# Portfolio optimization under solvency II: implicit constraints imposed by the market risk standard formula.

INTRODUCTIONThe new risk-based capital standards Solvency II, which are currently scheduled for implementation in 2016, aim to modernize and harmonize the regulation of insurance companies in the member states of the European Union (EU). Particularly for smaller firms with less sophisticated risk management and modeling capacities, the regulator provides standard formulae that allow the calculation of solvency capital requirements with regard to all kinds of different risk types. Although Solvency II is regarded as one of the most innovative regulatory frameworks, its market risk module is a quite basic stress factor approach. In this article, we examine the impact of the respective capital charges on a life insurer's strategic portfolio choice. Potential implications are far-reaching since inadequate constraints may give rise to asset allocations that are disadvantageous for both policyholders and shareholders. Apart from that, insurance companies are among the largest institutional investors in Europe, together holding some EUR 6.7 trillion of assets. Hence, major changes in their asset management practices could entail substantial consequences for the demand situation and pricing of asset classes in European capital markets (see, e.g., Fitch Ratings, 2011).

Despite the tremendous amount of literature on different aspects of Solvency II, only a few authors deal with the relationship of the market risk standard formula and an insurer's investment policy. Rudschuck et al. (2010) argue that the new framework will prompt insurance companies to reduce their equity holdings, thus rendering it more difficult to earn required returns in the current low interest rate environment. Bragt, Steehouwer, and Waalwijk (2010) show that portfolio structure and asset duration of an insurer have a considerable influence on its capital charges. Moreover, in their report on the impact of Solvency II on insurance companies' asset allocations, Fitch Ratings (2011) anticipates strong effects on European debt markets. Further practitioner studies published by Ernst & Young (2011, 2012) address opportunities for portfolio construction that will be brought about by the regulatory changes. Gatzert and Martin (2012) demonstrate that the asset allocation strongly influences the solvency capital requirements and point out that model risk may be an important issue. In addition, Hoering (2013) investigates whether the investment portfolios of insurance companies will be reshaped under the new framework. As the rating model of Standard & Poor's (S&P) seems to be more conservative than the Solvency II standard formula, Hoering does not expect a binding constraint for the asset management of insurance companies. Similarly, Fischer and Schlutter (2015) examine how the calibration of the equity risk module affects the regulatory capital and the investment strategy of an insurer that maximizes its shareholder value. Their findings indicate situations in which a more conservative stress factor leads to a reduction in both the firm's equity portfolio and capital buffer. Finally, Braun, Schmeiser, and Siegel (2014) compare the capital charges for private equity under the market risk standard approaches of Solvency II and the Swiss Solvency Test with results for an internal model and find that the former excessively penalize the asset class.

A rigorous examination of constraints to portfolio optimization under Solvency II has been conducted by Eling, Gatzert, and Schmeiser, (2009). (1) They propose an alternative standard model that determines firm-specific lower limits for investment performance, employing the ruin probability, the expected policyholder deficit, and the tail value at risk (VaR). Our work differs from the aforementioned study in at least three important aspects. First, in their empirical analysis, Eling, Gatzert, and Schmeiser provide for a skewed claims distribution but assume stochastic independence of assets and liabilities. Their results are therefore particularly insightful for nonlife insurers. In contrast, we consider a life insurance company where the duration gap and corresponding asset-liability dependence structure exhibits a crucial impact on the riskiness of the equity capital. Second, we improve on the underlying time-series data and add investment limits to the portfolio optimization algorithm. Since it is not clear yet whether regulatory ceilings on portfolio weights will be abandoned after the introduction of Solvency II, we believe that they should be taken into account at this point in time. Third, and most important, our analysis is centered on the idea of benchmarking the genuine standard formula with a parsimonious partial internal model that is well grounded in portfolio theory and, consistent with Solvency II, relies on the VaR measure with a confidence level of 99.5 percent. Using empirical data, we estimate risk-return profiles for the major asset classes held by European insurance companies and run a quadratic optimization program to derive nondominated frontiers with budget, short-sale, and investment constraints. In a next step, we calculate the corresponding market risk capital charges under the standard formula as well as the internal model to identify those efficient portfolios that are attainable for an exogenously given amount of equity. Furthermore, we systematically select inefficient portfolios and assess their admissibility, too. Our results highlight inherent shortcomings of the standard formula that interfere with economically sensible asset management decisions.

The rest of the article is organized as follows. In the next section, we present the two market risk models that underlie our analysis. The derivation of empirical risk-return profiles as a basis for the insurer's asset allocation as well as the calibration of the solvency models is then discussed in the third section. Moreover, in the fourth section, we run the quadratic programs for portfolio optimization with various constraints and calculate the capital requirements for the efficient as well as selected inefficient portfolios. The economic implications of our results are discussed in the penultimate section. Finally, in the last section, we draw our conclusion.

SOLVENCY MODELS

Solvency II Market Risk Standard Formula

In order to calculate the solvency capital requirement (SCR), the regulator provides insurance companies with standard formulae for different risk types that are calibrated on the basis of historical data to reflect a VaR with a confidence level of 99.5 percent and a time horizon of 1 year (see, e.g., EIOPA, 2012b). (2) At the heart of our analysis is the market risk module, which covers one of the most important risk categories in the insurance industry. Among European life insurers, for example, market risk accounts for almost 70 percent of the overall SCR (see, e.g., Fitch Ratings, 2011). The difference between the market values of an insurance company's assets and liabilities is termed basic own funds (BOF). (3) Changes in the BOF, denoted [DELTA]BOF, are induced by given stress factors that reflect shocks from the financial markets (see EIOPA, 2012b). The aggregate capital charge for market risk, [SCR.sub.Mkt], consists of several submodules (see, e.g., CEIOPS, 2009). In this regard we limit our analysis to interest rate risk, equity risk, property risk, and spread risk. (4)

Generally, both assets and liabilities of an insurance undertaking are interest rate sensitive. Thus, upward and downward shifts of the term structure may have a detrimental influence on the BOF, which is why the capital requirement for interest rate risk, [Mkt.sub.int], comprises two states (see, e.g., EIOPA, 2012b):

[Mkt.sup.Up.sub.int] = [DELTA]BOF[|.sub.up], (1)

[Mkt.sup.Down.sub.int] = [DELTA]BOF[|.sub.down]. (2)

[Mkt.sup.Up.sub.int] and [Mkt.sup.Down.sub.int] represent [DELTA]BOF caused by a rise and a fall in interest rates, respectively. In both cases, stress factors are applied to the prevailing yield curve as follows (see, e.g., EIOPA, 2012b):

[DELTA][r.sup.up.sub.t] = [r.sub.t] x (1 + [s.sup.up.sub.t]) - [r.sub.t] [for all]t, for the upward shock, (3)

[DELTA][r.sup.down.sub.t] = [r.sub.t] x (1 + [s.sup.down.sub.t]) - [r.sub.t] [for all]t, for the downward shock, (4)

where [r.sub.t] is the spot interest rate for maturity t and [r.sup.down.sub.t] as well as [s.sup.down.sub.t] equal the shocks for the upward and downward scenario. To translate these shocks into [DELTA]BOF values, a measure for the interest rate sensitivity of the assets and liabilities such as the well-known duration is needed.

In addition, CEIOPS defines equity risk exposure as those asset and liability positions whose values react to changes in equity prices. The capital requirement for equity risk, Mktgq, consists of two categories: "global equity" and "other equity." (5) The former contains stocks that are listed in an EEA or OECD country. Private equity, commodities, emerging market equities, hedge funds, as well as any other assets not covered elsewhere in the market risk module, in contrast, are assigned to the latter. (6) Due to this split, two steps are needed to calculate [Mkt.sub.eq]. First, the capital requirements for each category are determined based on preset stress factors (see, e.g., EIOPA, 2012b):

[Mkt.sub.eq,i] = max ([DELTA]BOF | equity [shock.sub.i]; 0), (5)

where i [member of] {global equity; other equity}. Second, the results are aggregated by means of a preset correlation matrix (see, e.g., EIOPA, 2012b):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

where i, j [member of] {global equity; other equity}, and [CorrIndex.sub.ij] denote the correlation coefficient between global and other equities.

