Financial Risk in Insurance

In this paper we shall illustrate how diffusion models can be used to extract the term structure of interest rates from observed prices and to determine theoric prices and risk measures of term-structure derivatives. We shall assume throughout that the risk of default can be ignored.

The issue of risk-based capital for financial institutions is of considerable current interest. We discuss how the option pricing paradigm provides a theoretical framework for the analysis of this topic. The modern approach to option pricing which was inspired by the seminal work of Black and Scholes has revolutionized our approach to both the theory and practice of many aspects of investment finance and corporate finance.

Actuarial Traditions. The management of interest rate risk is a problem with deep roots in the actuarial literature. The results by Redington (1952) and by Haynes and Kirton (1952) may be considered path-breaking on a problem that, even then, “for many years must have intrigued so many actuaries”¹. The purpose of their work is to analyse the financial structure of a life office and, in particular, the relationship between the assets and liabilities of a life assurance fund. The specific problem is how to determine the allocation of assets to make them, as far as possible, equally as vulnerable as the liabilities to those influences (typically the effects of fluctuations in the market rate of interest) which affect both. Redington adopts the word “immunization, to signify the investment of the assets in such a way that the existing business is immune to a general change in the rate of interest”². Haynes and Kirton use the word “insulation” in a similar way. It is remarkable to consider how closely both the authors agreed in their fundamental conclusions. But it is also relevant, for critical purpose and for the sake of theory, to remark the difference in the approach to the problem: Haynes and Kirton paper “dealt primarily with matching, valuation being a by-product”, while Redington’s paper “dealt primarily with valuation, matching being a by-product”³. The first approach led to the development of strategies for cash-flow matching⁴. The second, which the present paper is concerned with, recognizes the central role of bond pricing theory and of models of the term structure of interest rates, which have undergone a strong development in the last twenty years.

Financial intermediaries exist and were already studied well before the second half of our twentieth century. Yet in the last fifty years many new and more precise insights on financial institutions have been derived exploiting results obtained by researchers of the scientific area defined (more or less precisely) as modern-quantitative finance.