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Since its invention in Italy in the fourteenth century, marine insurance has provided merchants with capital protection in times of crisis, thus oiling the gears of trade and commerce. With a focus on customs, laws, and organisational structures, this book reveals the Italian origins of marine insurance, and tracks the spread of underwriting practices and institutions in Europe and America through the early modern era. With contributions from eleven leading researchers from seven countries, the book examines key institutional developments in the history of marine insurance. The authors discuss its invention in Italy, and its evolution from private to corporate structures, assessing the causes and impacts of various state interventions. Amsterdam and Antwerp are analysed as one-time key centres of underwriting, as is the emergence and maturity of marine insurance in London. The book evaluates an experiment in corporate underwriting in Cadiz, and the development of insurance institutions in the United States, before applying the metrics of underwriting to discuss commerce raiding in the Atlantic up to the nineteenth century.

"Halley's Comet has been prominently displayed in many newspapers during the last few months. For the first time in 76 years it appeared this winter, the nocturnal sky. This is an appropriate occasion to clearly visible against point out the fact that Sir Edmund Halley also constructed the world's first life table in 1693, thus creating the scientific foundation of life insurance. Halley's life table and its successors were viewed as deterministic laws, i. e. the number of deaths in any given group and year was considered to be a well defined number that could be calculated by means of a life table. However, in reality this number is random. Thus any mathematical treatment of life insurance will have to rely more and more on probability theory. By sponsoring this monograph the Swiss Association of Actuaries wishes to support the ""modern"" probabilistic view of life contingencies. We are fortu­ nate that Professor Gerber, an internationally renowned expert, has assumed the task of writing the monograph. We thank the Springer-Verlag and hope that this monograph will be the first in a successful series of actuarial texts. Zurich, March 1986 Hans Biihlmann President Swiss Association of Actuaries Preface Two major developments have influenced the environment of actuarial math­ ematics. One is the arrival of powerful and affordable computers; the once important problem of numerical calculation has become almost trivial in many instances."

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Financial and insurance calculations become more and more frequent and helpful for many users not only in their profession life but sometimes even in their personal life. Therefore a survey of formulas of ?nancial and insurance mathematics that can be applied to such calculations seems to be a suitable aid. In some cases one should use instead of the term formula more suitable terms of the type method, p- cedure or algorithm since the corresponding calculations cannot be simply summed up to a single expression, and a verbal description without introducing complicated symbols is more appropriate. The survey has the following ambitions: • The formulas should be applicable in practice: it has motivated their choice for this survey ?rst and foremost. On the other hand it is obvious that by time one puts to use in practice seemingly very abstract formulas of higher mathematics, e.g. when pricing ?nancial derivatives, evaluating ?nancial risks, applying accou- ing principles based on fair values, choosing alternative risk transfers ARL in insurance, and the like. • The formulas should be error-free (though such a goal is not achievable in full) since in the ?nancial and insurance framework one publishes sometimes in a h- tic way various untried formulas and methods that may be incorrect. Of course, the formulas are introduced here without proofs because their derivation is not the task of this survey.

I am pleased to participate in this Summer School and look forward to sharing some ideas with you over the next few days. At the outset I would like to describe the approach I will take in 1 presenting the material. I aim to present the material in a non­ rigorous way and hopefully in an intuitive manner. At the same time I will draw attention to some of the major technical problems. It is pitched at someone who is unfamiliar with the area. The results presented here are unfamiliar to actuaries and insurance mathematicians although they are well known in some other fields. During the next few minutes I will make some preliminary comments. The purpose of these comments is to place the lectures in perspective and motivate the upcoming material. After this I will outline briefly the topics to be covered during the rest of this lecture and in the lectures that will follow. One of the central themes of these lectures is RISK-SHARING. Risk-sharing is a common response to uncertainty. Such uncertainty can arise from natural phenomena or social causes. One particular form of risk-sharing is the insurance mechanism. I will be dealing with models which have a natural application in the insurance area but they have been applied in other areas as well. In fact some of the paradigms to be discussed have the capacity to provide a unified treatment of problems in diverse fields.

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