The Work of an Actuary: Mortality Tables and Life Insurance

1. The Work of an Actuary: Mortality Tables and Life Insurance By: Justine McAleese & Rita Kamau April 8, 2009

2. Introduction - Actuaries • An Actuary is a business professional who analyzes the financial consequences of risk using mathematics, statistics, and financial theory to study uncertain future events. • They use mathematical models that allow them to assess risks and price insurance products. • Historically, Actuarial work was concerned with life insurance • Nowadays, Actuaries are employed in the financial sector dealing with problems in the area of super annuation, health insurance, stock broking, banking and investments.

3. Probability • In general probability indicates how many times an event may be expected to occur out of a certain number of opportunities. • Probabilities are applied to Life Insurance by calculating the probabilities of living and dying • The most important probability in life insurance is the probability that a person will die within one year – rate of mortality • Rate of mortality is determined by observing all the people that a company has insured over a limited period of time.

4. Life Insurance Probability • Characteristics that affect the probability of dying are 1) Age ( the most important because a different probability exists for each age. 2) The person’s sex- males generally have a higher mortality rate than females 3)The health status at the time of insurance 4) Length of time since a person became insured • Four concepts that formulate the uncertainty of age-to-death in probability concepts. 1. Survival Function 2. Time- Until-Death for a person at age X 3. Curtate-Future-Lifetime 4. Force of Mortality

5. 1. Survival Function While dealing with mortality, the uncertainty is associated with the length of time that an individual can survive – shown by the Survival Function. Let x denote a person aged x exactly and, T(x) to be the length of the future lifetime of the individual now aged exactly x In general, T(x) is a continuous time until death random variable and is the basic building block. If we also define F(x) to be the differential function of the random variable X ( the Age-to Death random variable), then we can define the Survival Function as: For any positive x, s(x) is the probability that a newborn will survive to age x

6. 2. Time-Until-Death • We can use probabilities in terms of survival rate to determine when age at death. For instance, if we want to know the probability a newborn dies btn ages x and z (x<z) we calculate it as • Using Conditional probability we can determine the time until death of an infant btn ages x and z, given survival at age x we get: Which is the same as

7. 3. Curtate-Future-Lifetime • This is a discrete random variable that is associated with the future lifetime • The random Variable is usually denoted as K(x) • The random Variable K(x) can be interpreted as the number of completed future years lived by a person x also known as the Curtate Future Lifetime of x. • It follows that T(x) is the future lifetime of x and may be denoted as T.

8. 4.Force of Mortality What if there is an unexpected death? We use the same equation as in Time-until-death but this time Or, In this expression F’(x) = f(x) is the p.d.f of the continuous age-at-death random variable. Has a conditional probability density and is known as the Force of Mortality denoted as The relationship between Force of Mortality and Survival Function [Table 2] Graph of Force of Mortality

9. Laws of Mortality • Gompertz Law : Restrictions B > 0, c ≥ 1, x ≥ 0 • Makeham Law: Restrictions B > 0, c ≥ 1, x ≥ 0, A ≥ -B Also note that A,B and c are chosen constants with the above restrictions The Gompetz is a special case of the Makeham with A = 0 The Laws are used to support the Survival Function

10. Mortality Tables • A mortality Table is a tabulation of the probabilities of dying during the year at each age i.e., the rates of mortality • There are 2 principal types of Mortality Tables: • Tables derived form population statistics- obtained from the National Office of Vital Statistics and Census enumerations • Tables derived from data on insured lives – Obtained from Life insurance companies a) Annuity Mortality Tables - used with annuity contracts where benefits are payable only if the contract holder is alive b) Insurance Mortality Tables - used with Life insurance contracts where benefits are only payable if the contract holder dies. • Life insurance mortality tables exhibit higher mortality rates because companies do not want to pay death benefits sooner than expected.

11. Structure/Constructionof a Mortality Table • Four basic columns • Age (starting at zero-first year of life) • Rates of Mortality, qx • At some age the probability of dying is a certainty (equals 1) • Column for number of living, lx • Column for number of dying, dx The rate of mortality is calculated as : • Construction of the mortality table follows 4 easy steps: Note that the table should start form the youngest age (usually 0) to the oldest. • Determine the Initial value for lx • Calculate the number of deaths between this age (x) and the next (x+1). This is done using the equation: • Calulate the number living at the second age (x+1) with the equation: • Repeat steps 2 and 3 for higher ages.

12. Graphical representation of rate of mortality • Ages from 0 to 70 are shown along the bottom of the graph • For each age, the distance up to one of the lines indicates the value of qx at that age, as set forth in the particular mortality table • See Graph on Handout • Example of Mortality Table

13. Life Insurance • In taking out a Life Insurance policy, a person enters into an agreement with a Life Insurance company to make periodic payments of a specified amount to the company in return for which the company agrees to pay a specified amount at some future time under specified conditions. • The terms of agreement are stated in a written document – Life Insurance Policy • Policy – Contract between the person whose life is “insured” and the company • Policy Holder – Person in possession of the policy, insured, Premium Payer • Premium – Amount of money that the insured agrees to pay to the Life Insurance Company periodically • Face of the Policy – Also known as the Amount of Insurance – the amount of money that the company agrees to pay at a future time in accordance with the term of the policy • Beneficiary – person designated by the insured to receive payment of death benefit in accordance with the terms of Policy

14. Types of Life Insurance • Personal (typical) • A person applies for insurance on their life, they own and control it • On behalf of someone else • Person applies for insurance on the life of another for their own benefit • Must be insurable interest • Ex: spouse, creditor, other business partners

15. Risk • Insurance companies classify applicants into three categories: • Standard risk • Substandard risk • 6% of insurance in U.S. is issued to substandard applicants • Uninsurable • Insurer gets information of applicant from three main sources: • Application • Medical examiners report • Inspection report

16. Term Life Insurance • Agreement to pay a death benefit if the death of the insured occurs within a specified period of time • If insured survives term, the policy expires without payment • Uses: • Need for protection is temporary • Secure the greatest possible amount of coverage for the cost • Ordinarily required for a term over a year • Yearly renewable • Short/ intermediate-term • Contracts for longer terms and to ages 60, 65, 70 • Two types: • Renewal provisions • Permit the plan to continue for additional periods of the same length • Conversion provisions • Permit the exchange of term life insurance for contract on whole life

17. Net PremiumPremium for 1 year • Net Single Premium: present value of the benefits offered by a particular insurance policy • Three factors to consider: • Appropriate mortality table • Assumption: death claims will be in accordance with death rates • Rate of interest • Company assumes it will be able to earn on premiums it will receive • Interest acquired on holding payment for future claims will help to decrease the premium required • Load: expenses and contingency • Death claims and interest are based on assumptions • Company takes allowance for possibility that the death rates may be higher or the interest earned could be lower then originally assumed • The amount paid in: A= Svn • A= (Amount each pays in) * (lx) • S= (Amount paid out to beneficiary) (dx) • Vn=interest rate

18. Net PremiumTerm Insurance • Similar to Net Premium for 1 year • Net Premium= $coverage* ((dxv+dxv2+dxv3…)/lx) Using previous example: Insurance policy of 3 years with $1,000 for 25yr male: Net Premium= $1000 *[(d25v+d26v2+d27v3) /lx] Where: d25v=(18,481)(.970874) d26v2=(18,732)(.942596) d27v3=(18,981)(.915142) lx =9,575,636 =$1,000[(17,943+17,657+17,370)/9,575,636] =$5.53

19. Whole Life Insurance • Whole life policy can provide lifetime protection • Death benefit will be paid whenever the death occurs • period of years covered by the insurance extends to the end of the mortality table • Calculation of the net premium is the same as for term insurance except the years extend to the end of the mortality table • Two forms: • Single premium whole life • Paid for in one payment, rarely purchased • Limited payment whole life • Premiums paid either for a stated number of years or until insured reaches certain age • 20 payment whole life period are payable for 20 years, could be more than 20 payments if you chose to pay semiannually, quarterly or even monthly • Straight life (most popular) • Premiums are payable for the lifetime of the insured making it the least expensive

20. Works Cited • Atkinson, M. E., and David C. M. Dickson. An Introduction to Actuarial Studies (Elgar Monographs). Grand Rapids: Edward Elgar, 2000 • Bowers, Newton L., Hans U. Gerber, James C. Hickman, Donald A. Jones, and Cecil J. Nesbity. Actuarial mathematics. Itasca, Ill: Society of Actuaries, 1986. • Gerber, Hans U. Life insurance mathematics. Berlin: Springer-Verlag, Swiss Association of Actuaries, 1990. • Harper, Floyd, and Lewis Workman. Fundamental Mathematics of Life Insurance. New York: Life Office Management Association, 1970. • http://www.ssa.gov/OACT/STATS/table4c6.html