Analogous to the equity risk submodule, the capital requirement for property risk, [Mkt.sub.prop], reflects a prespecified decline in real estate prices that may have an impact on the asset and liability values of the insurance undertaking (see, e.g., EIOPA, 2012b):

[Mkt.sub.prop] = max ({DELTA]BOF | property shock; 0). (7)

Furthermore, [DELTA]BOF due to a widening of credit spreads is captured by the spread risk submodule. Apart from various types of traditional fixed income instruments, the respective capital requirement, [Mkt.sub.sp], covers all other spread-sensitive investments such as asset-backed securities, structured products, and credit derivatives. Yet, to simplify the analysis and ensure the availability of reliable data, we restrict ourselves to corporate bonds. (7) Again, a two-step approach is needed. First of all, the spread risk shock on bonds should be calculated as follows (see, e.g., EIOPA, 2012b):

spread shock on bonds = [n.summation over (j=1)] [MV.sub.i] x duration x [F.sup.up] ([rating.sub.i]), (8)

where [MV.sub.i] represents the exposure to bond i [member of] {1, ..., n}, duration equals the associated modified duration, and the stress factor Fup(ratingi) is a function of the security's external rating. Based on the shock, [Mkt.sub.sp] is defined as (see, e.g., EIOPA, 2012b):

[Mkt.sub.sp] = max ([DELTA]BOF | spread shock on bonds; 0). (9)

Finally, the total capital requirement for market risk, [SCR.sub.Mkt], can be derived by means of the following aggregation formula (see, e.g., EIOPA, 2012b):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where i, j [member of] {int; eq; prop; sp}. The superscripts indicate the upward and downward states of the interest rate risk submodule, and [CorrMkt.sup.up] and [CorrMkt.sup.down] are the entries of the corresponding correlation matrix, which can be found in Table A1 of the Appendix. (8)

Partial Internal Model for Market Risk

In addition to the standard formula, insurance companies have the option to employ a partial internal model for the calculation of their market risk capital charges. Below, we introduce a parsimonious asset-liability approach with roots in portfolio theory and structural credit modeling, which is assumed to be preapproved by the regulator. (9) In line with Solvency II, we consider a 1-year evaluation horizon and calculate the capital requirements based on the VaR measure with a confidence level of 99.5 percent. Default occurs when the value of the firm's liabilities exceeds the value of its assets. Based on discrete compounding, the value of the assets at time t = 1 can be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

where

[[??].sub.1] = stochastic market value of the assets at time t = 1,

[A.sub.0] = deterministic market value of the assets at time t = 0,

[[??].sub.A] = stochastic return on the assets between t = 0 and t = 1.

Consistent with Markowitz (1952), the stochastic asset return is assumed to be normally distributed: [[??].sub.A] ~ N([[mu].sub.A], [[sigma].sub.A]). Instead of considering individual investments, we facilitate the analysis by modeling aggregate asset class subportfolios. (10) Therefore, [[??].sub.A] is determined by the following weighted average:

[[??].sub.A] = [n.summation over (j=1)] [w.sub.i][[??].sub.i] = w'R, (11)

where

[w.sub.i] = portfolio weight for asset class i,

w = vector of portfolio weights,

[[??].sub.i] ~ N([[mu].sub.i], [[sigma].sub.i]) = normally distributed return of asset class i between t = 0 and t = 1,

R = random vector of asset class returns,

n = number of asset classes in the portfolio.

Owing to the normality assumption, the asset value distribution at time t = 1 is entirely described by its first two central moments, which evidently depend on E [[[??].sub.A]] and Var [[[??].sub.A]]. Based on Equation (11), it is straightforward to derive the required expressions

[[mu].sub.A] = E[[[??].sub.A]] = [[n.summation over (j=1)] [w.sub.i][[??].sub.i]] = [n.summation over (j=1)] [w.sub.i][[mu].sub.i] = w'M, (12)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)

where

[[mu].sub.i] = mean return of asset class i,

M = vector of mean returns,

[[sigma].sub.i] = return volatility of asset class i,

[[rho].sub.i,j] = correlation between the returns of asset classes i and j,

[summation] = variance-covariance matrix of returns.

The value of the insurance liabilities at time t = 0 is given by the discounted expected payments to the policyholders. When estimating the respective future cash flows, the insurer takes into account contractual obligations, the mortality of the insured, as well as embedded options. Under Solvency II, the discount rate for the technical reserves will be defined by EIOPA on a quarterly basis (see Omnibus II Directive). Analogous to the asset side, we define the value of the liabilities at time t = 1 as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

where

[[??].sub.1] = stochastic market value of the liabilities at time t = 1,

[L.sub.0] = deterministic market value of the liabilities at time t = 0,

[[??].sub.L] = stochastic growth rate of the liabilities between t = 0 and t = 1.

Just as [[??].sub.A], the liability growth rate is assumed to be normally distributed: [[??].sub.L] ~ N([[mu].sub.L], [[sigma].sub.L]). Since data on individual insurance product categories are quite difficult to obtain, we refrain from a further breakdown of [[??].sub.L].

Having specified the marginal distributions of [[??].sub.1] and [[??].sub.1], we additionally introduce a dependency structure between these two random variables. More specifically, let [[??].sub.A] and [[??].sub.L] adhere to a bivariate normal distribution:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (15)

with the two-dimensional mean vector [M.sub.A,L] = ([[mu].sub.A], [[mu].sub.L])', and the 2 x 2 variance-covariance matrix [[summation].sub.A,L]. In our setup, it is assumed that the common variation of asset and liability values solely arises due to the interest rate sensitivity of both magnitudes. Thus, the correlation [[rho].sub.A,L] will be modeled as a function of the asset portfolio composition (see the next section).

Finally, based on the stochastic assets and liabilities, it is possible to derive a distribution for [B[??]F.sub.1], that is, the basic own funds at the end of the year and, in turn, the change [DELTA]B[??]F between t = 0 and t = 1:11

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

Given the above-mentioned assumptions, [DELTA]B[??]F is also normally distributed (12) with mean

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

and variance

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

The VaR for a confidence level 1 - [alpha], [VaR.sub.[alpha]], is generally defined as the loss over a particular period that is only exceeded with probability [alpha]. For random variables that represent value changes, such as [DELTA]B[??]F, [VaR.sub.[alpha]] equals the [alpha]-quantile of the distribution. Once the cumulative distribution function (cdf) for [DELTA]B[??]F has been estimated, the risk measure of Solvency II, that is, [VaR.sub.0.5%] can be employed to determine [SCR.sub.Mkt]. Since [DELTA]B[??]F is normally distributed, the following closed-form solution applies:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)

where [z.sub.0.5%] represents the 0.5 percent quantile of the standard normal distribution.

Before we move on to the next section, a few remarks with regard to the presented model specification are due. We decided in favor of this parsimonious setup to minimize design-related deviations from the results of the unidimensional standard formula, which only acknowledges risk (stress factors). Owing to its simplicity, our benchmark approach could in fact very well be considered as an alternative standard model. An extension beyond the two dimensions expected return and standard deviation would naturally lead to further discrepancies in the capital charges based on statistical factors. (13) Having said that, the question how well the model approximates the true risks and thus the solvency capital requirements of a life insurer remains an empirical one. Although plenty of evidence suggests that return distributions are generally fat-tailed, the degree of kurtosis tends to differ by asset class.

