Basis Risk, Procyclicality, and Systemic Risk in the Solvency II Equity Risk Module

Transcription

1 Basis Risk, Procyclicality, and Systemic Risk in the Solvency II Equity Risk Module Martin Eling, David Pankoke 1 Abstract The purpose of this paper is to systematically analyze the equity risk module of Solvency II, the new regulatory framework in the European Union. The particularly interesting aspect of this module is that it contains a symmetric adjustment mechanism commonly known as equity dampener which shall reduce procyclicality and thus systemic risk in the insurance industry. We critically review the equity risk module in three steps: we first analyze the sensitivities of the equity risk module with respect to the underlying technical basis, then work out potential basis risk (i.e., deviations of the insurers actual equity risk from the Solvency II equity risk), and based on these results measure the impact of the symmetric adjustment mechanism on the goals of Solvency II. The equity risk module is backward looking in nature and a substantial basis risk exists if realistic equity portfolios are considered. Also, the adjusted capital stress substantially deviates from the proposed 99.5% confidence level. Our results are helpful for academics interested in regulation and risk management as well as for practitioners and regulators working on the implementation of such models. Keywords: Solvency II, equity dampener, procyclicality, systemic risk JEL classification: G22; G28; G32 1. Purpose and Motivation In light of the ongoing financial crisis, the scope and structure of insurance regulation is the subject of intense discussions, both in academia and practice. Regulators around the world are revising their regulatory frameworks, including in the United States (see Klein and Wang (2009)), the European 1 Martin Eling is professor of insurance management and director at the Institute of Insurance Economics at the University of St. Gallen, Kirchlistrasse 2, 9010 St. Gallen, Switzerland (martin.eling@unisg.ch). David Pankoke is a PhD candidate at the Institute of Insurance Economics at the University of St. Gallen (david.pankoke@unisg.ch).

2 Union (Eling, Schmeiser, and Schmit (2007)), and Switzerland (Filipović and Vogelpoth (2008)). A new and important aspect that has been added to the regulatory agenda is the question of whether the insurance industry exposes systemic risk and, if so, how regulation might help mitigating undesired outcomes arising from such risk (see Klein (2011), Cummins and Weiss (2011), Harrington (2009), Harrington and Miller (2011), Grace (2011)). This paper contributes to the discussion by focusing on the equity risk module of Solvency II, the new regulatory framework for insurance companies in the European Union. The module basically consists of a capital requirement for equities based on a standard capital stress scenario which is the 0.5% quantile of past returns and an embedded adjustment term to counteract systemic risk (see CEIOPS (2010b)). The idea of the adjustment term is that capital requirements are tightened or relaxed depending on the market environment. Neither the calculation of the standard capital stress nor the proposed mechanisms for counteracting systemic risk have been the subject of academic research to date. Thus, our work shall contribute to the academic discussion on the optimal design of insurance regulation as well as help practitioners in their efforts to develop a framework for a safe and sound insurance industry. The design of Solvency II and the models that are planned in this context have been the subject of a fair amount of research during the past few years. One prominent discussion is the option to choose between a regulatory standard model and an internal risk model. Both Liebwein (2006) and Albarrán, Marín, and Alonso (2011) argue that companies should use internal risk models since they better reflect the actual risk of the company than the standard formula. More technically, Christiansen, Denuit, and Lazar (2012) critically review the aggregation formula used to sum up the capital requirements for different risk classes; they find that the aggregation formula can be supported theoretically, but that the underlying correlation matrix is highly questionable. The aggregation formula requires risks to follow a multivariate normal distributions and further research should be done to discover if this assumption is valid. In this context Pfeifer and Strassburger (2008) show that if the individual risks are skewed, then the solvency capital requirements can be largely under- or overestimated. According to Savelli and Clement (2011) the proposed aggregation formula produces correct results only for a restricted class of independent distributions and can lead to an 1

3 underestimation of the diversification effect. As an alternative, they propose using copula functions to model the dependencies of distributions and to derive more appropriate capital requirements. Van Laere and Baesens (2010) discuss the calculation of the capital requirement for credit risk and suggest an approach similar to that of Basel II to predict credit ratings for non-rated companies. To date, the equity risk module has not been in the center of any quantitative analysis, nor have the interrelations between solvency regimes and systemic risk. Importantly, however, many authors have claimed that regulation can increase systemic risk (see, e.g., Keller (2011) and Huerta de Soto (2009) with regard to Solvency II), which is the motivation behind introducing the adjustment term. The typical argument is that in case of an economic downturn or a stock market crash, riskbased capital standards might force insurers to sell risky assets in crisis, which might cause a run in the market and thus aggravate the crisis (see Eling, Schmeiser, and Schmit (2007)). The goal of the paper is a thorough analysis of the equity risk module of Solvency II. We therefore use a three-step analysis design. In the first part we analyze the sensitivities of the equity risk module based on the indices used to calibrate the model. In the second part we consider more realistic investment portfolios in order to identify potential basis risk in the Solvency II model. In our context we define basis risk as the risk that the Solvency II risk measure deviates from the actual risk of the insurance company due to the simplified portfolio construction used in the standard formula. The goal of the third part is then to find out what impact the symmetric adjustment mechanism as a procyclicality-counter measurement has on the goals of Solvency II. In this analysis, we will use results derived in the first and the second parts. Our main results can be summarized as follows: The sensitivity analysis of the equity risk module strongly illustrates the backward looking nature of the new Solvency II capital requirements since the capital charges only reflect past crisis. Our results complement the already known shortcomings of the aggregation formula (Christiansen, Denuit, and Lazar (2012); Pfeifer and Strassburger (2008); Savelli and Clement (2011)) with a detailed analysis of the equity risk module. Our analysis of the basis risk shows that the proposed standard capital stress for equity risk can substantially deviate from the individual insurers portfolio risk (in our model cases the actual capital stress can be up to

