The Capital-on-Capital Cost in Solvency II Risk Margin

In recent times, practitioners have engaged in discussions about potential revisions to the calculation of the Solvency II risk margin (RM). This issue has been highlighted in reports by the British Treasury Committee BTC (October 2017) and the Actuarial Association of Europe AAE (December 2019). This paper aims to contribute to the literature on time consistent valuation of insurance liabilities and to the ongoing discussion on revisions of risk margin calculation, by formally defining the concept of capital-on-capital cost. We describe the capital-on-capital as the amount required to cover unexpected variations in future regulatory capitals from the current time to the liabilities maturity. Then, the capital-on-capital cost is the RM component dedicated to covering the risk of future RMs and not to covering variations in the best estimate of liabilities cash-flows. We mathematically formalize the capital-on-capital cost as the difference between two alternative time consistent valuation formulas for technical provisions (TPs). The first valuation formula for TPs is obtained through backward iteration of the one-period market-consistent valuation operator, and is based on the iteration of the Solvency Capital Requirement (SCR) risk measure. We propose a second alternative valuation formula for TPs, based on a new time consistent dynamic formulation of the SCR risk measure, which we call additive-SCR (ASCR). The ASCR represents the expected total capital requirement for the period from the current time to the liabilities maturity and can be decomposed into annual expected-SCRs (ESCRs). The ESCR is the best estimate of the capital required for a specific year of the product and is particularly intuitive and economically meaningful. We prove that the second valuation formula, based on ESCRs, is time consistent unless it is not based on iteration of the SCR risk measure. Finally, we present an illustrative example of the proposed approach for a portfolios of long term equity linked life-insurance contracts. We compare the two valuation formulas for liabilities and we estimate the capital-on-capital cost by calculating the difference between them. We obtain that for long-term liabilities with a maturity of 30 years, the capital-on-capital cost can reach 25% of the RM value obtained with the backward iteration procedure