Academic journal article Journal of Legal Economics

# A Perpetuity, Life Annuity, and Life Insurance Related to a Decomposition of 1

Academic journal article Journal of Legal Economics

# A Perpetuity, Life Annuity, and Life Insurance Related to a Decomposition of 1

## Article excerpt

(ProQuest: ... denotes formulae omitted.)

I. Introduction

This note is about the historical development of the expression (1 iax), where i is an annual interest rate and ax is a life annuity for apersonagex. This expression has meaning in its own right, but it also equals ð1 þ iÞAx, where Ax is the expected present value of insurance of 1 unit paid at the end of the year of death.1 The equation 1 iax ¼ (1 þ i)Ax provides a foundation for taxation in life estate and inheritance matters in which iax captures income from a life estate and 1 iax represents the remainder (i.e., the value of 1 in excess of the life estate). When iax is placed on the right side, the equation 1 ¼ iax þ (1 þ i)Ax states that 1 consists of life annuity and life insurance components.

Three problems play central roles in the development of the equation 1iax¼(1þi)Ax; one problem is due to Abraham de Moivre (1667-1754), another to Thomas Simpson (1710-1761), and the third to Richard Price (1723-1791). Sections II and III discuss these problems after brief sketches of de Moivre, Simpson, and Price. Section IV is a conclusion.

Abraham de Moivre was a Huguenot who fled France for London in 1687 shortly after Louis XIV revoked the Edict of Nantes. Despite being a prolific mathematician and probabilist of the first order, he earned much of his living working out of coffee houses where he solved business, probability, and actuarial science problems for pay. He counted luminaries such as Isaac Newton (1643-1727) and Edmond Halley (1656-1742) as friends. His Treatise on Annuities (1725) and Doctrine of Chances (1738) were seminal works in actuarial science and probability theory. The earliest development of the normal probability distribution is due to de Moivre in 1733 when he created a continuous approximation of the symmetric binomial distribution.2

Thomas Simpson was a largely self-taught mathematician. His name is attached to a rule for the approximation of definite integrals, although the technique was known before his work. His The Nature and Laws of Chance (1740) contained new results for approximating the non-symmetric binomial probability distribution. He was a successful author of books on differential calculus and actuarial science. Simpson and de Moivre had similar interests and were embroiled in an unfortunate priority and plagiarism dispute primarily over parts of Simpson's The Doctrine of Annuities and Reversions, Deduced from General and Evident Principles (1742) and de Moivre's The Doctrine of Chances (1738). See Hald (2003), Stigler (1986), and Haberman and Sibbett (1995) for Simpson's work in probability theory and actuarial science.

Richard Price was a dissenting Unitarian minister. His principal congregation was a Presbyterian ministry in Newington Green, part of modern day London. He was a civil libertarian, republican, and open supporter of American independence. He wrote in probability theory, demography, actuarial science, and moral philosophy. When Thomas Bayes (1701-1761), also a Presbyterian minister, died, his relatives asked Price to review Bayes' unpublished mathematical work. After writing an introduction and an appendix, Price read a paper by Bayes to the Royal Society in 1763. The paper contained Bayes' now famous treatment of inverse probability and was published in two parts in Philosophical Transactions of the Royal Society of London in 1764 and 1765.3 See Hald (1998), Stigler (1986), and Haberman and Sibbett (1995) for discussions of Price's work.

II. Payments Contingent on Death

In 1742, Thomas Simpson published a small book on annuities and reversions. The book contains problems, solutions, and rules for problem solving. Here is one of Simpson's problems and his solution; it deals with a contingent payment made upon the first death among three people.

PROBLEM XXI

Supposing any given number of lives P, Q, R and that A, or his Heirs, are to receive the sum S upon the first vacancy of any of their lives: To find the value of A's expectation in present money. …

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