Proposal for correction of the SCR calculation bias in Solvency II

Abstract

In the context of Solvency II the Solvency Capital Requirement (SCR) is a well known financial demand which will have to be fulfilled by all European insurance companies to assure a theoretical ruin probability of 0.005 or less.

A standard formula for the calculation of the SCR will be provided. Its current state is given by the Technical Specifications of the 5th Quantitative Impact Study. Every European insurance company will be obligated to use the provided standard formula if they do not legitimate an internal risk model.

The standard formula uses a lognormal distribution which is parameterized with a mean of 1 and a standard deviation parameter. The latter can be set corresponding to the market-wide estimations or corresponding to the data of the company.

We favor the possibility for insurance companies to take into account their individual risk situation and believe that the restriction of a mean of 1 is not appropriate. We therefore introduce a correction formula and propose its implementation into the formula for the undertaking-specific parameter. Using the correction formula leads to the same SCR as taking into account both the individual mean and the individual standard deviation.

Appendix

The existence of a real $\tilde{s}^{2}$ in Theorem 2 is important for the benefit from the correction formula. Therefore, we have to analyze for which m∈ℝ⁺ and s∈ℝ⁺ the relation

holds.

Using Proposition 1 and Proposition 2 we have $\ln(F^{-1}_{m,s^{2}}(u))=\sigma_{m,s} q_{u} + \mu _{m,s}$ with appropriate μ _m,s and σ _m,s , depending on m and s, and q _u defined as before. For simplicity in notation we write μ instead of μ _m,s and σ instead of σ _m,s . Then we have

(6)

For m≤1 and for all s∈ℝ⁺ the term in (6) is obviously greater or equal to zero and therefore $\tilde{s}^{2} \in\mathbb{R}$ for such m. Else (m>1) there exists an interval I, given by $I=(\max \{0,q_{u}-\sqrt{\ln(m^{2})}\},q_{u}+\sqrt{\ln(m^{2})})$, such that the term in (6) is smaller than zero iff σ∈I. Thus, $\tilde{s}^{2}$ is an element of ℝ iff

$$\sigma\in I^c$$

respectively, iff

$$s^2 \in J^c,\quad J:=\Bigl(m^2 \bigl(e^{\max\bigl\{0,q_u-\sqrt{\ln (m^2)}\bigr\}^2} -1\bigr),m^2\bigl( e^{\bigl(q_u+\sqrt{\ln(m^2)}\,\bigr)^2}-1\bigr)\Bigr).$$

In particular, for m≤3 and u=0.995 we get $q_{u}-\sqrt{\ln(3^{2})}>0$ and thus

Therefore, we have $\tilde{s}^{2} \in\mathbb{R}$ if $s\leq1.5 <\sqrt{a}$.