4 percentage points higher than the standard capital stress). Finally, we show how the symmetric adjustment mechanism systematically alters the regulators goal to set a confidence level of 99.5%. 2 The remainder of this paper is structured as follows. In Section 2 we shortly explain the calculations behind the capital requirement of the market equity risk module and the symmetric adjustment mechanism. In Section 3 we discuss the results of the sensitivity analyses of the capital requirements with respect to their technical basis. The basis risk is then evaluated in Section 4 and Section 5 focuses on procyclicality and systemic risk reduction. Section 6 concludes and indicates the path for further research. 2. Capital Requirements for Equity Risk and the Symmetric Adjustment Mechanism The calculation of the capital requirement for the equity risk module is set out in three publications from European supervisors. Directive 2009/138/EC, the bill passed by the European Parliament and the European Council (2009), sets the general outline of Solvency II. It determines the 0.5% risk level for capital requirements and the cap for the symmetric adjustment mechanism. The symmetric adjustment mechanism is the algorithm which determines the capital requirement according to the market environment. The QIS5 Technical Specifications (2010c) set out the guidelines for the fifth test run of the Solvency II capital calculations which took place in The Solvency II Calibration Paper (CEIOPS (2010b)) presents the reasoning behind algorithms set out in the specifications. The three mentioned publications are the latest publically available information about the application of Solvency II. However, discussions between European institutions are ongoing and further changes in the specifications as well as in the Directive itself are very likely (e.g., see proposed changes in the Directive by the European Commission (2011), called Omnibus II or the new time schedule for the introduction of Solvency II suggested by the European Commission (2012)). The standard capital stress shall be calibrated according to a Value at Risk measure with a confidence level of 99.5% (Article 104(4), European Parliament and European Council (2009)). It differentiates between two classes of equities. Equities listed in EEA or OECD countries are 2 The confidence level of the adjusted capital stress defines the probability that an actual yearly loss is below the potential loss considered in the adjusted capital stress (standard capital stress adjusted by the adjustment term) and therefore sufficiently covered by solvency capital. The European Directive 2009/138/EC passed by the European Parliament and Council (2009) demands a confidence level of 99.5%. 3

5 considered under the class global. Equities not listed in EEA or OECD countries, hedge funds, commodities, private equities as well as other alternative investments are categorized as other equities. Thus, the 0.5% quantile of annual returns from different benchmark indices are taken into account. For global equities the MSCI World Price index is used and for other equities the LPX 50 Total Return index, the HFRX Hedge Fund Total Return index, the MSCI BRIC Price index and the S&P GSCI Commodities Total Return index are considered. The calculations done by CEIOPS 3 are based on a rolling window of daily measured annual returns for the longest period from which data are available. 4 The capital requirement for equity risk (Mkt eq ) is calculated per equity category as follows: max ; 0 (1) where capital requirement for the equity category "global" capital requirement for the equity category "other" NAV net value of assets minus liabilities equity shock prescribed fall in the value of equities The symmetric adjustment mechanism is the name of the algorithm determining the adjusted capital stress. equity shock adjusted capital stress standard capital stress adjustment term (2), 0.1, 0.1 (3) where value of the MSCI World Price index at time t n number of days of the reference period 3 4 CEIOPS (Committee of European Insurance and Occupational Pensions Supervisors) was renamed in 2011 into EIOPA (European Insurance and Occupational Pensions Authority). The capital requirement for the equity risk class global is based on the MSCI World Price index. For this index daily data is available from January 1970 until January Capital requirements for other equities consider four indices approximating alternative investments: the LPX 50 Total Return index (Private Equity) from January 1994 to January 2012, the HFRX Hedge Fund Total Return index (Hedge Funds) from April 2004 until January 2012, the MSCI BRIC Price index (Emerging Markets) from June 1994 until January 2012 and the S&P GSCI Total Return index (Commodities) from January 1970 until All data can be obtained via Datastream. 4

