Interaction between asset liability management and risk theory

In order to apply the ALM model of Janssen (see also References 2 and 3), to insurance companies, we study an extension of the model in which the asset fund A takes into account fixed-income securities. Therefore, we model the rates of return of the portfolio by a Vasicek process. The liability process B is defined by a geometric Brownian motion with drift which may be correlated with the asset process. In this generalized Janssen model, we concentrate on the relations between the asset process Aand the liability process B in order to point out some management principles. More exactly, we study the probability that the assets and liabilities of a company have no good matching and we propose some indicators of the mismatching. Therefore, we look at the process a = (at, t ⩾ 0) defined by at = ln(At/Bt) and at the first mismatching time τ = inf{t:0 ⩽ t ⩽ T, a(t) ⩽ 0}. The determination of the probability of mismatching leads to the calculation of crossing probabilities P[τ < T]. Only in special cases, explicit results are obtained and we turn for the general cases to the approximations proposed by Durbin and Sacerdote and Tomasetti. A degree of mismatching follows from option theory. These results are important as they are useful to determine ALM-strategies for insurance companies.