The capital-on-capital cost in solvency II risk margin

This work contributes to the literature on time consistent valuation of insurance liabilities and to the ongoing discussion on revisions of risk margin (RM) 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 liabilities maturity. That is, the capital-on-capital cost is the RM component dedicated to cover the risk of future RMs and not to cover variations of the best estimate of liabilities cash-flows. We mathematically formalize the capital-on-capital cost as the difference between two alternative time consistent liabilities valuation formula. The first is obtained through backward iteration of the one-period market-consistent valuation operator, by iterating the solvency capital requirement (SCR) risk measure. We propose a second alternative valuation formula for liabilities, based on a new time consistent dynamic formulation of the SCR risk measure, called additive-SCR (ASCR). The ASCR represents the expected total capital requirement from current time to liabilities maturity. We prove that the second valuation formula, based on ASCR, is time consistent, unless it is not based on iteration of the one-period SCR risk measure. Finally, we apply the proposed approach to a portfolio of long term equity linked life-insurance contracts. In particular, we estimate the capital-on-capital cost by calculating the difference between the two valuation formulas. Numerical results show that for liabilities with long maturities the capital-on-capital cost is a non negligible component of the RM.