## JOHANN DE WITT AND LIFE ANNUITIES

March 23, 2011 at 1:40 am | Posted in Books, Financial, History, Research, Science & Technology | Leave a comment**Christiaan Huygens and Johann de Witt, two seventeenth century Dutchmen, were respectively instrumental in the emergence of modern physics and modern finance.**

**Johan de Witt**

**Johan de Witt****, heer van Zuid– en Noord-Linschoten, Snelrewaard, Hekendorp and IJsselveere ^{[1]} (Dordrecht, 24 September 1625 – The Hague, 20 August 1672) was a key figure in Dutch politics at a time when the Republic of the United Provinces was the dominant power in Europe, dominating trade routes and thus the wealthiest nation in the world. In the mid 17th century he controlled the Netherlands political system in close cooperation with his uncle Cornelis de Graeff.^{[2]}**

**Mathematician**

**Besides being a statesman Johan de Witt, also was an accomplished mathematician. In 1659 he wrote “Elementa Curvarum Linearum” as an appendix to his translation of René Descartes‘ “La Géométrie”.**

**In 1671 his “Waardije van Lyf-renten naer Proportie van Los-renten” was published (‘The Worth of Life Annuities Compared to Redemption Bonds’). This work combined the interests of the statesman and the mathematician. Ever since the Middle Ages, a Life Annuity was a way to “buy” someone a regular income from a reliable source. The state, for instance, could provide a widow with a regular income until her death, in exchange for a ‘lump sum’ up front. There were also Redemption Bonds that were more like a regular state loan. De Witt showed – by using probability mathematics – that for the same amount of money a bond of 4% would result in the same profit as a Life Annuity of 6% (1 in 17). But the ‘Staten’ at the time were paying over 7% (1 in 14).**

**The publication about Life Annuities is seen as the first mathematical approach of chance and probability.**

**The drop in income for the widows contributed no doubt to the “bad press” for the brothers De Witt. Significantly, after the violent deaths of the brothers the ‘Staten’ issued new Life Annuities in 1673 for the old rate of 1 in 14.**

**In 1671 De Witt conceived of a life annuity as a weighted average of annuities certain where the weights were mortality probabilities (that sum to one), thereby producing the expected value of the present value of a life annuity. Edmond Halley’s (of comet fame) representation of the life annuity dates to 1693, when he re-expressed a life annuity as the discounted value of each annual payment multiplied by the probability of surviving long enough to receive the payment and summed until there are no survivors. De Witt’s approach was especially insightful and ahead of its time. **

**In modern terminology, De Witt treats a life annuity as a random variable and its expected value is what we call the value of a life annuity. Also in modern terminology, De Witt’s approach allows one to readily understand other properties of this random variable such as its standard deviation, skewness, kurtosis, or any other characteristic of interest.**

**In addition, in his Elementa curvarum linearum, De Witt derived the basic properties of quadratic forms, an important step in the field of linear algebra.**

**References**

**1. **** www.herenvanholland.nl Johan de Witt at Heren van Holland (nl)**

**2. **** Andries Bickers Biographie at the DBNL**

**3. **** www.herenvanholland.nl Anna de Witt at Heren van Holland (nl)**

**4. **** Rowen, Herbert H. John de Witt, Statesman of the True Freedom (Camebridge University Press 1986, New edition 2002), page 220**

**5. **** Troost, 43**

**6. **** Kok, J. (1794) Vaderlandsch woordenboek; oorspronkelijk verzameld door Jacobus Kok. Deel 32, p. 352; Veeghens, D. (1884) Historische studien: Uitg. door J.D. Veegens. Eerste Deel, p. 48; the first name of Verhoeff was Hendrik according to Fruin, R. (1901) Robert Fruin’s verspreide geschriften, p. 374, fn. 2**

**Literature**

**Herbert H. Rowen,***John de Witt, grand pensionary of Holland, 1625-1672*. Princeton, N.J.: Princeton University Press, 1978, which is summarized in

**Herbert H. Rowen (1985).***Johan de Witt. Staatsman van de ‘Ware Vrijheid’*. Leiden.

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