risk area into the high risk area. The green area represents a low danger area, where only risks with low probability of occurrence and small amount of expected losses are located. The risks within this area do not really imply a high danger, but they should not move into other areas. In order to show the impact of operational risks more accurately we will model the equity of an insurer with and without inclusion of operational risks. 1,000.00 350.00 Equity 650.00 Liabilities Total 1,000.00 1,000.00 Total Assets Assets 1,000.00 336.74 Equity 650.00 Liabilities 13.26 Bad Debt Reserve Total 1,000.00 1,000.00 Total Assets Assets Liabilities Liabilities Figure 24: Balance Sheet Excluding & Including Operational Risks - 48 - At first glance it can be recognized that inclusion of operational risks as a bad debt reserve immediately leads to a lower actual capital of 336.74 compared to 350. All balance sheet positions will be simulated on the assumption of lognormal distribution with the following parameter: Expected Value Coeff. of Variation Standard Deviation Assets 1,000.00 10.0% 100.00 Liabilities 650.00 12.5% 81.25 Operational Risks 13.26 419.9% 55.64 The capital excluding operational risks results as difference between assets and liabilities, while the capital including operational risks is additionally reduced by the simulated risks. The results on the basis of 5,000 simulations are shown in the following table: excl. OR incl. OR in % Expected Values 351.97 339.24 96.38% Ruin Probabiity 0.67% 1.29% 192.54% Required Capital 369.22 423.22 114.63% Capital Distribution Figure 25: Simulated Capital & Ruin Probability The required capital under a VaR approach corresponds to the expected value minus the 0.5%-quantile. The inclusion of operational risks in our example is reflected in the increase of capital required - by 14.6% from 369.22 to 423.22 while the available capital decreases only by 3.6%. Thus, the inclusion of operational risks increases the ruin probability and the capital required disproportionally; operational risks have a considerable impact. 2.3.5 Non-technical Result – Extraordinary Tax Depreciation The extraordinary tax depreciation occurs only in extreme situations and has a very negative impact on the P&L result. If a company observes a loss, there is usually a "negative" tax burden in form of a “loss carried forward”. This loss can be balanced against future profits. In a market value model this can be treated as a deferred tax asset on the economic balance sheet. If there is no further future profit expected, then this deferred tax asset has to be written off extraordinarily. - 49 - Any internal model should include suitable management rules to treat such extraordinary tax depreciation. There is a “minimal rule” to write off if the capital is only covered by deferred tax assets. Compare the figure below. 300.00 Equity 50.00 Liabilities Deferred Tax 350.00 Total 350.00 350.00 Total Assets Liabilities Figure 26: Extraordinary Tax Depreciation In this scenario, the company owns “tax assets” of 350 covering an equity of 300. Given such a situation, the company is more or less insolvent so that deferred taxes of 350 have to be written off. The equity after depreciation equals -50; the company is insolvent. This example reflects the fact that tax effects do not prevent a ruin. In this case, the company won’t be saved from insolvency by the tax authority. Tax effects can only smooth the P & L results, but nothing more. The extraordinary tax depreciation may produce extreme non-linear effects. Thus, it is by no means clear how much capital a company has to inject (in case of a deficiency) or can extract (in case of a redundancy) according to solvency requirements. 2.4 Internal Models – Required Capital In the previous chapters, the most relevant mathematical and economic basic principles for an internal risk model have been developed. It has been explained how to model the individual components of a stochastic profit and loss account by Monte Carlo simulations. In this section, all information will be combined to an overall model. To obtain the required capital we have to perform the following steps: • Merging the individual model components to an overall model by using management rules. • Performing a simulation run to determine the empirical overall distribution. • Evaluation of the empirical overall distribution to determine the required capital by the Value at Risk (VaR) or the Tail Value at Risk (TVaR) Principle. • Allocation of the required capital to the risk influences (top-down approach). - 50 - As previously explained, the deterministic capital at the beginning of the period and the stochastic profit and loss account simulated by Monte Carlo simulations are used to calculate the capital at the end of the period where the choice of input parameter is fundamental in this context. 2.4.1 Complete Model & Capital Distribution The stochastic profit and loss (P&L) due to the basic equation discussed before consists mainly of stochastic profit & loss contributions and the respective input parameter but it is also determined by management rules. Management Rules Management rules are non-stochastic elements of an overall model that affect the income statement. They serve as a further basis for business decisions. In the modeling process the corporate strategy should be designed without unnecessary complexity. The following management rules were applied in all our calculations: • All assets like stocks are considered as accumulated without any liquid dividend outgo. • All liquid accruals are invested in short term risk-free papers until the end of the year. • Short loans to cover negative liquidity can also be performed on a risk-free base. • Dividends from subsidiaries or to parent companies are not taken into account. 29 It should be pointed out that the impact of management rules is not very strong in a short term calculation. But in consideration of several periods, management rules can represent significant factors which influence the results. Input Parameter With regard to the overall model, the following types of parameter have to be considered: • Market parameter (e.g. market interest rate). • Corporate parameter (e.g. tax rate). 29 Heep-Altiner, Erfolgsorientierte Unternehmenssteuerung, Vorlesungssskript, 2012. - 51 - • Profit & Loss specific parameter (e.g. assets, premium-income, claims reserve, reinsurance structure). • Correlation parameter (e.g. between risk-free rate and spread of fixed-income bonds) After all input parameter and management rules have been established, Monte Carlo simulations can be performed based on the calculation scheme. 2.4.2 Complete Model & Capital Distribution – Calculation Example In this section a simplified stochastic profit & loss account model will be established and used so that a capital allocation at the end of the period can be determined. Monte Carlo simulations are based on random experiments, which are carried out by using suitable random numbers. It should be noted that a sufficient number of simulations have to be generated, in order to produce stable results. Monte Carlo simulations establish an empirical distribution which serves as an approximation of the theoretical distribution. The quality of the approximation depends on the number of simulations. Based on the distribution, the capital needs of the company are determined. Finally the capital is allocated by using a top-down approach to individual model components. The following figure shows the input parameters for the example, which will be analyzed further in more detail. Parameter Average Coeff. of Variation Market Interest Rate 4% 10% Capital at Begin 500 Tax Rate 35% Op. Risk (in % of Premium) 5% 250% Premium 1,000 2% Cost Ratio 20% 10% Loss Ratio 70% 35% Figure 27: Input Parameter for the Calculation Example In the example described in this section the following components of a P&L account are modelled in a simplified way: - 58 - of capital than low-risk segments. In consequence, due to their higher capital volume, high-risk segments have to generate more profit (in absolute values). In the following sections, we would like to outline different mathematical methods to allocate the capital in an insurance company. However, before any method can be applied, we have to determine the required capital where two different approaches can be used according to the Value at Risk or Tail Value at Risk principle. The Value at Risk at a 99.5% security level is used in Solvency II. The Tail Value at Risk is used very often in internal models. Proportional Allocation The Proportional Allocation is the simplest approach to allocate the capital in a nonlife insurance company without large calculation effort, because the synergy effect is allocated proportionally. Stochastic properties are not considered in this approach. The capital is calculated by using the following mathematical formula: RC i,mod = RC i · RC ges / ∑ RC i where RC i,mod Capital Allocation per Single Risk RC i Required Capital per Single Risk (without synergy effects) RC ges Required Capital at Company Level ∑ RC i Sum of all Single Required Capital (without synergy effects) The disadvantage of this approach is that no risk structure and no dependence between single risks is considered. Adjustment of Risk-Level The basic assumption of this approach is the reduction of the security-level for single contributions so that the sum adds to the total capital requirement. With this approach in mind, the following formula applies: RC i,mod = RC i,β with ∑ RC i,β = RC ges,α RC i,mod Capital Allocation per Single Risk RC i,β Required Capital at a Security Level β per Single Risk ∑RC i,β Sum of Requ. Capital at a Security Level β for all Single Risks RC ges,α Capital Requirement of the company at a Security Level α - 59 - This approach of Risk-Level Adjustment takes stochastic properties into account. In comparison to a proportional allocation, the risk situation in the tail area is modeled more adequately. A disadvantage of this approach is that it is not linear. Covariance Algorithm For the Covariance Algorithm, the covariance contributions of the individual components to the overall variance are calculated with the aid of a correlation matrix. The capital is allocated according to the covariance contributions, see the following figure with an allocation algorithm on the base of the RC according to TVaR principle. Techn. Nonoperat. Capital Result Techn. Risk at End Result after Tax Required Capital 664.4 13.4 87.7 765.4 in % 86.8% 1.8% 11.5% 100.0% Figure 33: Covariance Algorithm All in all, the Covariance Algorithm represents a relatively simple and easily applicable method for the allocation of capital, which considers the risk in an adequate manner. A disadvantage is that this algorithm puts a disproportionate amount of weight on high risks. 30 Co-Measure Algorithm The Co-Measure Algorithm is based on the linearity of the conditional expected value so that the Algorithm is suitable when the capital requirement is determined by the Tail Value at Risk principle. The Co-Measure Algorithm is defined by the following formula: C 1 = C 0 + ∑ PL i E [C 1 ] = C 0 + ∑ E[PL i ] TVaR α [C 1 ] = C 0 + ∑ E[PL i | PL ≤ VaR α [PL]] RC α = ∑ (E[PL i ] - E[PL i | PL ≤ VaR α [PL]])= ∑ RC i,α C 1 Capital after one year E [C 1 ] Expected value of capital after one year TVaR α [PL] Tail Value at Risk with risk level α RC α Capital requirement with risk level α after one year 30 Nguyen 2008, Handbuch der wertund risikoorientierten Steuerung von Versicherungsunternehmen, p. 218. - 60 - The Co-Measure Algorithm is a modern statistical approach with good mathematical properties. A disadvantage is that it may allocate extremely high capital requirements to higher risks. Therefore alternative approaches should be considered if necessary, for example the Shapley Algorithm. 31 Shapley Algorithm The Shapley Algorithm is a game theoretical method which determines the capital need of a risk throughout the accession to an already existing collective. This method is a combinational procedure where all possible N! combinations of N risks are taken into account. In a portfolio with a wide number of risks this method causes enormous calculation effort. For clarification, the Shapley Algorithm will be explained with the following example given three risks X, Y and Z. In case of normally distributed risks the required capital is proportional to the standard deviation (STD) in such a way that we focus on this risk measure in the following. We have the following marginal contributions: 1. If X is considered as the first risk: M x = STD (X) 2. If X is considered as the second risk after the risk Y: M X|Y = STD (X+Y) – STD (Y) 3. If X is the last risk: M X|Y+Z = STD (X+Y+Z) – STD (Y+Z) 4. Combination of all risk contributions R X = ⅓ · M X + ⅓ · (½ · M X|Y + ½ · M X|Z ) + ⅓ · M X|Y+Z 5. The overall risk is described as followed: R X + R Y + R Z = R X+Y+Z = STD(X+Y+Z) Overall, the Shapley Algorithm receives a wide range of acceptance. Because of the enormous calculating effort due to the large number of risks, the practical application of this method is questioned. If we use the variance instead of the standard 31 Heep-Altiner; Haker; Lazic; Westermann et al. 2011, Internes Holdingmodell nach Solvency IISchritt für Schritt zu einem internen Holdingmodell, p.26-27. - 61 - deviation as a risk measure, then the Shapley Algorithm delivers the Covariance Algorithm. Comparison of Allocation Methods 32 The following table presents an overview of the main attributes of different allocation methods as well as their advantages and disadvantages. Allocation Method Advantages Disadvantages Proportional Allocation • Simple handling • No consideration of stochastic properties Adjustment of Risk Level • Consideration of stochastic properties • Relatively complex • No Linearity • Big risks demand high capital Covariance Algorithm • Consideration of stochastic properties • Genuine acceptance • Application in many standard models • Linear approach • Big risks demand high capital • Does not fit to the VaR or TVaR principle Co-Measure Algorithm • Consideration of stochastic properties • Linearity • Coherence • Big risks demand high capital • Low acceptance of results • Elimination of small risks • Fits only to the TVaR principle Shapley Algorithm • Intuitive allocation algorithm • Widely accepted • Equality principle • Highly complex calculation • Calculation time The Covariance Algorithm is a very manageable approach, because it presents a relatively simple and easily executable method of capital allocation that also con32 Heep-Altiner; Kaya; Krenzlin; Welter et al. 2010, Interne Modelle nach Solvency II - Schritt für Schritt zum internen Modell in der Schadenversicherung, 2010, p. 222. - 62 - siders the risk in an adequate way. However, the method only represents a linear dependency between the risks that is not adequate in every case. The use of the Proportional Allocation is very easy, but the dependencies between the risks and the risk situation in the tail area are not considered. With respect to the Adjustment of Risk-Level, stochastic properties are also considered and the risk situation in the tail area is indicated more accurately. However, this method is not linear. Other methods like the Co-Measure Algorithm or the Shapley Algorithm seem to be attractive approaches, but they are not always applicable, because business segments carrying big risks demand high capital (Co-Measure Algorithm) or because the method demands a great computing time in case of a high number of risks (Shapley Algorithm). Cost of Capital The required capital is the central input factor for the business model of insurance. In this section the determination of the required capital has been explained in more detail. Costs of Capital define the price for providing this input factor. In the following figure the mechanism to calculate the Cost of Capital is illustrated: Period Required Capital Extra Dividend Cost of Capital t = 0 t = 1 t = 2 ED(1) ED(2) RC(1) RC(2) RC(3) RC(n) ED(n) … … … t = n-1 t = n … … … ED(n-1) Figure 34: Cost of Capital (CoC) As the figure illustrates Cost of Capital can be defined as the present value of extra dividends (in the sense of a risk spread) on the Required Capital that is needed to secure the risk coverage. In the following sections the CoC will be described in more detail. - 63 - 3 Risk-Based Performance Measurement In the previous section we saw how an insurance company can determine its required capital and how this capital can be allocated to several risk influences. This chapter presents firstly the management of underwriting. Subsequently, it illustrates how insurance companies can control their total portfolio including the capital investment. The last section describes the performance optimization. To sum up, the following topics are treated: • Underwriting Performance, • Asset Performance. Furthermore, the section dealing with underwriting performance is separated into the following two different approaches: • Traditional Performance Measurement, • Risk-based Performance Measurement. In order to understand the difference between those two approaches, detailed examples are discussed. 3.1 Underwriting Performance Measurement One part of underwriting performance measurement consists in the definition of guidelines to subscribe the risk. Profitability analyses are used to verify the those guidelines. These analyses take place before the underwriting (new business) or afterwards during the execution (existing business). There are two perspectives: “A priori” in order to tariff a new business or “a posteriori” to control an existing business. The figure below illustrates the time horizon of a profitability analysis. Figure 35: New Business versus Existing Business t=0 t=1 t=n New Business A priori Existing Business A posteriori … - 64 - The following section focuses on the "a priori" underwriting analysis with respect to new business. There is a consideration of the target values at the beginning of the underwriting period. The following values have to be estimated: • The claims amount, • administration and other costs, • costs of capital, • risk-free interest rate and • required capital. With this input data we can determine the premium and check whether the segment is profitable or not. 3.1.1 Traditional Performance Measurement This section starts with the traditional approach of premium calculation. It is only based on the results of the underwriting process and does not include the expected investment income. This will be evaluated separately and does not influence directly the premium calculation. In practice, the premium calculation is influenced by more factors e.g. the impacts of competition policy. New Business According to the traditional approach the premium has to cover the administration costs, the ultimate claims amount and an additional profit margin. In non-life insurance it is assumed that there is usually a profit margin between two and three percent. 33 The following relation holds: Administration Costs + Ultimate Claims Amount + Profit Margin = Premium This premium is the basis for assessing profitability. In this assessment, usually the technical result or the combined ratio is calculated. These terms are explained later. 33 Heep-Altiner (2010), p. 45 - 65 - Existing Business: In order to assess profitability the underwriting result is determined. Additionally it might be considered that the expected profit margin could be fulfilled as calculated in the premium. The “a posteriori” underwriting result is defined as follows: Premium - Administration Costs - Claims Amount = Underwriting Result Another method of profitability assessment is the consideration of the combined ratio as a combination of loss ratio and cost ratio. Both, the cost ratio and the loss ratio are already used as an indicator for a portfolio assessment. The loss ratio is the relationship of claims payments to received premiums. The cost ratio represents the relation of administrative costs versus received premiums. The combined ratio is calculated as follows: Combined Ratio = (Administration Costs + Claims Amount) / Premium. In an underwriting perspective, the combined ratio should be less than 100% for delivering a return. In practice, the combined ratio varies widely between different branches. Both key indicators of the traditional approach are easy to determine and easy to understand. But just the underwriting is considered and not the capital investment. A consideration of the cash flows is usually not performed. But for the insurance business, it is characteristic that the payments have to be paid with a time delay to the premium income. Because of that the financial resources are not needed in total and can be invested in the capital market bearing interest. This can compensate a negative underwriting result. But the traditional approach does not consider this aspect adequately. Therefore the traditional performance measurement may not assess whether an achieved profitability is sufficient. The examples described in the following assume average claims and cost payments. These are only statistical parameters which may not realize in practice. If these variations cannot be compensated by the collective, the insurance company has to compensate an unfavourable claim experience by the provision of capital. The traditional approach does not show which level of risk should be secured by - 66 - capital and how much excess return the insurance company has to generate in order to use this capital. Thus, the traditional performance measurement does not consider all important aspects. 3.1.2 Traditional Performance Measurement – Calculation Example In this section an example of a liability segment is discussed with respect to the traditional performance measurement with the following input parameter: Premium 1,000.0 Cost Ratio 25.0% Loss Ratio 80.0% Duration 3 The premium income of 1,000 is received at the beginning of the first period. Additional premium payments do not occur. It is assumed that there are costs of 25% of the premium and a loss ratio of 80%. Due to a security principle the claim reserve is initially constituted with 900 (over reservation). The duration (e.g. the average payment duration) is 3 years. When the cost ratio and the loss ratio are summed up, it results a combined ratio of 105%. It will be checked if this business can be at all profitable or if the insurance company suffers a loss. After an example with respect to a single accident year, we consider a regular premium income resulting from an increasing or a decreasing portfolio over several accident years. Finally, we consider the impact of interests. Profit & Loss Effect – Single Accident Year This example assumes a constant portfolio on the base of a single accident year. The following table shows the development of the liability segment for the financial years 1 to 4 where the premiums are recorded as an income in the first year. An amount of 25% of the premium is subtracted immediately as costs. Also, a claims reserve of 900 is established. - 67 - 1 2 3 4 Inc. Exp. Inc. Exp. Inc. Exp. Inc. Exp. Inc. Exp. Premiums 1,000 0 0 0 1,000 0 Costs 250 0 0 0 0 250 Claim Payments 0 0 0 800 0800 Claim Reserves 900 0 0 -900 0 0 Sum 1,000 1,150 00000-100 1,000 1,050 Combined Ratio Financial Year 115.0% 105.0% Total Figure 36: Income & Expenses – Single Accident Year The result in the first year covers an income of 1,000 and expenses of 1,150 and results a combined ratio of 115% for this financial year. In the next two years there are no cash flows, so that the claims reserves remain unchanged until the fourth year. Because of the dissolution of the over reserved claim reserve in this year, the insurance company gets an income of 100. The example ends in the fourth year, because there are no additional incomes / expenses. In total, the insurance company receives an income of 1,000 and expenses of 1,050. There is a combined ratio of 105% in year 4. Without the consideration of investment income, a segment with a combined ratio above 100% can never produce a positive result. The example should be modified, because a constant portfolio for only one accident year is not typical for the insurance business. Profit & Loss Effect – Several Accident Years We consider now a regular premium income over several accident years which results in an increasing or a decreasing portfolio. Assuming an annual growth of 10%, we obtain the following table: Growth 10% Inc. Exp. Inc. Exp. Inc. Exp. Inc. Exp. Inc. Exp. Accid. Year 1 1,000 1,150 00000-100 1,000 1,050 Accid. Year 2 1,100 1,265 00001,100 1,155 Accid. Year 3 1,210 1,392 0 0 1,210 1,271 Accid. Year 4 1,331 1,531 1,331 1,398 Accid. Year 5 1,464 1,537 Sum 1,000 1,150 1,100 1,265 1,210 1,392 1,331 1,431 6,105 6,410 41 2 3 Combined Ratio 115.0% 115.0% Financial Year 115.0% 107.5% 105.0% Total Figure 37: Income & Expenses – Several Accident Years, 10% Increase The combined ratio in the first financial year is again 115%. Because of the constant increasing costs and premium income, the combined ratio does not change in the next two financial years. From the fourth year, when the first claims are settled, the combined ratio of the financial years decreases to 107.5% where the combined - 74 - In this example, a S&P-capital-allocation-model with the following input data is used: S&P Company Level 125.0% S&P Premium Rate 27.0% S&P Reserving Rate 10.0% Extra Dividend Rate 6.0% The allocation of capital demonstrates the risk-related capital demand for different lines of business. The amount of allocated capital depends on the considered segment and on the company’s target rating. The chosen multiplier of 125% is used for companies with a strong BBB-rating as target rating. Cost of Capital of 6% is required to compensate the risk bearing, analog to the Swiss solvency model. Period Premium Claims Reserve Prem. Fact. Res. Fact. Total Ent. Fact. 27.0% 10.0% 125.0% 1 1,000.0 0.0 270.0 0.0 270.0 337.5 20.0 0.0 800.0 0.0 80.0 80.0 100.0 30.0 0.0 800.0 0.0 80.0 80.0 100.0 40.0 800.0 800.0 0.0 80.0 80.0 100.0 50.0 0.0 0.0 0.0 0.0 0.0 0.0 Total 800.0 Base for Capital Allocation Capital Allocation due to S&P Figure 45: S&P Allocation of Capital for General Liability with CR = 105.0% The next step is to calculate the cost of capital. Costs of capital on the required capital provided are required at the end of a period. The CoC is obtained by multiplying the required capital at the beginning of a period with 6%. The discounted extra dividends add up to the Capital Costs in total. Period Accumul. Discount Required Amount Capital CoC Rate 6.0% of Cover. 4.00% Nominal Discounted 1 100.00% 337.5 735.4 2 96.15% 100.0 20.3 19.5 0.0 3 92.46% 100.0 6.0 5.5 0.0 4 88.90% 100.0 6.0 5.3 -697.4 5 85.48% 0.0 6.0 5.1 0.0 6 82.19% 0.0 0.0 Total 35.5 38.0 Cost of Capital with Begin of the Period Figure 46: Required Cost of Capital for General Liability with CR = 105.0% - 75 - The present value of amount of coverage and the sum of the discounted cost of capital are now known. They need to be compared in order to discover whether the analysed segment is profitable enough. Obviously, the present value of amount of coverage is higher than the required cost of capital. Thus, the segment is sufficiently profitable. In the next example the segment motor insurance – fire and theft is considered to illustrate the impact of a different cash flow structure. Partially Comprehensive In contrast to the liability segment, the main characteristic of the partially comprehensive segment is the low probability of late claims and the quick claim settlement. Therefore, there is just a short duration in the considered segment. To analyse this segment, the same input data as before is considered: Premium 1,000.0 Expense Rate 25.0% Combined Ratio 105.0% Market Interest Rate 4.0% The nominal view on this segment results in the same negative technical result of - 50 as before. The different cash flow structure does not play any role at this stage. Period CF in % Average Duration Premium Costs Result Incurred Future 1 80.0% 0.5 1,000.0 250.0 640.0 110.0 2 20.0% 1.5 0.0 0.0 160.0 -160.0 30.0% 2.5 0.0 0.0 0.0 0.0 4 3.5 0.0 0.0 0.0 0.0 50.0% 4.5 0.0 0.0 0.0 0.0 Total 100.0% 0.7 1,000.0 250.0 0.0 800.0 -50.0 Nominal Values Claims Figure 47: Nominal Cash Flow for Partially Comprehensive with CR = 105.0% In contrast to the liability segment, 80% of the claims payments are paid in the first year and 20% in the second year. The next table shows the impact on the amount of coverage by considering the present values. - 76 - Period Accumul. Discount Premium Costs Result 4.00% Incurred Future 1 98.06% 980.6 245.1 627.6 107.9 2 94.29% 0.0 0.0 150.9 -150.9 3 90.66% 0.0 0.0 0.0 0.0 4 87.17% 0.0 0.0 0.0 0.0 5 83.82% 0.0 0.0 0.0 0.0 Total 980.6 245.1 0.0 778.4 -43.0 Discounted Values Middle of the Period Claims Figure 48: Discounted Cash Flow for Partially Comprehensive with CR = 105.0% The consideration of the present values has a positive effect on the liquid result. However, in contrast to the general liability segment, the effect is not positive enough so the result is still negative. Consequently this segment is not profitable even under the modified perception. The calculation of the present values clarifies that the modified perception only has a low impact on segments with a short duration. The cash flow structure determines the profitability of a business. The next question is how the combined ratio should be changed to ensure sufficient profitability. The target combined ratio depends on the capital allocation policy of an entity and therefore it is different for different types of insurers. The following parameters in particular determine the target combined ratio: • Segment characteristics (like volatility or duration), • company’s security level (determining the S&P multiplier), • required extra dividend and • market interest rate. The table below indicates that a positive liquid result derives from a combined ratio of 99.6% in the partially comprehensive segment. Period CF in % Average Duration Premium Costs Result Incurred Future 1 80.0% 0.5 1,000.0 250.0 596.5 153.5 2 20.0% 1.5 0.0 0.0 149.1 -149.1 30.0% 2.5 0.0 0.0 0.0 0.0 4 3.5 0.0 0.0 0.0 0.0 50.0% 4.5 0.0 0.0 0.0 0.0 Total 100.0% 0.7 1,000.0 250.0 0.0 745.6 4.4 Nominal Values Claims Figure 49: Nominal Cash Flow for Partially Comprehensive with CR = 99.6% - 77 - The smaller combined ratio arises from a reduction of the expected claims payments (due to a relative increase in premium rates). Because of the lower combined ratio the technical result is now positive. The next table illustrates the effect of discounting. Period Accumul. Discount Premium Costs Result 4.00% Incurred Future 1 98.06% 980.6 245.1 584.9 150.5 2 94.29% 0.0 0.0 140.6 -140.6 3 90.66% 0.0 0.0 0.0 0.0 4 87.17% 0.0 0.0 0.0 0.0 5 83.82% 0.0 0.0 0.0 0.0 Total 980.6 245.1 0.0 725.5 9.9 Discounted Values Middle of the Period Claims Figure 50: Discounted Cash Flow for Partially Comprehensive with CR = 99.6% There is now an amount of coverage of 9.9. It should be checked whether this is enough to cover the required costs of capital. As before, this can be controlled by using the S&Pcapital allocation model. Therefore, in the first instance, the remaining reserve at the beginning of the second period needs to be calculated. Period Accumul. Discount Premium Reserve 4.00% Single Accum. 1 98.06% 1,000.0 596.5 596.5 0.0 2 94.29% 0.0 149.1 745.6 149.1 3 90.66% 0.0 0.0 745.6 0.0 4 87.17% 0.0 0.0 745.6 0.0 5 83.82% 0.0 0.0 745.6 0.0 Total 745.6 Base for Capital Allocation Middle of the Period Claims Figure 51: Base for Capital Allocation for Partially Comprehensive with CR = 99.6% The S&P and the CoC model can be used with the same data as before with the exception that there are other S&P factors for partially comprehensive. S&P Company Level 125.0% S&P Premium Rate 12.0% S&P Reserving Rate 12.0% Extra Dividend Rate 6.0% The next table illustrates the calculation of the allocated capital based on the premium and reserve factor and the entity factor defining the company’s security level. - 78 - Period Premium Claims Reserve Prem. Fact. Res. Fact. Total Ent. Fact. 12.0% 12.0% 125.0% 1 1,000.0 596.5 120.0 0.0 120.0 150.0 20.0 149.1 149.1 0.0 17.9 17.9 22.4 30.0 0.0 0.0 0.0 0.0 0.0 0.0 40.0 0.0 0.0 0.0 0.0 0.0 0.0 50.0 0.0 0.0 0.0 0.0 0.0 0.0 Total 745.6 Base for Capital Allocation Capital Allocation due to S&P Figure 52: S&P Allocation of Capital for Partially Comprehensive with CR = 99.6% Referring to the calculation of the present value of amount of coverage it has to be checked whether the required cost of capital is higher or lower than the amount of coverage, see the following table. Period Accumul. Discount Required Amount Capital CoC Rate 6.0% of Cover. 4.00% Nominal Discounted 1 100.00% 150.0 150.5 2 96.15% 22.4 9.0 8.7 -140.6 3 92.46% 0.0 1.3 1.2 0.0 4 88.90% 0.0 0.0 0.0 0.0 5 85.48% 0.0 0.0 0.0 0.0 6 82.19% 0.0 0.0 0.0 Total 9.9 9.9 Cost of Capital with Begin of the Period Figure 53: Required Cost of Capital for Partially Comprehensive with CR = 99.6% It can be seen that the present values of amount of coverage and of costs of capital are equal with respect to the chosen combined ratio. Consequently the business is profitable and 99.6% is the target combined ratio. 3.1.5 New Business versus Existing Business – Calculation Example In order to find out possible miscalculations in a segment and to start suitable countermeasures afterwards, it is important to compare the actual values with the target values. General Liability – New Business “A posteriori” it should be checked, if the parameter estimated “a priori” fit with the realized values up to the point in time t. The input values – such as average claims history or risk-free interest rates – may have evolved differently than predicted before. If there are negative deviations from the expected result, the insurance company should perform a detailed analysis, in order to prevent forecast errors in future. - 79 - In case of negative deviations it should distinguished whether underwriting risk or capital investment risk is attributable. Misjudgements of the underwriter regarding damage and loss experience is part of the underwriting risk and fall to the responsibility of the underwriter, but forecast errors in terms of investment should not be attributed to the underwriter. The figure below illustrates the “a priori” consideration where all future liabilities are only estimated and discounted to the starting point t = 0. Figure 54: New Business at t = 0 34 The following calculation example illustrates an “a priori“ consideration where only estimated values are used. At time t=0, the following data input are given: Market Interest Rate 4.0% Premium 1,000.0 Cost Ratio 25.0% Combined Ratio 97.5% S&P Company Level 150.0% S&P Premium Rate 27.0% S&P Reserve Ratio 10.0% Extra Dividend 12.0% 34 Heep-Altiner, Maria: Ausgewählte Aspekte der wertorientierten Unternehmenssteuerung in der Schadenversicherung; p. 63. 0 t t+1 T Assessment of Future Liabilities Discounting of Future Cash Flow - 80 - The required extra dividend of 12% and the capital allocation of 150% refer to the target values of an industrial insurer, because there are increased requirements with respect to the return in contrast to a mutual insurance company. The following table illustrates the estimated cash flow situation at the beginning of the consideration time period. Period CF in % Average Duration Premium Costs Result Incurred Future 1 30.0% 0.5 1,000.0 250.0 217.5 532.5 2 25.0% 1.5 181.3 -181.3 3 20.0% 2.5 145.0 -145.0 4 15.0% 3.5 108.8 -108.8 5 10.0% 4.5 72.5 -72.5 Total 100.0% 2.0 1,000.0 250.0 0.0 725.0 25.0 Nominal Values Claims Figure 55: Nominal Cash Flow at t=0 In the first step the nominal cash flow is calculated. Independent on the duration of the liabilities the total cash balance is 25.0. Period Accumul. Discount Premium Costs Result 4.00% Incurred Future 1 98.06% 980.6 245.1 213.3 522.2 2 94.29% 170.9 -170.9 3 90.66% 131.5 -131.5 4 87.17% 94.8 -94.8 5 83.82% 60.8 -60.8 Total 980.6 245.1 0.0 671.2 64.2 Discountend Values Middle of the Period Claims Figure 56: Discounted Cash Flow at t=0 In the second step the discounted cash flow is calculated. In this case (depending on the duration of the liabilities) the cash balance will increase up to 64.2. It must be checked whether the calculated cash balance is sufficient enough with respect to the extra dividend requirements. In the next table the capital allocation according to Standard & Poors model is calculated at time t=0. Period Premium Claims Reserve Prem. Fact. Res. Fact. Total Ent. Fact. 27.0% 10.0% 150.0% 1 1,000.0 217.5 270.0 270.0 405.0 2 181.3 507.5 50.8 50.8 76.1 3 145.0 326.3 32.6 32.6 48.9 4 108.8 181.3 18.1 18.1 27.2 5 72.5 72.5 7.3 7.3 10.9 Total 725.0 Base for Capital Allocation Capital Allocation due to S&P Figure 57: Capital Allocation at t=0 - 81 - The required costs of capital are calculated in the following table according to the required cost of capital ratio of 12% in order to check the target fulfillment. Period Accumul. Discount Required Amount Target Capital CoC Rate 12.0% of Cover. Fulfillment 4.00% Nominal Discounted in % 1 100.00% 405.0 522.2 2 96.15% 76.1 48.6 46.7 -170.9 3 92.46% 48.9 9.1 8.4 -131.5 4 88.90% 27.2 5.9 5.2 -94.8 5 85.48% 10.9 3.3 2.8 -60.8 6 82.19% 1.3 1.1 Total 64.3 64.2 100.0% Cost of Capital with Begin of the Period Figure 58: Target Fulfillment at t=0 The table shows the target fulfillment in the “a priori” consideration at about 100.0%. General Liability – Existing Business “A posteriori” the cash flows CF 1, CF 2,…, CF t for past liabilities have realized and the cash flows CF t+1, CF t+2,…, CF T for future liabilities have to be estimated according to a modified forecast. The past cash flows must be accumulated up to t where the future cash flows have to be discounted back to t, see the figure below. Figure 59: Existing Business at t 35 In the case that the realized and future cash flows are unknown or difficult to estimate, the following approximation scheme can be used: 35 Heep-Altiner, Maria: Ausgewählte Aspekte der orientierten Unternehmenssteuerung in der Schadenversicherung; p. 63. t-1 t+1 t 0 Assessment of Future Liabilities Valuation of Past Liabilities Accumulation of Past Cash Flow Discounting of Future Cash Flow - 82 - • The “a priori” estimated cash flow pattern until time t can be calibrated to 100% and used as cash flow pattern for the incurred liabilities. • The “a priori” estimated cash flow pattern starting from time t + 1 can be calibrated to 100% and used as cash flow pattern for the future liabilities. The realized cash flows until time t must be accumulated with the realized risk-free interest rates r 1, r 2,…, r t , and the estimated future cash flows from time t must be discounted with the estimated risk-free interest rates r t+1, r t+2,…, r T. In this context, it can be worked approximately with a fixed average interest rate for the past and a fixed average interest rate for the future. As an approximation, it is also possible to “fix” the capital allocation to the “a priori” allocation. In the example considered before, the claims and interest rate experience and estimation have developed at time t = 2 in the following way: Realized Market Interest Rate 3.75% Estimated Future Interest Rate 3.50% Realized Incurred Claims Payment 400.0 Estimated Future Claims Payment 350.0 At t=2 the realized market interest rate of 3.75% is different to the initially estimated risk-free market interest rate of 4%. The prospective market interest rate is estimated with 3.5%.The expected loss ratio over the total run-off period is estimated as 75%. Those developments will have a negative impact on profitability. In this example, the cash flow pattern will be approximated in the way previously described. The change in claims experience leads to nominal liquid balance of zero. Period Premium Costs Result Incurred Future Incurred Future 1 30.0% 1,000.0 250.0 218.2 531.8 2 25.0% 181.8 -181.8 3 20.0% 155.6 -155.6 4 15.0% 116.7 -116.7 5 10.0% 77.8 -77.8 Total 55.0% 45.0% 1,000.0 250.0 400.0 350.0 0.0 Cash Flow in % Nominal Values Claims Figure 60: Nominal Cash Flow at t=2 In a second step the incurred values will be accumulated until t=2 with the realized market interest rate of 3.75% as following: - 83 - • For period 1: (1+0.0375) 1.5 = 1.0568, • For period 2: (1+0.0375) 0.5 = 1.0186. The estimated future cash flows will be discounted to t = 2 with the estimated future risk-free interest rate of 3.50% as follows: • For period 3: (1 + 0.035) -1/2 = 0.9892, • For period 4: (1 + 0.035) -3/2 = 0.9497, • For period 5: (1 + 0.035) -5/2 = 0.9176. Accumulation of incurred past cash flows and discount of estimated future cash flows results to an overall discounted cash balance of 41.7. Period Accumul. Discount Premium Costs Result 3.75% 3.50% Incurred Future 1 105.68% 1,056.8 264.2 230.6 562.0 2 101.86% 185.2 -185.2 3 98.29% 152.9 -152.9 4 94.97% 110.8 -110.8 5 91.76% 71.4 -71.4 Total 1,056.8 264.2 415.8 335.1 41.7 Discountend Values Middle of the Period Claims Figure 61: Accumulated / Discounted Cash Flow at t=2 Although the liquid balance after accumulation and discounting is positive, it must be checked however, whether the liquid balance is sufficient with regards to the required capital costs. The following table illustrates the reserve at the beginning of the period, which is needed as a base for the capital allocation. Period Premium Claims Reserve Prem. Fact. Res. Fact. Total Ent. Fact. 27.0% 10.0% 150.0% 1 1,000.0 218.2 270.0 270.0 405.0 2 181.8 531.8 53.2 53.2 79.8 3 155.6 350.0 35.0 35.0 52.5 4 116.7 194.4 19.4 19.4 29.2 5 77.8 77.8 7.8 7.8 11.7 Total 750.0 Capital Allocation due to S&PBase for Capital Allocation Figure 62: Capital Allocation at t=2 - 90 - Obviously the combination with 90% of Asset 1 generates a minimal risk (expressed in terms of the standard deviation). The figure above shows a typical Markowitz allocation with the efficient boarder and one inefficient combination. Only the allocations on the border are efficient. Calculation Example - Preference-Systems There are a lot of different efficient portfolios. However, which allocation should a company choose? The chosen portfolio should fit with the individual company preferences. A proper allocation can be determined by using preference functions. Two classic preference functions (in the context of value based management), which were introduced in the first chapter, will be determined after one year and will be used to find out the best solution: RORAC = Expected Value / Required Capital ≈ E / (t α · STD), EVA = Expected Value – Capital Costs ≈ E – k ·t α ·STD with α the risk level and E the expected value. In the following, a private lines insurer will be examined by using these two preference systems. This insurer invests in the two asset classes as before and concentrates only on risk life insurance. In this case, there is a (relatively) safe outflow of liabilities with the amount of 1,100 and a standard deviation near to zero. We also assume a cost of capital ratio of 7.5% and a BBBrating (conforming to the Solvency II – Security Level of 99.5%). Share STD Asset 1 Assets Liabilities Result 100% 1,100 1,100 0 100 90% 1,110 1,100 10 98 80% 1,120 1,100 20 105 70% 1,130 1,100 30 119 60% 1,140 1,100 40 139 50% 1,150 1,100 50 163 40% 1,160 1,100 60 188 30% 1,170 1,100 70 215 20% 1,180 1,100 80 243 10% 1,190 1,100 90 271 0% 1,200 1,100 100 300 Expected Value Figure 70: Risk / Return Analysis given two Assets (2) - 91 - In this case, the risk return profile is quite similar to the risk return profile of the investment company only investing in two assets with the difference that all expected cumulated values are reduced by 1,100, where, with respect to the combination of 0% Asset 1 and 100% Asset 2, we obtain the following results: RC = 2.58 · 300 = 773 RORAC = 100 / 773 = 12.9%, EVA = 100 – 7.5% · 773 = 42 In the following table, all RORAC and EVA combinations are listed in such a way that an optimal value can be derived. Share STD RC RORAC EVA Asset 1 99.50% 7.50% 100% 100 258 0.0% -19 90% 98 252 4.0% -9 80% 105 270 7.4% 0 70% 119 308 9.8% 7 60% 139 359 11.1% 13 50% 163 419 11.9% 19 40% 188 485 12.4% 24 30% 215 554 12.6% 28 20% 243 625 12.8% 33 10% 271 699 12.9% 38 0% 300 773 12.9% 42 Figure 71: Capital Allocations – Private Line Insurer Although different methods were used, the same optimal results can be observed in this calculation example. According to RORAC, the allocation with 0% of Asset 1 and 100% of Asset 2 is the most profitable with 12.9% extra dividend. The maximum EVA is also observed given 100% Asset 2. RORAC as well as EVA provide preference systems in order to find an optimal decision among all efficient portfolios. 3.2.2 Asset & Underwriting Performance Looking at a private line insurer with two Assets and two lines of business (LoB) in this section, the following input situation can be assumed: - 92 - Correl. 10% expected STD Asset 1 1,100 100 Asset 2 1,200 300 after one year Correl. 10% expected STD LoB 1 1,075 100 LoB 2 975 300 after one year Figure 72: Input Data given two Assets and two LoB A correlation of 10% is given between the assets as well as between the liabilities, excluding a correlation between assets and liabilities. Because of four variables, there are many different possible combinations. Due to this fact, we will only take into account allocations with 50% steps. In the following table the expected values and standard deviations are listed but only for 0% 50% and 100% combinations: Exp. Asset 1 LoB 1 Assets Liab. Total Result 100% 100% 100 100 141 25 50% 100% 163 100 191 75 0% 100% 300 100 316 125 100% 50% 100 163 191 75 50% 50% 163 163 230 125 0% 50% 300 163 341 175 100% 0% 100 300 316 125 50% 0% 163 300 341 175 0% 0% 300 300 424 225 Share STD Figure 73: Risk / Return Analysis given two Assets and two LoB Given a y% share of asset 1 and a x% share of liability 1 we obtain the following formulas: E = (y · 1,100 + (1 – y) · 1,200) – (x · 1,075 + (1 – x) · 975), VAR(A) = (y · 100) 2 + ((1 – y) · 300) 2 + 2 · 10% · y · 100 · (1 – y) · 300 VAR(L) = (x · 100) 2 + ((1 – x) · 300) 2 + 2 · 10% · x · 100 · (1 – x) · 300 VAR = VAR(A) + VAR(L) Given a combination with 50% of Asset 1 and 50% of LoB 1 we obtain the following results: - 93 - E = (50% · 1,100 + 50% · 1,200) – (50% · 1,075 + 50% · 975) =125 STD = (163 2 + 163 2 ) 0.5 = 230 Please notice that the standard deviation of 163 given a combination of 50% Asset 1 has been calculated before. (The same calculation applies to the standard deviation given a combination of 50% LoB 1.) Calculation Example – Private Line Insurer In the following section, different business models will be checked with respect to the given input data – a private line insurer and a reinsurer. The results for the private line insurer are listed below with respect to all combinations of 0%, 50 % and 100 % of Asset 1 or LoB 1. Exp. RC RORAC EVA Asset 1 LoB 1 Result 99.50% 7.50% 100% 100% 25 364 6.9% -2 50% 100% 75 492 15.2% 38 0% 100% 125 815 15.3% 64 100% 50% 75 492 15.2% 38 50% 50% 125 593 21.1% 81 0% 50% 175 879 19.9% 109 100% 0% 125 815 15.3% 64 50% 0% 175 879 19.9% 109 0% 0% 225 1,093 20.6% 143 Share Figure 74: RoRAC and EVA Optimum – Private Line Insurer The RoRAC optimum is achieved in “the middle” given a combination of 50% Asset 1 and 50% LoB 1 where the EVA optimum is achieved at “the boundary” given 0% Asset 1 and 0% LoB 1. This is a quite extreme combination with a high capital requirement and it is not clear, if such a high capital amount is available. In total, the situation is much more complex and less transparent than in the case of two assets. One the one hand it is still possible to exclude inefficient allocations but on the other hand an efficiency curve cannot be easily identified. Looking at the risk return profiles of the combinations previously analyzed, “visually” efficient combinations can be identified, but there are still a lot of inefficient combinations. In the figure below all combinations of 0%, 25%, 50%, 75% and 100% Asset 1 or LoB 1 are listed. - 94 - Exp. RC RORAC EVA Asset 1 LoB 1 Result 99.90% 15.00% 100% 100% 25 437 5.7% -41 50% 100% 75 590 12.7% -14 0% 100% 125 977 12.8% -22 100% 50% 75 590 12.7% -14 50% 50% 125 711 17.6% 18 0% 50% 175 1,055 16.6% 17 100% 0% 125 977 12.8% -22 50% 0% 175 1,055 16.6% 17 0% 0% 225 1,311 17.2% 28 Share 0 50 100 150 200 250 0 100 200 300 400 500 Risk Return Figure 75: Risk / Return Profile given two Assets and two LoB In the next calculation example the RoRAC and EVA optima for a reinsurer with different CoC parameter is analyzed. Calculation Example - Reinsurer The results for the reinsurer are listed in the table below. As previously stated, only combinations with 0%, 50% and 100% of Asset 1 or LoB 1are shown in the figure. The RORAC optimum is achieved for a combination with 50% Asset 1 and a 50% LoB 1. The EVA optimum is achieved for the “extreme” combination with 0% Asset 1 and 0% LoB 1. The optimal combinations are the same as before whereas the optimal values are different. Figure 76: RoRAC and EVA Optimum – Reinsurer - 95 - A portfolio with two risky branches and two risky asset classes creates a different return structure than a portfolio with two risky assets and one more or less risk-free liability. The situation in the second case reflects the classical Markowitz approach whereas the (more complex) situation in the first case is more realistic. Calculation Example - Conclusion The capital cost rate is predefined by the management. But there is an uncertainty about the achievability of this goal. The decisive factor is the market, which can be hardly influenced by the insurer. In order to achieve a specified target, the insurer would have to increase the premium or the capital market returns. But the insurer has to focus on the market prices and the competitors in order to be competitive. Nevertheless, a relatively low capital cost rate would be unattractive for potential investors. As a consequence, companies are forced to set almost unachievable goals. Let’s have a look at the previously examined reinsurer with a security level of 99.9% and a capital cost rate of 15%. It is questionable whether this capital cost rate is appropriate. At this security level the insurer expects one default within 1,000 years. If you compare that fact with the relativity high capital cost rate the proportionality between risk and return is doubtful. The rate of return defined by the management is often outside the efficiency curve. There are two possible measures to “produce” achievable combinations: Firstly, the insurer can reduce the aimed extra dividend. However, the insurer is in a competitive situation and investors could be dissatisfied with the return on investment. Therefore, it is not so easy to reduce the rate of return. Secondly, they could take more risks, but then the targeted security level would not be reached. By using the EVA method some questions occur. Is it reasonable to define a negative EVA value as a destruction of capital? In order to specify this issue one could look at the figure above. An extra dividend of 12.8% (additionally to the risk-free return) with negative EVA is observed. The RORAC produced is quite high, so the interpretation as capital destruction seems to be doubtful. 3.2.3 Asset & Underwriting Performance – Separate Treatment In the previous section it was demonstrated how the simultaneous optimization of underwriting and capital investment can be managed in an insurance company. Due to the fact that insurance is a co-product, you can virtually split an insurance company into two parts: Underwriting and asset management. Regarding the asset management, the following assets are assumed in the following: - 96 - Return Accum. CV Absolute Asset 1 4.0% 1,040 0.0% 0 Asset 2 10.0% 1,100 30.0% 330 Expected Std. Deviation Figure 77: Input Data – Available Asset Portfolio It is obvious that Asset 1 represents a risk-free asset. Therefore no capital is needed to secure this asset. In contrast to this, Asset 2 is a risky asset which demands capital to secure the asset. With this in mind, two strategies will be checked: In the first case, the insurer only invests in the risk-free asset (Strategy 1). Alternatively, the insurance company invests also in the risky asset (Strategy 2). Comparing those two strategies enables a proper steering of the portfolio according to underwriting and investment impact on the risk. These strategies will be discussed on the basis of the following questions: • What is the required capital of both strategies at a default level of 0.2 %? • Which strategy provides a higher return on Risk Adjusted Capital? • Which strategy is the best? Taking into account a risk-free investment of the required capital at the beginning of the period as well as the expected result due to investment and underwriting, we obtain the following relationships for both strategies: RC = (t · STD – (E(A) – E(L)) / (1 + r) STD the overall standard deviation, E(A) the expected value of the assets at the end of year, E(L) the expected value of the liabilities at the end of the year and r the risk-free interest rate. These relationships are derived from the distribution of the capital at the end of the period under an assumption of normally distributed assets and liabilities. Calculation Example – Strategy 1 with risk-free Assets As previously stated, the insurance company invests solely in a risk-free asset, for example a government bond with a default risk of almost zero. We assume that the total premium income is at the beginning of the year. Furthermore all cost expenditure is at the end of the year. As a result of this assumption, the premium income can be fully invested over a period of one year. (This assumption can be achieved in any case by consideration of suitable present values.) - 97 - CV Absolute Asset 1 100.0% 4.0% 1,040 0.0% 0 Asset 2 0.0% 10.0% 0 30.0% 0 LoB 1 100.0% 98.5% 985 15.0% 148 Total 55 148 Expected Std. Deviation Figure 78: Input Data – Strategy 1 For Asset 2 and the line of business, a normal distribution is assumed. Moreover