# A Treatise on the Valuation of Annuities and Assurances on Lives and Survivors

• Joshua Milne
Part of the Biomathematics book series (BIOMATHEMATICS, volume 6)

## Abstract

Milne’s notation and principal equations translate into modern form as:
$$^n \ell = l_n = \,\text{survivors}\,\text{to}\,\text{exact}\,\text{age}\,n\,\text{from}\,\text{among}\,l_0 \,\text{births}\,\text{at}\,\text{obseved}\,\text{age - specific}\,\text{death}\,\text{rates}$$
$$S^n \ell = \sum\limits_{x = n}^\omega {l_x }$$
$$\mathop \text{L}\limits^n = P_n = \text{observed}\,\text{population}\,\text{at}\,\text{ages}\,n\,\text{to}\,n + 1$$
$$\mathop D\limits^n = D_\text{n} = \text{annual}\,\text{deaths}\,\text{to}\,\text{persons}\,\text{ages}\,n\,\text{to}\,n + 1$$
$$\delta \text{ = }d_n \text{ = deaths}\,\text{to}\,\text{persons}\,\text{ages}\,n\,\text{to}\,n + 1\,\text{in}\,\text{the}\,\text{life}\,\text{table}$$
$$\frac{{S^n \ell - \mathop {\frac{1}{2}}\nolimits^n \ell }}{{^n \ell }} = \frac{{\sum\limits_{x = n}^\omega {l_x - \frac{1}{2}l_n } }}{{l_n }}\dot = \mathop e\limits^ \circ _n = life\, expectancy\,at\,age\,n$$
$$\frac{{n\ell \mathop D\limits^n }}{{\mathop L\limits^n + \frac{1}{2}\mathop D\limits^n }} = l_n \left( {\frac{{D_n }}{{P_n + \frac{1}{2}D_n }}} \right)\dot = d_n .$$

## Keywords

Geometrical Progression Annual Death Annual Average Number Annual Birth Absolute Fecundity
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Joshua Milne (1815) London. Excerps are from pages vi–xii, 89–91, 97–100, 487–489, 582.Google Scholar