With respect to the liabilities, in contrast, the case is less clear-cut. In this regard, it needs to be taken into account that potential mortality-related higher moments are explicitly not part of our analysis. Instead, we exclusively focus on market risk, the largest part of which arises due to the interest rate sensitivity of the life insurance liabilities. Typical duration measures imply a symmetric reaction to shifts in the yield (see, e.g., Briys and De Varenne, 1997). Moreover, widespread term structure models, such as the one by Vasicek (1977), rely on a symmetrically distributed short rate.14 Hence, although the normality assumption may initially appear to be quite strong, there are theoretical considerations concerning the interest rate risk of life insurance liabilities that lend support to it. In addition, there is a nontrivial implementation problem in that the scarcity of data for the liability side considerably hinders calibration efforts. The same arguments apply for the linear correlation between assets and liabilities. To sum up, while insurance companies should strive to apply the most adequate distributions and dependency structures possible, we deem the aforementioned simplifications acceptable for the purpose of our analysis. At the expense of comparability to the standard formula, ease of calibration, and the presented closed-form solutions, the model can of course be adjusted to provide for complete flexibility with regard to the characteristics of the relevant random variables.

DATA AND CALIBRATION

Empirical Risk-Return Profiles

Prior to running a portfolio optimization, we need to define the set of assets that the firm can choose from. Consistent with the design of the solvency models discussed in the previous section, the analysis will be conducted on the level of subportfolios for whole asset classes instead of individual securities. In other words, we suppose that the firm optimizes its strategic asset allocation whereas the composition of each subportfolio has already been determined by means of a bottom-up selection process. Our categorization is based on EIOPA (2011) as well as Fitch Ratings (2011) and comprises the following six generic asset classes in which insurance companies typically invest: stocks, government bonds, corporate bonds, real estate, hedge funds, and money market instruments. (15) Each asset class subportfolio will be represented by a common benchmark index whose returns are assumed to be normally distributed. Due to the recession following the collapse of the new economy in 2001 and the global financial crisis of 2007-2009, the world's major equity markets exhibited negative mean returns throughout the last decade. To overcome this problem, we estimate expected values, standard deviations, and correlations based on monthly index return time series over a 20-year time horizon ranging from January 1993 to December 2012. As a consequence, our calibration spans several business cycles as well as different interest rate environments. All figures have been obtained from Bloomberg or Datastream.

With regard to the insurer's equity portfolio, we draw on the EURO STOXX 50, a popular underlying index for a broad range of investment products such as exchange-traded funds (ETFs) and various types of derivatives. This index covers 50 blue-chip firms from 12 Eurozone countries, thus being a suitable benchmark for the European stock markets. In addition, we proxy the firm's government bond holdings through the REX Performance Index (REXP), which reflects the total return of 30 idealized German debt securities with maturities of 1-10 years and three different coupon types. Given the relative size and importance of the German government bond market, such instruments play a vital role in the asset allocation of most European insurance companies. Furthermore, due to the scarcity of suitable time-series data for European corporate debt prior to 1998, we decide to capture the risk-return characteristics of the corporate bond asset class by means of the Barclays U.S. Corporate Bond Index, which consists of investment-grade fixed-income securities that meet specific maturity and liquidity requirements. Despite its different geographic focus, we deem this widely used benchmark to be an appropriate representation of an investment-grade corporate bond portfolio. (16) For the insurer's real estate exposure, we resort to Grundbesitz Europa, an open-end, actively managed fund that invests in commercial and residential property across Europe. Its return data are preferred to that of real estate investment trusts (REITs), since they are based on valuations of the underlying buildings and should thus be less contaminated by general stock market fluctuations. Before running our analysis, we adjusted the respective time series for dividends, which are distributed to the fund investors on an annual basis. Apart from the aforementioned asset classes, the insurance company's portfolio may also comprise a certain percentage of hedge fund investments, which are represented by the HFRI Fund Weighted Composite Index (HFRI), a widely used industry-level performance benchmark. (17) Finally, money market investments are represented by the 1-month Euro Interbank Offered Rate (EURIBOR). (18) Figure 1 illustrates the empirical risk-return profiles of these asset classes as well as the corresponding fitted normal distributions.

Calibration of the Market Risk Standard Formula

We use the parameter values from the relevant Solvency II directives (see CEIOPS, 2010a, 2010b, 2010c) and the latest EIOPA proposal (see EIOPA, 2012b), which includes insights from the financial crisis. (19) Concerning interest rate risk, CEIOPS derived stress factors based on EUR and GBP government zero-bond yields as well as the corresponding LIBOR swap rates. For practical reasons, we assume that the term structure of interest rates is flat and that the insurance company under consideration exclusively invests in Euro-denominated assets. (20) The mean of the AAA-rated Eurozone zero-bond spot yield curve on December 31, 2012, amounting to 0.92 percent, serves as the unstressed interest rate. Consistent with this proceeding, we average the given interest rate stress factors for all maturities. As a result, a single upward shock of +45 percent and a single downward shock of -40 percent is obtained. Since the corresponding absolute changes in both scenarios should be at least 1 percentage point, we manually adjust them as prescribed by the regulator (see EIOPA, 2012a, 2012b). The sensitivity of the firm's assets and insurance liabilities to these shifts is reflected by their respective durations DA and DL. The changes in basic own funds due to interest rate movements can then be calculated as follows:

[DELTA]BOF[|.sub.up] = (- [A.sub.0] x [D.sub.A] x [DELTA][r.sup.up] - (- [L.sub.0] x [D.sub.L] x [DELTA][r.sup.up]), (21)

[DELTA]BOF[|.sub.down] = (- [A.sub.0] x [D.sub.A] x [DELTA][r.sup.down] - (- [L.sub.0] x [D.sub.L] x [DELTA][r.sup.down]). (22)

As mentioned before, the equity module comprises two different risk categories. To estimate the shock for the "global equity" portfolio, CEIOPS drew on historical returns of the MSCI World Developed Equity Index (see CEIOPS, 2010a). Their analysis resulted in a suggested stress factor of 39 percent (see EIOPA, 2012b). Moreover, with regard to the category "other equity," index time-series data for the asset classes hedge funds, commodities, private equity, and emerging market stocks are considered (see CEIOPS, 2010a). Despite the substantial heterogeneity of the category's constituents, CEIOPS proposes a single stress factor of 49 percent. Finally, the correlation coefficient between global and other equity is set to 0.75 (see EIOPA, 2012b). We abstract from the symmetric adjustment mechanism that allows to change the basic stress factors for the equity risk submodule in line with the current economic environment. For further information refer to EIOPA (2012b). (21)

The Solvency II property risk submodule has been calibrated by means of the Investment Property Databank (IPD) real estate index for the United Kingdom, which consists of income figures and appraisal-based valuations submitted by institutional investors, property companies, and mutual funds (see CEIOPS, 2010c). Due to the fact that the different market segments such as retail, office, industrial, and residential are quite similar with respect to their empirical return distributions, CEIOPS refrained from a further breakdown and suggests that a uniform shock of 25 percent should be applied (see EIOPA, 2012b).

Finally, for the spread risk module, CEIOPS primarily analyzed a set of corporate bond indices from Merrill Lynch, covering different maturities and rating buckets (see CEIOPS, 2010c). The resulting stress factors are classified according to both the duration and the rating of the security under consideration (see EIOPA, 2012b). Since the corporate bond portfolio of the hypothetical insurance company underlying our analysis is represented by the Barclays U.S. Corporate Bond Index, which exhibits a modified duration of 7.09, we take the average across all investment-grade rating buckets (i.e., AAA to BBB) for the duration category of between 5 and 10 years and thus obtain a single shock of -9.10 percent. Table 1 summarizes the interest rate, equity, property, and spread risk parameter values for the Solvency II standard formula that enter our calculations.

Calibration of the Partial Internal Model

At first glance, the origin of the parameter values for the standard formula appears to be fairly transparent. However, a lack of more detailed information with regard to their derivation and the fact that certain indices used by CEIOPS are not publicly available obstruct the use of the exact same data basis for the internal model. In addition, several aspects of the calibration process, including statistical methodology and benchmark selection, have been heavily criticized in the extant literature (see, e.g., Mittnik, 2011). Owing to these considerations as well as its architecture based on portfolio theory, the asset part of our internal model will be directly calibrated with the empirical risk-return profiles that are presented in Figure 1. (22) Table 2 contains the descriptive statistics for all six asset classes that enter our calculations. (23) The associated variance-covariance matrix (E) can be found in Table A2 of the Appendix.

