wmmm'^i''^^ ^, ^ IMAGE EVALUATION TEST TARGET {MT-3) Z^ % /, // :^\^ /. is \ Z/. ^ i 1.0 I.I m. 125 £f b£ 12.0 u I' i L25 iu 6" I 16 i^lographic Sdences .Corporalion ^ \ 33 WIST MAIN ST^HT \yiBSTIR,N.Y. I45M> (71«) •73-4503 '^ ij.-* • *■^A'^^*^•^ iV"? CIHM Microfiche Series (i\/lonog raphe) ICMH Collection de microfiches (monographles) ■ • y Canadian Institute for Historical Microraproductions / institut Canadian da microraproductibns historiquas \ ^ O^ "*.-. ,■; «jU'ityio-is ■J-.i.i »t K VJ Technical and Bibliographic Notes / Notes techniques et bibliographiques The Institute has attempted to obtain the best originat CQPV available for filming. Features of this copy which may be bibliographically unique, which may alter any of the images in the reproduction, or which ma>i ^ significantly change the usual method of filming^ *n checked below. L'Institut a microfilm* le meilleur exemplaire qu'il lui a M possible de se procurer. 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'■. / .- ' . / Un dee aymbolas suivehts spperettre sur le dernlire image da cheque microfiche, selon le cas: li symbols «-^ siftnifie "A SUIVRE". le symbols ▼ signifie "FIN". / / Les cartes, ptonches. tableeux. etc.. peuvent itre filmas a des taux de riduction diffirents. Lorsque le document est trop grand pour itra reproduit en un aaul clicha. il est film* a partir de I'engle sup#rleur geuche. de gauche i droita. et de haut fn baa. en prenant le nombre d'imeges nacesseire. Les dlsgrammas suivsnts illustrent le mithode. \. 2^ ' \j 6 32 X .Ws'Mk>2;^i&~.i.u^Mi.i4l 1 ,•_ A ..u, i -ai-.*' '■\^L:^jm^'^ t , a^^jiL >(lm* £f^. >j&jS&M!fi^jM!.>^LkJ^j''^rf t WS^' 'I *''%^,'£|5fT.rv'«# '\ WA-FsdN'S • J r .^ .. COPOOKTXrNtERE&T AND ANNUITY, " • ^ Loan and t^timTioN Tables •>^^ '^1 MORTGAGES, BONDS, pEBg|rUR£Sl|R/NNUITIES.T . '-{■liV-Jv. J:. mw nftnoir, obuatlt fHiaesD. ■!.''i* » i fiW .»Y- ,WILLrAj\| E *■ t ¥ lA_ 't ■- .4- 4 4 Ft# <:»«'»** *«^ # '^sT ^^' * \ ' ■ v. V ■* . V ■% ^ Entered according to the Act of the Parliament o( Canadafin the year one thouaand elyht hundred and eighty-four, by Winr^« E. Wawon, In the Office of the Hialater of Agriculture. OUIM.EV ft auntm s^fei.'tii."' .5'- /^*H 0-. PREFACE \ ■\ >-' «>»So«- The following Tables embrace those prepared by James Watson, Manager of The Peo})le's Loan and Deposit Company, the first edition of which has been exhausted, hut the stereotype plates of which have been secured by the compiler of this edition. , The former edition contained 22 rates, but the present work has been ex- U tended so as to embrace pi rates, and the new, together with the old rates, in seveVal of the Tables, have been extended to 45 years on the half-yearly, and 70 years on the yearly basis. '' There have been also added two additional tables, showing the amount of $1, and the amount of an annuity of $1, for 50 half-years, and 50 years; and a complete list of all tire nominal yearly, and half-yearly rates, showing the true rates of these, convertible yearly, half-yearly, quarterly and monthly, and the Logarithms of these true rdites, to 10 places of decimals by the use of which arty item in the Tables may be verified, and new problems beyond their limits solved with the greatest accuracy. There have been added besides, complete formulae for the calculation of nearly all questions which can arise involving compound interest and annuities, together with examples showing the practical use of Logarithms in such calcu- lations. ' • The compiler hopes that the work will thus be found a useful manual for all institutions or individuals dealing in Bonds, Mortgages or Annuities- Life or otherwise. The Tables have