6 β regression coefficient in the OLS regression of the MSCI World Price index on the weighted average index level 5 standard capital stress 39% for equities listed in EEA/ OECD countries, for other equities 49% However, the final standard capital stress is not the result of a clear algorithm, but determined by CEIOPS. According to the type of equity CEIOPS proposes a basis capital stress of 39% for global equities and 49% for other equities as mentioned in QIS5. In contrast to that, in the Solvency II Calibration Paper CEIOPS concludes that a 45% capital stress would be more compliant to the goal of achieving a 99.5% confidence level. Procyclicality and the risk of asset price contagion in the market equity risk are addressed by Solvency II via an adjustment term (Equation (3)). The standard capital stress and the adjustment term together constitute the adjusted capital stress, which determines the stress scenario and thus the capital requirement. These calculations have to be done separately for global equities and other equities, so that for both equity classes capital requirements are generated. In order to derive the capital requirement for the equity risk module, the capital requirements for global and other equities are aggregated as shown in Equation (4). 2 (4) where overall capital requirement for the equity risk module c constant for approximating the diversification effect, set to 0.75 by CEIOPS A constant is used to consider the diversification effect between the two equity categories. It is based on the correlation between the different benchmark indices, but finally determined by CEIOPS. The following evaluations are based on the same indices as used by CEIOPS to develop the specifications for Solvency II. 5 The regression equation is as follows: is the MSCI World Price index at time t, the constant and the regression coefficient. n is the length of the reference period and the error term. For the regression analysis the time period from January 1971 until January 2012 is considered. If not otherwise indicated, we assume a of one in this paper for further analysis, since in all regressions it is close to one regardless of the reference period. For more details see the analysis about the length of the reference period and its impact on the symmetric adjustment mechanism in the Appendix. 5

7 3. Sensitivity Analyses The purpose of this section is to critically review the calculation of the equity risk module. We therefore look at the assumptions behind the standard capital stress (see Equation (2)), the symmetric adjustment mechanism (Equation (3)), and the aggregation formula (Equation (4)). Numerous other aspects could be critically questioned. In the presentation of the paper we restrict ourselves to the above mentioned three important aspects while results for other possible tests (definition of returns, risk measures and β calculation) can be found in the appendix of the paper. The calculation of the standard capital stress is based on a predefined time period. We analyze the impact of the chosen time horizon in order to work out the backward looking nature of the model. CEIOPS uses the full period of data available as basis for the standard capital stress. We work out the time dependency of this approach in Figure 1. That is, for each trading day from the 1st of January 1970 to the 31st of December 2011 the standard capital stress based on the longest time period available on that specific date is given on a rolling basis. 6 An important result from Figure 1 is that the recent financial crisis significantly increased the standard capital stress which would have been much lower if Solvency II would have been introduced before This emphasizes the backward looking nature of the model since only past risks are considered. In fact, the Quantitative Impact Studies (QIS) done for Solvency II exactly reflect this problem. In QIS 4, the capital stress for global equities was set to 32% and for QIS 5 it was already set to 39%. The capital stress for QIS 4 was published in March 2008, the one for QIS 5 in March Further information regarding the results of this analysis, if time windows are considered instead of increasing time horizons, can be found in Table A1 in the first row in the Appendix. 6 For Private Equity, Hedge Funds and Emerging Markets less data is available and therefore lines have a later starting point in Figure 1 than Global and Commodities. Data based on the LPX 50 Total Return index is available since January 2000, based on the HFRX Hedge Fund Total Return index since April 2004 and based on the MSCI BRIC Price index since June

8 80.00% 70.00% 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% Global Private Equity Hedge Funds Emerging Markets Commodities Figure 1: Standard Capital Stress over Time We empirically compare the proposed correlations within the aggregation formula with actual correlations of the different asset classes. 7 In the Quantitative Impact Study 5 a correlation between global and other equities of 0.75 is proposed and considered in the aggregation formula in order to consider the diversification effect. Empirically we find that the correlations range from 0.09 to In order to illustrate the time-varying nature of the correlation, Figure 2 shows the correlation between the MSCI World Price index and the other four indices used to define the standard capital stress. The horizontal black line indicates the assumed correlation between the equity class global and others in the aggregation formula. Notable is the extreme variation for the commodity index and the MSCI World. From July 1990 to July 1995 the correlation has been lowest with a coefficient of and it has been highest from March 1977 to March 1982 with a coefficient of These results clearly illustrate that the assumption of a fixed correlation of 0.75 which is not time-varying is not an optimal solution. 8 Further information about correlations can be found in the third row of Table A1 in the Appendix. 7 8 Returns are calculated annually based on a one year rolling window with daily data. In addition correlation coefficients of returns are shown over time based on a 5 year rolling period. Another important aspect which can be observed in Figure 2 is that in times of crisis the correlations are higher. 7