On the liability side, in contrast, we are faced with a lack of suitable time-series data for parameter estimation. Consequently, it is necessary to resort to approximations for the first two moments of the normally distributed liability growth rate gL. With regard to the mean, we draw on the current technical interest rate for life insurers in Germany, [i.sub.tec], which is published by the German Federal Financial Supervisory Authority on a regular basis (see BaFin, 2012): (24)

[[mu].sub.L] = [i.sub.tec] = 0.0175. (23)

Moreover, in line with the assets, we assume that the firm's insurance contracts are entirely Euro-denominated such that their values exclusively react to changes in the eurozone term structure. We then follow Braun, Schmeiser, and Siegel (2014) and estimate [[sigma].sub.L] based on the standard deviation of the monthly absolute changes in the euro interest rate between January 1995 and December 2012, [[sigma].sub.EUR], as well as the modified duration of the liabilities, [D.sub.L]. (25) The former equals 69 basis points and the latter is set to 10, in accordance with practitioner estimates for the German life insurance market (see, e.g., Steinmann, 2006): (26)

[[sigma].sub.L] [approximately equal to] [[sigma].sub.EUR] x [D.sub.L] = 0.0069 x 10.00 = 6.9%. (24)

The duration of the asset side, [D.sub.A], depends on the composition of the firm's investment portfolio and can be calculated as follows:

[D.sub.A] = [k.summation over (i=1)] [D.sub.i] x [w.sub.i], (25)

where [w.sub.i] is the portfolio weight of asset class i, and [D.sub.i] denotes the respective modified duration. As can be seen in Table 2, only government and corporate debt securities are sensitive to shifts in the term structure of interest rates. Thus, [D.sub.A] ultimately depends on the percentage shares of the company's bond portfolios.

Finally, [[rho].sub.A,L], that is, the correlation between the aggregate asset return [[??].sub.A] and the growth rate of the liabilities [[??].sub.L] is estimated as follows (see Braun, Schmeiser, and Siegel, 2014): (27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

Hence, we link the asset-liability interaction of the internal model governed by [[rho].sub.A,L] to that of the standard formula, under which [D.sub.A] and [D.sub.L] translate interest rate shocks into [DELTA]BOF values. The presented relationship relies on the assumption that equity, property, and spread risks on the asset side are stochastically independent from the liability side. If this holds true, the common variation of the firm's assets and liabilities can only arise due to the fact that they both react to interest rate movements. We therefore map the prevailing duration gap into a correlation value for the internal model. (28) The smaller the duration gap of the life insurer, the higher the value of [[rho].sub.A,L]. Although it could theoretically take up a value of one, meaning that assets and liabilities change in lockstep, the practical maximum of [[rho].sub.A,L] in the context of our analysis equals 0.7. This is due to the fact that the highest achievable asset duration equals 7.09 and would be obtained for a portfolio that exclusively consists of corporate bonds (see Table 2). Equation (25) illustrates that the inclusion of any other asset class will lead to a lower value of DA and, in turn, [[rho].sub.A,L].

NUMERICAL ANALYSIS

The Portfolio Optimization Problem

Portfolio theory assumes that the investor's expected utility can be perfectly described by a preference function V(x) based on the mean and the variance of the return (or wealth) distribution. Consider the following Taylor expansion around the mean of [[??].sub.A] of some utility function U(x) with positive marginal utility (U > 0):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Taking expectations, the second term of Equation (27) disappears and we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

Because our model framework builds on multivariate normally distributed asset returns, the third and higher central moments are zero. (29) In addition, utility as an ordinal indicator is invariant to positive monotonic transformations. Therefore, the insurance company's equivalent preference function V([[mu].sub.A], [[sigma].sub.A]) may be expressed as

V V([[mu].sub.A], [[sigma].sub.A]) = [[mu].sub.a] - [kappa]/2 [[sigma.sup.2.sub.A], (29)

where [kappa] > 0 implies U" < 0 and thus measures the degree of risk aversion. When drawing on V([[mu].sub.A], [[sigma].sub.A]) to select from the efficient set, one precisely maximizes the true expected utility. (30)

Now, as indicated in the previous section, suppose the insurance company would like to construct efficient portfolios consisting of the six asset classes shown in Table 2. Since it is a risk-averse investor, it aims to minimize the variance (standard deviation) for a given expected return. Hence, it faces the well-known quadratic optimization problem (see, e.g., Kroll, Levy, and Markowitz, 1984)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

subject to w'M = [bar.[mu]].sub.A], (31)

w'1 = 1, (32)

[w.sub.i] [greater than or equal to] 0, (33)

and [w.sub.i] [less than or equal to] [u.sub.i] i [member of] {1,2, ..., 6}. (34)

The first three auxiliary conditions are common. Equation (31) introduces the fixed mean [bar.[mu]].sub.A]; Equation (32) is the budget constraint, which excludes the possibility of borrowing and ensures that the individual asset weights always add up to 100 percent; and Inequality (33) rules out short sales. (31) Furthermore, by adding Inequality (34), we acknowledge the fact that the restricted assets backing the insurer's technical provisions usually need to comply with strict investment limits. These upper bounds for each asset class i are denoted [u.sub.i]. Since the corresponding jurisdictions differ across EU member countries, we decide to align our analysis with the constraints faced by German insurers, which are published in the so-called "Investment Regulation" (in German: Anlageverordnung, AnlV). (32) Accordingly, the subportfolios for corporate bonds, stocks, and hedge funds may jointly not exceed a maximum of 35 percent of the firm's restricted assets. Furthermore, the weight of the category hedge funds is limited to 5 percent and the firm's real estate holdings are capped at 25 percent. In contrast, investment limits do not apply for government bonds and money market instruments (see BMJ, 2011, 2012). Taking these considerations into account, we define the upper bounds u; as shown in Table 3.

Efficient Frontiers and Aggregation

Since the investment limits do not apply with regard to the free assets, that is, the proportion of the portfolio that corresponds to the firm's equity capital, the optimization will be carried out in two steps: (1) including and (2) excluding the inequality constraint (34). Hence, our analysis accounts for the fact that insurance companies are quite flexible concerning the deployment of their shareholders' contributions but must adhere to strict rules when investing policyholder money. Owing to the differing complexity involved in solving the respective convex quadratic programs, we draw on the dual algorithm of Goldfarb and Idnani (1983) for the free assets and the primal dual interior point method of Ferris and Munson (2003) for the restricted assets. As a consequence, we obtain two separate efficient sets, which then need to be aggregated to form the overall asset portfolios. For this purpose, we rearrange Equation (29) to construct indifference curves in the mean-variance space. Subsequently, we systematically vary [kappa] and determine the associated tangency points on both of the nondominated frontiers. (33) In doing so, we are able to identify two corresponding portfolios for each possible degree of risk aversion, which are then combined as follows:

[w.sub.i] = [[alpha].sup.FA] [w.sup.FA.sub.i] + (1 - [[alpha].sup.FA])[w.sup.RA.sub.i], (35)

where [[alpha].sup.FA] is the ratio of free to total assets and [w.sup.FA.sub.i] as well as [w.sup.RA.sub.i] denote the weight of investment category i in the free and the restricted assets portfolio, respectively. In line with empirical evidence for the German life insurance market, we set [[alpha].sup.FA] to 12 percent (see BaFin, 2011).

Figure 2 illustrates the asset class subportfolios and the results of the described optimization procedure in the [mu]-[sigma] space. (34) The uppermost efficient set (solid gray) results when only the budget constraint (32) is included in the optimization. Adding the short-sale constraint (33) yields the solid black curve, comprising those portfolios that may be constructed with the company's free assets. Finally, when incorporating the investment limits (34), one obtains the dot-dashed nondominated frontier that reflects the restricted assets part of the balance sheet. In addition, we have drawn two exemplary indifference (or iso-expected utility) curves for the same value of k. A life insurer exhibiting this degree of risk aversion would choose the free asset portfolio located at the tangency point of the upper indifference curve and the solid black frontier. Similarly, the corresponding restricted asset portfolio can be found at the tangency point of the lower indifference curve and the dot-dashed black frontier. Both portfolios are then merged as implied by Equation (35). Through a repetition of this procedure for [kappa] starting at zero and tending to infinity, it is possible to generate the efficient frontier for the insurer's total asset portfolio, which, for reasons of clarity, has not been depicted in this graph. (35)

Overall, we compute an aggregated frontier consisting of 34,885 portfolios, whose compositions, that is, the weights vectors w, are depicted in Figure 3 with both expected returns and standard deviations increasing from left to right. A closer look reveals three distinct kinks at which the weights change quite substantially. Based on these reference points marked by vertical dotted lines, we define four areas. The very low-risk portfolios in area I as well as the riskiest ones in area IV are not likely to be observed in practice. Areas II and III, however, comprise quite realistic asset allocations. This is underlined by the fact that they resemble the average portfolio composition of European insurers as reported by Fitch Ratings (2011).

Admissibility of Efficient Portfolios

We now calculate the market risk capital requirements for the portfolios on the aggregated efficient frontier. Before doing so, however, the balance sheet total of the considered insurance company has to be defined. Consistent with a short- to medium-term decision horizon, we treat the firm's equity capital as exogenous. To ensure a certain alignment with possible real-world scenarios, we analyze data published by BaFin and, based on a sample of 92 life insurers in the fourth quarter of 2012, decide to set the balance sheet total to EUR 10 bn. (36) In combination with the previously defined [[alpha].sup.FA] = 0.12, this implies [BOF.sub.0] = EUR 1.2 bn. Figure 4 shows the results for the Solvency II standard formula as well as the partial internal model. (37) Again, the portfolios are sorted in order of growing risk and expected return and the graph has been split into four areas reflecting distinct changes of w. It is immediately recognizable that the two solvency models lead to completely different shapes of the capital charges. For an asset allocation to be feasible under the new regulatory framework, the following condition must hold: [BOF.sub.0] [greater than or equal to] [SCR.sub.Mkt]; that is, the firm needs to possess more equity than prescribed. Hence, the admissibility of the efficient portfolios can be easily assessed by means of the dashed horizontal line. The four areas indicated by the dotted vertical lines correspond to those in Figure 3. To further facilitate the interpretation, Figure 5 illustrates the impact of the capital charges on the aggregated frontier in the [mu]-[sigma] space.

Standard Formula. The results for the standard formula are driven by two major aspects: the stress factors assigned to the different asset classes on the one hand as well as the size of the asset-liability duration gap associated with each portfolio composition on the other hand. When comparing the shape of the respective [SCR.sub.Mkt] curve in Figure 4 with the development of the weights vector w in Figure 3, we notice quite similar patterns. Therefore, the standard formula reacts very sensitively to changes in the proportions of those asset classes that have been assigned a relatively high stress factor by CEIOPS. The duration gap, in contrast, appears to be of lesser relevance. To underline this notion, we take a closer look at the four areas in Figure 4.

At the beginning of area I, the capital requirements are very low and all portfolios are admissible. This can be explained by the fact that they predominantly consist of money market instruments, which are free of charge. Hence, [SCR.sub.Mkt] is mainly based on the small additional fractions of real estate, government bonds, hedge funds, and corporate debt (see Figure 3). At the same time, the lack of long-term bonds in these portfolios causes a large duration gap between assets and liabilities, which we have plotted in Figure 6(a). Throughout area I, we then observe a substantial increase in [SCR.sub.Mkt] that may be attributed to the fact that the money market subportfolio receives less weight, whereas the investments in real estate, government bonds, and hedge funds gradually rise. Interestingly, the simultaneous reduction of the duration gap does not seem to notably curb the upturn in the capital charges. Area II, on the other hand, is characterized by stagnating high values of [SCR.sub.Mkt,] which clearly exceed the insurer's equity holdings. It can be shown that this phenomenon occurs for two reasons. First of all, Figure 6(a) illustrates that the continuing growth of the government bond subportfolio helps to reduce the duration gap. Thus, it exhibits a dampening effect on the capital charges. Apart from that, the real estate subportfolio weight increases only marginally in the first part of area II before it slowly begins to shrink (see Figure 3). (38) Consistent with this observation, [SCR.sub.Mkt] also starts to decline around portfolio number 16,000. The subsequent small spike at the very end of area II is brought about by a notable expansion of the firm's stock and hedge fund subportfolios (see Figure 3). Hence, those asset classes with high stress factors have a much more pronounced impact on the outcome than the interest rate risk submodule and the duration gap. (39) Unfortunately this causes a lot of the rather well-diversified portfolios in area II to be disallowed. Furthermore, in area III the fraction of corporate bonds remains virtually constant and the allocations to stocks and hedge funds grow rather slowly. In contrast to that, the real estate holdings are successively lowered to zero and mainly substituted by further government bond investments (see Figure 3). Since the standard formula penalizes the latter much less than the former, [SCR.sub.Mkt] diminishes notably. Similarly, the further reduction of the duration gap as observed in Figure 6(a) helps to lower the capital charges. Nevertheless, fewer than a thousand portfolios in this area are admissible under Solvency II. Finally, area IV is characterized by another substantial increase of [SCR.sub.Mkt]. Here the firm is no longer invested in real estate, the size of the corporate bond portfolio continues to stay constant, and government bonds are starting to be replaced by stocks and hedge funds (see Figure 3). Again, Figure 6(a) illustrates that this effect is reinforced by a widening duration gap.

Partial Internal Model. Turning to the [SCR.sub.Mkt] curve for the internal model shown in Figure 4, we witness a completely different shape than for the standard formula, which does not resemble the pattern of the weights vector w in Figure 3. Results of an unreported sensitivity analysis with another set of benchmark indices show that the slight difference in the databases used for calibration explains only a small bit of the observed deviations. The main reason in fact lies in the more reasonable model architecture. Due to its portfolio-theoretic foundations, the internal model captures the first two central moments of the empirical return distribution for each asset class, not just simple stress factors. This ensures that the impact of portfolio rebalancings on the capital charges is less dramatic, as long as they coincide with a sufficiently beneficial risk-return trade-off. Instead, the duration gap, which governs the asset-liability correlation [[rho].sub.A,L] via Equation (26), becomes much more influential. To illustrate these observations, we again consider the four areas in Figure 4.

Interestingly, those asset allocations at the left end of area I exhibit the smallest return standard deviations but the most conservative capital charges. Up until portfolio 8,000, the latter are even higher than for the Solvency II standard formula. Although this seems contradictory at first glance, it can be well explained by the economic intuition underlying our internal model. As already mentioned above, those compositions close to the minimum-variance portfolio comprise rather small fractions of government and corporate bonds. Hence, in Figure 6(a), we see a substantial duration gap, translating into a low [[rho].sub.A,L]. The consequence is an insufficient asset-liability hedge that leads to riskier basic own funds and, in turn, higher capital charges. This reasoning is illustrated by Figure 6(b), which shows the probability density functions for [DELTA]B[??]F that correspond to the minimum and the maximum-variance portfolio. Unsurprisingly, the latter distribution has a larger mean ([[beta].sub.[DELTA]B[??]F]), since the underlying asset allocation places more weight on investments with relatively high expected returns (see Figure 3). At the same time, however, the smaller duration gap leads to a lower standard deviation ([[beta].sub.[DELTA]B[??]F]), despite the substantially riskier asset side. For these reasons the minimum-variance portfolio is associated with higher capital charges than the maximum-variance portfolio. To assess this graphically, compare the 0.5 percent quantiles that are highlighted by the vertical lines in Figure 6(b).