been prepared and revised with great care, and may be confidently relied on. \i.j&?.,o. \i IP- ':■ "" J ^ EXPLANATIONS AND EXAMPLES. INTEREST. The number of rates pf interest included in the present edition Is 38. and comprises whole ' ^1^ :T T '""^'^^ '° "^' •^^•'"'^'"S ''y -'-^- -<^ 'h' results of the Tate compounded year/y and half-ytarly, are shown. • whi Jt^ '"" T^ of these nominal annual rates are exhibited in a classified form on page 17, <«dh of these 38 nom.na rates. Along with these true rates is shown the L<,garith,« of eaph ^1J:2: '. - TJri'tl '"'""'* '"'* "" '*'' "^"^'"""^ °'"'"' "'" increased by u^ty v^r/vtrTHTT""*^"^ >i«//^>'«rO' are equivalent to one-half of these rates compounded >«r/, for double the time as regards Tables I and VII. Thus 6 months at 3% half-yearly is rrppira^uar^ -' ^^^ ^^"'^' -^ • -^^ ^^'--^^^^ '« ^ — -- "- In Table IV. Half-yearly f I I I ^ . P*' '"'"■y'=-- half the given rate compounded yearly. " ^^"^ ^^"' *' TABLES. Table I-Gives the /»r««,/ va/u, of a ««f/, >y,;^«/ ,/ ^,, due at tht end of any month Irom I to 240 {20 years), at the various rates of interest designated at the top of each page, computed yearly, and half-yearly. It also shows the /r««/ va/u,U of a smg/, paymmt 0/ Si, due at the end of any half-year, from 40 to 90 half- ' TJ^ f 1° ^^ ''''*"^' *°'""' half-yearly, and of a sinole payment of $,, due at thfe end of any year from 20 to 70 years, interest yearly, at the various rates mdicated. . Table II-Gives the presmt value of a monthly instalment of$i, payable at the end of each month from i to 240 months at same rates. id of I Table III-Gives the present valut of a quarterly insalment of $1, f,ayable at the end of each quarter from i to 80 (20 years), at the various rates given, when the first instalment becomes due at the end of the ist, 2nd or'3rd month, and the second, or corres- ponding mstalment, at the end of the 4th, 5th or 6th month, and so on. Table IV-Gives the/r««,/ value of a half-yearly instalnunt of $1, payable at the end of each half-year from i to 40 {20 years), at same rates as above, compounded half-yearly and yearly, when the first instalment becomes dtie at the end of the ist, 2nd 3rd ^ 4th, sth or 6th month, the second at the end of the 7th, 8th, 9th, loth, i nh or 12th '"O""' respectively, and so on; and is continued showing the present value of in-„ ^_ stalmema «r the «wfwf/ji' of each half-year, from 40 to 90 half-years, at rates' com- pounded half-yearly. USES OF THE TABLES. g Table V— Gives the prcntt value of a vearh in^inln,^, .f « P^'d at the end line ^ipV^^^'':::j^:TrS:X:^il^T^^^ T^I^K^'"'' -•" •^^ - '»•« sa^e and multiply this factor by the sum of the Mortis ""^^ ''^ "^^ '*^''"'''"' ^^''^ °f '"'^^est, value is desired. ' °' '"* Mortgage, or other payment, of which the present at ^oiZ^^^f^Ztyt^'X''^:^^^^^ 7 months hence, interest .24098 X iooo=$240.98, the present value! . ^* ^'^^'^ ^ months is .24098 and after!-nu^brr^V5LroTl;tk^?;e^^^^^^^^^ mtertst, compounded half-yearly or yerrl^ ^ * '"°"'*''' "nd bearing any rate of and millSylhe pr^pZ't'^: FJ^T xtrbv^lahT Vv''^ ^ «^"'* P^"-' ^"l"''-'. the interest or coupons are payable ha f^earlVor'v.Jrlfi*'^^ ^' *' "'*' ^''^ ""y ^i: thai half-yearly. • y*^*"^ »"«" 2 months hence; to pay 9%. compounded ^' "J 4^'^""^ P"""' ^'^'"^ °f $' ''^^ '34 months hence at 9%=.374.7 ^"'ra'tUtleJ'^ 'Theirs^. \"-^^»'-nts of $, ^^^ ^.,e ,„,« and ^'^^'^^ ' 14.56^x1^= 1^ interest on $4000 at 7%=$i40. and ao3<) 66 The present value of the Mortgage is $3536.34 '' >' USES OV THE TABLES. \ . ■ interfsfar6y%™„„teri'r "^ 5!'o°'i'»^i"g '9 years and 2 months to run. and bearing ByXable I, present *alue of $i, due 230 months hence, at 5%=. 