there is no correlation between assets and liabilities. The required capital is calculated according to the previously specified formula where for a security level of 99.8 % the factor t equals to 2.88 so that RC = (2.88 · 148 – 55) / 1.04 = 357 RORAC = (E(A) – E(L)) / RC = 55 / 357 = 15.4% Total Return = r + RORAC = 4.0% + 15.4% = 19.4% The insurance company needs a required capital of 357 based on a default risk of 0.2 %. An average total RoRAC of 19.4% is achieved. Calculation Example – Strategy 2 with risky Assets In this case the insurance company invests 65% in the risk-free asset and 35% is invested in the risky asset class. The data needed for the following calculations are listed in the table below. CV Absolute Asset 1 65.0% 4.0% 676 0.0% 0 Asset 2 35.0% 10.0% 385 30.0% 116 LoB 1 100.0% 99.0% 990 15.0% 149 Total 71 188 Expected Std. Deviation Figure 79: Input Data – Strategy 2 For completeness it should be noted that in strategy 2 there is also no correlation between capital investment and the line of business. Due to the risky asset, the company has to provide more capital. Furthermore, the additional costs for the complex asset management are reflected in the combined ratio, which is 0.5 percentage points higher than for strategy 1. - 98 - Based on those results it is possible to calculate the required capital and the return on investment. Concerning the previously mentioned formula, we can calculate with respect to the 99.8% security-level as follows: RC = (2.88 · 188 – 71) / 1.04 = 452 RORAC = 71 / 452 = 15.7 % Total Return = 15.7 % + 4.0 % = 19.7 If the results of the different strategies are compared, the following differences can be observed. Following strategy 1, the company has to provide capital of 357 and achieves a total RoRAC of 19.4%. Under strategy 2, a higher capital of 452 must be provided because of the risky assets. This is rewarded with a marginally better return on investment of 19.7%. Does this result imply that strategy 2 is better than strategy 1? In the following section it will be verified whether the results stay valid when the parameters are changed. Calculation Example – Impact of Parameter Change To analyze the impact of parameter change, the principal scenarios for both strategies are maintained, but the following parameters are changed: • Reduction of the risk-free interest rate. • Increase of the Combined Ratio. • Reduction of the expected return of asset 2. • Increase of the standard deviation of asset 2 • Increase of the standard deviation of the LoB. Due to those changes, both strategies must be analyzed with respect to total RoRAC and the default risk. The following table shows the impact of the parameter changes (with unchanged capital at start): - 99 - Parameter RoRAC Ruin Old New Prob. Risk Free Interest Rate 4.0% 3.0% 15.6% 0.26% Combinded Ratio 98.5% 99.5% 16.6% 0.26% Expected Return Asset 2 1,100 1,075 19.4% 0.20% Std. Deviation of Asset 2 330 550 19.4% 0.20% Std. Deviation of Branch 15.0% 20.0% 19.4% 1.52% Value Figure 80: Impact of Parameter Changes (1) Due to the fact that strategy 1 invests only in risk-free assets, there is no impact on the RoRAC and the default risk when the expected return decreases and the standard deviation increases with respect to asset 2. Relative to this scenario, only the increase of the LoB volatility produces crucial results. In the following table the impacts of the parameter changes are illustrated for the second strategy: Parameter RoRac Ruin Old New Prob. Risk Free Interest Rate 4.0% 3.0% 17.3% 0.24% Combinded Ratio 99.0% 100.0% 17.5% 0.25% Expected Return Asset 2 1,100 1,075 17.8% 0.30% Std. Deviation of Asset 2 330 550 19.7% 1.30% Std. Deviation of Branch 15.0% 20.0% 19.7% 0.91% Value Figure 81: Impact of Parameter Changes (2) Strategy 2 is highly affected by the risky asset, in such a way that any increase of the standard deviation produces another crucial impact in this case. Before the consideration of parameter changes, strategy 2 seemed to provide a (slightly) better performance than strategy 1. After the consideration of parameter changes strategy 2 seems to be more volatile and to produce more crucial situations with respect to the solvency requirements. Asset & Underwriting Performance – Separate Treatment of Performance Investment exclusively in risk-free assets is not in any case satisfactory for insurance companies and their ambitious return targets. Due to this fact, a company must sometimes invest in risky asset to increase the profitability due to synergy effects. In such a case, it is important to differentiate between the performance of the underwriting and of the asset management. In the following it will be discussed how underwriting and asset management contribute to an overall performance, see the following table. - 106 - ρ 00 … ρ 0n φ 00 … φ 0m … ρ ik … … φ il … ρ n0 … ρ nn φ n0 … φ nm φ 00 …φ 0n ψ 00 …ψ 0m … φ jk … … ψ jl … φ m0 … φ mn ψ m0 … ψ mm Figure 85: Correlation Matrix 37 We can trace the general case to the already considered cases by decomposing the symmetric correlation matrix C to C = M T · D · M (Cholesky decomposition) with M an upper triangular matrix and D a diagonal matrix. 0 0 0 0 0 0 0 0 0 0 = x 0 0 0 0 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Figure 86: Cholesky Decomposition of the Correlation Matrix 38 All variables will be uncorrelated when we transform all assets and liabilities according to the Cholesky decomposition. The transformed combined ratios can be interpreted as combined ratios for a linear combination of the liabilities, but the transformed returns are mixed terms now. Concerning the general case, we will concentrate on the solutions with respect to the RORAC, because the solutions with respect to the EVA are quite complex. In many cases such a solution does not exist. Regarding the RORAC optimization in the general case, the following formulas are obtained concerning the partial derivatives of S by x i or y j : ∂S/∂x i = (∑(ρ ik · σ i – ρ 0k · σ) · x k · σ k + (∑(φ il · σ i – φ 0l · σ) · y l · τ l ) / S =: X i */ S ∂S/∂y j = (∑(φ jk · τ j – φ 0k · τ) · x k · σ k + (∑(ψ jl · τ j – ψ 0l · τ) · y l · τ l ) / S =: Y j */ S 37 Heep-Altiner, 2011, Performanceoptimierung des (Brutto) Neugeschäfts in der Schadenversicherung, p. 9. 38 Heep-Altiner, 2011, Performanceoptimierung des (Brutto) Neugeschäfts in der Schadenversicherung, p 10. - 107 - The formulas in the uncorrelated case are simplified versions of the general formulas. In the general case, we obtain for all i, j, k, l > 0 X i */∆r i = X k */∆r k = –Y j */∆c j = –Y l */∆c l E · X 1 * = ∆r 1 · S 2 where the first equations are linear ones. Substituting S 2 in a suitable way, another linear equation is obtained (r – c) · X 1 * = ∆r 1 · (σ · (∑ ρ 0k · x k · σ k + ∑ φ 0l · y l · τ l ) + τ · (∑ φ 0k · x k · σ k + ∑ ψ 0l · y l · τ l )). In total, together with the two normalization equations ∑ x i = 1 = ∑ y j we now obtain (m + n + 2) linear equations with (m + n + 2) variables x, x 1 , …, x n and y, y 1 , …, y m , which are solvable for the general case (i.e. with the exception of singular parameter constellations). Only in special cases can the solution be described explicitly. A solution is not automatically feasible because some coefficients may not be between zero and one. In some special cases, simplified solutions will be obtained that can be treated in EXCEL. In the next section, some results concerning special cases are discussed. 3.3.4 Calculation Examples This section discusses some examples. In this context, the term “feasible” optimum implies that a local maximum exists in which all coefficients are between zero and one. Maximum values at the boundaries of allowed combinations are not considered at this point. Calculation Examples - Uncorrelated Assets and Liabilities For the first example, the assumption is to invest the new business premium into two risky asset classes, which are not correlated with each other. The following input data is assumed: Standard Deviation Cumulated Return Asset 1 10.0% 107.5% Asset 2 20.0% 112.5% Figure 87: Cumulated Returns - 108 - For simplicity, only the underwriting of a single risk-free line of business with a combined ratio of 100.0% is considered (underwriting just a relatively low risk category in a very large collective.) Under these assumptions, the model provides as follows, the classical Markowitz approach for assets: Standard Deviation Return Asset 1 10.0% 7.5% Asset 2 20.0% 12.5% Figure 88: Uncumulated Returns Concerning the uncumulated returns, the following table provides the RORAC and the EVA optimum given a capital cost parameter of c = 30.0% with respect to EVA. Standard Deviation Return RORAC Optimum EVA Opt. c = 30.0% Asset 1 10.0% 7.5% 70.6% 35.3% Asset 2 20.0% 12.5% 29.4% 64.7% Figure 89: RORAC and EVA Optimum given Risk-free Liabilities With respect to the EVA optimum there is significantly more investment in the riskier asset class than with respect to the RORAC optimum. Furthermore, only the absolute value is optimized without consideration of the capital requirements. In the specific case, the capital requirement given the EVA optimum is significantly higher than the capital requirement in case of the the RORAC optimum. Possibly, the company is not able to deposit the required capital for the EVA optimum. The RORAC optimum can be illustrated as a tangent point on the risk / return curve with a line through the origin as follows: - 109 - 0% 5% 10% 15% 0% 5% 10% 15% 20% Risk Return Figure 90: RORAC Optimum The EVA optimum is defined as a tangent point on the risk / return curve with a line whose slope equals the cost of capital parameter c. Because the slope of the risk / return curve does not fall below an asymptotic value, such a tangent point does not exist for small parameter c, see the following figure. 0% 5% 10% 15% 0% 5% 10% 15% 20% Risk Return 0% 5% 10% 15% 0% 5% 10% 15% 20% Risk Return Figure 91: EVA Optimum for c = 30.0%, but no EVA Optimum for c = 10.0% If the cost of capital parameter c is too small then only a maximum value exists at the boundary with 100% investment in the risky asset – regardless of the amount of capital required. - 110 - Now the example will be extended by underwriting two lines of business (uncorrelated with each other and with the asset classes) with the following risk / return profile in the optimization approach: Combined Ratio Standard Deviation LoB 1 97.0% 20.0% LoB 2 98.5% 10.0% Figure 92: Combined Ratios The following table illustrates the asset and liability shares with respect to the RORAC optimum: STD Cum. Return Share STD Comb. Ratio Share Asset 1 10.0% 107.5% 64.5% LoB 1 20.0% 97.0% 24.7% Asset 2 20.0% 112.5% 35.5% LoB 2 10.0% 98.5% 75.3% Figure 93: RORAC Optimum given Risky Assets and Lines of Business The shares of the asset classes have changed with the inclusion of the lines of business in the optimization, because the optimization approach (even with uncorrelated assets and liabilities) is not independent of the asset and liability risk / return profiles. In this particular case, there is a maximum because the returns are positive. With the inclusion of risky lines of business, there is a higher weight with respect to the risky asset due to a higher degree of diversification. An even stronger shift with respect to the weight of the risky asset and liability classes are observed after EVA optimization – regardless of the amount of capital required. STD Cum. Return Share STD Comb. Ratio Share Asset 1 10.0% 107.5% 12.9% LoB 1 20.0% 97.0% 40.1% Asset 2 20.0% 112.5% 87.1% LoB 2 10.0% 98.5% 59.9% Figure 94: EVA Optimum given Risky Assets and Liabilities for c = 30.0% - 111 - The tables show that there are significant differences between the EVA and RORAC optimum. For the EVA optimization approach there are higher weights for the riskier positions. It has already been mentioned that in the present model approach the consideration of risk-free assets is no problem because there is usually risk in the liability positions. In fact, it is an independent business decision, to invest the cash flows resulting from the technical result in risky or riskless investments. In this case a risky investment is not necessarily “better” than a risk-free investment. Thus, the example is extended by including a risk-free asset with an interest rate of 4.0% in order to check how the weights of asset and liability classes will change: STD Cum. Return Share STD Comb. Ratio Share Risk Free 0.0% 104.0% 22.4% LoB 1 20.0% 97.0% 40.1% Asset 1 10.0% 107.5% 12.9% LoB 2 10.0% 98.5% 59.9% Asset 2 20.0% 112.5% 87.1% Figure 95: RORAC Optimum given a Risk-free Asset The risk-free asset only receives a relatively small weight because of the high diversification. With respect to EVA, the results are quite complex. There is no solution for the (quite realistic) capital cost parameter c = 30.0%. To obtain a solution in the classical sense, the parameter has to be significantly increased, e.g. c = 60.0%. STD Cum. Return Share STD Comb. Ratio Share Risk free 0.0% 104.0% -119.8% LoB 1 20.0% 97.0% 31.7% Asset 1 10.0% 107.5% 136.8% LoB 2 10.0% 98.5% 68.3% Asset 2 20.0% 112.5% 83.1% Figure 96: EVA Optimum given a Risk Free Asset for c = 60.0% As can be seen in the table above, there is a solution for the EVA approach but it is not feasible in the sense that each share is positive and less than one. There is such a dramatic restructuring of assets that in principle the recommendation is not to buy the risk-free assets, but rather to borrow it, in order to take even more of the - 112 - riskier asset classes to the