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 "xed-income securities. Therefore, we model the rates of return of the portfolio by a Vasicek process. The liability process B is de"ned 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 A and 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"(a R , t*0) de"ned by a R "ln(A R /B R ) and at the "rst mismatching time "inf+t : 0)t)ยน, a(t))0,. The determination of the probability of mismatching leads to the calculation of crossing probabilities P[ (ยน]. 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.