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

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


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


Geometrical Progression Annual Death Annual Average Number Annual Birth Absolute Fecundity 
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  1. Joshua Milne (1815) London. Excerps are from pages vi–xii, 89–91, 97–100, 487–489, 582.Google Scholar

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© Springer-Verlag Berlin · Heidelberg 1977

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  • Joshua Milne

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