A game-theoretic approach to non-life insurance market cycles

In this paper, we develop a noncooperative game to model a non-life insurance market. Our first goal will be to analyze the effects of competition between insurers through different indicators: the solvency level, the market share, the underwriting results. Secondly, we will seek to further understand the genesis of insurance market cycles. Insurance market cycles have troubled actuaries and academics for decades: this game-theory focus will allow us to shed a different light on the subject. A game-theoretic approach to non-life insurance market cycles By Christophe Dutang Institut du Risque et de l’Assurance, Université du Maine, Le Mans Winner of the French Actuarial Prize, 2012 Acknowledgment: This paper is based on a on-going work with Hansjoerg Albrecher from Université de Lausanne and Stéphane Loisel from Université Lyon 1. September 2014 N°29 Texts appearing in SCOR Papers are the responsibility of their authors alone. In publishing such articles, SCOR takes no position on the opinions expressed by the authors in their texts and disclaims all responsibility for any opinions, incorrect information or legal errors found therein. Fair Valuation of risks Under Solvency II, the Market Value Margin (MVM) is meant to bring technical provisions to a fair value, and is to be computed using the Cost of Capital approach. In the background lies the Market Consistent economic balance sheet which reflects what Solvency II seeks to achieve: a fair valuation of risks. Limiting ourselves to the reserve risk only – as will be done in the rest of this note – the following graph shows that the Capital should be sufficient to restore the balance sheet to a fair value of liabilities after a 1 in 200 event: 2 For Solvency II, the Solvency Capital Requirement (SCR) is meant to cover one year of deterioration, meaning that only “shocks” applied to the following year are considered. The graph depicts, on the liability side of the economic balance sheet, how the capital funded at time t=0 is adequate to restore the balance sheet to a fair value of liabilities at the end of a distressed first year, where both the Best Estimate of Liabilities (BEL) and the MVM are subject to a distressed scenario. Cost of Capital approach The CoC approach takes the perspective that sufficient capital is needed to be able to run-off the business. Here, the risk margin is estimated by the present value of the expected cost of current and future SCRs for non-hedgeable risks to support the complete run-off of all liabilities. Schematically, the MVM calculation can be carried out in 4 steps: First, project the expected SCR until all liabilities run-off. This puts into the equations the fact that an undertaking taking over the portfolio has to put up future regulatory capital SCR(1), SCR(2), ... , SCR(n‐1) until the portfolio has run-off completely at time t=n; Second, multiply all current and future SCR by the Cost of Capital rate (c or CoC). This captures the fact that the insurer selling the portfolio has to compensate the insurer taking over the portfolio for immobilizing future capital requirements; Third, discount everything to time 0; The sum then gives the CoC risk margin. SCOR Paper n°18 Calculations under the SII CoC approach 1. Background