MATHEMATICS OF LIFE INSURANCE CHAPTER I INTRODUCTION Operations in actuarial science, in so far as they depend upon nathematics, rest upon relatively few fundamental principles. Fhese will be reviewed briefly in the following pages. 1. Compound Interest.—Financial operations are usually per- ormed under the assumption of compound interest; that is, nterest upon interest. The annual rate of interest is indicated )y the letter i. Thus the amount of $1 invested now at rate i or 1 year is 1 + i; for 2 years is (1 + z)2; for 3 years is (1 + i)3; ind so on. In general, the amount of $1 at rate i invested for n years is (1 + O". The expression (1 + i)" is often called the accumulation factor or compound interest. The values of (1 + i)" for various rates if interest and for a series of years have been computed and tabulated for convenience of computation (Table I). The principal is the amount invested. If the principal is indicated by P and the amount by A, we have the fundamental Formula , , A = P(l + i)". W The interest earned is the amount less the principal invested. If I indicates the interest earned, then 7=P(l+i)»-P=.P[(l+i)"-l]. (2) The expression (1 + i> - 1 may be called the interest factor for compound interest. 1 2 . MATHEMATICS OF LIFE INSURANCE [§2 From Eq. (1) the value of any one of the numbers P, i, re, A can be found when the other three are given. Thus,
2. Present Value.—The expression -,—,— is the present value r of $1 due in 1 year, or the amount that must be invested now to yield $1 in 1 year. The present value of $1 due in 1 year is indi- cated by v. Hence, "=r:h- w The present value of $1 due in n years is "" = (i-^)-n = (1 + *)-». (2) Eq. (3) of the preceding section may now be written P = Ay. (3) The process of finding the present value P of an amount A due at a future time is called discounting. The factor y is often called the discount factor for compound interest. The values of »" for various rates of interest and for a series of years have been computed and tabulated for convenience of computation (Table I). Exercises 1. Find the amount of $342 in 10 years at 5 per cent compound interest. 2. How much must be invested now at 4 per cent compound interest to yield $1000 in 10 years? 3. In how many years will a sum of money double itself at 4 per cent compound interest? At 5 per cent compound interest? 4. At what .rate of interest will money double itself in 10 years if compounded annually? 6. What is the present value of $1000 due in 8 years at 5 per cent compound interest? Due in 50 years at the same rate? ] INTRODUCTION 3 6. How much money must be invested now at 4 per cent compound terest to amount to 8100,000 in 50 years? 3. Annuities.—An annuity is a sum of money payable at stated tervals of time and for a period of years. Usually an annuity is lyable annually and at the end of each year. If the annuity payable at the beginning of each year, it is called an annuity- te. Annuities fall into two distinct classes, namely, Annuities-certain. Contingent annuities. An annuity-certain is certainly payable irrespective of the hap- 'ning of any event. A contingent annuity is payable only in case a rtain event happens, usually that some person is living to receive ie payment or to make the payment. Thus the annual premium larged for a life insurance policy is a contingent annuity-due, nee it is payable at the beginning of each policy year and only if ie insured is living. Again, if $1000 is invested now at 6 per cent, the interest will mount to $60 at the end of each year. This is an annuity-certain F $60 per year. The annual rental or annual yield is $60. Conversely, an annuity is equivalent to interest receivable on an ivested principal. Thus, an annuity of $1000, payable at the end f each year, is equivalent to the annual interest at 5 per cent on a invested principal of $20,000. In general, an annuity whose annual rental is R, payable at the nd of each year, is equivalent to the annual interest at rate t n an invested principal of R/i. If the annuity is payable only for a limited number of years, it is ailed a term annuity. If the annuity is payable without limit, it is ailed a 'perpetuity. If the payments on an annuity are not withdrawn as they fall lue, but are kept invested, the annuity is said to be forborne. Exercises 1. A wills his wife the use of his estate valued at $100,000. If his estate can be kept invested at 41.^ per cent, what is the widow's .nnual income? 4 MATHEMATICS OF LIFE INSURANCE [§4 2. B has a legacy of $1000 payable at the end of each year. This annuity is equivalent to what invested principal, assuming money is worth 4 per cent? 4. Amount of a Forborne Annuity.—When the payments on an annuity are kept invested for a period of years, the total amount due at the end of the period is called the amount of the forborne annuity. For example, suppose an annuity of $1000, payable at the end of each year, is forborne for 10 years and the payments are kept invested at 5 per cent. Since this annuity is equivalent to the annual interest at 5 per cent on an invested principal of $20,000, the amount at the end of 10 years is the accumulated interest on $20,000 at 5 per cent for 10 years. Making use of the interest factor (Sec. 1, Eq. (2)), the required amount is $20,000 [(1.05)10 - 1] = $12,577.892. In general, if the annual rental of a forborne annuity is R, payable at the end of each year, the amount at the end of n years at rate i is i =^[(1+1)"-I], (1) I/ since the annuity is equivalent to the annual interest at rate i on an invested principal of R/i. If R = 1, that is, if the annual rental is 1, payable at the end of each year, the amount at the end of n years is indicated by the symbol s—, and we have the important formula s,,i=^[(l+z)»-ll. (2) Eq. (1) can now be written A = R s^. (3) The advantage of Eq. (3) lies in the fact that the values of s,, can be computed for various rates of interest and for various periods of time and tabulated once for all. If the annuity is not forborne, that is, if the payments are with- drawn as they fall due, the annuity is kept exhausted and has no amount. I INTRODUCTION 5 6. Present Value of a Term Annuity.