portfolio. This very simple example shows that the EVA optimization may be much more crucial than the RORAC optimization. Calculation Examples – Correlated Assets and Liabilities In this section, the effects of correlations will be analyzed . Only the RORAC optimization will be analyzed because of the high complexity of the EVA optimization in connection with very crucial solutions. For this purpose, the case of the two previously introduced risky assets and lines of business is considered, where it is now assumed that the two assets and liabilities are correlated with 50% to each other. This example already contains essential features of the general case: STD Cum. Return Share STD Comb. Ratio Share Asset 1 10.0% 107.5% 63.0% LoB 1 20.0% 97.0% 11.1% Asset 2 20.0% 112.5% 37.0% LoB 2 10.0% 98.5% 88.9% Figure 97: RORAC Optimum given Risky Assets & Liabilities with 50.0% Corr. The use of correlations changes the weights, such that the lower-risk positions obtain a higher weight. If the correlations are too high, it is no longer a feasible solution. In a final step, this example is extended even more by including the risk-free asset with an interest rate of 4.0%. This example will contain the whole complexity of the general case except the correlations between assets and liabilities. STD Cum. Return Share STD Comb. Ratio Share Risk Free 0.0% 104.0% 36.4% LoB 1 20.0% 97.0% 9.1% Asset 1 10.0% 107.5% 33.3% LoB 2 10.0% 98.5% 90.9% Asset 2 20.0% 112.5% 30.3% Figure 98: RORAC Optimum with a Risk Free Asset with 50.0% Correlation. If a risk-free asset in the RORAC optimization is involved (assumed the asset and liabilities are correlated to each other) this asset obtains a high weight. It is also remarkable to see that the inclusion of a risk-free asset influences not only the weight of the assets, but also the weight of the liabilities. - 113 - 3.3.5 Conclusion Under some simplified model assumptions, there is always an optimum for risky investments with respect to the RORAC. In the special case of two risky investments, one can represent this as tangential point on the risk / return curve with a line through the zero point. If the EVA as a preference function is used, then in the simple case of two risky investments there will not always be a feasible solution. In such a case a solution is obtained as a tangential point on the risk / return curve with a line Yield = constant + c · STD. This tangent point does not exist if c is too small. Thus, the optimum is the maximum at the boundary – regardless of the capital requirement. Higher cost parameter c produce solutions, but may be unrealistic. This result indicates some doubts concerning the convenience of the EVA as a useful KPI, because even in the simplest case a solution depends on the choice of the capital cost parameter. The fact that only very high capital cost parameters produce an optimum, should be critically evaluated. A (simplified) RORAC optimization will generate solutions, which are acceptable but not necessarily in the sense that all parameters are between zero and one. If there are just correlations between assets and liabilities, we will immediately recognize that optimal combinations of assets will be influenced by the characteristics of the assets themselves and the characteristics of the liabilities (and vice versa). It is possible to summarize groups or segments to have a smaller dimension in the optimization approach in order to obtain feasible solutions. In general, there is a solution for the RORAC optimization problem. It is, however, very difficult to interpret. As has already been mentioned, the considerations outlined here are not necessarily suitable for the optimization of complex reinsurance structures. The approach presented here can only be used to deliver a simplified model to have better starting values for alternative calculations in internal models. Nevertheless, some implications for value-based management can be identified. - 114 - 3.4 Treatment of Extra Dividends In this chapter, the treatment of extra dividends will be discussed. Therefore, it is good to keep in mind the following equivalence equation, which reflects all relevant aspects for tariff rating before a contract is written: Present Value of Premiums = Present Value of Claims + Present Value of Costs + Present Value for Extra Dividends. This chapter concentrates on the correct distribution of dividends to the shareholder after having underwritten the contract. Costs of capital included in the premium should only be distributed to the shareholder when they have been earned. This will be analyzed by means of some calculation examples in the following sections. 3.4.1 Cost of Capital and Target Premium Extra dividends are defined as the return on required capital above the Risk-free interest rate. The Cost of Capital as present value of all extra dividend depends on the capital allocation at the beginning of a period as well as on the required extra dividend at the end of a period. We have analyzed in a previous section how target combined ratios and thus target combined premiums depend on different levels of risk-free interest rates with respect to different CoC models reflecting different insurance markets. We did not consider any change in the CoC requirements given different levels of risk-free interest rates, compare the table below. Market Interest Rate 2.0% 4.0% 6.0% Extra Dividend 13.0% 11.0% 9.0% Total Yield 15.0% 15.0% 15.0% Present Value Claims 88.7 79.6 72.0 Present Value Costs 29.1 28.3 27.6 Present Value Extra Dividends 31.7 24.2 18.0 Present Value Premium 149.5 132.1 117.6 Premium 151.0 134.7 121.1 Figure 99: Target Premiums given a Fixed Total Yield - 115 - If the total yield is fixed than there is strong dependency on the different levels of risk-free interest rates with respect to the target premiums. Market Interest Rate 2.0% 4.0% 6.0% Extra Dividend 8.0% 11.0% 14.0% Total Yield 10.0% 15.0% 20.0% Present Value Claims 88.7 79.6 72.0 Present Value Costs 29.1 28.3 27.6 Present Value Extra Dividends 19.5 24.2 28.0 Present Value Premium 137.3 132.1 127.6 Premium 138.7 134.7 131.4 Figure 100: Target Premiums given a Variable Total Yield The model with variable total yields results less-volatile target premiums in the case of a change in the risk-free interest rate. This concept could be used to stabilize the premium calculation. 3.4.2 Extra Dividends According to Underwriting Performance The treatment of extra dividends will be explained by a calculation example. Firstly, the underwriting performance will be taken into account; afterwards the impact of risky investments will be analyzed. Both sides should be considered separately, because underwriting and asset management act independently from each other in an insurance company. For this example an industrial insurer was chosen. Normally, an industrial insurer has higher yield expectations than an insurer in a personal lines business – for example to get a good “A” rating. The model calculation is based on the following input parameter: Market Interest Rate Premium Cost Ratio Claims Ratio Entity Factor Premium Factor Reserve Factor CoC Rate 4.0% 1,000.0 25.0% 73.3% 150.0% 22.0% 12.0% 10.0% The next figure illustrates the a-priori underwriting cash flows with a target fulfillment of 100.0%. The capital allocation of 330 at the beginning is obtained by multiplying the premium of 1,000 with the entity factor of 150% and the premium factor - 122 - 4 Embedded Value as Fair Value Approach The on-going change to a value based management requires appropriate key figures and steering systems. Internal models in non-life insurance are usually based on the economic capital after one year as stochastic target function according to the immediate realization of assets and liabilities at market values. That does not always constitute a realistic hypothesis. The direct liquidation of all assets and liabilities would result in high discounts on the assets respectively in high surcharges on the liabilities. Especially the existence of market values for loss reserves seems illusionary. In addition, the tensions in the financial markets lead to distortions in market prices (e.g. in form as a liquidity premium). These circumstances do not reflect adequately the medium to long term value situation of an insurance company. 39 Given the Embedded Value EV (as an alternative approach for evaluating corporate economic capital) the fair values of assets and liabilities will be realized only over time according to a virtual run-off. Thus, “modelling” of a virtual external investor of the insurance portfolio is not requested. This approach leads to the following advantages and disadvantages: • As a consequence of frictional and other costs, the EV is lower than the directly attributable economic value. • The EV is more realistic because there would be discounts in the case of selling the portfolio. • The EV reacts less sensitively to market price fluctuations. Therefore, the EV-approach is well established in the typical long term business of life insurance. In non-life insurance, however, this concept is actually not well established, although there are first applications within the integrated steering of the whole business. Thus, it is consequent to consider also the EV within the value based management in non-life insurance at a middle-term perspective. This will result in a coherent view on risk steering at group level. Especially within the framework of Solvency II the insurance groups are interested in a consistent company steering system. Therefore a harmonization of modelling approaches between the life and non-life segments is required. 40 This chapter describes in what way and to what extent the EV concept could apply to non-life insurance. First, for a better understanding, the methodology and devel39 Heep-Altiner, Krause (2012), p. 2. 40 Heep-Altiner, Berg (2012). - 123 - opment of the EV in life insurance will be explained. After that, an approach to transfer the idea to non-life insurance is presented. The following example of the fictional property/casualty insurer named “Feldafinger Brandkasse” will illustrate an EV calculation. Finally, based on the results a conclusion and outlook is given. The explanations and descriptions in this chapter are mainly based on the research results of the cooperation between the working group “Embedded Value Non-life” of the German Association of Actuaries (Deutsche Aktuarvereinigung) and the master students at the Institute of Insurance Studies in Cologne (Institut für Versicherungswesen der FH Köln). 41 4.1 Embedded Value in Life Insurance The return profile in life insurance distinguishes from other lines of business because of its long-term character. Typically the high acquisition costs at the beginning of a contract will be amortized over time by the profits in future years, see the figure below. Figure 110: Annual Profit of a Life Insurance Contract 42 Therefore, life insurers in a period of growth show an operating loss due to its high rate of new business. It concludes that the annual reported gain from income statements does not reflect the adequate value of the life insurance portfolio. Thus, future cash flows have to be taken into account for the valuation of contracts. The EV considers the present value of all future profits of the insurance portfolio and regards the long-term nature of the business. To summarize this, the EV is an indicator of the prospective earnings potential of a life insurance company and is the key performance figure for the shareholders and potential investors. 41 Heep-Altiner (2012). 42 Gürtler (2012), p. 7. - 124 - 4.2 Historical Development James Anderson published the basic conceptual idea in the year 1959. 43 Based on his isolated projection of future cash flows the EV approach evolves constantly over time. Today the EV is a generally accepted indicator in life insurance and this is why most companies publish an additional EV report beside the legal reporting requirements. 44 The EV estimates the value of the company, based on its current net worth plus the present value of future profits minus costs. The estimation of future cash flows requires an extensive set of assumptions. For example, the future interest rates, inflation, policyholder behaviour and mortality have to be considered. Attempts to harmonize and improve the initial concept of the traditional Embedded Value (TEV) led to the concept of the European Embedded Value (EEV) and finally to the Market Consistent Embedded Value (MCEV). 45 Traditional Embedded Value (TEV) The TEV corresponds to the value of the adjusted equity (net asset value) plus the Present Value of Future Profits (PVFP) for the covered business minus the Cost of Capital (CoC). Figure 111: Traditional Embedded Value 46 The separate components of the traditional Embedded Value are explained in more detail below. Net Asset Value (NAV) The NAV is the book value according to generally accepted accounting principles (e.g. German GAAP) of the equity adjusted with valuation reserves (difference between the market values and the accounting values) and the dividends for the shareholders which are included in the balance sheet profit. The NAV is divided into the Required Capital (RC) and the Free Surplus (FS). The RC is demanded for example as a solvency capital by the insurance supervision or by the rating agencies. 43 Anderson (1959). 44 PWC, p. 1. 45 Heep-Altiner; Krause (2012), p. 7. 