Actuarial Mathematics of Life Insurance

Transcription

1 Actuarial Mathematics of ife Insurance How can calculate premium in life insurance? The ratemaking of life insurance policies (i.e. calculation premiums) is depending upon three elements, they are: i) Mortality rates: These rates mean probabilities of death and survival of life insurance policies. They can be calculated by Mortality table. ii) Interest (i.e. interest rate that by which the premiums are invested) iii) Sum insured (i.e. the face value of the policy) Using the foregoing elements, the fundamental relationship between the net premium and the future benefits for any type of life insurance policies may be epressed on the purchase date as follows: Present value of net premiums = present value of future benefits (on purchase date) What is the mortality table? The answer: The mortality table is "a statistical table by means of which the probabilities of death and survival may be measred at any given age for a group of individual". As shown from the following table Commissioners 1958 standard Ordinary CSO table of mortabity Male Age Femal Number of living () Number of dying (d) Probability of death (q ) Probability of survival (P)

2 1- Construction The Mortality Table After study of mortabity table and its contents the following question may be raised: How can we construct the mortabity table? The answer: The mortality table, may be constructed according to the following steps: Determining the radi of the table. Using the foregoing relationships, in particular, the relationships q = 1 p and p = 1 q for calcutating probabilities of death and survival Finding number of living for all ages using the relationships + 1 =. P, + 2 = + 1. P+1. and so on Finding number of dying for all ages using the relationships d = +1, d+1 = and so on 2- Epectation of life What is the epectation of life? Epectation of life at age () is meant," the average number of year to be lived in future by persons now aged()" Epectation of life is classified into two types, they are: i)curtate epectation of life (e): For calculating the curtate epectation (e) we assume that: 1) We have a group of individuals () aged () as indicated in the following diagram w - 1 w X X + 1 X + 2 W-1 w =

3 e 2) All deaths that occur for the individuals in any year take place at the beginning of that year (i.e. fractional parts of years are neglected). e is The curtate epectation with negligence of the fractional parts of years. ii)the complete epectation of life ( e ) Given that deaths that occur for individuals in any year take place at the end of year, Hence w1 e 1... w1 the complete epectation of life will equal the arithmatic average of formulas e 1 e 2 w wt 2 t1 A Notice:The complete epectation of life takes into consideration the fractional parts of years. Moreover, it is useful in making comparison between the various of mortality tables. Eample 1: Solved problems - 3 -

4 Given that mortality rates of population in TANTA over the ages 30,31,32,33,34 and 35 as follows: q30 = , q31 = , q32 = q33 = , q34 = , q35 = Required: Construction of a mortality table in TANTA, if you know 30 = 1000,000 individuals solution A life table may be constructed according to the following steps. First step: Finding survival rates by relationship P = 1 q, P = 1 q P30 = = and so on for the net ages as indicated in the following table: d q P , Second step: Finding number of living () by relationship +1 =. P 31 = 30. P30 For net age as indicated int the following table: - 4 -

5 d q P , Third step: Finding number of dying (d) by relationship d = -1 D30 = = 1000, = and so on for net ages as indicated in the following table d q P ,

6 d q P Solution Using the relashinships that have already studied, we may complete the previous table, age by age, as follows: 1) age 40: = +1 + d 40 = 41 + d40 = = 100,000 d Also q40 = So, P40 = 1 q 40 = ) age 41: d41 = = = 399 d q41 = P41 = = = 42 d42 = = ) age 42: d43 = 44 + d43 = = q42 = = = 427 d q42 = )age 43 and 44: - 6 -

7 d q43 = P43 = = q44 = = d44 = 44 q44 = = 492 Hence, the mortality table will be completed as follows: d q P A notice: The preceding table may be constructed by other method by completing first, then d, then q and p.. try by yourself. Eample 3: Calculate both the curtate epectation of life and the complete epectation of life for the ages in the following table: e e Solution The curtate epection of life (e) can be constrocted by the following relationship: - 7 -

8 1... w1 e w Then, the complete epectation of life can be constructed by the relationship 1 e e as shown below: e e Also : e e and so on for the net ages (97 100), then the table will become as follows: e e Probabilities of Death and Survival for a Person 3.1 Probability of death (q ): d q 1-8 -

9 P Probability of survival (P): 1 P As we have already seen,the heart of the morality table is, q, that is called probability of death. This probability has a vital role in calulating the net premium for any life insurance policy where, The premium equals the probability (q or p) multiplied by sum insured. 3.3-Probability of death for a person aged () over n year( n q ) The symbol n q means the probability that a person aged () will die before reaching age (+n). In other words, the probability that a person will die over n years. That is between age () and age (+n) as indicated in following diagram n d + 1 d +1 X X + 1 Consquently, nq d d d... d 1 2 n1 n (6.10) +n-1 +n Probability of survival for a person aged () n years ( n p ) The n p means the probability that a person aged () will live to reach age (+n). That is out of the persons alive at age () there are +n survivors at age (+n) as indicated in the preceding diagram. Hence, np n Probability of survival of a person n years and his death over 1 year ( n /q ) (6.11) - 9 -

10 The sympol n /q means the probability that a person aged () will live to reach age (+n), then die between age (+n) and age (+n+1) as indicated in the following diagram. n n - 1 d + n + n + n + 1 X X + 1 +n-1 +n +n+1 Hence, n / q n n1 (6.12) or n / q d n (6.12) repeated probability of survival of a person n year and his death over m years ( n / m q ) This symbol n / m q means the probability that a person aged () will live to reach age (+n) then die between age ( + n) and age ( + n + m) as indicated in the following diagram: m n n + n + m X X + 1 +n-1 +n +n+m Hence n / m q d d d n n1 n2 nm1 (6.13)... d or n / m q n nm = n p n+m p ( 6.13) repeated

11 In conclusion by contemplating the preceding notation, it should be noted that: a) The letter P with the proper subscripts is used to denote the probability of a person living a given period b) The letter q is used to denote the probability of a person dying during a given period Solved problem Eample 4 Interpret in words the following symbols, then calculate their values using the American life table (1958 CSO) a) q, P60 b) 5 P30, 7 q27 c) 6 /q35, 7 / 5 q solution a) q: means a probability that a person aged () will die over one year. That is between age () and age (26) q or d q P60: means a probability that a person aged (60) will live to reach age (61) P b) 5 p30: means a probability that a person aged (30) will live to reach age (35) 5 P

12 7 q 7 q: means probability that a person aged () will die over seven years. That is between age and 1ge c) 6 /q35: means a probability that a person aged (35) will live to reach age (41), then die between age (41) and age (42): / q or d / q / 5 q: means a probability that a person aged () will live to reach age (32) then die between age (32) and age (37) 7 / or 7 / 5 5 q q d d d 34 d 35 d