74 MATHEMATICAL THEORY OF INVESTMENT Dividing this equation through by (1 +1')2, we obtain X(l + i~)~i + X(l + i)-1 =1000 (l+^)-i+ 500 (1+»)-2, which is identical with equation (1). Again, if we multiply equation (1) by v100, we obtain an equation of value in which all sums have been discounted to a date 100 years ago. The equation would be unchanged, however, except in form. The equation of value in its most general form would be an equation expressing the fact that a series of sums X^, X, - ; X^, due in n^, n^ ., n^ years, respectively, is equivalent to another set Yy 3^, ; Y,, due in m^ m^, ; m, years. Discounting all sums to the present, we have for the equation of value ;Civ»i 4. Xsv"^ + + XyV"^ = Yiv^ + rgt'"^ + ... + TsV"1!. (2) The principle involved in the equation of value should be thoroughly mastered, for it forms the basis for the solution of a very considerable number of important problems. 29. Equation of payments. The term equation of payments, or equation of accounts, is used to denote a rule for determining the time at which several debts, due at different times, can be equi- tably discharged by the payment of a single sum equal in amount to the sum of all the debts. The rule is found by solving an equation of value. The time thus found is called the equated time. PROBLEM. To determine the equated time for the payment of several debts due at various times. Let s^, s^, Sy, .,_ s,., denote the sums due at later dates, and »i, n^, n^ ; w,., the times to elapse before they fall due. If n denote the equated time, and all sums be discounted to the present, the equation of value will be (8! + ^ + + ^O v" = slv"l + ^ + V"' + - + Sr^- (1) Solving this equation for v", we find ^ «it>"' + s^v"' + + s^ «i + s, + + s, INTEREST 75 Solving this exponential equation for n, we find „ _ ^g C8!^' + ^ + + srv"r]- log [«! + ^ + + «J ,^ logv If we replace vby its value (l+z')~1, equation (2) may be written in the form 10gl»l + 83 + +«r] — log [8iV»l+ .S3V»i + + Sr^M /q^ w, ^s ' ' ^^^^ ^^^"^ " '—'' '""' < 0 ) log(l+») v y The equated time will be computed by means of formula (2) or formula (3). The ordinary rule for the equation of paymentsis obtained by sub- stituting in equation (1) approximate values for v", v\ v'\ , v"r. It is stated as follows: Multiply each sum by the time to elapse before it becomes due, addthe products, and divide the sum by the sum of all the amounts to fall due. To obtain the formula which gives the rule, consider the expansions ^(l+^)-n=l-n{+^-n\^n-l\2+..\ -L 2t ^(1+ z)-.=l-^-+ (" wl)("nl-v) ^+ » 1 it ^^(l+,)-.^l-^+<-.^<-Mr-l\2+. -L u Dropping powers of Ihigher than the first, and substituting for v", v"i, r\, v"'-, the approximate values, 1—m, 1—n^i, 1—n^i, , 1—n^i, in equation (1), we obtain the equation (^+ »,+ 8,+ ... + «,) (1- ni)=8i(l- n,z)+«,(l- n/) +...+8,(l-n,i). 76 MATHEMATICAL THEORY OF INVESTMENT Solving this equation for w, we find _ HI^I + U3«3 + + nr»r_ ,^ »1 + «8 + + «r v / Equation (4) gives the ordinary rule that is used in practical work. The ordinary rule favors the debtor by slightly increas- ing the equated time, though it is reasonably accurate where the periods of time involved are short. (See Todhunter, " Institute of Actuaries' Text-Book," Chap. II, Art. 8.) EXAMPLES 1. What sum due 9 months hence without interest will be the equiva- lent of three debts, of $500, $400, and $700, due 6, 8, and 12 months hence, respectively, when money is worth 5% ? 2. A man owes the following sums: $500 due in 6 months without interest, $700 due in 1 year without interest, $600 due in 2 years with interest at 5%, payable annually, and $300 due in 1 year with interest at 4^%, payable annually. He wishes to arrange to discharge his indebted- ness by three equal annual payments, the first to be paid 1 year hence. If money is worth 5%, what will be the annual payment ? 3. A man is offered $2000 cash and $1000 at the end of each year for 3 years for a house and lot, or $1250 cash and $1250 at the end of each year for 3 years. Which is the better offer if money is worth 5% ? 4. Find the difference between the equated times by the ordinary rule and by the exact rule for the following sums, due without interest, when money is worth 5%: $600 due in 1 year, $700 due in 3 years, $400 due in 2 years, and $1000 due now.