1. World of Uncertainty The future is full of uncertainty. In the course of our lives we may encounter undesirable events that can impact us both emotionally and financially. While extra care can be taken to reduce the likelihood of some undesirable events such as accidents attributed to carelessness, there are some events that are simply inevitable. Death is a prime example. People have little control over when they die (barring suicide) but an untimely death presents a major risk to any family. Consider the impact on a familys economic wellbeing if the family loses its sole breadwinner to an untimely death. To reduce the financial impact of an untimely death and safeguard economic security, families and individuals have the option to purchase life insurance. For many life insurances, the size and timing of premium payments depend greatly on the time of death of the insured. Unfortunately the insurance companies do not know precisely when death will occur, but mathematics and statistics do offer tools for making viable predictions. Survival analysis is concerned with the estimation of survival probabilities in a population. Typical questions one may encounter in survival analysis are: What fraction of a population will survive past a certain time? Given survival to a certain age, what is the rate at which members of a population will die? How do particular characteristics and circumstances increase or decrease the odds of survival? Survival Analysis and Actuarial Life Tables Linus Waelti 07 Swarthmore College, Department of Mathematics & Statistics How well can you predict my schedule? A Theorem of Expected Values If T is a continuous random variable with c.d.f. G(t) such that G(0) = 0 and p.d.f. G(t) = g(t), and z(t) is such that: it is a nonnegative, monotonic, differentiable function E[z(T)] exists, then In the context of our life table, this theorem enables us to express the complete-expectation-of-life in simpler terms. The complete-expectation-of-life in actuarial notation is denoted by which is equivalent to E[T(x)]. By the definition of the expected value we have: Applying the theorem using and yields:, a much simpler expression to solve. For broader applications, this theorem is useful in that it gives us a way to find the expected value of some continuous random variable, T, when a well-defined p.d.f. g(t) is not readily available. Actuarial life tables In actuarial science, a life table or mortality table is a table of statistics that provides information on the average probability of survival or death at different ages, the remaining life expectancy for people of different ages and the proportion of the original birth cohort still alive. In terms of the random variable X (age-at-death) the life table summarizes the distribution of X. Life table are usually constructed separately for men and for women and other characteristics can also be used to distinguish different risks, such as smoking-status, occupation and socio-economic class. Life tables find applications in a multitude of fields including engineering, biostatistics and demography. In life actuarial practices, actuaries use life tables to build models for their insurance systems designed to assist individuals facing uncertainty about the times of their deaths. Random Survivorship Group Life tables seem to describe the mortality experience of a group of newborns. It is common practice in actuarial life tables to set Does this mean that the lives of 100,000 newborns are observed until all are dead? Yes, in the case of cohort life table, but no in the more readily available period life tables. Period life tables do not represent the mortality experience of an actual birth cohort. Instead, the table presents what would happen to a hypothetical cohort if it experienced throughout its entire life the mortality conditions of particular period of time. For example: the life table above assumes a hypothetical cohort subject throughout its lifetime to the age-specific death rates prevailing for the actual population in 2003. The 100,000 members therefore constitutes a random survivorship group. The expected number of survivors to age x from the newborns is related to the survival function by: The function,, is interpreted as the expected density of deaths in the age interval (x, x + dx). The Curve of Deaths The Survival Function, s(x) Since the age we reach when we die is an uncertain figure we call this age-at-death a random variable, denoted by X. X has cumulative density function, c.d.f. (lifetime distribution function) which tells us the probability that a newborn dies by age x: The survival function, s(x), is the complement of and is defined as the probability that a newborn will attain age x: Properties of s(x) s(x) is non-increasing s(0) = 1 s(x) tends to 0 as x approaches infinity Note: It common practice to define a limiting age ω such that s(x) = 0 for all x ω. For human lives, there have been so few observations of age-at-death beyond 110 thus it is assumed that there exists a limiting age that nobody survives beyond. Probability Density Function (p.d.f.) and s(x) The p.d.f. is the derivative of the c.d.f. so: Conditional Probability The conditional probability that a newborn will die between the ages x and z, given survival to age x, is Time-until-Death, Force of Mortality and some whacky actuarial notation Consider a person of age x. This persons future lifetime is described by a continuous random variable T (x). Since the future lifetime of this person assumes he has already survived to age x, we have To make probability statements about T(x), we have the following actuarial notation: The symbol can be interpreted as the probability a person of age x will attain age x + t. The symbol can be interpreted as the probability a person of age x will die within t years. In the special case of a newborn, we have and Force of Mortality The probability of death within the next instant of time given survival to age x is: The force of mortality, µ(x), is the value of the conditional p.d.f. of X at exact age x, given survival to age x: Finding the p.d.f. of T(x) and X So, for the special case of a newborn, 0 x X = x + T(x) References and Acknowledgements Arias, E. United States Life Tables, 2003. National Vital Statistics Reports, Vol. 54, No. 14. April 19, 2006. Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. 1986. Actuarial Mathematics, 1 st Edition, edited by M.M. Treloar. The Society of Actuaries, Itasca, IL. Klein, J.P. Survival Distributions and their Characteristics. A contribution to the Encyclopedia of Biostatistics. Accessed 2006, Oct 7. Shand, K. Survival Distribution and Life Tables. Warren Center for Actuarial Studies and Research. Accessed 2006, Oct 7. Webshots. Grim Reaper. Accessed 2006, Nov 20. Wikipedia. Life Tables. Accessed 2006, Oct 7. Wikipedia. Survival Analysis. Accessed 2006, Oct 7. I am grateful to my fellow MATH 97 classmates and especially Walter Stromquist for valuable comments made in the course of preparing this poster. Age