Niveles de contaminación del aire elevados y en aumento, debido a:

To test if our results are sensitive to the composition of the liability portfolio we will vary with the distribution of the insured members and compare the allocation of SCR(VaRα)

with the allocation of SCR(ESθ).

We will compare the allocation of SCR(VaRα) with the allocation of SCR(ESθ) for

the following liability portfolios:

• Young life annuity insurance company: Insured members with an average age of 35;

• Average life annuity insurance company: Insured members with an average age of 50 (base scenario);

• Old life annuity insurance company: Insured members with an average age of 65. The annuity that will be paid does not differ from the annuity described in Chapter 4. All other named assumptions made in Chapter 4 do still hold. For a precise description of the liability portfolios see appendices C, E and G. The number of insured members

Solvency II SCR based on Expected Shortfall — Wouter Alblas 31

Figure 5.13: Comparing the allocation of SCR(VaRα) with the allocation of SCR(ESθ) for the fictitious young life annuity insurance company, where the insured members have an average age of 35. The horizontal axis represents the confidence interval α used for VaR in the calibration method. The vertical axis represents the change in allocation of the SCR’s when the stress scenarios are calibrated on ES instead of VaR.

per liability portfolio is chosen based on matching the Best Estimate of the liabilities to the Best Estimate of the liabilities of the base scenario.

The asset portfolio will consist of 25% stocks and 75% government bonds, similar to the base scenario. Duration matching is used to assign the amounts invested to the 5- year and 30-year UK Gilts. To match the duration in the case of the young life annuity insurance company Canadian 50-year government bonds are added to the portfolio, these are 2.96% coupon bonds with a remaining time to maturity of 50 years.

Figures 5.13 and 5.14 show the change in allocation of the total SCR over the three different risk modules when ES is used to calibrate the shock scenarios instead of VaR for the different liability portfolios. To see how sensitive our results are to the composition of the liability portfolio we can compare these figures to each other and the base scenario in Figure 5.5.

Figure 5.5, 5.13 and 5.14 have a most things in common. Namely, the change in allocation of the total SCR is very small for a confidence interval α of 99.5%, this difference becomes more significant and is maximized for α = 98.5%, down to 97.5% the difference grows smaller again. We see that for all liability portfolios that longevity risk is underestimated and equity risk overestimated when VaR is used. For a confidence interval of 98.5%, longevity SCR would grow with over 4% and equity SCR would decline with close to 2% if ES was used instead of VaR for all three different liability portfolios. The only thing that differs for the different liability portfolios is interest rate SCR, if you compare Figure 5.13 with 5.14 you can clearly see that interest rate SCR changes are dependent on the composition of the liability portfolio.

We can conclude that for different liability portfolios it still holds that the difference in SCR allocation is maximized at α = 98.5% and that longevity risk is underestimated and equity risk overestimated when VaR is used. Also can be concluded that interest rate SCR is dependent on the composition of the liability portfolio.

Figure 5.14: Comparing the allocation of SCR(VaRα) with the allocation of SCR(ESθ) for the fictitious old life annuity insurance company, where the insured members have an average age of 65. The horizontal axis represents the confidence interval α used for VaR in the calibration method. The vertical axis represents the change in allocation of the SCR’s when the stress scenarios are calibrated on ES instead of VaR.

Chapter 6

Conclusion

This research paper examines the consequences for an average life annuity insurance company if the Solvency II SCR estimation is based on a more appropriate risk measure instead of Value-at-Risk. This Chapter consists of a summary of the findings and a discussion.

6.1

Summary of the findings

In the first part of this paper a literature study is undertaken which deals with finding an appropriate risk measure to estimate sufficient capital. Since Value-at-Risk does, among other things, not consider the tail beyond the confidence level, it is not a very suitable risk measure for a capital requirement calculation. The Expected Shortfall does not only consider the tail beyond the confidence level, it also satisfies the four axioms of translation invariance, subadditivity, positive homogeneity and monotonicity and can therefore be called a coherent risk measure. In this research we therefore chose to calibrate the SCR estimations on Expected Shortfall instead of Value-at-Risk and compare the differences.

The quantitative research of this paper consist of two parts: The first focusses on calibrating the SCR stress scenarios for equity risk, interest rate risk and longevity risk based on Value-at-Risk and the stress scenarios based on Expected Shortfall. The second part consists of comparing the SCR(VaRα) with the SCR(ESθ) for a fictitious

life annuity insurance company.

Since θ is chosen such that the total SCR(VaRα) equals SCR(ESθ), we focused on

the allocation of the total SCR over the three risk modules: Equity SCR, interest rate SCR and longevity SCR. By comparing these allocations for using VaR and ES, we saw if the current method, which uses VaR, underestimates or overestimates certain risks. This was done for different confidence intervals α used for VaR. For the confidence interval α used in Solvency II, namely 99.5%, the difference in allocation is very small between SCR(VaRα) with the SCR(ESθ). When the confidence interval decreases down

to 98.5%, the differences are maximized. Then it is shown that by using the VaR to determine the shocks, longevity risk is significantly underestimated and equity risk is overestimated. This conclusion can be drawn because when ES is used to determine the shock, the longevity SCR would grow and the equity SCR would decline if compared to VaR is used. Down to the confidence interval of 97.5% the differences become smaller again.

These results are very solid in terms of variation with the composition of the asset and liability portfolio of the life annuity insurance company. If the percentage of stocks in the portfolio declines, the wight given to the overestimation of equity risk increases and the weight given to the underestimation of longevity risk declines. Interest rate SCR is very sensitive to changes in the liability portfolio and that is one of the reasons that we can not draw one uniform conclusion about the behaviour of interest rate SCR. To test

the dependence of the results on the estimation methods used by EIOPA, we compared the results with the differences in allocation when the stress scenarios were determined in an empirical way. There we found that when empirical stress scenarios were used, the difference in allocation is also maximized for α = 98.5%, and compared to estimation methods used by EIOPA the difference in allocation increased, meaning longevity risk is more underestimated and equity risk is more overestimated when empirical stress scenarios are used.

Overall, we can conclude that for EIOPA’s current confidence interval α of 99.5%, the allocation of SCR(ESθ) is very close to SCR(VaRα). This can be explained by noticing

that the VaR99.5% is already captured in the “unexpected” long tail for the current

data. Might EIOPA choose to calibrate the stress scenarios on a smaller confidence interval, underestimation of longevity risk and overestimation of equity will most likely take place if the current data is used. This under- and overestimation is maximized for α = 98.5%. If EIOPA chooses to use different calibration data then is used now, then a similar research like this will be able to show if any underestimation might take place.