Model of Survival
- Consider a future lifetime of a newborn, \(T \in [0, \omega]\), and a future lifetime of a life aged \(x,\quad T_x\in[0, \omega - x]\)
\[ F_x(t) = P[T_x \leq t] = P[T\leq x+t|T>x] = \frac{S(x)-S(x+t)}{S(x)} \]
\[ S_x(t) = P[T_x > t] = 1-F_x(t) = P[T > x+t|T>x] = \frac{S(x+t)}{S(x)} \]
\[ f_x(t) = \frac{dF_x(t)}{dt} \]
Actuarial Notation
Actuarial Notation
Force of Mortality (at age x)
Expectation of Life
Complete Expectation
Curate Expectation
Life Table
Select Life Table
Intial and Central Rates of Mortality
- We know \(q_x \leq m_x\) as \(l_x \geq \int^{x+1}_x l_zdz\), as the average live cohort over \(x,x+1\) will be smaller or equal to the live cohort at the start of the period as the cohort is non increasing.
Methods for Non-integer Ages
- Only integer values of \(l_x\) are typically avaliable within a life table, therefore we need a method to get values for \(l_{x+s}\), where \(0\leq s\leq1\)
Uniform Distribution of Deaths
Constant Force of Mortality
Balducci Assumption
Comparing Assumptions
- Evolution of \(l_{x+s}\):
- Evolution of force of mortality:
