TEMPORARY LIFE ANNUITIES

Life annuities may be either temporary life annuities or whole life annuities. Both kinds have many practical uses in actuarial calculations. In a temporary life annuity, each payment is made only if a designated person is then alive, but the payments are limited to a fixed number of years. In a whole life annuity, the payments continue for the entire lifetime of a designated person.

PROBABILITIES INVOLVING MORE THAN ONE EVENT

Probabilities involving more than one event may be calculated using the Addition Rule and Multiplication Rule. Those are as follows:

PRESENT VALUE OF A LIFE ANNUITY

As stated earlier, an annuity is a series of payments and the present value of the annuity is the total of the present values of each of the individual payments. To find the present value of a series of payments where each payment is made only if the designated payer or recipient is alive to pay or receive it, we can now use the method for calculating the present value of a single payment.

For Example: The present value at age 35 of a life annuity of $100 per year for three years (with the first payment due at age 36) can be represented by the following line diagram:

LIFE ANNUITIES

In this topic, you will begin to learn how life insurance companies make important calculations which combine the principles of compound interest and probability. In a life insurance company, annuity contracts represent payments being made only if a person is alive, while life insurance contracts represent payments being made only when a person dies. We will explain how to calculate the present value and the accumulated value of life annuity payments, that is, payments which are contingent on a designated person being alive.

Present and accumulated values for life annuities can be calculated in a manner quite similar to the method for annuities. We will begin by considering the present value and accumulated value of a single payment.

THE CENSUS OPERATION

In any country census is conducted within the frame work established by registration. Whether a census covered the entire population of a nation or on the some segments there off, it involve the following step.

DEMOGRAPHY

Demography is the science of human population. A Belgian name of ACHILE GLUILLARD first used the term in 1855 when he published his book in FRENCH named “Elements of Human Statistics or Comparative Demography”. He defines it as the natural and social history of the human species or the mathematical knowledge of population, of their general changes and of their physical civil intellectual and moral conditions. Population analysis is more or less used as synonymous to demography. However population analysis has more mathematical and numerical analysis. The term demography has been derived from two Greek words DEMOS meaning to human population and GRAPHY means to measured. Demography has further defined in different senses and levels. We have narrowest sense, broadest sense and widest sense.

PROBABILITIES OF DYING

The probability that one of the events will happen is the total of the probabilities of each individual event happening.

(Probability of living 1 year) + (Probability of Dying Within 1Year) = (Probability of Either Living or Dying That Year)

Symbols can be substituted for each of the above expressions, as follows:

Substitute p_x for (Probability of Living 1 Year)

Substitute q_x for (Probability of Dying within 1 Year)

Substitute 1 (certainty) for (Probability of Either Living or Dying That Year)

Consequently, the equation is

PROBABILITIES OF LIVING:

The probability that a person age “x” will die in the next year is represented by the symbol q_x. The probability that a person age “x” will live to reach (x + 1) is represented by the symbol “p_x”.

That is. P with a subscript x. it is read “p sub x” or simply “p x”. An example would be p₃₄ which is read “p sub 34” or “p 34”. It means the probability that a person age 34 will live to reach age 35, that is, will be alive for at least one whole year.

In general terms, it may be said that if the number living at age (x + 1) is divided by the number living at age x, the result will be the probability that a person age x will live to reach age (x + 1). In equation form, this is written as:

RATES AND RATIO

 RATIO:

 Ratio measures change in “X” per unit change in”y”.

Ratio = x/y

There are different types of ratio:

i)                    When we have numerator and denominator belongs to the same population.

ii)                   When we have numerator and denominator belongs to the different population (universe). We call the ratio as proportion (specific name of ratio).

                                Proportion= x/x+y

Proportion tends is equal to zero if x=0 and proportion is equal to 1 if y=0.

Ratio indicates the relative magnitude of a numerator and denominator.

MORTALITY & FERTILITY

MORTALITY:

Mortality rate (word mortality comes from mortal, which originates from Latin Mors, means death) is the number of deaths (from a dieses or in general).Per thousand people and typically reported on an annually basis. It is distinct from mortality rate, which refers to the no of people who have a dieses compared to the total no of people in a population. The United Nations and World Health Organization have proposed following definition of death or mortality.

“The mortality rate is the ratio of the number of deaths during a given time period per 100,000 live births during the same period of time”.

LIFE TABLE

A life table presents a set of tabulation that describes the probability of dying, the death rate, and the number of survivals for each age or age group. Accordingly, life expectancy is at word is an outward of a life table.                                     

In actuarial science a life table (also called the mortality table or actuarial table) is a table which shows for a person at each age, what is the probability is that they die before their next birthday. From this starting point, a no of statistics can be derived and also included in the table.

ERROR IN AGE REPORTING AND THEIR METHODS

In demographic data the most important and common error is committed in recording of ages. Ages are recorded in whole numbers in completed years. Extract date of birth are seldom asked, even if exact date of birth are known, the date is tabulated with a reference to date in incomplete year which creates biased over above ages are misstated due to conscious acts and same times unconsciously wrong ages are recorded. Studies of various ages data revealed that following type of misstatement of ages are usually recorded.

MEASUREMENT OF ERRORS IN DEMOGRAPHIC DATA

Methods of measurements of errors have been developed for census statistics; however can equally be used in vital statistics registration.

Response error is a broad term which includes both errors of coverage and error of content. The sampling theory assumed the census or survey or VSR records collected at time period are regarded as one of the series of trials, collection of responses which would vary from trial to trial. This hypothesis further assumed that the trials have been conducted under the same conditions and that they are independent of each other.

Various indices, measuring levels of errors can be calculated through this table: