Abstract

Citation

Researchers should cite this work as follows:

• Steffensen, M. (2023). 2007-Mogen_Steffensen-Differential Equations in Finance and Life Insurance. SIMIODE, QUBES Educational Resources. doi:10.25334/SRM5-FZ08

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Article Context

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Steffensen, Mogen. 2007. Differential Equations in Finance and Life Insurance. In: Jensen, B.S. and Palokangas, T. (2007) Stochastic Economic Dynamics. CBS press.

From the opening of the paper,

“The mathematics of finance and the mathematics of life insurance were always intersecting. Life insurance contracts specify an exchange of streams of payments between the insurance company and the contract holder. These payment streams may cover the life time of the contract holder. Therefore, time valuation of money is crucial for any measurement of payments due in the past as well as in the future. Life insurance companies never put their money under the pillow, and accumulation and distribution of capital gains were always part of the insurance business. With respect to the future, appropriate discounting of contractual obligations qualities the estimates of liabilities.

“Financial contracts specify an exchange of streams of payments as well. However, while the life insurance payment stream is partly linked to the state of health of the insured, the financial payment stream is linked to the 'state of health' of an enterprise. That could be the stream of dividends distributed to the owners of the enterprise or the stream of claims contingent on the price of the enterprise paid to the holder of a so-called derivative. The discipline of personal finance is particularly closely linked to life insurance. Decisions on e.g. consumption, investment, retirement, and insurance coverage belong to some of the most substantial life time financial decisions of an individual.

“Valuation of payment streams is probably the most important discipline in the intersection between finance and life insurance. Various valuation dogmas are in play here. The principle of no arbitrage and the market efficiency assumption are taken as given in the majority of modern academic approaches to valuation of financial contracts. Life insurance contract valuation typically relies on independence, or at least asymptotic independence, between insured lives. Then the law of large numbers ensures that reasonable estimates can be found if the portfolio of insurance contracts is sufficiently large. Both dogmas reduce the valuation problem to being primarily a matter of calculation of conditional expected values.

“Conditional expected values can be approached by several different techniques. E.g. Monte Carlo simulation exploits that conditional expected values can be approximated by empirical means. Sometimes, however, one can go at least part of the way by explicit calculations. E.g. if a series of auxiliary models with explicit expected values converges towards the real model in such a way that the series of explicit expected values converges to the desired quantity. A different route can be taken when the underlying stochastic system is Markovian, i.e. if given the present state, the future is independent of the past. Then solutions to certain systems of deterministic differential equations can often be proved to characterize the conditional expected values. This is the route taken to various valuation problems and optimization problems in finance and life insurance in this exposition. Here, we just state the differential equations and do not discuss possible numerical solutions to these, though.”

Keywords: differential equation, model, finance, life insurance, insurance, Thiele’s equation, Black-Scholes equation, disability, mortality, rick, partial differential equation