—By the present value of a rm annuity is meant a sum invested at the beginning of the term compound interest which will just suffice to furnish the pay- 'nts on the annuity as they fall due. Such a sum must, viously, furnish the amount of the annuity in case it is forborne ring the term. We shall find the present value of the annuity, erefore, by discounting the amount back over the term. Thus, the example considered in the preceding section, the amount was ind to be $12,577.892. Discounting this amount for 10 years rate 5 gives $7721.734 as the present value of the given annuity. this present value is invested at 5 per cent compound interest, will furnish $1000 at the end of each year for 10 years. Or, if e payments are not withdrawn at the end of each year, the fund II amount to $12,577.89 at the end of the tenth year. The present value of an annuity of 1, payable at the end of ch year for a term of n years, is indicated by the symbol a-,. ice the amount of this annuity, if forborne for the term, is , we have ^ = Sn] "" = j [(1 + O" - 1] v" = ^(1 - v")- W the annual rental is denoted by R, the present value is 7?.ff,.y=^(l-.»). (2) The longer the term of an annuity the more closely does its esent value approach R/i, since the fraction v" approaches zero n increases in value. The expression R/i is, therefore, the esent value of a perpetuity of annual rental R. 6. Alternative Proofs.—The proofs of the formulas for s,, d a^, are often based upon the formula for the sum of a geometric agression; that is, upon the formula Ir — a ^T^l- lere I stands for the last term of the progression, a for the first m, and r for the common ratio. Thus, if the payments on an nuity of 1 are kept invested from the time they fall due until 6 MATHEMATICS OP LIFE INSURANCE [}7 the end of the nth year at rate i, they will yield the following amounts: (1 + i)"-1, (1 + iV-\ (1 + i)"-', . . . , 1. Taken in reverse order, we have a geometric progression in which ; = (1 + i)"~1, a = 1, and r = 1 + i. Hence the sum, by the above equation, is „ .d±^l (D Again, if each payment as it falls due is discounted to the present, we have the following present values: v,v'',v3, . . . ,v", or a geometric progression in which I = V, a == v, and r = v. The sum is, therefore, ti"+1 - v _ v - v"+1 _ 1 -v" _ 1 -v" "I ~ v-1 ~ l-v ~ Y^ ~ ~T~' w Equations (1) and (2) agree with the corresponding equations in Sec. 4 and 5 as they should. Exercises 1. What is the present value of an income of $1000 payable at the end of each year for 10 years, money being worth 3% per cent? Payable for 50 years? 2. If the annual income in the preceding exercise is kept invested at 4 per cent, what will be the amount at the end of the tenth year? At the end of the fiftieth year? 3. How much money must be invested now at 5 per cent compound interest to yield a perpetual annual income of $1000, first payment at the end of 10 years, provided the fund can be kept invested at 4 per cent from then on? 4. How much money must be invested now at 5 per cent compound interest to yield $5000 at the end of each period of 5 years? 7. Present Value of an Annuity-due.—An annuity-due was defined in Sec. 3 as an annuity whose payments fall due at the beginning of each year, the first payment being due immediately. 8] INTRODUCTION 7 The present value of an annuity-due of 1 is indicated by the toman "full-faced " letter a. Thus a, denotes the present value f an annuity-due of 1 payable annually for a period of n years. Obviously, ^ = i +a^ w ince a—r,i is the present value of 1 payable at the end of each ear for a term of n — 1 years. We shall have much to do with annuities-due in what follows, ince, as has been remarked, the annual premium charged for a ife insurance policy is a contingent annuity-due. It is well, herefore, to distinguish carefully between the symbols a and a. Exercises 1. Determine the present value of an annuity-due of $1000 for 50 ears, money being worth 4 per cent. 2. If the annuity-due in the preceding exercise is forborne, deter- aine its amount at the end of the 50 years. 8. General Remark.—The foregoing sections constitute in >arest outline the application of mathematics to financial opera- ions. For details and for further applications, the interested eader should consult a text devoted to the subject.1 9. Factorials.—Many questions in the application of mathe- natics to statistics and financial operations involve the use of actorials. The continued product 1-2-3-4-5-6. . .nis ailed factorial n and is indicated by the symbol n!. Thus, n\ = 1-2-3-4-5 . . . n. (1) n particular, I! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120, and so on. Fhe symbol 0!, or factorial 0, is defined to be 1. The product of a succession of consecutive integers can be very onveniently represented in terms of factorials. Thus, ,,„ 1.2.3.4.5-6 6!
