Chapter 9 [Measurement of Risk and Mortality Table].pptx

Chapter 9

Measurement of Risk and Mortality Table

Slide prepared by: Abdullah Al Yousuf KhanAssistant Professor IUBAT 9

ChapterMcGraw-Hill/IrwinCopyright 2006 by The McGraw-Hill Companies, Inc. All rights reserved.Measurement of Risk and Mortality Table

The risks are measured or evaluated for fixation of premium to be charged by the insurer. There are two methods of calculation of premium;Value of Service Cost of Servicemethods of calculation of premium;Value of Service; It determines the rate of premium according to the utility or value to insurance to each proponents.Since the value or utility varies with each person, the premium rates will also vary. But this principle can not be used because its utility to each individual cannot be determined. The value is higher to the richer section of the society and to the head of large family.Moreover, high premium will not attract business.So the Value of Service cannot be used due to its impractibility.methods of calculation of premium;Cost of Service; Premium should be charged according to the cost to the insurer.In insurance, demand side does not play important role.That is why it is called, an insurance is not bought, it is sold.So, the insurer must fix the cost of the premium to be charged on a particular risk or policy. The cost includes all expenses of the business plus a small profit margin. Above the profit margin, the insurer is not expected to gain. The insurance business is expected run on no loss, no gain basis.The premium charged to meet the amount of claim, is called net premium.Administrative cost is another type of cost;Fixed costRecurring cost

Cost of ClaimsThe claims may arise at the death or at the end of the policy.In annuity contract, the payment shall continue to death, therefore, the expectation of survival will be the basis of cost. In life insurance, payment of claims depends upon the death. The death is certain but not the time. Therefore, the main problem is to decide when the death will take place. Forecasting of death is very important facto to decide the period and amount of claim. If these are decided, the premiums can easily be calculated.The forecasting of death can be done on;The experience on medical science, andThe experience of past recordsForecasting the DeathThe death of one life cannot be forecasted but the expectations of number of deaths from a group of the same age can be forecasted on the basis of;Theory of Probability and The Law of Large NumberForecasting the DeathTheory of Probability;It reveals the chances of death of a person out of a group. Can be three types;Certainty; expressed as one. It means the chance of happening a certain event, say death, is 100 percent. Simple probability; when the events are mutually exclusive. e.g. if at the age of 40, 2 persons die out of 10,000. the probability of a death of a person can be expressed as; 2/10,000=.02 percent or 0.0002 of units.Compound probability; is combined with two persons simple probability. (see example)Estimation of ProbabilityCan be estimated on the;Priori basis; Estimated merely on the basis of knowledge, not by experiment or practice. E.g. it is well known fact that the death rate after the age of 50 will continue to increase year after year.Posteriori basis; Calculated on the basis of experiment. Estimations involves a large number of data to be accurate. But 100 percent accuracy and universal experiment is not possible. Law of Large NumbersThe accuracy of probability depends on two factors;Accuracy of dataThe large number of unitsIt has been observed that the larger the number, the lesser the deviation between actual and probable estimation.

