In the following simulation study we want to give an example how a multi-year internal model can be used in strategic ERM.
The internal simulation model presented here is based on a five-year period using 100,000 simulations. We illustrate the applicability of the model using real-world data and an internal model actually in use at a medium-sized German non-life insurance company. To protect the anonymity of the company, we transformed all data so as to change the absolute values but not the underlying risk structure. In our application, we rely on the internal model presented in Diers6 and extend this model as described in the section “Model approach for measuring multi-year risk capital” above.Footnote 21
Two types of assets (high- and low-risk investments) and three types of claims (catastrophe, large and attritional (base) claims) are integrated into the model, which can be simulated according to adequate probability distributions, subject to a given dependency structure. We consider four insurance segments: storm, flood, hail, which are dominated by natural catastrophes, and “All other Lines of Business”, as well as the investment segment.
We define multi-year risk capital using simulated multi-year insurance and investment results via multi-year development of economic capital as described in the section “Model approach for measuring multi-year risk capital”.
For modelling natural catastrophes such as storms see Diers.Footnote 22 The time at which extreme events take place has been simulated in order to address the potential need for short-term liquidity in capital investment strategy. In the asset model for each scenario management rules (sales priorities) define which assets have to be sold depending on the level of liquidity required.
In the case of scenarios with adverse developments on the capital markets or natural catastrophes, the liquidity requirement may not be fulfilled. One of management's major responsibilities is to ensure that these scenarios are minimised by adequate underwriting policy, sufficient reinsurance protection and appropriate asset allocation. In the following we want to give an incentive to use internal models to analyse the different strategic instruments in the context of an ORSA process by quantifying the effects on the risk-adjusted key performance indicators of the example company.
Our example company mainly underwrites insurance policies with private and low industrial businesses. We have assumed a consistent underwriting policy and asset allocation within the five-year period. In addition we assumed that the claims that will occur in the following five years simulated will be stochastically independent of one another (concerning the different years).
At time t=0, the example company has economic capital EcCap
0 amounting to €135 million. In an ORSA process, management decides to take TVaR at a confidence level of 99.8 per cent as a risk measure for one year risk-capital requirement, and TVaR at a lower confidence level of 99.5 per cent in the five-year risk-capital requirement. We have set capital costs r
=15 per cent and only refer to this for risk capital, resulting in a risk-adjusted EVA_ra. Additional risk-free interest on economic resources has already been deducted from the investment result for benchmarking purposes. Therefore the investment result has been reduced by risk-free interest (on economic capital and liabilities). Risk-free interest represents a benchmark for investors, risk-free interest on economic capital has been given in a separate position not considered here.
We have used the TVaR allocation principle to identify risk factors (see the section “Capital allocation for performance management in ERM”). Table 2 shows risk and return indicators in the insurance segments shown as gross, that is, before reinsurance. We have assumed a stochastic independence between results from the different insurance segments (lines of business) and between insurance and investment results with the exception of hail and storm, where we have assumed a slight tail dependency.
Before reinsurance, the storm segments constitute the dominating risk factor affecting the company. According to TVaR allocation, storm segments need a one-year risk capital of €296 million where all other segments together need a negative risk capital of −€5 million. This shows that the other segments do not contribute to the risk capital of the company, because storm is the dominating risk. Only hail shows a positive allocated risk capital because of the slight tail dependency concerning storm and hail results.
We use Figure 4 to clarify this effect. It shows the dependency structure in storms and company-level results using 10,000 simulations. Note that the storm results are part of the company-level results. One can see that the worst company results occur together with the worst storm results, so storm risk dominates company risk. The diversification effect between the four lines of business (storm, flood, hail and others) and investment results is relatively low at 39 per cent due to the dominance of storm risk. The shortfall probability is very high at 1.9 per cent for one year, and extremely high over five years (9.0 per cent), as the €135 million in economic capital will not be enough to cover the company's risk-capital requirement for one year and for five years (€370 million). According to Solvency II the one-year shortfall probability has to be less than 0.5 per cent. Therefore the actual strategy does not only fail internal management requirements defined in an ORSA process, but also regulatory requirements following Solvency II.
Our aim is to test how risk-lowering strategies will affect the risk-return situation in our example company. Table 3 gives a survey of the strategies applied.
In Strategy 1, all other lines of business that are not influenced by natural catastrophes such as storm, hail and flood should be extended by 20 per cent (increase in the number of contracts) in order to generate increased returns and to benefit from increased diversification effects.Footnote 23 To reduce risk capital requirements a €250 deductible is applied to policy-holders in storm insurance segments, where the reduction in claims and regulation costs due to the deductibles are completely passed on to the client by premium adjustments (an alternative might be here to assume that only a portion of the reduction is passed on to the policy-holder, which would ceteris paribus enhance the risk and return situation of the insurer). This leads to an unchanged (stand-alone) RoRAC=11.3/300=8.5/225=0,038 in storm insurance (see Table 4 where return decreases from €11.3 million to €8.5 million and risk capital decreases from €300 million to €225 million).