38808 x 100 And by Table IV, present value of ha'f-yearly instalments of $1 for same time— 25. 141 X 3 (value of half-yearly covpon^= • . • . $38,808 ' V. The present value of the Debenture is $114,231 ner aljf/^^r""' "^ l'"" ^^ *"'' ^ "^^^ °' Debenture may be calculated to pay the purcha- wUl bi^stn St "? P";«=»'"=^*! -"""^y. notwuhstanding the rate of the security being different. It l«s than fu ki purchase, gets a h.gher rate than that b.,me by the security, he will pay ifthe ra.i .^r^ . '';.*"'L'^ ^^ '^'''"^f \'°*'^^ •"*'•= •"= *^"' P''/ ^'^"^ 'han the face value oh.-in/h T ^ "^"^ Mortgage or Debenture be the same as that which the purchaser obtams, he wll pay for ,t the par value. This m.y be shown by .he tables as in the above cas«! l,.„i„^^*" '■''''■ ^rl.^'''"' '* the present value of a Debenture of $100 due 18 years hence having coupons of $8 each payable yearly, to pay 8% yearly? ^ " And by Table V, p. v. of yeariy instalments of $1 for same lime=9.37i9 x 8= 7J.975 The present value of the Debenture is $100,000 months. ^° ^"^ ^^^ *""""' ^° ***'"*' ^"^ *"*" "^""'"^ accumulate after a given number of reauiS'^^'Th^n?"; ^^. "'f. T'^u"' ^'''"^ °^^' '^''^ ''' ^^^ «^"d "^ '^'^ '"">. ""d at the mte required. 1 he quotient will be the amount. amoum r Is yeTr?an"d'j montht? *'°°' "°" """•=' *' '°^° '""=■■"' compounded half-yearly, answM ^"'''^ ^' '''* P""^^*"' ^^'"« of $'. due 223 months hence = . 16310 .-. -2^= $613. 12, For any even number of half-years or years up to 50 respectively, Table VII eives the amount of $., and this factor multiplied by the given^um'will gl^e its amount for samf^riod 4. To find the present value of ^ny Instalment or Annuity, payable yearlv hatf-veariv Ton h^ '^ 7 monthly, during a given number of years, or a broken ^riod TyVaVs and som^ months, at any rate of interest given. ' Find the present value of an Instalment of $1 for the proper time and rate in the Table corresponding to the periodic payment, and muhiply this factor Cy the given Insralment A r^.^^u^,^^^ i.-A Mortgage payable by monthly instalments of $20 each, has 8 years and 4 months to run What is its present value, interest 10%, convertible half-yeariy ? ^ 20=$f,36t2VanIw'er^''''*'"' ' "" '"^'''''"'="' °f $' f^' i°° months at .o%=$68.,64x . ^^^.^1,'^.*"''"''' ^T^ft *^°"g^e. payable by quarteriy instalments of $25 each, has 8 yeirs and I ™o"th .0 ^u„ What IS Its present value, interest 9%, and convertible half-yearly By I able III, the present value of a quarterly instalment for 8 years and I month lie 11 instalments, first due one month hence), at 9% =$23-545 x 25 = $588. 6 "answer. ^^ to n.n^^"'".';'' ^T^ Lease, payable by half-yearly rents of $60 each, has 7 years and^ months" it ro%!ron'vLrible y^;'? '^" '""^ '"'^'^ '""""'^ •*•=""'• ^'^^^ '^ ''^ P^-' value.^nteres? at i^Iyl:S;2${^)\r:ZV£l$;^^^^^^^^ "'^•^'"^'^"' °^ $■ f- 7 years and 3 months Example 4.— A Mortgage, payable by yeariy annuities of $210, has l? years and 1 By Table V, (he present value of 16 yearly instalments of $1, last instalnient due L months hence, at 9}i%, annually=.$8 6301 x 2io=$,,8i2.j2i. answer '"''*'"'*^"' '"«' '^3 5%. c^omp"unde Tables Iir, IV an4 V the present value of a quarl^erly, hall yearly or yearly payment of Rent, or Interest on Mortgages or Debentures, can be determined to pay *hy of the rates pven, and in the case of Mortgages or Debentures the present value of the prirt^ipal may be found by Table I, and added to that of the interest as in the examples of § 2. 