History
The instrument's evolution has been long and continues as part of actuarial science. Medieval German and Dutch cities and monasteries raised money by the sale of life annuities, and it was recognized that pricing them was difficult. The early practice for selling this instrument did not consider the age of the nominee, thereby raising interesting concerns. These concerns got the attention of several prominent mathematicians over the years, such as Huygens, Bernoulli, de Moivre and others: even Gauss and Laplace had an interest in matters pertaining to this instrument. It seems that Johan de Witt was the first writer to compute the value of a life annuity as the sum of expected discounted future payments, while Halley used the first mortality table drawn from experience for that calculation. Meanwhile, the Paris Hôtel-Dieu offered some fairly priced annuities that roughly fit the Deparcieux table discounted at 5%. Here is a quick comparison table of early life annuity prices:
| Head age (x) | Value of a unit annuity | |||||
| Ulpian
ca. 200 AD |
de Witt
1671 |
Hôtel Dieu
ca. 1680 |
Halley
1693 |
Deparcieux
1746 |
||
| 1 | 30 | 16 | n/a | 10,28 | n/a | |
| 10 | 30 | 15,19 | n/a | 13,44 | 16,25 | |
| 20 | 28 | 13,83 | 20 | 12,78 | 15,58 | |
| 30 | 22 | 12,22 | 20 | 11,72 | 14,84 | |
| 40 | 19 | 10,39 | 15 | 10,57 | 13,62 | |
| 50 | 9 | 8,68 | 12 | 9,21 | 11,58 | |
| 60 | 5 | 6,70 | 10 | 7,60 | 9,24 | |
| 70 | 5 | 3,77 | 8 | 5,32 | 6,36 | |
| 80 | 5 | 0 | 8 | 3,05 | 3,86 | |
| 90 | 5 | 0 | 6 | 1,74 | 1,58 | |
| 95 | 5 | 0 | n/a | 1,02 | 0 | |
| Values are approximated | ||||||
Continuing practice is an everyday occurrence with well-known theory founded on robust mathematics, as witnessed by the hundreds of millions worldwide who receive regular remuneration via pension or the like. The modern approach to resolving the difficult problems related to a larger scope for this instrument applies many advanced mathematical approaches, such as stochastic methods, game theory, and other tools of financial mathematics.
Read more about this topic: Life Annuity
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