9 Figure 2: Pearson-Correlations over Time for Rolling Windows of 5 Years between MSCI World Price Index and Indices considered for the Equity Category Other Another crucial part of the market equity risk module is the length of the reference period in the symmetric adjustment mechanism. We analyze the impact of different time horizons on overall capital requirements. This analysis is motivated by ongoing discussions between regulators as to which time horizon is most appropriate. We compare the most discussed time horizons as mentioned in the calibration paper by CEIOPS (2010a) 1 year and 3 years and analyze their impact on the confidence level of the adjusted capital stress (results for other time horizons, which were also discussed, i.e. 1 month and 4 months, can be found in Table A1 in the Appendix). We analyze the symmetric adjustment mechanism and the confidence level at a specific date taking all data into account available until this specific date. In this way the development over time is shown on a rolling basis. The argument of regulators concentrated on the question if one year (260 trading days) or three years (780 trading days) as reference period is appropriate. As CEIOPS (2010b) points out in its calibration paper the longer the reference period, the more frequently the 10% band is hit and the risk sensitivity is reduced. 9 They conclude that a longer reference period on the one hand alters the 9 Indeed, a longer reference period considered in the calculation of the moving average leads to a more frequent application of the maximum or minimum adjusted capital stress for all indices. Especially, the maximal cap is reached more often in comparison to the minimal one for all indices if longer periods are considered. Taking the MSCI World Price index as an example one can see that a standard capital stress level of 49% is applied 56.6% of the time and a level of 29% only in 13.0% of all cases if a reference period of three years is employed. As can be expected, this leads to a lower empirical default probability for longer reference periods, too. The confidence level for the MSCI World Price index is beneath 99.5% during 25.3% 8

10 empirical default probability and on the other one leads to lower capital requirements in falling markets which could create moral hazard. Insurance companies might shift their investments from asset classes without an adjustment mechanism to equities. Therefore the majority of regulators proposes a one year reference period. However, a minority still argues that a three year reference horizon is more appropriate, because capital requirements fluctuate a lot if a comparable short reference period is chosen. This would significantly complicate the steering of the asset mix for insurance companies. Also, it is not the goal of the symmetric adjustment mechanism to respond to temporary market movements. Figure 3 shows the impact of the chosen reference period on the adjusted capital stress. Both, the adjustment term as well as the standard capital stress, are calibrated based on the MSCI World Price index. Also, the index itself is shown. It can be seen that a longer reference period has two effects. First, on average higher adjusted capital stresses are applied and second, the adjusted capital stress becomes binominal either the highest or the lowest adjusted capital stresses are applied. For example, if a three year reference period is applied and the MSCI World Price index is considered, in 56.6% of the time an adjusted capital stress of 49% is applied and only 13.0% of the time an adjusted capital stress of 39% is applied (further details can be found in Table A1 in the Appendix). of the time if a reference period of one year is employed and 20.7% of the time if the adjustment term considers a period of three years. Even if longer reference periods lead to fewer deviations from the proposed confidence level other shortcomings have to be considered, too. Binominal capital requirements imply unpleasant market distortions, if insurers reallocate their investments each time when capital requirements switch from the maximum to the minimum or vice versa. It is important to note that such a regulation could lead to fire sales which themselves influence market prices. So, any analysis based on historical data, when the regulation was not in place has to be interpreted with caution. 9

11 Figure 3: Adjusted Capital Stress for 1 and 3 Year Reference Periods for the Moving Average Figure 4 illustrates the impact of the reference period used by the adjustment term on the confidence level of the adjusted capital stress. The standard capital stress is set according to a 99.5% confidence level as mentioned in Section 2. However, in order to generate the adjusted capital stress, the standard capital stress is altered by the adjustment term. Therefore, the adjusted capital stress is not in line with the proposed confidence level anymore. Figure 4 shows these deviations with regard to the length of the reference period considered in the adjustment term. It can be seen that a longer reference period leads to a higher confidence level and fewer deviations from the proposed confidence level. If a reference period of one year is employed the confidence level of the MSCI World Price index is beneath 99.5% during 25.3% and only 20.7% of the time if the adjustment term considers a period of three years. 10

12 Figure 4: Confidence Level of Adjusted Capital Stress based on MSCI World Price Index and varying Reference Periods for the Moving Average While the sensitivity analysis presented in this Section are not more than a what if sensitivity analysis, we believe that the results are important especially to illustrate the dynamics of the modeling approach chosen for Solvency II. One of the drawbacks of the new Solvency II regime is that it has not been tested over time. Our results illustrate how the equity risk model would have been developed over time if Solvency II was already running for years: It would result in a backward looking adaption to historical crisis (time dependence), an insufficient approximation of the true correlations with an underestimation of the risk especially in crisis (aggregation), and risk insensitive and binomial capital requirements if the reference period is very long or substantial deviations from the proposed confidence level if it is very short (moving average). 4. Solvency II Basis Risk Under Solvency II the MSCI World Price index is used to set the standard capital stress for equities listed in EEA or OECD countries (class global ). This standard capital stress has to be applied by all insurance companies irrespectively of their actual portfolio composition. A natural question to ask is how good this approximation is and how substantial deviations from this proxy are if we consider more realistic portfolios. Depending on the true portfolio composition of the individual insurer the standard capital stress based on the MSCI World might substantially deviate from the actual one and thus basis risk emerges. Our interpretation of basis risk in a Solvency II context is thus deviations of the actual insurers portfolio risk from the risk measured by the standard regulatory model. 11

