Definitions

# Life table

In actuarial science, a life table (also called a mortality table or actuarial table) is a table which shows, for a person at each age, what the probability is that they die before their next birthday. From this starting point, a number of statistics can be derived and thus also included in the table:

• the probability of surviving any particular year of age
• remaining life expectancy for people at different ages
• the proportion of the original birth cohort still alive
• estimates of a cohort's longevity characteristics.

Life tables are usually constructed separately for men and for women because of their substantially different mortality rates. Other characteristics can also be used to distinguish different risks, such as smoking-status, occupation, socio-economic class, and others.

Life tables are also used extensively in biology and epidemiology. The concept is of importance, as well, to product life cycle management.

## Insurance applications

In order to price insurance products, and ensure the solvency of insurance companies through adequate reserves, actuaries must develop projections of future insured events (such as death, sickness, disability, etc.). To do this, actuaries develop mathematical models of the causes of these events, as well as the amount and timing of the events. They do this by studying the incidence and severity of these events in the recent past, developing expectations about how the drivers of these past events will change over time (for example, whether the increase in life expectancy that has been experienced by most generations over prior generations will continue) and, accordingly, develop an expectation for what the timing and amount of such events will be into the future. These expectations usually take the form of tables of percentages indicating the number of such events that will occur in a population, usually based on the age or other relevant characteristics of the population. More specifically, they may be referred to as mortality tables (if they provide rates of mortality, or death), morbidity tables (if they provide rates of disability and recovery), or by other names if they cover other decrements.

The invention of computers and the proliferation of data gathering about individuals has led to fundamental changes in the way actuarial tables are computed for different uses, and a variety of emerging methods factor a range of non-traditional behaviors (e.g. gambling, debt load) into specialized calculations utilized by some institutions for evaluating risk.

## The mathematics

To give an indication of how life tables are used, here are a few sample calculations. These samples may not be obvious to someone who has never studied probability theory, but are intended to introduce new ideas to people who have some understanding of discrete probability theory.

• $,q_x$: the probability that someone aged exactly $,x$ will die before their $,\left(x+1\right)$th birthday
• $,p_x$: the probability of surviving from age $,x$ to age $,\left(x+1\right)$

$,p_x = 1-q_x$

• $,l_x$: the number of people who survive to age $,x$

note that this is based on a starting point of $,l_0$ lives, typically 100,000
$,l_\left\{x + 1\right\} = l_x cdot \left(1-q_x\right) = l_x cdot p_x$
$,\left\{l_\left\{x + 1\right\} over l_x\right\} = p_x$

• $,d_x$: the number of people who die aged $,x$

$,d_x = l_x-l_\left\{x+1\right\} = l_x cdot \left(1-p_x\right) = l_x cdot q_x$

• $,\left\{\right\}_tp_x$: the probability that someone aged exactly $,x$ will survive for $,t$ more years, i.e. live up to at least age $,x+t$ years

$,\left\{\right\}_tp_x = \left\{l_\left\{x+t\right\} over l_x\right\}$

• $,\left\{\right\}_\left\{t|k\right\}q_x$: the probability that someone aged exactly $,x$ will survive for $,t$ more years, then die within the next $,k$ years

$,\left\{\right\}_\left\{t|k\right\}q_x = \left\{\right\}_t p_x cdot \left\{\right\}_k q_\left\{x+t\right\} = \left\{l_\left\{x+t\right\} - l_\left\{x+t+k\right\} over l_x\right\}$

## Biology

When biologists use life tables, they will normally also include fertility for each age. The extra parameter used is

• $,m_x$: expected number of progeny for an individual aged $,x$