Mortality TableMortality table is such data which records the past mortality and is put in such form as can be used in estimating the course of future data. Past deaths recorded to predict future mortality.A large number of persons are selected and served for death and survival rate till all of them is dead.It is also described as the picture of a generation of individuals passing through time.Features of Mortality TableObservation of Generations; persons of a generation is selected and observed till death.Start from a point; and will continue up to the point till everyone dies.Yearly estimation; records the yearly death or survival rate. Each year is considered. Mortality and Survival rates; any table giving mortality rates is not mortality table unless mortality rate of a generation is calculated each year. Each years number of living is the previous years number of living minus previous years number of dying. Therefore, as persons go on dying year after year the number o living goes on shrinking till it is reduced to zero and mortality tables ends there.Construction of Mortality TableAttained age; Means age nearer to birth date - should be selected.E.g. persons of age of 19 years 6 months to 20 years 5 months 29 days will be treated as the age of 20 years. The selected persons of the attained age will be observed and the number of deaths will be recorded during a year till the persons selected are dead. The number of death in a year is deducted from the number of living at the beginning of year to get the number of living in the beginning of the next year. Criticism of Mortality Table A large number of persons or an attained age is difficult to get.Constant watch (monitoring) is not possible.Requires long period to construct the table.Waste of money and time to record.Even though its constructed, it will be of no use due to the changes might have occurred over the years.Construction of Death Rates on Yearly BasisTo avoid the above difficulties, death rate is calculated on a yearly basis. The death rate is calculated for every age.Separate sample is taken for each age.The number of living in each age is observed and the number of death during the year is recorded.The death rate for the age is calculated by dividing number of deaths by number of living in each age.A year is selected because a year constitutes various types of weather, therefore, low, high, and normal mortalities are averaged in the year.

Example of Construction of Death Rates Age Number of LivingNumber of DeathDeath Rate Survivors Rate201,000,0002,0000.0020.99821998,0003,0000.0030.99722995,0004,0000.0040.996Construction of Death RateTo avoid the difficulties in calculating death rates, it is calculated on yearly basis.It is calculated for every age with separate sample taken from each age. The number of living in each age is observed and the number of death during the year is recorded.The death rate for a given age is calculated by dividing number of death by the number of living in each age.A year is selected because a year constitutes various types of weather, therefore, low, high, and normal mortalities are averaged in the year. Premiums are quoted on yearly basis so the cost depending on mortality shown also be based on yearly basis.Sources of Mortality InformationFor construction of a mortality table, number of living in the beginning of each year and the number of deaths during the year are required. The mortality table must be constructed as accurately as possible to represent the past experience.The sources of mortality can be obtained from;Population Statistics accuracy may of question Records of Insurers very accurate Construction of Mortality Table; Example10,000 persons are taken at the age of 20, 20,000 persons at age 21, 5,000 persons at age 22, 10,000 persons at 23, 20,000 persons at age of 24. The number of deaths observed at these ages are 20, 80, 15, 60, and 100 respectively. Therefore the death rates will be 0.002, 0.004, 0.003, 0.006, 0.005 respectively at this stage. The death rate is calculated by the following formula;

Table 9.1 Yearly DeathAge Number of LivingNumber of DeathDeath Rate Survivors Rate2010,000200.002?2120,000800.004?225,000150.003?2310,000600.006?2420,0001000.005?Crude and Graduated Mortality RatesCrude Death Rates;The yearly death rates may be different because;The generation of an age is not observed. Different persons at different ages are observed.There may be a large number for observation.The date may be cent percent correct.Graduated Mortality Rates;By smoothing the fluctuations of crude rates with the help of interpolation and graphical methods. Crude and Graduated Mortality RatesTable 9.2 Mortality Table Age Number of Living Number of Deaths Mortality Rate Survival Rate x lx dx qx px=1-qx201,000,0002,0000.0020.99821998,0002,9940.0030.99722995,0063,9800.0040.99623991,0264,9550.0050.99524986,0655,7160.0060.994The number of death at a particular age is calculated by multiplying the number of living at the age with the mortality rate.

Thus, dx= lx qx. Here at the age 20 the d20 = l20 q20 = 1,000,000 0.002 = 2,000.

Number of living persons is calculated as follows;l21=l20 d20 = 1,000,000 2,000 = 998,000

Example The death rate of a person aged 20 after 3 years can be calculated as follows;

A Complete Mortality TableAge No. of Living at xNo. of Deaths between x and x+1Death RateSurvival RateForce of Mortality at age xNo. of Survival at min age lx and lx +1Total Survival No.Complete Expectation of Life at age xAgexlxdxqx= dx/lxPx=1-qxUxLx= (lx+x+1) Tx=Lxx=Tx/Lxx2096,0615480.005720.994280.0055095,7874,044,23842.101202195,5135820.006080.993920.0059295,2223,948,45141.339212294,9316090.006430.993570.0062994,6263,853,22940.590222394,3226310.006680.993320.0065994,0073,758,60339.849232493,6916470.006910.993090.0068296,3673,664,59639.11424