From areas I to III, we then observe a continuous decline of [SCR.sub.Mkt] that is clearly attributable to the increasing share of bonds in the portfolio and the resulting improvement in the asset-liability matching shown in Figure 6(a). This effect is not dominated by the inclusion of more volatile asset classes such as hedge funds or stocks, since the model acknowledges that those are also associated with higher expected returns. As the insurer moves along the (concave) efficient frontier, however, it receives less and less additional [[mu].sub.A] per unit of [[sigma].sub.A], implying that the asset risk is gradually developing a greater impact on the capital charges. For portfolio number 32,644, the duration of the asset side reaches its maximum of 4.19, which corresponds to the highest possible asset-liability correlation in Figure 6(a) and the lowest capital requirements in Figure 4. Finally, [SCR.sub.Mkt] begins to rise again in area IV, although much less pronounced than for the standard formula. This is due to the fact that government bonds are now beginning to be replaced by hedge funds and stocks such that the duration gap widens again. At the same time, the risk-return trade-off is becoming less beneficial as the efficient frontier flattens out to the right.

Admissibility of Selected Inefficient Portfolios

Below we calculate the market risk capital requirements for a systematically selected batch of inefficient portfolios and assess their admissibility, too. In this context, inefficient means that a portfolio is mean-variance dominated; that is, there is another one with the same [[sigma].sub.A] but a higher [[mu].sub.A]. If the solvency models are designed consistently, more efficient asset allocations should be preferred to less efficient ones. Otherwise, the insurer might ceteris paribus be incentivized or even forced to invest in portfolios that provide a suboptimal expected return for any given level of risk. Per definition, the inefficient portfolios of both the free and the restricted asset categories lie below the respective efficient sets shown in Figure 2. Since they form whole arrays instead of distinct frontiers, it is not possible to match and merge corresponding portfolios by searching for tangency points of the iso-expected utility curves. Therefore, we decide to exclusively focus on the restricted assets, which account for the major proportion of the insurer's balance sheet. (40) To cover the space of inefficient portfolios as comprehensively as possible while limiting computational intensity, we fix a discrete increment of 2.5 percent and form all possible combinations of the six weights in w that fulfill the budget, short-sale, and investment constraints. (41) In doing so, we obtain a total of 40,875 inefficient portfolios. Subsequently, their capital charges associated with both solvency models are calculated and compared with the cutoff value of EUR 1.2 bn. Figures 7(a) and (b) show all portfolios in the [mu]-[sigma] space, indicating their admissibility under the standard formula and the internal model, respectively. Due to the investment limits, the inefficient portfolios are concentrated in concave arrays below the efficient frontier for the restricted assets. By means of Panels (c) and (d), we magnify the central areas of Panels (a) and (b) to provide a more detailed illustration of the corresponding patterns.

Standard Formula. Concerning the admissibility of the efficient portfolios for the restricted assets as depicted in Figure 7(a), we notice an overall pattern comparable to that of the aggregated frontier in Figure 5(a). The points at the beginning of the curve are allowed, followed by a large number of disallowed portfolios. Further to the right of the efficient set, we then have a second section of permitted portfolios. Finally, those asset allocations with the highest attainable expected returns and standard deviations toward the far end are inadmissible again. This is a strange outcome, since per definition, no point along the efficient frontier is dominated by another one: as the insurer decides to bear more risk, it is rewarded with a higher expected return. Therefore, in the absence of an assumption about preferences, the only sensible measure to differentiate between economically better or worse efficient portfolios in our context is their interaction with the liability side of the insurer's balance sheet. Yet, from the analysis in the previous section, we know that the effect of the duration gap is in many cases crowded out by the heavy impact of the stress factors on the capital charges. In other words, the standard formula does not consistently promote those efficient portfolios that provide a better asset-liability hedge. When turning to the inefficient portfolios, we gather an even more surprising picture. Figure 7(a) shows that broad areas of disallowed portfolios draw through the [mu]-[sigma] space, generating an almost zebra-like pattern. Consequently, for most values of [[sigma].sub.A], the standard formula does not promote more efficient over less efficient portfolios. The corresponding magnification of the central part of Figure 7(a) as depicted in Figure 7(c) underlines this observation. All efficient and near-efficient portfolios in the area are inadmissible. Consider a vertical slice for a standard deviation of 2 percent as an example. Given this level of risk, the insurer is not permitted to select from the entire group of asset allocations that offer the highest possible expected returns. Instead, two ranges of portfolios with different degrees of inefficiency are admissible.

Partial Internal Model. In contrast to the standard formula, the internal model produces the coherent outcome that we see in Figure 7(b). With regard to the efficient frontier for the restricted assets, we find that the insurer may not chose points in the wider proximity of the minimum-variance portfolio, whereas a large range of points from the first third toward the right end of the curve are admissible. The same was true for the aggregated frontier shown in Figure 5(b). Now, recall from the previous section that the low-volatility portfolios mainly consist of money market instruments and are therefore associated with a wide duration gap, whereas those asset allocations further to the right contain much larger fractions of government and corporate bonds, thereby leading to higher asset-liability correlations. Consequently, the internal model acknowledges that along the nondominated frontier, larger standard deviations are generally coupled with higher expected returns and consistently prefers those efficient portfolios that help to reduce the riskiness of the insurer's equity capital. Furthermore, the pattern that can be observed below the restricted asset frontier is highly sensible, too. Only the most inefficient points in the [mu]-[sigma] space are disallowed. These findings are substantiated by the magnification in Figure 7(d): the insurer may select all efficient and near-efficient portfolios for a given level of risk. Accordingly, there is not a single [[sigma].sub.A], for which the capital requirements induce the insurer to select an inefficient portfolio over the corresponding efficient one.

ECONOMIC IMPLICATIONS

Our analysis reveals substantial weaknesses of the Solvency II standard formula for market risk. Due to its one-dimensional design, which is centered around stress factors for different asset classes, it exclusively evaluates portfolios based on their riskiness as judged by the regulator. The role of expected returns, on the contrary, is disregarded completely and diversification effects are only marginally accounted for, implying that the approach is unable to distinguish investments on the basis of risk-return and correlation profiles. Consequently, the capital charges are heavily dominated by the size of the stress factors and lead to an unsystematic pattern with regard to the admissibility of different asset allocations. This has several highly problematic implications. First, portfolios that contain stocks, corporate bonds, real estate, and hedge funds, are disproportionately penalized under the Solvency II standard formula so that many well-balanced asset allocations are only attainable with large amounts of equity capital. Particularly for life insurers, this is a serious issue, as the expected return of portfolios that mainly consist of money market instruments and government bonds may be too low to cover their interest rate guarantees in the long-term. Second, although it was intended to take the asset-liability matching aspect into account, the standard formula seems to be incapable of consistently promoting those efficient portfolios that lead to a tighter duration gap. Therefore, insurers may be encouraged to select asset allocations that ultimately increase instead of decrease the volatility of their equity capital. Third, efficient portfolios are not systematically preferred to inefficient portfolios, meaning that some companies may need to expose themselves to a higher than necessary degree of market risk to earn their costs of capital.

The last aspect is particularly important from a shareholder value perspective. At first glance, it may not seem to be a severe problem that efficient portfolios are more costly to form than their inefficient counterparts. As long as the gain in the expected return on assets exceeds the increase in the weighted average costs of capital due to the additional amount of equity required, it will be beneficial to raise the corresponding funds. However, recall from the previous section that we treat the firm's available equity as exogenous, since, in most real-world scenarios, the capital base cannot be adjusted merely for the purpose of portfolio rebalancing. Consequently, the insurer is forced to select the inefficient portfolio in the short run. In addition, without knowing the insurer's cost of capital function, this cannot be assumed to be the general case. It could in fact very well be the other way round: the losses in expected return that are incurred by foregoing an efficient portfolio may be overcompensated by the cost of capital that is saved through the associated reduction in equity. In this case the shareholder value-maximizing firm would be clearly incentivized to select the inefficient asset allocation. Yet, it does unfortunately not represent the economic optimum, solely because it is the best feasible choice under the standard formula.

