[Paper Review] On a capital allocation principle coherent with the Solvency 2 standard formula

Fabio Baione, Paolo De Angelis+1 more|arXiv (Cornell University)|Jan 26, 2018

Insurance and Financial Risk Management7 references3 citations

TL;DR

This paper proposes a closed-form capital allocation formula for Solvency II's Standard Formula that is fully coherent with the Euler principle, enabling precise top-down allocation of the Solvency Capital Requirement (SCR) across risk modules and lines of business. The method uses the square root aggregation structure of the Standard Formula to derive risk contributions that reflect diversification effects, ensuring consistency with regulatory capital calculations and providing a transparent, axiomatic approach to capital allocation.

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 permits a capital reduction due to 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 SCR. The aim of this paper is to provide a closed formula for capital allocation fully coherent with the Solvency II Capital Requirement assessed by means of Standard Formula. The solution proposed permits a top-down approach to assess the allocated SCR among the risks considered in the multilevel aggregation scheme established by Solvency II. Besides, we demonstrate that the allocation formula here proposed is consistent with the Euler's allocation principle

Motivation & Objective

• To address the lack of a standardized, coherent capital allocation method for the Solvency Capital Requirement (SCR) under the Solvency II Standard Formula.
• To develop a closed-form allocation formula that respects the square root aggregation structure of the Standard Formula and ensures consistency with the Euler principle.
• To enable top-down allocation of SCR across risk modules and lines of business (LOBs) without relying on proxies or approximations.
• To support risk-adjusted performance measurement and strategic capital management by providing accurate capital absorption figures per risk component.
• To demonstrate the method’s practical applicability through a numerical case study using data from a non-life insurance company.

Proposed method

• The paper derives a closed-form solution for capital allocation based on the Euler principle, applied to the multilevel square root aggregation formula used in the Solvency II Standard Formula.
• It uses the partial derivative of the SCR with respect to each risk component’s capital requirement to compute the marginal contribution of each risk to the total SCR.
• The allocation formula is applied at both the risk module level (e.g., CAT, premium risk) and the sub-risk level (e.g., windstorm, fire), ensuring consistency across the entire aggregation hierarchy.
• The method leverages the square root formula’s structure: SCR_total = sqrt(Σ(SCR_i)^2) for uncorrelated risks, allowing analytical derivation of risk contributions.
• For allocation across lines of business, the paper uses the amount insured as a risk driver to distribute allocated SCR from sub-risks to LOBs.
• The approach avoids proxies by directly computing allocations from the regulatory formula, ensuring full coherence with the Standard Formula’s diversification logic.

Experimental results

Research questions

• RQ1 How can a capital allocation method be developed that is fully coherent with the Solvency II Standard Formula’s square root aggregation structure?
• RQ2 Can the Euler principle be applied in a closed-form manner to the multilevel aggregation of the Standard Formula to allocate SCR across risk components?
• RQ3 How does the proposed allocation method compare to alternative allocation principles in terms of consistency and transparency?
• RQ4 What is the practical impact of this allocation method on capital absorption and risk-adjusted performance measurement across lines of business?
• RQ5 Can this method support strategic capital management decisions such as RORAC optimization and risk appetite alignment?

Key findings

• The proposed allocation formula is mathematically consistent with the Euler principle and provides a closed-form solution for SCR allocation across all levels of the Solvency II risk aggregation hierarchy.
• For the natural catastrophe risk, the allocation to windstorm and flood perils was 13% and 22% respectively, reflecting their relative contributions to the total SCR.
• In the man-made catastrophe risk, fire perils contributed 49% of the allocated SCR, highlighting their dominant risk impact despite being a single sub-risk.
• The total allocated SCR for CAT risk was 6,158,875, with 54% allocated to man-made catastrophes and 25% to natural catastrophes, reflecting their relative risk contributions.
• The allocation to individual lines of business was derived using the amount insured as a risk driver, enabling a market-driven, transparent distribution of capital across LOBs.
• The method enables precise capital absorption tracking, supporting strategic decisions such as RORAC optimization and risk appetite management.