PDF Ebook From Actuarial to Financial Valuation Principles
In recent years, there has been an ever increasing trend to bring finance and insurance closer together. The motivation for the present paper could almost be quoted from Hans Bühlmann (1997) who in this context also coined the term “actuary of the third kind” in Bühlmann (1987): “ ... finance and insurance mathematics should be presented to today’s students as one discipline”. In accordance with this, our starting point is the observation that the valuation of random amounts is an important topic in both actuarial and financial mathematics and has been studied extensively in both fields. Almost any textbook includes a treatment under headings like premium principles or derivative pricing. We attempt to bring these approaches together by embedding an actuarial valuation principle in a financial environment.
The basic idea is simple. We begin with an a priori valuation rule which assigns a number (“premium”, “price”) to any random payoff from a suitable class. Typically, this rule is given or motivated by an actuarial premium principle. But the payoffs we consider do not exist in a vacuum; they are surrounded by a financial environment described by the outcomes of trades available to participants in a financial market. Such trades can be used to reduce the risk one has contracted by the sale or purchase of some random amount like an insurance claim or a financial obligation. To value a given payoff in this environment, we compare two procedures. One is to ignore the payoff completely and simply trade on the financial market in a subjectively optimal way. More precisely, one tries to obtain via trading from a given initial capital a final outcome with maximal value, where the value is computed according to the given a priori rule. Alternatively, one can first sell the payoff under consideration to increase one’s initial capital and then look for a trade whose resulting net final outcome (trading outcome minus payoff) has again maximal value. The selling price for the payoff is then defined implicitly by equating these two maximal values; it thus compensates for the payoff since one becomes indifferent between optimal trading alone and the combination of selling and optimal trading inclusive of the payoff. The resulting a posteriori valuation is called the financial transform of the a priori valuation rule.
Of course, this abstract program is too general to be feasible. We therefore specialize the financial environment to a frictionless market modelled by a linear subspace G of L2 with a riskless asset B. We consider two specific examples of actuarial valuation principles and explicitly determine their financial transforms. As a whole, the paper is a joint venture between finance and insurance. Finance makes explicit the transformation mechanism and insurance provides the input on which the mechanism can operate. In particular, an actuarial justification for the choice of one particular a priori valuation could via this approach lead to a foundation for pricing in an incomplete financial market.
An alternative economic approach to price insurance contracts has recently been suggested by Kliger/Levikson (1998). Their idea is to determine the premium for an insurance product by maximizing some form of expected profit that involves in particular a specific type of insolvency costs. However, their method does not include any genuine financial component and thus is rather different from what we propose here.
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PDF Ebook From Actuarial to Financial Valuation Principles
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