Hm Maikaham Graduation Mortality Table

Description of the TableColumn (x). Column x denotes the age of the prospect. The mortality table can start from any age and continue to 100 to 120 years, as required by the insurer.Column (lx) This column indicates the number of living persons at the beginning of each year.The table starts from age 0, the number in the column lx against for this age group l0, is, say e.g. 9,000,000 or any number that represents the number of persons age 0.If the table starts with the age 30, the starting value for lx will be l30Column dx;Number of persons dying between age x and x+1 are shown in this column.The difference of lx - lx +1 is the number of persons die between ages x and x+1.Thus dx = lx - lx +1 Column (qx= dx/ lx);this column indicates the probability of death.It gives for a successive values of x the probability that a person aged x dies within one year, i.e. before reaching the age x+1. Thus the death rate or mortality at age x is equal to the number of deaths at age x divided by the number of living at age x or

Column Px;The column (Px) indicates the rate of survival. It gives for successive ages the probability that a life aged x survives to age x+1. thus, the probability of survival (Px) is equal to number of survivors to age x+1 divided by total number of living age x. Hence, Px= lx+1/ lx It is also known as Px=1-qx

Force of Mortality (Ux);Te force of mortality at age x is denoted by UxThe force of mortality at age x can thus be defined as the limiting value of the nominal yearly rate of mortality at age x, over a small interval of time dt as the length of interval dt tends to zero. In practice the smallest interval we can consider is one day i.e., dt = 1/365 days of a year.The death between age x and are and the rate of mortality at age x per day is The corresponding yearly rate is

No. of survival at mid age x and x+1(Lx)The number of persons in the population between ages x to x+1 is denoted by Lx.Thus

Complete Expectation of Life at age x (x)

This column is known as the complete expectation of life denoted by the symbol x.The expectation of life is the average number of complete years of life lived by each person aged x, after reaching age x.It is an average obtained by dividing by lx, the total number of futures years of life lived by the lx persons.

Types of Mortality TableAggregate TableA table constituted without distinguishing the select , and ultimate lives. The lives from which the mortality rates of the aggregate lives are derived being a mixture of the select lives and ultimate live, the aggregate rates lie between the select and ultimate rates for the same age attained. Also called Mixed and General Mortality Table. The Select TableMortality table giving rate depending on both age and duration elapsed since entry are called select mortality tables.

Table 9.3 Mortality Rate/ 1000 (dx)Age Years of Insurance 6 years and overAge attained 12346202.733.593.803.964.134.3125212.783.663.864.014.184.3526222.833.723.914.074.214.3827232.863.763.064.084.244.4128Formation of a Select tableTable 9.4 Select Mortality Table

AgexNumber of Livinglz Number of Deathlx Death Rate per Thousandl 35100,0003163.1635+199,6844284.2935+299,2564544.5735+399,8024744.8036100,0003233.2336+199,6774114.42Ultimate Mortality TableA mortality table in which the rates in the select period are omitted and only the ultimate rates are tabulated is called an ultimate mortality table. Table 9.5 Ultimate Mortality Table

Age at entry6 and over Age attained204.3125214.3526224.382728234.4128Interest FactorThe second factor after death rate is interest factor;For calculating net premium Because the premium is obtained in advance, andClaim is paid subsequently on a later date when claims made.So during this period the insurer can earn certain interest.Since the insurer can earn additional amount on the premium collected, its benefit should be given to the policyholder. Questions AgeNumber of Living Number of DeathsProbability of DeathProbability of Survival(x)(lx)(dx)(qx)(Px)201,000,0000.00409210.00370220.99653230.99658240.00342250.00335Fill up the blanks