CHAPTER III THE MORTALITY TABLE 20. Definition.—The probability of living, or of dying, at a given age is measured by a mortality table. The fundamental column in a mortality table exhibits the number of people left alive at each age out of an assumed number alive at a given early age until all are dead. Thus, the American Experience Table exhibits the number of persons left alive at each age out of 100,000 assumed alive at age ten. It shows that all of these persons are dead at age ninty-six (Table II). The number assumed to be living at the beginning of a mortality table is called the radix of the table. The radix of the American Experience Table is 100,000. The last age in a mortality table is denoted by the Greek letter co. In the American Experience Table, u = 96. The difference &> — x is called the complement of life at age x. The complement of life at age fifty in the American Experience Table is 96 - 50 = 46. Mortality tables are constructed in various ways. A mortality table for a particular locality may be constructed from census returns for that locality covering a period of years and then graduated, or smoothed, either mechanically or by some mathe- matical device. The American Experience Table was constructed by Sheppard Homans and first published in 1868; It was probably based upon the mortality experience of the Mutual Life Insurance Company of New York supplemented by results from other sources. It is used very extensively by life insurance companies in this country. For theoretical purposes, the symbol li is used to denote the number of persons alive at age x, and the symbol d, to denote the 27 28
MATHEMATICS OF LIFE INSURANCE
§21
number of persons dying during the year from age x to age x + 1. Hence we have the fundamental equation d, = L - ;*+i. (1) Thus (American Experience Table), dso = lw — In = 92,637 — 91,914 = 723. The symbol (x) is used to denote a person of age x. 21. The Life Curve.—Figure 1 represents the values of L plotted against the age x for the American Experience Table.
Fio. 1. It is known as the life curve for this table. It should be regarded, however, as rather the lower limit of a fluctuating curve represent- ing actual mortality. If actual mortality rates in this country, say, for all ages from ten to ninety-five were reduced to a radix of 100,000 at age ten and
i22] THE MORTALITY TABLE 29 >lotted for a single year, the result would be an irregular curve ying somewhat above the curve in the figure. This irregular :urve would fluctuate from year to year and so would come to cover , somewhat irregular band, of which the curve in the figure would orm approximately the lower edge. 22. Probability of Living or Dying.—The probability that (a;) rill live 1 year is denoted by px, and the probability that (x) will lie during the year from age x to age x + 1 is denoted by q,. Since we may regard the event of living 1 year as having iccurred Ix+i times in Ix trials, and the event of dying during the 'ear as having occurred dx times out of Ix trials, we have (Sec. 16) Ix+i dx ,1-, Px = -7-' I" = 7- (1) vx vx dearly, px + qx = 1, or q. = 1 - p.. (2) The probability that (a) will live n years is denoted by np» ,nd the probability that (x) will die before the expiration of n rears is denoted by \nqx. Hence, f-x4-n 11 vx "— vx+n -I /0\ »p, = —" and \nqx = ————- = 1 - npx. (3) (x (x rhus, iop2, = I30 = |i|§ = 0.92232, and |^o = 0.07768. The symbol « n?* indicates the probability that (x) will live m rears and then die within the next n years. Hence, m\nqx = mPx — m+nPt. (4) ?or Iz+m persons are alive at age x + m and L+m+n are alive at ige x + m + n. Hence the number of persons who live m years md die within the next n years is ;,+m — Ix^m+n and the prob- ibility that any one person will live m years and die within the iext n years is, therefore, , li-l-m— It+m^-n _ ll+m Ix+m+n _ _ m[nC[x — ————,—————— — , i — mPx m+nrx- is' l'XVX An important special case of Eq. (4) is »-i i<[x= »-ip» — npx.(5) 30 MATHEMATICS OF LIFE INSURANCE [§23 This equation gives the probability that (.x)will die during the nth year from the present, that is, during the year from age x+ n — 1 to age x+ n.For example, the probability that a person of age twenty will die between the ages sixty-nine and seventy is 40,890 - 38,569 „ „„, , 49P20 - 50P20 = 92 637——— = 0025+ Exercises 1. Compute the numerical values of icpao and 110520 (Table II). State in words the meanings of the results. 2. Compute the numerical values of io| 20930 and 2o!io3ao. State in words the meanings of the results. 3. Compute the numerical values of 29) 1331 and 2»p3i. What do the results signify? 