46 Gürtler (2012), p. 8-10. - 125 - Cost of Capital (CoC) The CoC corresponds to an adequate interest on the RC. For the purpose of providing capital for the insurance company the shareholders demand an appropriate return on the invested capital (Risk Discount Rate, RDR). The actual investment income on the RC is usually lower than the expected risk discount rate (RDR). Furthermore, the participation of the policyholders as well as the taxes on the investment income on the RC should be considered. Present Value of Future Profits (PVFP) An essential element of the TEV is the deterministic PVFP. For the calculation, the following assumptions are used: • The insurance portfolio is in run-off. • The profit and loss account and the balance sheet will be projected over the predefined projection period. • The future new business will not be considered. • The investment income on equity is not taken into account. As a result, the future profits are determined. The following discounting calculation uses the RDR and the PVFP is then identified and quantified. European Embedded Value (EEV) An earlier lack of clear guidelines for the determination of the TEV made comparability between the different companies complicated for investors and shareholders. In the year 2004 the so called CFO forum, comprising the 21 chief financial officers of the most important European insurance groups, established the European Embedded Value Principles (EEVP). The EEVP set down 12 general binding rules. 47 For instance, beside the three components of the TEV, the EEV considers the Time Value of Options and Guarantees (TVOG) as an additional factor, see the figure below. Figure 112: European Embedded Value 48 47 CFO Forum, European Embedded Value Principles. 48 Heep-Altiner (2012), p. 18. - 126 - The guarantees mainly refer to fixed promised financial guarantees. An example for options is the right of cancellation for policyholders or the lump sum option in annuity insurances. Therefore, the deterministic perspective of capital market scenarios is insufficient for assessing the TVOG appropriately. Especially the evaluation of financial guarantees needs a stochastic asset / liability projection model to reflect the volatility of the financial markets. 49 It is necessary to develop management rules, e.g. for determining the participation of the policyholders on investment incomes and for an assumption of future policyholders’ behaviour. Furthermore, the EEVP requires a consistent calculation for the RDR and homogeneous publication standards. 50 Market Consistent Embedded Value (MCEV) The components of the MCEV align with the EEV. Furthermore, costs of nonhedgeable risks must be taken explicitly into account. Because the EEVP did not solve the problem of an appropriate and objective RDR sufficiently, the CFO Forum published the Market Consistent Embedded Value Principles (MCEVP) in June 2008 in order to bring greater consistency and improved disclosure to the EEV. The MCEVP include 17 “Key principles”, 145 “Areas of guidance” 51 and a “Commentary on Principles & Guidance (Basis for Conclusions).” 52 The MCEV is currently the most sophisticated and harmonized EV concept. It values assets and liabilities on a market-consistent basis. Assets are valued at the amount for which they can be sold at the time of valuation. The liabilities, which are not traded and illiquid, are valued by a replicating portfolio or other adequate mathematical techniques. The MCEVP also require a consistent valuation for the TVOG similar to the pricing of options and other derivatives on capital markets (Black & Scholes). Furthermore, costs of non-hedgeable risks must be taken explicitly into account. But the discussion about the right methodology and assumptions still continues. In October 2009, the CFO Forum published an amendment to the MCEV Principles to allow for the inclusion of an illiquidity premium. Furthermore, in December 2011 a press release was issued by the CFO Forum to take account of the current sovereign debt market conditions in EV reporting as an initial step towards the expected convergence of MCEV with the developing Solvency II regulatory framework. 53 However, the discussion is still on-going as to how and to which products such illiq49 CFO Forum, Basis for Conclusions European Embedded Value Principles, p. 15. 50 Gürtler (2012), p. 10-11. 51 CFO Forum, MCEV Principles & Guidance. 52 CFO Forum, MCEV Basis for Conclusions. 53 www.cfoforum.nl /embedded_value.html. - 127 - uidity premiums should be applied as well as how sovereign debt market conditions should be taken into account under Solvency II. 54 4.2.1 Application of Embedded Value The Embedded Value applies in the following areas of the insurance business: 55 Evaluation of a Company: The EV is an alternative approach for the evaluation of a life insurance company with a more significant expressiveness than the classical figures. Therefore, it is the main component in the negotiation process of mergers and acquisition transactions. Company steering: As the material part of internal life insurance models, the EV determines the required risk capital and is therefore a main part of the risk management, especially considered in the framework of Solvency II. The required capital is calculated based on sensitivities, stress scenarios and the required security level. Movement Analysis: With a movement analysis as a tool of a value-added analysis, the separate impacts for a change in the EV could be examined ex post. The Movement Analysis is an important tool for the performance measurement and the evaluation of the management of a life insurance company. Therefore, the MCEV at the end of the year (EoY) will be compared a posteriori with the MCEV at the beginning of a year (BoY). The reasons for the change of the value will be analysed individually. The following figure shows a schematic example for a Movement Analysis. 54 Munich Re, Market Consistent Embedded Value Report 2011, p. 3. 55 DAV, Embedded Value in der Schadenversicherung, p. 6. - 128 - Non-financial Experience variances MCEV BoY Opening adjustments Non-financial assumption changes Mismatching profit Return on assets not backing Liabilities: “unwind” New Business Unexplained MCEV EoY Dividends to shareholders market-consistent compensation for taking ALMrisk: eliminate for measuring return Figure 113: Movement Analysis 56 Different factors lead to an increasing / decreasing MCEV. For example, the change of the non-financial assumptions with regard to the future has reduced the value. On the other hand, factors like the deviation between the realized and the estimated non-financial assumptions, the overperformance of investment earning and the value added by new business, lead to a higher MCEV at the end of the year. 4.2.2 Market Consistent Embedded Value The MCEV is the present value of shareholders’ interests in the earnings distributable from assets allocated to the covered business after making sufficient allowance for the aggregate risks involved. When calculating the MCEV the following principles have to be considered: 57 Closed Fund Projection: In opposite to the Appraisal Value 58 (AV) the EV does not consider future new business. The EV and the existing insurance portfolio will be projected in run-off. Best Estimate: The calculation is based on realistic assumptions. Going Concern: All assumptions base on a continued business operation. Further Consideration Regulatory and legal frameworks and continuous management rules have to be taken into account. 56 DAV-Arbeitsgruppe EV Sach: Embedded Value in der Schadenversicherung. Bericht an den Ausschuss Schadenversicherung DAV. Stand 16. September 2010. 57 Gürtler (2012), p. 7. 58 The AV can be interpreted as EV plus Goodwill. - 129 - The MCEV components correspond essentially with the EEV but use a closer classification level for the single components. The MCEVP distinguish between the following components of EV. 59 Figure 114: Market Consistent Embedded Value 60 The different components of the Market Consistent Embedded Value are described in more detail in the following: Net Asset Value The Net Asset Value is divided into the components Required Capital (that has to be kept within the company) and Free Surplus (that can be paid out). Required Capital RC is the market value of capital allocated to the covered business. It equals at least the regulatory solvency capital, but may be higher to meet internal risk capital models or rating targets. RC is tangible and may be distributed over time as liabilities run-off. Free Surplus FS is the market value of capital allocated to the covered business but not required to support the in-force covered business at the valuation date. FS is tangible and may be distributed immediately. The FS is a residual amount. To calculate the FS, an analysis of the whole equity is needed. 59 CFO Forum, MCEV Principles & Guidance and Munich Re (2011), p. 19 ff. 60 Heep-Altiner, Jutzi (2012), p. 23. - 130 - Equity of an insurance company is defined as the difference between all assets and liabilities. FS equals the subtraction between equity and RC (under consideration of tax and shareholder dividends). For a German insurance company, the components FS and RC can be concluded from the German GAAP balance sheet. But an adjustment of the balance sheet positions is necessary to get to the required market value view. Value of In-Force The Value of In-Force covered business (VIF) consists of the PVFP, TVOG, FCRC and CRNHR. Present Value of Future Profits The PVFP is the present value of future local statutory (e.g. German GAAP) shareholder after-tax profits emerging from the business covered on the condition that all economic and non-economic assumptions are met. Therefore, the following factors are essential to determine the future insurance portfolio development: • Investment Income, • development of cost and claims, • cancellation behaviour of policyholders, • dynamics on financial markets, • reimbursement from reinsurance and • risk discount rate. The assumptions based on the Best Estimate principle have to be made for each line of business and product individually. Furthermore, the calculation of future profits considers the going concern assumption. The assumptions made are assumed to be adequate for the future as well on an inflation-adjusted basis. Based on the assumptions, the calculation procedure for the PVFP follows these steps: 1. Determination of the net profit before tax, based on the underwriting and investment results. 2. Determination of the net profit after tax for each period under review. 3. Discounting with the RDR to the beginning of the projection. - 131 - Time Value of Financial Options and Guarantees Participating life business is generally characterized by options and guarantees, which are strongly dependent on the financial markets (e.g. a minimum interest rate or a minimum level of bonus is guaranteed to the policyholder). The participating features are usually a combination of contractual or legal constraints and management discretion that has to take competitive pressure or market practice into account. The calculation of TVOG should be based on a stochastic variation of future economic conditions using methods and assumptions consistent with the underlying EV. Frictional Costs of Required Capital FCRC reflect the taxation costs for risk-free investment on the assets backing required capital as well as the costs for the investment management for those assets. Cost of Residual Non-Hedgeable Risks CRNHR are Cost of Capital for all (residual) risks that have not been considered in the market value of a risk component yet. The following figure illustrates the range of the MCEV components. RC FS PVFP CoC FC TVOG MCEV Figure 115: Components of the MCEV 61 61 Own figure based on Munich Re (2011), p. 4. - 138 - Figure 119: In-Force Business and Renewals 67 In the first case, the contract is completed before the balance sheet date and thus before the date of the MCEV approach. Therefore, this insurance contract is attributed to the In-Force portfolio until the end of the reporting year. As can be seen in the second timeline, there is a renewal on the balance sheet date. The contract from the previous year is continued automatically, which implies that this contract is assigned to the renewals and influences the value of the continuing business. Looking at the third example, the signing of the contract took place before the balance sheet date. The contract is therefore attributed to the In-Force business, beginning at the balance sheet date and ending at the maturity date. After that the contract will be allocated to the renewals. This distinction of the existing contracts is of enormous importance for the portfolio development in non-life insurance. For example, the portfolio value is influenced by the appropriate assumptions regarding the lapse rate and claims cost. These assumptions must be made individually for every line of business, which also means a high reliance on uncertain planning assumptions and thus a dependency on the business policy of a company. In motor insurance the loss ratio may increase due to the loss of good risks. Political factors could also lead to wrong assumptions, as they are derived from the past 67 Heep-Altiner (2012), p. 48. - 139 - and therefore have no validity for the future. The scrapping premium of 2009 led to a significantly higher number of car sales. This would mean a higher portfolio loss as it was calculated for the Embedded Value. Model Shocks Model shocks are isolated changes in individual input parameters. They serve for the comprehension (sensitivity) and testing (plausibility) of the calculated MCEV. Especially sensitivities of the MCEV give a good first impression of its value drivers and critical success factors. For parameters that are strongly influenced subjectively, such as the Cost of Capital this is vital. Especially, sensitivities have an added significance as they are published in the IFRS consolidated financial statements 68 . In life insurance, predefined model shocks have to be applied, indicating a change in single calculation parameters. For non-life insurance, such model shocks are also necessary to achieve a better understanding of the dependency of the Embedded Value to the various input parameters. Possible model shocks (among others), that could have a significant impact on the MCEV, are listed in the following: • Increase of Costs of Capital, • increase of Tax rate, • Increase of Cost Ratios, • Change in the Risk Discount Rate, • Increase of Administrative Costs, • Premium reductions as well as • Change in Claims Reserves. The general approach to calculate an Embedded Value in non-life insurance is explained in the next section. 