13 To analyze basis risk, we model the investment portfolio of 16 insurance companies from 16 European countries. Rather than analyzing 16 real portfolios, we have set up 16 stylized country portfolios which proxy the typical allocation of insurers from these countries. To keep the analysis simple and comprehensible, the 16 country portfolios are equally composed of the MSCI country index, the MSCI Europe index excluding the respective country and the MSCI World index excluding Europe (only price indices are considered). 33.3% of each portfolio is thus invested in the home market, 33.3% in Europe outside the home market and 33.3% worldwide outside Europe. Due to the home bias for investment decisions (see, e.g., Tesar and Werner (1995)) we believe these portfolios might better approximate the actual equity allocation of insurers in Europe than the MSCI World Price index. For these 16 country portfolios we calculate the standard capital stress over time and compare it with the Solvency II standard capital stress which only considers the MSCI World index. For each portfolio we neglect the standard capital stress of the first four years so that the results are not driven by the low number of data points. Figure 5 illustrates the results of this analysis for the German, Greek, Irish and Austrian country portfolios for the last twelve years. The black thick line illustrates the Solvency II standard capital stress based on the MSCI World Price index (it corresponds to the line global in Figure 1). The other lines represent the results, if the capital stress is based on the country portfolios. We see that the risk of the country portfolios can substantially deviate from the one proposed by Solvency II. For example, the standard capital stress based on the MSCI overestimates the risk of the German portfolio, but underestimates the one of the Greek portfolio. We also see that for all portfolios the risk significantly increased after As illustrated in Figure 5, these effects can be very substantial and they can occur in both directions (over- and underestimation of the actual risk). For example, on the 19 th December 2000, the proposed standard capital stress is 29.7 percentage points higher than the suitable standard capital stress of the Irish portfolio. 10 On the 30 th October 2008 the Greek portfolio was underestimated by percentage points. In general the standard capital stress based on the MSCI World index seems to overestimate the risk in normal market conditions and underestimates it 10 The relatively low standard capital stress for Ireland might be related to the fact that the data for the Irish Portfolio starts in the 1990s in which stock markets where booming. The data for Finland, Greece, Ireland and Portugal starts in the 1990s or later while the time period for the other countries are longer. For all countries we find significant deviations from the standard capital stress based on MSCI World. 12

14 in times of crisis. This makes sense, since the MSCI World portfolio is more diversified than the individual country portfolios. In contrast, individual country portfolios rely more on a specific geographic area and thus inhibit idiosyncratic risks attached to single European countries, which were subject to specific crisis during the investigation period (especially Ireland and Greece). A regulatory question that thus arises is which of these two alternatives a global standardized view or the more country specific one - is more adequate to account for the equity risk of insurance companies. Figure 5: Standard Capital Stress over Time based on the 0.5% Quantile for Different Country Portfolios Table 1 shows the results for all 16 country portfolios. The second column shows the standard capital stress set by CEIOPS as described in QIS5. Since all country portfolios only invest in OECD countries, for all portfolios a standard capital stress of 39% would apply. The third column shows the standard capital stress based on the 0.5% quantile of the returns based on the MSCI World portfolio. In order to focus only on the basis risk we use the figures in the third column as a benchmark. In the fourth column the standard capital stresses calibrated according to the country portfolios are shown. 11 In column five and six the average differences between the standard capital stress of the MSCI Portfolio and the country portfolios are shown. In column five we consider the full time period available and in column six we consider the time period after the bankruptcy of Lehman Brothers (September 2008 January 2012). It is important to note that column four cannot just be subtracted 11 The observed time horizon ranges from January 1974 until January Different maximal time periods are used in the case of Finland (March 1993 January 2012), Greece (March 2006 January 2012), Ireland (May 1998 January 2012) and Portugal (December 2002 January 2012). 13