4. Given np2o = 0.75, find n. 6. Given n{3o = 0.5, find n. 23.Population.—By a stationary community is meant one unaffected by emigration or immigration and in which the number of births each year is constant. Such a community is obviously ideal, but for many statistical purposes is important. If we assume that the deaths in a stationary community are distributed uniformly throughout each year, it is clear that the number of persons arriving at age x is constantly the same; that is, there will be always;, persons alive at age x. Thus, by the American Exper- ience Table, there will be always 100,000 children arriving at age ten and 99,251 arriving at age 11. Hence, there will be %(100,000 + 99,251) = 99,652 children living between ages ten and eleven or, more exactly, at age 10%. The symbol L, means the number of persons living in a station- ary community of age between x and x + 1. Hence, Lr = }(;, + ;,+i). (1) The value of Lx can be computed for each age and tabulated, if desired. Thus, 1-96 == W» + W = K3 + 0) = 2 Z-91 = K?M + ;95) = M21 + 3) = 12 Ln = K^a + l'n) = i(79 + 21) = 50 Z,92 = ^92 + ;93) = ^(216 + 79) = 148 24] THE MORTALITY TABLE 31 The total population of age x and over in a stationary com- aunity is denoted by T,. Thus, T'96 = Z/95 + LM = 2 TM = Z-94 + T95 = U Tn = 1-93 + T94 = 64 T92 = Z;»2 t' ^93 = 212 In general, n=a—x \=L,+L.+i+L^t+L^s+ ...+£»=' ^ ^to- (2) n"0 The death rate per thousand of population at age x and over is 1000;,
Tx ince the lx persons alive at age x must necessarily die out of the iopulation Tx. At age ten, Tx = 4,872,000 and lx = 100,000 American Experience Table). Hence the death rate per thou- nd at age 10 and over is 100,000,000/4,872,000 = 20.53. 24. Central Death Rate.—The ratio of deaths at age x to popu- »tion at age x is called the central death rate and is denoted by ix. Hence dx 2di (-> mx = T~ = n -i- 1—T '1/ Lix (/» + lx+v leplacing d, by its value lx — L+i, we get 2^1-^ ,^^__^,i___J ^ ix T lx+l ,, t»+l 1 + —,—— lx lemembering that -^tl = px, we have the central death rate in is erms of the probability of living 1 year, 2(1 - p,) , , mx = "(TTpT (3) Solving Eq. (3) for px, we get (2 - roj ,,,, p^=(2Tm^ (4) 32MATHEMATICS OF LIFE INSURANCE [§25 It is possible to obtain the central death rates for a stationary or quasi-stationary community, from census returns. Suppose the value of mx for each age x has been found in this way and has been corrected for variations in birth rate and for emigration and immigration. Equation (4) will give the value of p» for each age x. Since px = li+i/lx we have, L+i = px Ix. Hence, if we agree to start with a radix, say, of 100,000 at age ten, we can derive at once ^11= pio. 100,000 ln'= Pii In lis= pit lis lx+1 = PX . Ix. In this way an ideal mortality table can be constructed for the community in which the m,'s have been determined as above. Exercises 1. Complete the tables in Sec. 23for Lx and T. as far back as age eighty, and illustrate the results graphically. 2. Compute the numerical values of OTio, m-so, mac. 3. Given ntx = 0.0100, compute px. 4. At approximately what age (American Experience Table) is m. = 0.0100 ? 25. Expectation of Life.—The average number of years lived by persons of age x is called the expectation of life. The symbol Ix+i may be thought of as expressing the total number of years lived by the Ix persons alive at age x during the year from age x to age x + 1. Similarly, lx+i expresses the total number of years lived by the Ix persons during the year from age x + 1 to age x + 2. In this way we see that the total number of years lived by the Ix persons from age x to age w is expressed by
n "»w—x lx+1 + lx+1 + L+3 + . . + la != ^, lx+t n-1
<.'/ 25] THE MORTALITY TABLE 33 'he average number of years lived by persons of age x is, lierefore, n ==&?— x _ l^.i + l^-t + . . .la, _ 2, l^" (2) el ~ —————L————— - "-i The symbol e, is called the curtate expectation of life because, in eriving its value expressed in Eq. (2), no account was taken of ie fractional part of a year any individual of age x may live eyond his last birthday. The assumption is made that, on the verage, a person will live half a year beyond his last birthday. 'he complete expectation of life is then defined to be half a year inger than the curtate expectation of life, and is denoted by the fmbol Ci. Hence, e. = ex + i. (3) leometrically, if we construct a rectangle whose length is AE = I, Fig. 1) such that its area is equal to the area under the life curve .EGHD, then the width of this rectangle, AB represents ?.. or the sum L+i + Ix+i + Ix+s + . + L the sum of the areas of a series of rectangles, each of width nity, whose lengths are, respectively, the above numbers. If to lis sum we add the sum of the areas of the curvilinear triangles hich lie between the curve and the tops of the rectangles, we lall have the area under the curve. The altitudes of these iangles are, respectively, the numbers dying from age x to age &». 