Notwithstanding its parsimony, the portfolio-theoretic architecture of our internal model, in contrast, is predestined to capture diversification effects and the associated risk-expected return trade-off. More specifically, the impact of each investment on the capital requirements is driven by its main distributional characteristics, not just blunt stress factors. Hence, the internal model gives insurers more leeway to draw on asset classes that exhibit larger standard deviations but also higher expected returns. Furthermore, it produces lower capital charges if the insurer's assets and liabilities are matched. In doing so, it promotes those efficient portfolios that are associated with smaller duration gaps and thus less default risk. Finally, it coherently favors efficient over inefficient portfolios, meaning that for a fixed level of asset risk, the insurer may always select the portfolio with the highest expected return. (42) For these reasons, the internal model, unlike the standard formula, is also fully compatible with the goal of shareholder value maximization. More specifically, it does not require the insurer to trade off expected return against costs of capital, because efficient portfolios are strictly associated with lower capital charges than the corresponding inefficient ones. Therefore, the efficient asset allocation for a certain degree of market risk (standard deviation) will generally be the one that maximizes shareholder value.

SUMMARY AND CONCLUSION

We consider the issue of optimizing a life insurance company's asset allocation in the context of portfolio theory when the firm needs to adhere to the market risk capital requirements of Solvency II. The discussion starts with a brief review of the standard formula and the introduction of a parsimonious partial internal model. Subsequently, we estimate empirical risk-return profiles for the main asset classes held by European insurers and run a quadratic optimization program to derive nondominated frontiers with budget, short-sale, and investment constraints. We then compute the capital charges under both solvency models and identify those portfolio compositions that are permitted for an exogenously given amount of equity. Finally, we consider a systematically selected set of inefficient portfolios and check their admissibility, too.

Our results have profound implications. Since the one-dimensional stress factor approach underlying the Solvency II standard formula is unable to properly account for expected returns and diversification effects, it does not produce economically sensible results. More specifically, it is found to heavily restrict the choice of investments, to be relatively insensitive with regard to the duration gap between assets and liabilities, and to promote inefficient over efficient portfolios. Hence, in many cases, insurance companies may find it difficult to fulfill their contractual obligations, while being encouraged to select portfolios that raise the volatility of their equity capital and contain an unnecessarily high level of market risk for the expected return they offer. Clearly, such severe asset management biases lead to less default risk protection for the policyholders as well as a nonnegligible reduction in shareholder value. Furthermore, given the importance of the European insurance sector among institutional investors, a widespread restructuring of investment portfolios, triggered by the introduction of Solvency II in its current form, could have a substantial impact on the demand and pricing for certain asset classes. As the Solvency II project enters its final development phase, regulators, industry professionals, and stakeholders require a comprehensive understanding of the possible effects of the new capital charges on asset allocations. By means of our analysis, we hope to sharpen the awareness of persisting shortcomings of the standard formula that necessitate a fundamental revision and, at the same time, promote the proliferation of sound internal solvency models within the industry.

APPENDIX: INPUT DATA

TABLE A1 Correlation Matrices for the Solvency II Standard Formula Panel A: Upward Stress Scenario [CorrMkt.sup.up] Equity Interest Property Spread Equity 1.00 Interest 0.00 1.00 Property 0.75 0.00 1.00 Spread 0.75 0.00 0.50 1.00 Panel B: Downward Stress Scenario [CorrMkt.sup.down] Equity Interest Property Spread Equity 1.00 Interest 0.50 1.00 Property 0.75 0.50 1.00 Spread 0.75 0.50 0.50 1.00 TABLE A2 Annualized Variance-Covariance Matrix of Returns (01/01/1993-12/31/2012) (1) (2) (3) Stocks (1) 0.0371 -0.0014 0.0016 Government bonds (2) -0.0014 0.0011 0.0008 Corporate bonds (3) 0.0016 0.0008 0.0031 Real estate (4) -0.0001 0.0001 0.0000 Hedge funds (5) 0.0094 -0.0005 0.0011 Money market (6) 0.0000 0.0000 0.0000 (4) (5) (6) Stocks (1) -0.0001 0.0094 0.0000 Government bonds (2) 0.0001 -0.0005 0.0000 Corporate bonds (3) 0.0000 0.0011 0.0000 Real estate (4) 0.0003 0.0000 0.0000 Hedge funds (5) 0.0000 0.0050 0.0000 Money market (6) 0.0000 0.0000 0.0000

DOI: 10.1111/jori.12077

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Alexander Braun, Hato Schmeiser, and Florian Schreiber are at the Institute of Insurance Economics, University of St. Gallen, Tannenstrasse 19, CH-9000 St. Gallen, Switzerland. The authors can be contacted via e-mail: alexander.braun@unisg.ch, hato.schmeiser@unisg.ch, florian.schreiber@unisg.ch

(1) Beyond the insurance literature, Kocagil and Klein (2013) of the investment manager PIMCO calculate mean-tail risk-efficient portfolios for the banking sector, taking into account the risk-weighted asset charges of Basel III.

(2) The SCR is the key magnitude of Solvency II pillar one. Please refer to EIOPA (2012b).

(3) Note that the BOF were called net asset value (NAV) under QIS5.

(4) According to CEIOPS (2009), these submodules account for approximately 80 percent of the overall market risk.

(5) In the latest official publication on the Solvency II market risk module, these categories have been renamed as "type 1 equity" and "type 2 equity" (see EIOPA, 2012b). Since many market participants have meanwhile become accustomed to the terms global and other equity, however, we decided to stick with the original terminology throughout this article.

(6) Although controversially discussed, the adequacy of the standard formula calibration for private equity investments has been vindicated in a recent technical report (see EIOPA, 2013).

(7) The spread risk submodule does not require a capital charge for government bonds emitted by EEA member states. Nevertheless, credit risk of government bonds needs to be accounted for within the "Own Risk and Solvency Assessment (ORSA)" of Pillar 2.

(8) Note that [Mkt.sub.eq], [Mkt.sub.prop], and [Mkt.sub.sp] are the same in both interest rate scenarios.

(9) Structural credit models date back to the seminal work of Merton (1974) and are well established in the literature (see, e.g., Leland and Toft, 1996).

(10) Note that given the availability of sufficient data for the calibration, it would be straightforward to adopt a much more fine-grained categorization of the asset side, even down to the individual security level.

(11) Christiansen and Niemeyer (2012) point out that the directive of the European Parliament (2009) remains unclear regarding application and choice of proper discount factors for the calculation of the solvency capital requirement. Although the Omnibus II Directive that was been ratified in March 2014 specifies the Solvency II Directive from 2009 in this regard, we refrain from discounting [B[??]F.sub.1]. Due to the 1-year time horizon and the current low interest rate environment, this decision will have a minor impact on the results.

(12) Since [DELTA]B[??]F represents changes in the basic own funds, it is reasonable to employ a probability distribution that allows for positive and negative values.

(13) As emphasized by Mittnik (2011), the standard formula is "only valid for elliptical and thus symmetric distributions."

(14) A corollary is that the short rate can become negative, which has regularly been criticized in the literature and led to the development of supposedly more sophisticated approaches. Yet, this view might have to be reevaluated as of late, since the European Central Bank (ECB) introduced negative rates for central bank deposits in June 2014.

(15) In contrast to Fitch Ratings (2011), we merge "mortgage" and "property" into the asset class "real estate."

(16) A pure investment-grade index is a reasonable choice in our context, since European insurance companies allocate about 87 percent of their corporate bond portfolio to the three highest rating classes of S&P (see CEIOPS, 2010c).