15 from column three in order to derive the average difference, because the standard capital stresses are not averages. Finally, the maximal deviation of the standard capital stress of a country portfolio in contrast with the MSCI Portfolio is shown in column seven. Proposed CEIOPS Standard Capital Stress (max period) Standard Capital Stress based on MSCI Portfolio (max period) Standard Capital Stress based on Country Portfolio (max period) Average Difference in Percentage Points over Time between MSCI and Country Portfolio (max period) Average Difference in Percentage Points over Time between MSCI and Country Portfolio (after September 2008) Maximal Deviaton from MSCI Standard Capital Stress in Percentage Points (max period) Date of maximal Deviation (max period) Austria 39% 44.25% 48.91% April 1981 Belgium 39% 44.25% 48.73% December 2008 Denmark 39% 44.25% 41.69% October 2007 Finland 39% 44.25% 46.62% March 2001 France 39% 44.25% 41.28% January 2007 Germany 39% 44.25% 41.88% April 1981 Greece 39% 44.25% 51.30% October 2008 Ireland 39% 44.25% 52.24% December 2000 Italy 39% 44.25% 43.43% April 1981 Netherlands 39% 44.25% 42.69% August 2008 Norway 39% 44.25% 45.07% August 2008 Portugal 39% 44.25% 45.96% November 2008 Spain 39% 44.25% 41.50% April 1981 Sweden 39% 44.25% 41.40% April 1981 Switzerland 39% 44.25% 42.62% August 1988 UK 39% 44.25% 42.86% August 1988 Table 1: Deviation in the Standard Capital Stress between the Country Portfolios and the MSCI Portfolio Just focusing on the MSCI index as the basis for the calculation leads to a standard capital stress of 44.25%. 12 Taking into consideration the maximal time period from January 1974 to January 2012 it can be seen that for most country portfolios (11 out of 16) the standard capital stress would be below the one based on the MSCI index. Focusing on the Austrian or Swedish country portfolio would lead to a standard capital stress 8.5 percentage points below the actual one at hand. However, if the Greek portfolio would be used as a basis, the standard capital stress would be 5.3 percentage points higher. Looking only at the time interval after the bankruptcy of Lehman Brothers, all country portfolios except for France, the United Kingdom, the Netherlands and Switzerland became more risky in comparison to the MSCI portfolio. This effect is most severe for the Austrian and the Irish portfolio. Taking the maximal time period into account the standard capital stress of Austria is on average 8.5 percentage points below the one based on the MSCI index. However, focusing just on the period from September 2008 onwards the standard capital stress of Austria is 3.8 percentage points 12 This result is in line with the evaluations of CEIOPS. As mentioned in Section 2 their analyses lead to a similar standard capital stress of 45%. However, finally in QIS 5 a standard capital stress of 39% is proposed. In order to avoid that our results represent this misfit of underestimation instead of the basis risk, we focus in this section on the standard capital stress of the MSCI portfolio as a benchmark. 14

16 above the one for the MSCI World index. For Ireland the shift is from 6.8 percentage points below to 8.2 percentage points above the standard capital stress. Looking at the maximal deviations between the standard capital stress of country portfolios and the one of the MSCI index reveals two insights. First, deviations can be quiet large. Second, in all cases when the maximal deviation is negative it took place before the financial crisis. Based on these results it can be argued that the standard capital stress based on the MSCI World price index misestimates the true risk of equity portfolios in Europe. Also, the results show that the risk approximation by the MSCI World index can lead to over- and underestimations of the actual risk of the same portfolio over time. The results of this analysis might be relevant, e.g., for the Own Risk and Solvency Assessment (ORSA) prescribed in Pillar 2 of Solvency II. Under this provision, insurance companies are obliged to report systematic deviations of their true risk from the Solvency II standard model. Our results emphasize that the deviations can be very substantial. 5. Procyclicality In this section we analyze the extent to which the symmetric adjustment mechanism as a procyclicality-counteracting measure affects the predefined goal of Solvency II of a 99.5% confidence level. Relaxing capital requirements in bad markets will systematically decrease the confidence level, while raising capital requirements in good markets should systematically increase the confidence level. We are especially interested in the possible range of outcomes; for the overall goals of Solvency II (e.g., creating a safe industry) it might be relevant to know if this range is between 99 and 99.9% or between 90 and 99.99%. We do this analysis first for the MSCI portfolio and second for all 16 country portfolios as defined in Section 4. We calculate the impact of the symmetric adjustment mechanism on the confidence level of the as follows. First, we take the standard capital stress which is calibrated according to a 99.5% confidence level based on the MSCI World Price index. Second, we calculate the adjusted capital stress according to the symmetric adjustment mechanism as described in Equation (2) and (3) in Section 2. For the adjustment term we take a reference period for the moving average of one year into account as set by CEIOPS. Third, we derive the confidence level of the adjusted capital stress. For 15

17 each point in time, the confidence level is simply the percentage of how many annual losses of the benchmark portfolio so far are lower than the adjusted capital stress valid at this point in time. Since only the standard capital stress is calibrated according to a 99.5% confidence level and based on the MSCI portfolio as a benchmark, we expect deviations from this threshold for the adjusted capital stress. Figure 6 shows the confidence level of the adjusted capital stress for the MSCI portfolio over time. 13 The total observation period ranges from January 1975 January However, data is available for five more previous years, but is omitted, due to the fact that the calculation of the confidence level makes only sense if a sufficient amount of observations is available. Otherwise, losses exceeding the adjusted capital stress early in the observation period lead to an underestimation of the confidence level at that point in time. Figure 6: Confidence Level of the applied Adjusted Capital Stress over Time for the MSCI Portfolio It can be seen in Figure 6 that most of the time a confidence level of 1 is reached with temporary deviations from this level. However, the confidence level never falls below 97.26%. After 2008, the confidence level of 1 is not reached anymore (99.91% is the maximum). These patterns can be explained by the characteristics of the adjustment term. Before 2008, there is no such incident where the MCSI portfolio exceeds an annual loss of 49%. So, when the maximum adjusted capital stress of 49% is employed, the confidence level is 1 per definition. Only when the adjusted capital stress turns out to be below the maximum, the confidence level sometimes cannot meet the 99.5% 13 In the case illustrated in Figure 6 the portfolio is invested in the MSCI World price index. Since the adjusted capital stress is calibrated to the MSCI World price index as well, there is no basis risk in this case. 16

