A discrete martingale model of pension fund guarantees in

In this paper we present a solution to the problem of pricing guarantees of minimum returns on pension fund contributions. These guarantees exist by law in Colombia and cover all pension fund contributions made to the country's private pension fund administration companies (AFPs). As of September 1997, the funds were collecting contributions of 2.5 million affiliates with an accumulated capital of 1.8 billion dollars starting from zero in 1994. Two types of guarantees exist: on obligatory contributions and on voluntary contributions. The solutions are based on a discrete martingale approach . We show that both guarantees are equivalent to an ''option to exchange.'' However, in the case of voluntary contributions a ceiling on the payoff s must be added. Using a discrete martingale framework and a binomial solution we develop all aspects of the model that are necessary for its practical application in the context of the pension fund guarantees. Binomial formulas are obtained for both forms of guarantees. Besides solving the problem of pricing the guarantees offered by insurance fund, the contributions in terms of options theory of this paper are: i) we adapt the binomila model of options to exchange to relate the relevant parameters of the same to a continuous-time lognormal process; ii) we provide a binomial solution to the problem of an option to exchange with a ceiling. We then investigate the incentives that the current fixed-price system introduces and propose possible systems of incentives that can be used to encourage higher-risk investment by the AFP's and a shift of the fund's portfolio to risky equity and debt. Given the country's effort to encourage capital markets development and the financing of the real sector via private financial markets, this strategy appears to be desirable from the social and economic point of view.