Time Consistent Solvency Capital Requirement Under Solvency II.

Abstract

In this paper, we analyse the current dynamic definitions of the Solvency Capital Requirement (SCR) and we propose a new dynamic and time consistent formulation of the SCR, that is compliant with the Solvency II directive. In case of a single liability cash-flow at maturity, Devolder and Lebégue (2017) analyse the time-consistent dynamic formulation of the SCR using iterated risk measures. Firstly, we extend the iterated-SCR (ISCR) formulation to the case of multiple liability cash-flows. However, the iterated formulation of the SCR presents some drawbacks. Devolder and Lebégue (2017) show that using iterated risk measures, the ISCR becomes quite expensive for long term liabilities. Moreover, we show that in case of multiple liability cash-flows, the ISCR is not completely compliant with the Solvency II directive. In fact, ISCR does not answer to a fundamental regulator request: the capital to be held by insurance company to meet obligations over the following year. Then, starting from the static definition in Christiansen and Niemeyer (2014), we propose a dynamic version of the SCR, that is time consistent and encompasses the drawbacks of the iterated formulation. Our proposed dynamic definition of the SCR, called additive-SCR (ASCR), indicates the expected total capital requirement for the period from time zero to the liabilities maturity. Moreover, the ASCR can be decomposed in the annual expected-SCRs (ESCR), which represent the expected regulatory capital for the next year, not only at time zero but also at the futures payment dates. Hence, we build a term structure of expected annual SCRs.