The Life Table

The life table is one of the most important tools of the actuary. Its primary use is for the calculation of life insurance and annuity products issued by financial institutions such as insurance companies. It summarises the mortality experience of a population as determined by historical data and prior opinion concerning that population. There are various columns in a standard life table. The notation used is Standard International Actuarial Notation and will be found in all actuarial literature written over the past few decades.

NOTATION:

x: this is the column indicating the age to which the data refers.

l_x: the expected number of lives alive at age x.

d_x: the expected number of deaths between the ages of x and x+1.

q_x: the probability that a life aged x will die before reaching age x+1. This can be calculated as simply d_x / l_x

p_x: the probability that a life now aged x will survive to age x+1. This can be calculated as l_x+1 / l_x or since we know that for a given year a life must either die or survive, p_x = 1 - q_x

mu_x: the force of mortality at age x + 0.5. This is usually estimated separately from the other quantities, a discussion of which will not be entered into here.

e⁰_x: the complete future expectation of life; that is, the number of years including part years that it is expected a life aged x has in front of them.

Using the life table, one can do simple calculations useful in determining life insurance premiums/benefits. For example, suppose we have the following extract from a life table:

x | l_x
------------------
24 | 39526.235
25 | 39411.123
26 | 39280.377
27 | 39129.458
28 | 38970.819
29 | 38772.697
30 | 38553.148
31 | 38304.964

We wish to calculate the probability that a life aged 24 will die before reaching age 31. In Actuarial Notation this could be written as ₇q₂₄ and is read "seven-q-twenty-four": the probability that a life aged 24 dies within the next seven years. The easiest way to calculate this from the supplied data is to calculate the probability that a life aged 24 survives seven years and subtract this from one, since the life must either die or survive. Now, the expected number of people alive at age 31 is 38304.964 (note that this is not a whole number which seems absurd, but remember this is merely the number we expect on average (the mean)). The expected number alive at age 24 is 39526.235. Logically then, the probability of survival is the ratio of these two numbers. That is, what is the pecentage of of those who we started with whom are still alive at the end?
Answer: l₃₁ / l₂₄ = 38304.964/39526.235 = 0.96910 (5 dec. pl.) or roughly a 97% chance.

An example of a Term Assurance Benefit
A term assurance is a life insurance contract in which the life assured pays premiums to secure a benefit with a limited life (say 5 years, though it could be much shorter or much longer). A common policy is that on death within the term, the life office (insurance company) will pay a benefit at the end of the year of death but will pay nothing if the life survives the term.

Consider a particular assurance policy where the benefit is $1, the policy starts and the premium is payable at exact age 25 and the term is 5 years. Assume that mortality follows that of the extract from a life table above. For the purposes of this example interest and expenses will be ignored, though in reality these two components are vital to producing accurate premiums. In calculating this premium the principle of equivalence will be used which states that the expected present value of the death benefit must be equal to the expected present value of the premium income. Now we wish to find the premium, call it P.

CALCULATIONS: we have P = Expected Death Benefit. Now using the notation above, P = $1*{₅q₂₅} = $1*( 1 - l₃₀ / l₂₅ = $1*{1 - 38553.148/39411.123} = $0.02177. This implies a fair premium under the above assumptions is 2.177 cents for every dollar assured. A death benefit of $100,000 would require $2177 to be paid at age 25.

The introduction of interest requires an understanding of present values, discounting etc. Using a rate of interest in the above calculation would require discounting each expected benefit (benefit amount times probability of payment) by the accumulated interest to the time at which the benefit would be paid. It would result in a smaller premium since any benefit would be paid at ages 26, 27, 28, 29 or 30, and discounting the benefit back to age 25 when the premium is paid results in a lower expected present value of benefits and hence lower premium.