de Moivre's law

For the identity connecting complex numbers and trigonometric functions, see de Moivre's formula.

De Moivre's Law is a survival model applied in actuarial science, named for Abraham de Moivre.^[1][2][3] It is a simple law of mortality based on a linear survival function.

Definition

De Moivre's law has a single parameter ω called the ultimate age. Under de Moivre's law, a newborn has probability of surviving at least x years given by the survival function^[4]

$S(x) = 1 - \frac{x}{\omega}, \qquad 0 \leq x < \omega.$

Under this model, the conditional probability that a life aged x years survives at least t years is^{[clarification needed]}

${}_t p_x = \frac{S(x+t)}{S(x)} = \frac{\omega-(x+t)}{\omega-x}, \qquad 0 \leq t < \omega-x,$

and the future lifetime random variable T(x) therefore follows a uniform distribution on $(0, \, \omega-x)$. The force of mortality (hazard rate or failure rate) for a life aged x is

$\mu(x+t) = \frac{1}{\omega - (x+t)}, \qquad 0 \leq t < \omega-x,$

which has the property of increasing failure rate (IFR) with respect to age that is usually assumed for humans, or anything subject to aging.

De Moivre's law is applied as a simple analytical law of mortality and the linear assumption is also applied as a model for interpolation for discrete survival models such as life tables.

Linear assumption for fractional years

When applied for interpolation, the linear assumption is called uniform distribution of death (UDD) assumption in fractional years and it is equivalent to linear interpolation. If $\ell_x$ denotes the number of survivors at exact age x years out of an initial cohort of $\ell_0$ lives, the UDD assumption for fractional years is that

$\ell_{x+t} = (1-t) \ell_x + t \ell_{x+1}, \qquad 0<t<1,$

or equivalently, that

$S(x+t) = (1-t) S(x) + t S(x+1), \qquad 0<t<1.$

Under the UDD assumption, the probability _tq_x that a life aged x fails within (0,t), is tq_x, and $\mu(x+t) = \frac{q_x}{1-tq_x}$, for $\, 0<t<1$.

Notes

1. ^ Abraham de Moivre (1725) Annuities upon Lives. The second edition of Annuities upon Lives was published in 1743.
2. ^ Abraham de Moivre (1752) A Treatise of Annuities on Lives.
3. ^ Geoffrey Poitras (2006). "Life annuity valuation: from de Witt and Halley to de Moivre and Simpson". In Geoffrey Poitras. Pioneers of Financial Economics: Volume I, Contributions Prior to Irving Fisher. ISBN 978-1-84542-381-0. 
4. ^ Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1997). Actuarial Mathematics (Second Edition), Schaumburg, Illinois, Society of Actuaries.