18 threshold. After 2008, the maximal loss exceeds 49% and consequently, as seen in Figure 6, a confidence level of 1 cannot be reached anymore. Next, we analyze the adjusted capital stress in the cases where the 16 country portfolios are considered as the benchmark portfolios. In these cases the standard capital stress and the adjustment term are calibrated according to the MSCI World price index as before, but the considered benchmark portfolio is not based anymore on the same index. Instead the country portfolios are used. Table 2 shows the maximal and minimal confidence level for each country portfolio. The time horizon for the analysis of the minimal and maximal confidence levels is split into two periods. One covers the time before the bankruptcy of Lehman Brothers (January 1975 September 2008) 14 and the other one focuses on the period after the insolvency (September 2008 January 2012). In this way the effect of the financial crisis can be estimated. The calculation of the adjusted capital stress at a specific date, though, takes all data into account which are available before this date. So, the splitting of the time period refers only to the optima. January 1975 September 2008 September 2008 January 2012 min max min max Austria 99.18% % 97.67% 99.77% Belgium 96.82% % 96.40% 99.51% Denmark 96.82% % 96.40% 99.51% Finland 94.57% % 92.67% 99.79% France 96.06% % 96.30% % Germany 98.11% % 96.42% % Greece 94.29% % 85.89% 98.42% Ireland 98.19% % 93.21% 98.40% Italy 96.82% % 96.76% 99.95% Netherlands 96.81% % 96.26% % Norway 95.36% % 96.21% 99.90% Portugal 95.40% % 91.63% 99.97% Spain 98.39% % 97.44% % Sweden 98.22% % 96.63% % Switzerland 96.07% % 96.87% 99.95% UK 95.28% % 96.75% % Table 2: Confidence Level over Time for Country Portfolios The results support the indication of Figure 6. Before 2008 the maximal adjusted capital stress can compromise 100% of the empirical observed annual, negative returns. However, from time to time the confidence level for all country portfolios can deviate largely from the set goal of 99.5% if 14 Different maximal time periods are again used for Finland, Greece, Ireland and Portugal. 17

19 capital requirements are relaxed. The country portfolios of Finland and Greece build the bottom line with a minimal confidence level of 94.57% and 94.29%. Differences in the analysis of the MSCI portfolio and the country portfolios emerge when the period after the bankruptcy of Lehman Brothers is taken into account. In contrast to the analysis based on the MSCI portfolio, for some country portfolios the confidence level cannot be reached anymore even if the maximal adjusted capital stress of 49% is applied. The Greek and Irish country portfolios reach at the most a confidence level of 98.42% respectively 98.40%. Also, Belgium and Denmark only slightly reach the proposed level with 99.51% at the most. The lowest confidence levels reach the Greek country portfolio with 85.89% and the Portuguese one with 91.63%. The results can be interpreted as follows. The proposed confidence level is reached most of the time for all country portfolios. However, the confidence level of all country portfolios from time to time substantially deviates from the required confidence level when capital requirements are relaxed. Especially, during the financial crisis the goal of Solvency II to ensure that insurers can meet their obligations with a 99.5 % confidence level would have been violated if certain portfolio compositions would have been considered. That the symmetric adjustment mechanism would temporarily violate the set confidence level of Solvency II could be expected. However, it should raise concern that even if the maximal adjusted capital stress is applied, for some portfolios it is not possible anymore that the capital requirements reach the proposed confidence level. 6. Conclusion and Future Research This is the first paper to systemically analyze the equity risk module of Solvency II. The main goal of Solvency II is the protection of policy holders and beneficiaries. (Article 16, Directive of European Parliament and European Council (2009)) Therefore capital requirements should ensure that insurance companies possess enough economic capital so that they are able with a probability of at least 99.5 %, to meet their obligations to policy holders and beneficiaries over the following 12 months. (Article 64, Directive of European Parliament and European Council (2009)) Our results cast some doubt that the planned application of Solvency II will achieve this goal for the equity risk module. 18

20 By backtesting Solvency II under varying assumptions on empirical data we find that the following issues will complicate the achievement of a 99.5% confidence level. Firstly, the standard capital stress is heavily influenced by the time period considered. The set 39% standard capital stress is below the appropriate standard capital stress at the moment of 48%. Secondly, the aggregation formula currently underestimates the true risk due to the used correlation matrix. Fixed correlation coefficients are problematic in general, because equity correlations are not stable over time. Thirdly, longer reference periods for the moving averages lead to fewer deviations from the 99.5% confidence level. Fourthly, regulation is not calibrated to equity portfolios of insurance companies which do not mirror the MSCI World Price index. So, capital requirements will constantly over- or underestimate the risk according to the asset allocation of the insurance company. Overall, the results of the analysis demonstrate the inherent drawback of a one size fits it all standard regulatory approach, since it only might serve as a rough approximation of the actual risk that the individual insurers have taken. The next step in our analysis is to take a closer look on procyclicality and test the approach of Solvency II to handle systemic risk. Does the symmetric adjustment mechanism really work in counteracting the procyclicality issue? If it successfully mitigates procyclicality, does it automatically reduce systemic risk, too? The results of our analysis will be relevant for academics interested in Solvency II, other regulatory frameworks, and risk management. Also, regulatory bodies and the risk management departments of insurance companies will benefit from our results. For example our results can be helpful for insurers in Own Risk and Solvency Assessment (ORSA) prescribed in Pillar 2 of Solvency II Solvency II. 19