In storm insurance, even low deductibles of €250 lead to a loss reduction of about 25 per cent. This significant loss reduction is a result of the enormous number of small claims that occur due to storm events. Therefore deductibles in storms have a remarkable effect on risk capital. Introducing deductibles tends to be unpopular with policy-holders, resulting in probable cancellations and cross-cancellations. We have not considered these effects in our calculations, but they should not be neglected in practice. For modelling deductibles we rely on Diers.22
Despite expansion in the non-catastrophe segments (all other LoB), risk-capital requirement hardly increases in the stand-alone view from €44 million to €49 million, because of the high diversification effects within and between these lines of business.
The somewhat lower risk situation in storm results in increased diversification effects to 48 per cent. Since the risk capital in our example company is dominated by storm risk, the conditions in Remark 1 (see the section “Capital allocation for performance management in ERM”) are fulfilled for all strategies unaffected by this dominance, which even applies to Strategy 1. Remark 1 therefore implies that strategies which increase the segment-level EVA also increase the EVA for the company.
In our example, increasing the segment EVA for storm and all other lines of business leads to an increase in EVA for the company, but this is still strongly negative at −€10.4 million. The one-year risk-capital requirement of €214 million is still substantially higher than the company's economic capital of €135 million, and the one-year shortfall probability is therefore still high, so that the Solvency II requirements are not fulfilled. The same holds for the five-year risk capital of €276 million and the five-year shortfall probability.
Strategy 2 is based on Strategy 1. In addition to Strategy 1, the storm segments are reinsured by an event-excess-of-loss contract. For modelling reinsurance contracts we rely on Eling and Toplek.7 We have calculated the reinsurance premiums using technical pricing methods (see Diers)6.
Reinsurance in the storm segments substantially lowers one-year risk-capital requirement in the storm segments to €80 million. The allocated risk-capital requirement reveals that now the other segments also contribute to the company's risk-capital requirement. However, storm risk still dominates the risk-capital requirement at company level (Table 5).
Figure 5 shows 10,000 simulation results of economical results in storm vs. economical results of the company. We see that by following Strategy 2 many of the worst scenarios of the company still fall together with the worst scenarios in storm insurance, which shows the further domination of storm risks in these scenarios (see black marking). Comparison of Figures 4 and 5 reveals a changed dependency structure, showing that the domination of storm risks following Strategy 2 is much less in comparison to the actual strategy.
The substantial reduction in risk-capital requirement in storm segments leads to a substantial increase in diversification effects to 66 per cent. Now the one-year risk-capital requirement of the company, €90 million, can be covered by the company's economic capital of €135. This leads to a one-year shortfall probability fulfilling the regulatory requirements (Solvency II). Apart from that, company EVA turns positive at €2.2 million.
The effect of reinsurance agreements can be assessed in Remark 3 together with Eq. (18) in the section “Capital allocation for performance management in ERM”. Using the definitions of the section the Strategy 1 describes the situation before reinsurance (“gross”) and Strategy 2 after reinsurance (“net”). The reinsurance contract has a positive EVA(R) and so leads to an increase of EVA on company level:
However, the company still has to cover its five-year risk-capital requirement from its own economic capital at t=0, enabling the company to remain over five years without external capital supply. We see that the €125 million in risk-capital requirement for the next five years are covered by the economic capital of the company, €135 million, too.
In order to increase the average return and in order to benefit from further diversification potential we define a Strategy 3 changing the asset allocation in the way that the part of high-risk investments is raised from actual 5 per cent to 10 per cent. Background for this strategy is the fact that the five-year risk-capital of €125 million requirement lies below the economic capital of the company, €135 million, so that we have a further €10 million for our multi-year risk position. This should be used in order to increase EVA on the company level.
Strategy 3 leads to an increase in one-year risk capital for investment results from €45 million to €61 million (Table 6). This leads to a low increase in one-year risk capital on the company level from €90 million to €92 million resulting from a further increase in diversification. The company can still cover its five-year risk-capital requirement from its own economic capital at t=0, because the €132 million in risk-capital requirement for the next five years are covered by the economic capital of the company (€135 million).
The simulation study shows that the TVaR allocation principle using TVAR as risk measure can be used for identifying risk dominating positions in the portfolio, because it allocates the amount of risk capital to each segment as it contributes to the whole risk capital. In this context it should be used in practice. Moreover the TVaR allocation principle can serve as an important base for strategic management decisions in the following sense (if the assumptions in Remark 1, section “Capital allocation for performance management in ERM”, hold): Increasing the segment-level EVA leads to an increase of EVA for the company. This means that defining risk-adjusted performance indicators on segment level using TVaR allocation principle can give the “right” incentives in strategic ERM.
On the other hand we can conclude from the simulation study and from the section “Capital allocation for performance management in ERM” that generally we neither have the “right” risk measure nor the “right” allocation method for all portfolio structures and problems. This also holds for coherent risk measures and allocation methods. Table 2 can serve as an example that shows the capital allocation (TVaR principle) following the actual strategy, allocating more than the total risk capital to the storm segment and so leading to a negative allocated risk capital for the other segments. This allocation method quantifies the risk dominance of storm risks in an adequate way and can be used for management decisions as described above, but negative allocated risk capital can cause a problem in practice in the context of risk limitation on segment level.