Capital allocation and RORAC optimization under solvency 2 standard formula

Abstract

Solvency II Directive 2009/138/EC requires an insurance and reinsurance undertakings assessment of a Solvency Capital Requirement by means of the so-called “Standard Formula” or by means of partial or full internal models. Focusing on the first approach, the bottom-up aggregation formula proposed by the regulator allows for a capital reduction due to the diversification effect, according to the typical subadditivity property of risk measures. However, once the overall capital has been assessed no specific allocation formula is provided or required in order to evaluate the contribution of each risk source on the overall Solvency Capital Requirement. The aim of this paper is twofold. First, we provide a closed formula for capital allocation fully compliant with the Solvency II Capital Requirement assessed by means of the Standard Formula. The solution enables a top-down approach to assess the allocated Solvency Capital Requirement among the risks considered in the Solvency II multilevel aggregation scheme; we demonstrate that the allocation formula adopted is consistent with the Euler allocation principle. Second, a solution is found as a result of an optimum capital allocation problem based on a Return On Risk Adjusted Capital measure; we establish the equivalence between the Return On Risk Adjusted Capital optimization, when the risk adjusted capital is calculated according to the Standard Formula, and the Markowitz mean-variance optimization.

A standard formula properties and remarks

The above reported aggregation formulas (Eqs. (14) and (15)) call for further considerations, which may be illustrated as follows:

• (i) As stated by EIOPA the overall SCR shall correspond to a specific risk measure, the Value-at-Risk (VaR), subjected to a confidence level of 99.5% over a one-year period. Thus considering the intention of EIOPA, it seems acceptable to write $SCR\approx VaR_{99.5\%}$.

• (ii) In EIOPA (2014a) has specified that the correlation matrices used for the aggregation of sub-risks (Eq. (14)) and risk modules (Eq. (15)) respectively are estimated to minimize the aggregation error through the following formulation:

  $$\begin{aligned} \left| \min _{\rho } VaR(X+Y)^2 - VaR(X)^2 - VaR(X)^2 -2 \rho VaR(X) VaR(Y)\right| \end{aligned}$$

  (52)

  where X and Y are r.v.s that represent two different risks.

• (iii) Each $SCR_{ix}$ is calculated by means of specific methodologies stated by EIOPA. The general principle for the calculation of a single sub-risk capital requirement $SCR_{ix}$ is to apply a set of shocks to the risk drivers and calculate the impact on the value of the assets and liabilities. The calibration objective - i.e. the calibration using Value at Risk subject to a confidence level of 99.5% over a one-year period - is extended to each individual risk module and sub-risk.

Remark 1

As well known, the structure with a square root of a quadratic expression and the use of correlation matrices produce a correct aggregation of quantiles in case of any centered elliptical distribution, such as the (multivariate) normal distribution³.

Remark 2

In the SCR calculation EIOPA does not put forward assumptions for the distribution of the losses of each risk class and/or sub-risk, but the underlying assumption of linear correlation and elliptic distribution are implicit and necessary for the correctness of the aggregation formulas.

Remark 3

The bottom-up aggregation approach proposed by EIOPA does not represent a ‘genuine’ bottom-up approach to risk aggregation in the sense of Filipović (2009). By nesting Eq. (14) in Eq. (15):

$$\begin{aligned} BSCR = \sqrt{\sum _{i=1}^{n} \sum _{j=1}^{n} \left[ \sqrt{\mathbf {SCR_{i\bullet }^{\mathrm {T}}} \cdot \mathbf {P_i} \cdot \mathbf {SCR_{i\bullet }}}\right] \cdot \left[ \sqrt{\mathbf {SCR_{j\bullet }^{\mathrm {T}}} \cdot \mathbf {P_j} \cdot \mathbf {SCR_{j\bullet }}}\right] \cdot \rho _{i,j}} \end{aligned}$$

(53)

we obtain an Eq. (53) that is inconsistent with any multivariate distribution of risks. Following Filipović a genuine bottom-up model uses a full base correlation matrix $\mathbf{B }:M \times M \rightarrow \mathfrak {R}$ (where $M=m_1+m_2+\cdots +m_n$) that aggregates all risk types, across risk classes, together:

$$\begin{aligned} SCR=\sqrt{\mathbf{A }^{\mathrm {T}} \cdot {\mathbf {B}} \cdot {\mathbf {A}}} \end{aligned}$$

(54)

where, ${\mathbf {A}}=[\mathbf {SCR_{i\bullet }}, \ldots , \mathbf {SCR_{n\bullet }}]$ is the vector of all sub-risk capital requirement vectors.

Nevertheless, the only available information about correlation is contained in each matrix $\mathbf {P_i}, i=1,2,\ldots ,n$ but is limited to the correlation coefficients among sub-risks referred to the risk modules themselves. The missing correlation coefficients in ${\mathbf {B}}$ are referred to sub-risks belonging to different risk modules (e.g. equity risk and lapse risk) whose estimate is an arduous task.

Finally, the risk aggregation bottom-up approach provided by EIOPA has the following properties and shortcomings:

• the overall SCR is based on a VaR risk measure so it involves all the coherent risk measure properties⁴ excluding sub-additivity;

• the implicit elliptical distribution assumption underlying Eqs. (14) and (15) involves the sub-additivity property;

• the nested aggregation formula (53) is homogeneous of the first degree;

• the two-step aggregation formula proposed by EIOPA is inconsistent with any multivariate probability distribution and does not represent a genuine bottom-up approach as stated by Filipović (2009).

The SCR computed using the Standard Formula should be interpreted as a risk indicator that, given the formal inconsistencies of the aggregation approach based on a unique standardized methodology allows for a proxy of the VaR for the unexpected loss only ideally. It is suitable to represent the overall solvency condition of an insurance undertaking because its value is coherent with the nature of risks assumed by the undertaking and, moreover, it increases (or decreases) according to higher (or lower) risk levels assumed. It is worth recalling that the Standard Formula approach can be completely or partially replaced by an Internal Model reflecting several of the previously addressed bottom-up considerations.