21 Appendix Further Sensitivity Analysis MSCI World Price index LPX 50 TR index (Priavte Equity) HFRX Global Hedge Fund TR index MSCI BRIC Price index S&P GSCI TR index (Commodities) Standard Capital Stress based on Time Period Standard Capital Stress based on Return Definition Linear Return Correlation between MSCI World Price and other Indices Linear Return Correlation between ES and VaR % n.a. n.a. n.a % % n.a. n.a. n.a % % n.a. n.a. n.a % % 74.82% n.a % 61.80% Daily Data 44.39% 73.34% 23.18% 62.56% 59.43% Monthly Data 43.79% 72.76% 22.98% 62.09% 58.36% Yearly Data 39.52% 63.68% 22.60% 59.77% 44.55% Proposed Total Period Maximum Minimum Total Period Maximum n.a Minimum n.a Max. (22 days) 0.09% 1.64% 0.00% 1.81% 0.66% Min. (22 days) 0.47% 2.73% 0.00% 4.10% 0.80% Frequency of Adjusted Max. (90 days) 4.12% 5.60% 0.00% 9.16% 10.64% Capital Stress reaching the Min. (90 days) 3.33% 5.12% 0.45% 7.04% 5.02% Maximum and Minimum Max. (260 days) 22.66% 18.13% 0.00% 17.07% 34.52% according to Reference Min. (260 days) 9.11% 9.59% 1.34% 10.13% 9.47% Period Max. (780 days) 56.61% 21.67% 2.04% 18.13% 54.13% Min. (780 days) 12.98% 11.25% 2.23% 10.30% 12.53% Table A1: Sensitivity Analysis of the Market Equity Risk Module The second row of Table A1 considers the definition of returns. The calculations done by CEIOPS are based on a rolling window of daily measured annual returns, i.e. 1. We analyze whether different definitions of returns lead to alternative outcomes. We look at a rolling window of monthly measured annual returns ( 1) and yearly data ( 1., and denote the current index value at a specific date, month or year. 15 For all indices, fewer data points lead to a reduced standard capital stress. The maximum difference is observed for the LPX 50 index. A one-year rolling window of daily data leads to a standard capital stress of 73% and yearly data leads to 64%. We suppose this is due to the calculating method used by CEIOPS. Annual returns 15 For the MSCI World Price and TR index as well as the S&P GSCI Commodities TR index data from January 1973 until December 2009 is used. For the LPX 50 Total Return index data from January 2000 until December 2009 is used. Calculations regarding the HFRX Global Hedge Fund Total Return index take the period from April 2004 until December 2009 into account. For the MSCI Emerging Markets BRIC Price index the period from June 1995 until December 2009 is considered. 20

22 are calculated based on a one year rolling window of daily index values. In this way all fluctuations are considered, whereas by using only annual data points fluctuations within a certain year are ignored. So, the method used by CEIOPS seems to be appropriate since neglecting fluctuations within a year and within a month would mean underestimating the volatility of equity prices. Solvency II considers a 0.5% quantile for the risk factors, which corresponds to the Value at Risk (VaR) at a 99.5% confidence level as a risk measure. We are motivated to look further at this issue by other regulatory approaches using different risk measures. For example, the Swiss Solvency Test employs the Expected Shortfall at a 99% confidence level. Also, the fact that companies might use other risk measures for their internal decision making makes the issue worth to consider. We test whether the Expected Shortfall (ES) at a 99.5% confidence level leads to comparable results. We calculate the differences in the standard capital stress for each index by using the ES and VaR measure for the time period from December 1975 until January The standard capital stress is calculated according a 5 year rolling period. Our results show that using Expected Shortfall (ES) as a risk measure instead of Value at Risk (VaR) leads to very comparable results (see Figure A1). The standard capital stress increases about 6 7 percentage points on average and extreme stock price movements are anticipated more quickly. Both could be expected due to the fact that ES considers all tail values and not, like VaR, only the threshold. For all equity classes the correlation over time between VaR and ES are close to 1 over the total period as shown in the fourth row of Table A1. Figure A1: VaR in Comparison to ES for the MSCI World Price Index 21

23 We analyze the impact of the reference period on β of the adjustment term, which is derived by a regression of the actual index level on the weighted average index level. Table A2 shows the results. As CEIOPS reports we find betas for the MSCI World indices and the S&P GSCI commodities Total Return indices are for all reference periods close to 1. Therefore we approximate beta in this paper by one. Reference period MSCI World Price MSCI World TR GSCI TR 1 month (22 days) 4 months (90 days) 1 year (260 days) 3 years (780 days) Table A2: Adjustment Term Betas 22