'he sum of the altitudes is, therefore, Ix. The base of each iangle is unity, and therefore the sum of the areas of the triangles approximately lx/2. Hence, the area under the life curve from ge x to age m is very nearly l.+i+l^+l^s+ . . . +1 +t|- lut this is also, by definition, the area of the rectangle ABFE. fence, ,,, Area, ABFE , 1 . AB = ——AE—— = ex + 2 = ^ 34 MATHEMATICS OF LIFE INSURANCE [§26 From the equations in Sec. 23,it can be shown that T, = e, k. (4) Hence the area of the rectangle ABFE represents geometrically the population in a stationary community of age x and over. 26.Probable Lifetime.—If the probability that (x) will live n years is one-half, then n years is called the probable lifetime of (a;). It is an even chance if (x) survives his probable lifetime or fails to do so. The probable lifetime of (a;) can be found from a mortality table by finding n from the equation ^n=|- (1) Thus (American Experience Table), the probable lifetime of a child of ten is very nearly 55 years, since lie = 100,000 and les = 49,341. By interpolation between lm and lw, we find the probable lifetime is 54.65 years. The expectation of life and the probable lifetime are convenient for the purpose of comparing one mortality table with another, but neither is ever used for computing premiums chargeable for life insurance policies or for contingent annuities. In Fig. 1, AC represents the probable lifetime of a person thirty years of age, since CH is AE/2. Exercises 1. Check the value of Sso as given in Table II. 2. Prove Tx = e, lx from equations in Sec. 23. 3. Compute Tio, T,s, Tw. 4. Compute the death rate per 1000 of population at age twenty and over and at age thirty and over. 5. Compute the probable lifetime at ages twenty, thirty, and forty. Illustrate results, referring to Fig. 1. 27. Mathematical Expectation.—The term "mathematical expectation" arose in connection with games of chance and comes down from the middle of the seventeenth century. It is still tacitly assumed to refer to the obtaining of money, or its equivalent, provided a certain event occurs. This event may be the winning of a game, the election of a certain person to an office, the con- tinuance of a life for a period of years, or any chance matter whose 28] THE MORTALITY TABLE 35 robability of occurrence may, or may not, be known. Thus, if is to win $6 in case a die falls ace uppermost in a single trial, A's cpectation is said to be $6 X H = $1. This does not mean could afford to pay $1 for every such chance, but only that, if s did so, he would probably neither gain nor lose in the long run. gain, if A is to receive $1000 provided he lives 10 years, his cpectation is the present value of $1000 due in 10 years multi- lied by the probability he lives 10 years. In general, if a sum of money S is to be received in n years rovided a certain event happens whose probability is p, the resent value of the mathematical expectation is S v" p. Exercises 1. A stake of $1080 will be won, provided a count of 9 is thrown a single trial with three dice. What is the mathematical expecta- an of the person who throws the dice? 2. If on an average two out of every 1000 frame dwelling houses im annually, what is the amount of risk assumed by insuring a iime dwelling house for $5000 for 1 year? 3. One thousand dollars is to be received by A, provided he lives I years. He is now twenty years of age and money is worth 4 per nt. What is the present value of A's expectation? 4. Of a certain class of ships, ninety-nine out of every hundred turn safely to port. What is the risk assumed by insuring a ship this class and its cargo for $500,000 for a single voyage? 28.Biometric Functions.—The various expressions considered this chapter are called biometric functions from the Greek bios life and metron = a measure. Each of these expressions pends upon the age x of an individual, and its numerical value ,n be found directly from a mortality table. The mathematical theory of life contingencies and death bene- .s deals with the properties of biometric functions coupled with a te of interest. The expression <S y" p of the preceding section an example, if p refers to the probability of living or dying. In the following pages expressions like S v" p, involving terest and the probability of living or of dying, will be called ntingent functions.