Borch's Theorem

Abstract

Borch's theorem, the most important application of utility theory and game theory in insurance, characterizes the set of Pareto optimal treaties in an exchange of risk among policyholders, insurers, or reinsurers. Concepts from game theory, such as the core or a value for bargaining games, narrow down the set of acceptable treaties.

Pareto Optimality and Equilibrium in an Insurance Market

“…A characterization of equilibrium in the reinsurance market was developed in Buhlmann (1980Buhlmann ( , 1984 and later in Aase (1993Aase ( , 2002 where risk allocation problems in financial markets were also studied. Actually, a particular case of the model suggested in the paper, where risk is allocated between insurer and the only insured (n = 1), has been studied in the frame of the Borch's model of reinsurance market by several authors, e.g., Lemaire (1990), Aase (1993Aase ( , 2002. The presented work is expected to be consistent with the known results for the case n = 1.…”

“…This concept implies that contracts made in the market are Pareto-optimal and, which is of importance, premiums of the contracts are determined by the market conditions as a whole, not only by covered risks. Borch's theorem on Pareto-optimal risk exchanges in the model of a reinsurance market was extended to a constrained case in Gerber (1978), an overview of applications of the Borch's theorem can be found in Lemaire (1990). A characterization of equilibrium in the reinsurance market was developed in Buhlmann (1980Buhlmann ( , 1984 and later in Aase (1993Aase ( , 2002 where risk allocation problems in financial markets were also studied.…”

Founding Works of Insurance Economics

“…Táto teoréma hovorí, že dosiahnutie stavu Paretovho optima na trhu zaistenia si vyžaduje, aby všetci zaisťovatelia držali pomerný podiel "trhového portfólia" poistného rizika. Charakterizuje teda súbor Pareto optimálnych dohôd pre prenos rizika medzi klientmi, poisťovateľmi a zaisťovateľmi (Lemaire, 2004) a ukazuje, ako môže byť mechanizmus transferu rizika realizovaný v praxi. Borchova teoréma transferu rizík pri dosiahnutí Paretovho optima naznačuje, že rozdeľovanie rizika je založené na rizikovej tolerancii subjektu (Wilson, 1968).…”

Pareto Optimality and Equilibrium in an Insurance Market

“…This concept implies that contracts made in the market are Pareto-optimal and, which is of importance, premiums of the contracts are determined by the market conditions as a whole, not only by covered risks. Borch's theorem on Pareto-optimal risk exchanges in the model of a reinsurance market was extended to a constrained case in Gerber (1978), an overview of applications of the Borch's theorem can be found in Lemaire (1990). A characterization of equilibrium in the reinsurance market was developed in Buhlmann (1980,1984) and later in Aase (1993Aase ( , 2002 where risk allocation problems in financial markets were also studied.…”