4.5-6 =
1-2-33!
* For example, SKINNER, "The Mathematical Theory of Invest- nent," Ginn & Company. 8 MATHEMATICS OF LIFE INSURANCE [§10 or, in reverse order, „ . A 6-5-4-3-2-1 6! . 65'i=——3^-1——-3-!- In general, n{n - l)(n. - 2)(re - 3) . . . (w - r + 1) ^ n(n - l)(n - 2)(n - 3)(n - 4) . . . (n - r + l)(n - r)(n - r - 1) . . .1 ,(n - r)(n - r - 1) ... 1 _ "' ,„, - (n - r)!' (2) Exercises 1. Express the continued product 6 7 8 9 10 as the ratio of two factorials. 2. Express n(n — 2) (re - 3) ... (n - K) as the ratio of two factorials. 3. Find the numerical values of 7!, 10!/6!, 85!/83!. 10. Independent Events.—Events are said to be independent if the occurrence of one or more of them has no influence upon the rest. Independent events can happen together or in sequence; and the law governing the combination of two such events may be stated as follows: If one event can happen in a ways and a second independent event can happen in b ways, the two events can happen together, or, in sequence, in a b ways. With each way the first event can happen, there are b ways the second event can happen and, therefore, a b in all. Thus, if one has two coats and three hats, he can dress for the street in 2X3=6 different ways. Again, if two dice are thrown, each die can fall in six different ways, and together in 6 X 6 = 36 different ways. Or if one die is thrown twice in succession, it can fall in six different ways the first time and in six different ways the second time, making thirty-six different ways in all. The above law, stated for two independent events, can be at once extended to the case of several independent events. Thus: The number of ways in which a series of independent events can happen simultaneously, or in sequence, is the product obtained by multiplying together the ways in which each event, taken by itself, can happen. §11] INTRODUCTION 9 11. Arrangements or Permutations.—A concept of importance in connection with the theory of probability concerns the number of ways in which a series of n different objects can be arranged, or permuted, when taken r at a time, where r can have any integral value from 0 to n. To illustrate, suppose there are three letters A, B, C and we wish to take them two, at a time. Clearly, they may be arranged in six different ways, namely, AB, AC, BA, BC, CA, CB. Each of these ways is called an arrangement or permutation. The number of arrangements of n objects taken r at a time is denoted by the symbol nAr. On the basis of the preceding section we can prove the following formula: ..A, = re(re - l)(n - 2)(re - 3) . . . (re - r + 1) = -1——1 (1) (n — r): (Sec. 9, Eq. (2)), for we can select the first object in re ways (since there are re objects); and, after the first object has been selected, the second object can be selected in re — 1 ways; the third object in n — 2 ways; and, finally, the rth object in re — r + 1 ways. Consequently, the total number of ways of selecting r objects out of the re different objects is given by the formula. As important special cases of Eq. (1) we have the following: A -rol _ i A _ re! _ """"(re-O)!"^ "Al- (re-l)!""' ^^(n^T! ="("-!); A"=(^),=W! (2) The last formula gives the number of arrangements of re different objects when taken all at a time. Exercises 1. Find the numerical value of icoAs. 2. In how many ways can the letters of the alphabet be arranged (without repetitions) when taken five at a time? 3. How many numbers between 100 and 1000 can be formed from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, without repetitions / ' In case the objects are not all different, this formula must be modified. Cf. any text on college algebra. 10 MATHEMATICS OF LIFE INSURANCE [§12 4. In how many ways can six boys and seven girls be arranged in a class, provided all the boys sit at the left of the instructor and all the girls at the right? 12. Combinations or Groups.