(17) Due to the fact that the HFRI encompasses over 2,200 constituent funds, its performance might, to a certain extent, exceed that of less diversified hedge fund portfolios. However, other commonly used proxies for this asset class, such as the HFRX Global Hedge Fund Index, do not offer return data back to the early 1990s.

(18) For the 6-year time period from 1993 to 1999 the predecessor rate FIBOR is employed.

(19) This proposal contains a number of adjustments compared to the CEIOPS directives and the Fifth Quantitative Impact Study (see EC, 2010). In addition to EIOPA (2012b), the errata document EIOPA (2012a) is taken into account.

(20) Consequently, foreign exchange (FX) risk is irrelevant in the context of our analysis.

(21) The calibration of the Solvency II standard formula attracted a lot of scholarly criticism. Identified issues include skewness-related stability problems, inadequate benchmark indices, and the amalgamation of very heterogeneous asset classes in the single category "other equity." Yet, the most fundamental shortcomings have been revealed in a recent study by Mittnik (2011), who documents that the derivation of the parameter values for the equity risk module "is seriously flawed" since CEIOPS relied on an overlapping rolling-window annualization of daily returns that gives rise to "highly unreliable and erratic VaR and correlation estimates."

(22) As we discuss in the next section, the slightly different databases used for calibration have a minor impact on the capital charges generated by the two models. This becomes already quite apparent when considering the high-return correlation between the two global equity risk benchmarks MSCI World and EURO STOXX 50.

(23) It should be emphasized that the internal model captures both the spread risk and the interest rate risk inherent in corporate bonds through the standard deviation of the Barclays U.S. Corporate Bond Index.

(24) The technical interest rate represents a lower bound for the mean growth rate of the liabilities that lends itself in the context of solvency models, since life insurers may refrain from paying nonguaranteed benefits in distressed situations.

(25) Euro yield curve time-series data are unavailable before January 1995.

(26) This approximation exploits the fact that the variation in the market value of the liabilities is mainly attributable to interest rate movements. Due to the linearity of the duration concept, it will be less reliable for large interest changes.

(27) In the context of our analysis, [D.sub.A] and [D.sub.L] are strictly limited to positive numbers.

(28) The chosen approximation will hold well for life insurers with sufficiently small correlations between noninterest rate risks on the asset, and the liability side.

(29) The same is true if the utility function approximated by the Taylor series is quadratic.

(30) Several authors have asserted that full compatibility of expected utility theory and mean-variance analysis requires either a quadratic utility function or normally distributed returns. As shown by Levy and Markowitz (1979), however, even if these conditions are not fulfilled, expected utility may be well approximated by means of Taylor series around specific points. Hence, without knowing his exact utility function in terms of Von Neumann and Morgenstern (1944), an investor is likely to maximize the true expected utility when selecting his preferred mean-variance efficient portfolio.

(31) Due to the difficulties involved in shorting most of the asset types in our analysis as well as the corresponding legal restrictions faced by insurance companies in regulated markets, this is a realistic constraint.

(32) The future of the "Investment Regulation" is currently being discussed among regulators and industry professionals. In the absence of explicit evidence suggesting that they will be completely abandoned after the introduction of Solvency II, we deem them an important part of the analysis. As a corollary of these additional constraints, the optimization algorithm returns more realistic portfolio compositions.

(33) If [kappa] is almost zero, the indifference curves exhibit a very low curvature and touch toward the right end of the efficient frontier. For extremely high values of [kappa], in contrast, the tangency points are located near the minimum-variance portfolio.

(34) With an expected return of 9.21 percent p.a. and a standard deviation of 19.26 percent p.a. (see Table 2), the stock subportfolio is located outside the boundaries of Figure 2. Moreover, note that the minimum-variance portfolio on all three frontiers actually lies slightly to the upper left of a pure investment in money market instruments.

(35) By definition, it would be located between the solid black and the dot-dashed black curve.

(36) Statistics on the German insurance market are available at http://www.bafin.de.

(37) In order to illustrate that we actually consider a discrete set of points rather than a continuous curve, every thousandth portfolio has been highlighted.

(38) Note that the investment limit for real estate as shown in Table 3 is slightly exceeded in area II. However, it should be taken into account that we consider the firm's overall efficient

(39) This finding can be shown to be robust against the chosen calibration. Even if the insurer were able to invest in government bonds with a much higher duration, the general shape of the [SCR.sub.Mkt] curve in Figure 4 would remain the same. frontier, for which the restricted and the free assets portfolios have been aggregated. Hence, this observation does not constitute a regulatory violation.

(40) An extension to the free assets is straightforward yet computationally more intense.

(41) For the asset class hedge funds, for example, this means that the insurer can choose to invest 0, 2.5, or 5.0 percent. Note that all investment limits are divisible by 2.5 (Table 3).

(42) Any other economically substantiated model that is more realistic than the presented two-dimensional approach can be expected to further accentuate these aspects.

Caption: Figure 1 Illustration of the Empirical Risk-Return Profiles

Caption: Figure 2 Efficient Frontiers With Budget, Short-Sale, and Investment Constraints

Caption: Figure 3 Compositions of the Portfolios on the Aggregated Efficient Frontier

Caption: Figure 4 Capital Requirements Under the Standard Formula and the Internal Model

Caption: Figure 5 Admissibility of Portfolios on the Aggregated Efficient Frontier

Caption: Figure 6 Properties of the Internal Model

Caption: Figure 7 Admissibility of Efficient and Inefficient Portfolios (Restricted Assets)

TABLE 1 Input Data for the Solvency II Standard Formula Submodule Shock % Interest rate risk -40.00/+45.00 Global equity -39.00 Other equity -49.00 Property risk -25.00 Spread risk -9.10 Notes: This table contains the shocks that will be used in combination with the Solvency II standard formula. All values are based on the suggestions in EIOPA (2012b). The differentiation of up and down state exclusively applies for interest rate risk. In this context, we use single stress factors that equal the mean of the CEIOPS values for each maturity. Similarly, the spread risk shock is obtained by averaging the parameter values across the investment-grade rating buckets for a maturity between 5 and 10 years. More information on the empirical derivation of the stress factors and the employed input data can be found in the relevant CEIOPS directives (see CEIOPS, 2010a, 2010b, 2010c). TABLE 2 Descriptive Statistics for Monthly Return Time Series (01/01/1993-12/31/2012) No. Asset Class Benchmark Index 1 Stocks EURO STOXX 50 (TR) 2 Government bonds REX Performance Index 3 Corporate bonds Barclays U.S. Corporate Bond Index (TR) 4 Real estate Grundbesitz Europa Fund (TR) 5 Hedge funds HFRI Fund Weighted Index 6 Money market 1 Month FIBOR/EURIBOR No. [[mu].sub.i] [[sigma].sub.i] Duration 1 9.21% 19.26% -- 2 5.96% 3.34% 4.92 3 6.99% 5.55% 7.09 4 4.81% 1.76% -- 5 9.65% 7.08% -- 6 3.14% 0.50% -- Notes: This table contains mean (m;) and standard deviation (a;) of the monthly index return time series representing the stock, government bond, corporate bond, real estate, hedge fund, and money market subportfolios of the insurance company under consideration. Each benchmark measures the total investment returns (TR) for its asset class (including coupons and dividends where applicable). All figures have been annualized. For the government and corporate bond portfolios, the modified duration as of January 31, 2012 is displayed as well. TABLE 3 Investment Limits for the Six Asset Classes Under Consideration Asset Class Investment Limit Stocks 20% Government bonds -- Corporate bonds 10% Real estate 25% Hedge funds 5% Money market -- Notes: The table contains the stylized investment limits in percent of the insurance company's total assets that will be applied in the context of our portfolio optimization. All values have been derived based on the current legislation in Germany (see BMJ, 2011).

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Author: | Braun, Alexander; Schmeiser, Hato; Schreiber, Florian |
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Publication: | Journal of Risk and Insurance |

Article Type: | Report |

Date: | Mar 1, 2017 |

Words: | 12871 |

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