Time Horizon and Solvency Capital within a Brownian Framework Partially Modulated by a Continuous-Time Markov Chain

A quite important issue in life insurance or concerning pension benefits is the determination of the “right” or “fair” minimum level of capital needed by the insurer to be solvent. Especially under the future so called “Solvency II framework” for insurance undertakings in the European Union which is designed to be more risk-sensitive. The particularity stands in the long-term characteristic of these products, and it may be important to take into account the time horizon effect of these products in the computation of the solvency capital, which is the purpose of this paper. We consider here the case of a pension fund. We take into account two market risks: the interest rate risk (with an affine term structure model) and the equity risk (with a Markovmodulated geometric Brownian motion, which incorporates a kind of economic risk with a continuous-time Markov chain which modulates the parameters of the geometric Brownian motion). Then, a portfolio is built in this market composed of three assets (a bank account, a zero-coupon bond and a stock) which satisfies the constant portfolio allocation hypothesis. We neglect the mortality and underwriting risks by assuming that the pension fund is fully hedged against these risks. The pension fund offers a fixed guaranteed rate on a certain time horizon. The probability of default is first considered, and then the issue of the determination of the solvency capital is addressed by means of the value at risk and the tail value at risk measures. A comparison is made with the Solvency II framework