4.3.3 General Approach In the figure below it is illustrated how the MCEV can be derived as a balance sheet projection over the total projection horizon. The illustration is based on German GAAP, but it is applicable to other generally accepted accounting principles, too. 68 See IFRS 4, 39A. - 140 - Figure 120: German GAAP Balance Projection for the MCEV Calculation 69 In a first modelling step, starting with the balance at t = 0, a Free Surplus or in a worst case scenario a Free Deficit, is realized as an immediate extraordinary payout. Thus, the company keeps the Required Capital at Market Value afterwards. The extraordinary payout takes place through a realization of hidden reserves or liabilities affecting the net income as well as a withdrawal or injection of capital with respect to the difference between balance equity and Required Capital without affecting the net income. After the extraordinary payout, the remaining Required Capital is invested risk-free with the result that no hidden reserves will exist in the following periods. In transition to every further period, the balance will be adjusted due to changes in the Profit and Loss Account and the withdrawal of free Required Capital. At the starting point of the projection no extraordinary payout happens for potential hidden reserves on liabilities as these are disclosed over time. Due to the projections the Required Capital as well as the liabilities are reduced over time. 69 Heep-Altiner (2012), p.54. - 141 - To get to the Embedded Value, the present value of all profits and losses and all capital withdrawals will be calculated on the basis of the interest rate curve adjusted by the CRNHR. 4.4 Embedded Value in Non-life Insurance – Calculation Example The aim of this section will be a presentation of the methodical approach to the determination of the MCEV using a fictitious insurance company - the so-called “Feldafinger Brandkasse” (FBK). Initially, the fictitious model-company will be introduced including its balance and all relevant input parameters, which are needed to determine the MCEV. Followed by a few calculation examples, the transition to the MCEV will be outlined using the calculated key ratios. A comparison between the MCEV and the economic capital will sum up this section. 4.4.1 Example Company Starting point of the fictitious insurance company is the following German GAAP balance for the FBK. Assets Liabilities Book Values Assets 236,139 48,236 German GAAP Equity Assets backing SHE 48,236 Assets backing Liab. 187,903 187,903 Book Values Reserves 153,952 Claims Reserves 33,951 Equalization Reserve Tax Receivables 0 0 Tax Reserve 236,139 236,139 Figure 121: German GAAP Balance at t = 0 70 All investments are split virtually into Assets Backing Liabilities (ABL) and Assets Backing Shareholders Equity (ABSE). The book values of ABL with an amount of 187,903 cover the technical provisions. The ABSE amounts to 48,236 and cover the German GAAP equity. Both, ABSE and ABL are assumed to be invested in riskfree zero bonds with redundancies / deficiencies according to the selected yield curve. Moreover, the FBK is subject to a tax rate of 32%. The German GAAP balance sheet of the FBK serves as the starting balance for the projection of surpluses in the projection model. The next figure lists all relevant input data, such as reserves and premiums, which are crucial for the MCEV determination. It also displays the German GAAP balance 70 Heep-Altiner (2012), p.57. - 142 - with book values at t = 0. Besides this, the fictitious company has only two lines of business, third-party motor vehicle insurance and homeowners insurance. Position Third Home Total Party Owners Earned Premiums 92,218 37,485 129,703 Book Value Claims Reserve 142,839 11,113 153,952 Best Estimate Claims Reserve 88,331 7,043 95,374 in % of Booked Claims Reserves 61.8% 63.4% 62.0% Book Value Equalization Reserve 26,863 7,088 33,951 in % of Booked Claims Reserves 18.8% 63.8% 22.1% Book Value Technical Reserve 169,702 18,201 187,903 Book Value Assets 236,139 Redundancy / Deficiency in % 2.0% German GAAP Equity 48,236 Figure 122: Input Data – Example Company 71 Concerning the Best Estimate Reserves of the existing business (evaluated by suitable mathematical algorithms) and the claims experience of the new business the following cash flow assumptions apply: 123456 Old Reserve 26.76% 20.02% 14.56% 10.59% 7.70% 5.60% New Business 68.54% 10.62% 7.04% 4.66% 3.09% 2.05% Cash-Flow in % after … Years Figure 123: Cash Flow Pattern at t = 0 72 Additionally, there is a further need of input parameter to carry out a MCEV projection, especially • global parameters, • projection information and • Required Capital information including Costs of Capital information. 71 Heep-Altiner (2011), p.124. 72 Heep-Altiner (2012), p.59. - 143 - The global parameters comprise a risk-free yield curve where the implicit forward rates can be deducted from the spot rates as illustrated in the following table. 0 1 2 3 Spotrates 3.92% 4.70% 4.53% 4.51% Forwardrates 4.70% 4.36% 4.48% Duration Figure 124: Yield Curve at t = 0 73 Concerning the asset structure at t = 0, it is assumed that the FBK has only invested in risk-free zero bonds with the following characteristics: average duration of fixed income securities 4,57 average interest rate of fixed income security 5.00% hidden reserves in the book values at t=0 2.00% investment costs in % of the market values 0.20% The hidden reserves of 2.00% result with respect to the chosen yield curve. The percentage of hidden reserves would change, if it were based on a different yield curve with different interest rate structures. Further input parameters are needed to determine the Required Capital, which is needed to generate the MCEV of the fictitious company. The following assumptions are made: • Parameter with respect to the SCR calculation, • 175% coverage due to Rating Requirements, • CoC Ratio of 6% with respect to the Solvency Capital. The determination of the SCR is based on the QIS 5 study 74 . Premium risk, reserve risk and the correlation between both risks depend on the internal model of the FBK. In the following, the MCEV projections are carried out only for the existing business in order to be consistent with the usual definition of economic capital in non-life in73 Heep-Altiner (2012), p.61 74 For more information see https://eiopa.europa.eu/consultations/qis/insurance/quantitative-impactstudy-5/index.html - 144 - surance. Furthermore, we consider a “virtual” run-off (e.g. within other business operations) such that only claims regulation costs (covered in the Best Estimate Reserves) and investment costs occur, but no operational costs for new business. Operational costs are not included in the projections. The projected development of the Claims Reserves (German GAAP as well as Best Estimate) and the Equalization Reserves is illustrated in the following figure. Position 0 1 2 3 Total Payments 25,518 19,095 13,887 BE-Reserve 95,374 69,855 19,095 36,873 German GAAP Reserve 153,951 112,760 50,761 59,520 Operational Expenses 0 0 0 0 Equalization Reserve 33,951 24,867 18,070 13,126 Value Figure 125: Projection of Reserves without Renewals 75 The projected German GAAP Reserves result from the projected Best Estimate Reserve (according to its cash flow pattern) after application of the initial overreserving percentage (according to the defined management rules), especially German GAAP Reserve (t) = BE-Reserve (t) · Over-reserving-Rate (t) The projected Equalization Reserve results from the projected German GAAP Reserve after application of the Equalization Rate, especially: Equalization Reserve (t) = German GAAP Reserve (t) · Equalization Rate (t) After those reserve projections the projection of the RC and the CRNHR can be performed. 4.4.2 Net Asset Value In this section, the Net Asset Value (as a sum of the Required Capital and the Free Surplus) will be calculated on the base of the previously specified input parameter of the FBK. 75 Heep-Altiner (2012), p.73. - 145 - Required Capital According to the management rules of the FBK, the Required Capital is the maximum of the • Required Capital to cover the SCR & MCR according to Solvency II and the • Required Capital to cover the solvency margin according to Solvency I with a required 175% overload. The figure below shows the projection of the Required Capital for several years: Position 012 (1) Market Value ABL 191,666 139,997 101,092 (2) Discounted BE Reserve 83,454 61,263 44,426 (3) Projection of Risk Margin 3,454 2,514 1,815 (4) SCR incl. 175% Overload 32,130 23,586 17,104 (5) Required Capital = MAX[(2)+(3)+(4)-(1);0] 0 0 0 Period Figure 126: Required Capital without Renewals (1) 76 Required Capital to cover the SCR is only needed if the hidden reserves are insufficient to cover the Solvency II requirements. As shown above the ABL market values are sufficient to meet Solvency II requirements such that there is no capital required due to this aspect. Furthermore, Required Capital to cover the MCR should be calculated. The MCR is set as 50% of the SCR based on Solvency II: See the next figure. Position 012 (1) SCR 18,360 13,478 9,774 (2) MCR = (1) · 50% 9,180 6,739 4,887 (3) Required Capital = (2) 9,180 6,739 4,887 Period Figure 127: Required Capital without Renewals (2) 77 As a next step, the Required Capital according to Solvency I in combination with a required coverage of 175% is calculated as 76 Heep-Altiner (2012), p.77. 77 Heep-Altiner (2012), p. 78. - 146 - Required Capital = 175% · MAX [Premium-Index; Claims-Index; 2,200]. The projections of the Required Capital due to Solvency I with a coverage of 175% are listed in the figure below. Position 012 (1) Premium Index 21,814 0 0 (2) Claims Index 24,236 15,327 6,337 (3) Solvency Margin = MAX [(1);(2);2,200] 24,236 15,327 6,337 (4) Required Capital = 175% · (3) 42,412 26,823 11,090 Period Figure 128: Required Capital without Renewals (3) 78 Finally, all the three steps have to be combined in order to determine the Required Capital in total to fulfil all solvency and rating requirements of the company: See the figure below. Position 012 (1) Required Capital SCR 0 0 0 (2) Required Capital MCR 9,180 6,739 4,887 (3) Required Capital Silvency I 42,412 26,823 11,090 (4) Required Capital = MAX[(1);(2);(3)] 42,412 26,823 11,090 Period Figure 129: Required Capital without Renewals (4) 79 The value of the total Required Capital decreases quickly with respect to the given run-off Scenario. At t = 3 the value of the Required Capital already amounts to the minimum. As a result, the fictitious insurance company needs a Required Capital of 42,412 for all underwritten risks in t = 0. Free Surplus The Free Surplus is the second component determining the Net Asset Value. In the balance at t = 0, an initial German GAAP equity of 48,236 is given. From this starting point, the Free Surplus can be calculated by the following approach: 78 Heep-Altiner (2012), p. 78. 79 Heep-Altiner (2012), p. 79. - 147 - • The initial German GAAP equity includes 2% of hidden asset reserves (= 996). By realizing those reserves a tax of 32% has to be paid, which results in an after-tax value of 657. • The difference between the initial German GAAP equity and the Required Capital can be treated as a tax-free capital withdrawal. The difference between both values is 5,824 = 48,236 – 42,412. If we combine all calculations carried out in this section we obtain the following Net Asset Value of the FBK: Free Surplus = 5,824 + 657 = 6,481 Required Capital = 42,412 Net Asset Value = 48,893 Next, the Value of In-Force Business of the FBK will be calculated by carrying out the German GAAP balance sheet projections over the projection period. 4.4.3 Value of In-Force Business Calculating the Value of In-Force Business depends on different economic parameters. One way to calculate the VIF is to calculate the Present Value of Future Profits and subtract the sum of Costs of Residual Non-Hedgeable Risks, the Time Value of Options and Guarantees and the Frictional Costs. Another way is shown in the following figure illustrating the projection results at t = 1. Position Value (1) Total Result after Capital Removal 40,156 (2) Cost of Capital 1,102 (3) Free Surplus 0 (4) Reproduction of Required Capital 17,581 (5) VIF = (1) - (2) - (3) - (4) 21,474 Figure 130: VIF Result at t = 1 without Renewals 80 The projection of all discounted VIF results over the projection period results in a total VIF of 67,527. 80 Heep-Altiner (2012), p.105.