—A set of objects taken together without regard to the order in which the objects occur is called a combination or group. Thus, AB and BA are two different arrange- ments but the same combination or group. If three different objects are taken two at a time there are six different arrange- ments, as we have seen in the preceding section, but only three different combinations, namely, AB, AC, BC. Each of these combinations gives rise to two arrangements, hence six arrangements in all. The number of combinations of n different objects taken r at a time is denoted by the symbol nCr. We can now prove that C -——n[——. (\\ ""' ~r!(n-r)! '/ For »Ar is composed of all the arrangements that can be formed from each of the nCr combinations when the r objects contained in nCr are taken all at a time. Hence, A - r A nr r -nAr- n[ »A, - n^r ,A,, or „(.,. - ^ - ^ _ ^, As special cases of Eq. (1) we have - n*__ _ , -, _ n! _ ., _ n! Bc<) =0\(n - or. ~ l;nLl- l!(ra-l)l - ; " 2 - 2!(n - 2)! -^^-nl^!-1- ^ From Eq. (1) we see at once that »Cr i= nC/n—r. (o) Exercises 1. How many committees of five can be selected from ten men if three of the ten can serve only as chairman? §13] INTRODUCTION 11 2. In how many ways can a crew of eight men and a hockey team of five men be selected from twenty candidates? 3. If ten points are placed in a plane, no three in a straight line, how many straight lines can be drawn joining them two and two? How many different triangles can be constructed with vertices at the given points? 4. In how many different ways can the letters of the word mobile be arranged in a straight line? Around a circle? 13. Binomial Coefficients.—The numbers nCr are binomial coefficients, since they occur in the formula for the expansion of (P + ?)" when n is an integer, for (p + g)" = (p + q)(p + q) (p + q) . . . to n factors must contain the product 'pry', since we can select p's from r of the n factors and q's from the remaining n — r factors. But this selection can be made in exactly as many ways as there are combinations of n things taken r at a time; that is, in nCr ways. Hence the expansion of (p + ?)" contains the term nC'rp'y and must consist of a sum of such terms where r is an integer ranging in value form 0 to n. We indicate this fact symbolically thus r==n (p + <?)" = S^w. (i) r=0 Replacing nCr by its value (Sec. 12, Eq. (1)), we have (P + 9)" = S^K^^- (2) Expanding the right member of Eq. (2) by giving to r all the integral values from 0 to n and adding, (p + 9)" = 9" + »W~1 + n n p2(^'-2+ +wp'-lg+p". This is the usual form of the binomial formula, except the terms appear in reverse order. The terms nC'r-lp'-1?"-^1, nCr^y, and nCr+lP'^'g"^-1 12 MATHEMATICS OF LIFE INSURANCE [§13 are consecutive terms in the bionomial Eq. (3). The ratios of the first and the last of these to the middle term are important in what follows.
nCr-l?''"^""^1 ^ ? j? nCrVy n - r + 1 ' p nC'r+iy'^'g""'"1 _ n - r p_
(4) (5)
nCrpV r + 1 q The numerical values of the bionomial coefficients for the first few values of n are given conveniently by the so-called Pascal's triangle. n = 0 1 n = 1 11 n= 2 121 n = 3 1 3 3 1 n = 4 1 4 6 41 ra = 5 1 5 10 10 51 n = 6 1 6 15 20 15 61 n = 7 1 7 21 35 35 21 7 1 Any number in the triangle (except the first and the last in each row which are always units) is the sum of the two numbers above and immediately to the right and left of it. Thus 35 = 20 + 15, 20 = 10 + 10, and so on. The triangle is easily extended as far as may be desired. If r (or n — r) is a small number, the value of the corresponding binomial coefficient can be computed without much labor. Thus, „ _ 100! _ 99 100 _ .„,„ lvs 2 - 2W ~ —2—— - 100!" 98 99-100 _ 100^97 = QTlql = —————fi———— — ^Oli'Ov. Exercises 1. Write the eighth term in the development of (p + g)20. 2. Find the ratios of the seventh term in the development of (p -{ q)30 to its two consecutive terms. 3. Which term of the development of (% + %)7 has the greatest numerical value? Which has the least numerical value?