5. Assuming that a Mortgagor has arranged with the Mortgagee to prepay his Mortgage, or a portion of same, in addition to his usual annuity. To find how such a payitfent would affect equital)ly the subsequent annuities, as to amount, or as to time. . , ■■ Example i.— A Mortgage, payable by monthly instalments of $20 each, yields io>i% interest, convertible half-yearly, and has 7 years and 5 months to run. The borrower wishes to pay down $600, and to find how long his instalments of the same amount must continue to pay off the debt. By Table II, the present value of a monthly instalment of $1 for 89 months =62.101x20 $I24a.oa Deduct 600.00 _,,.,. Balance $642.02 Dividmg t IS by the amount of the. instalment, viz. $20, will give the value of an instalment of $1 for the necessary time- -642.02 -r- 20= . . 32. 10 - And p. v. of a monthly instalment of $i for 37 month8= . . . . 31.59 (nearest amt. below) Difference on $ I instalment = 0.51-* jo=$io.20 The time therefore would hi 37 months, and $10.20 additional cash to be paid now ; or if postponed till 38 months would be (3) $io.20-r.72320=$i4.io, to be paM as a last instalment. Example 2.— A Mortgage, payable by quarterly instalments of $&> each, '■ Deduct 300.00 I Present value of balance = $717.30 and this amount, divided by the present value of quarterly instalments of $1 for 5 years4tnd 5 months, viz. ii.Vn=$42-3''; «?, Second method. — Divide the amount paid down by the present value of an instalmjnt of $1 ' for the period to run, and deduct the quotient from the former instalment for the new instalment. Thus, in the above example, amount paid down\= $300, present value of instalmenU of $1 for 5 years and 5 months=$i6.9SS ; then 300-f i6.95S=$i7.69 ; and 60- I7.69=$42.3i. 6. The a<^teP °^ * percentage to the amount loaned for the whole term, and that amount divided «»pe number of the instalments to be made during this period, yields a variable rate of interest, mfCdrding to thft time for which the loan is made, and the number of instalments; —monthly yielding a belter n»te than quarterly, and quarterly than half-yearly. Example i.— -A Borrower receives $1000 cash, at 6% for 10 years, to be repaid by mftnthly instalments. To the $1000 there is added interest at 6% per annum for 10 years=$6oo-|- 1000 = i6ix), and this amount is divided by the number of payments, 1600-7-120=13.34. It is required to determine the rate of iirterest half-yearly which this investment yields, ByHTable VI an instalment of $13.11 will repay $iooo in 10 years at 10% half-yearly, while 10 "4% would require an instalment of $13.37 The rate would therefore be between 10% and io>i%. Example 2. — A Loan of $4000 on same terms is made for 5 years. Required the rate this investment produces. To 4000 add 5 years' interest at 6%=$i20O-H4O0O=52oo-r 6o=$86.67 monthly, or per $iooopiSo" X 86 67=$2i. 67. By Table VI, instalment to repay $1000 in 5 years, at 11%, haif yearly=%2\.6l, an^at =Mji%.>i%. yearly . '$3. 7453. Vhi\'aTeT?h'u^'4>ii; JeaHy. '^"' "" '""°""' °' ^' ^"^ ^o years a.. sum.®a, a'S'ven'te'"' -' """^ "''""'^ ''" "^^ '^'^"'^'^ ^^ '» accumulAe to 'a larj^er given .r .L nearest give^l^i'lH^r^r^ Slcr::^^^:^;^- ^^ ^ ^ — • yearly.ToTl'um tXlTSfslo?' '?^x?i°°°: '-»""««"--•« ^t 'he rate of 6%. convertible ^r $ra. 6%. yea'ri;. ^'^t^Li^:!^^ ^^:^^^;^;^ ^^^'« VII. the a.o.nt ' ^cJlTel^ST^'r ""^I w"'C.^'^""1 '''^ "'"'^^ °f *hese Lies, or a more accurate result be required by additional decimals, the. following formulae are added a^d the true rafes for one year, half year, quarter annonth of aTl the . noratna rates of interest, compounded half-yearly And yearly, are given on u^L^e 17. together with, their logarithms, to 10 places of decimals ^ ^^ 9- Formula: of Compound Interest, with or without the u§e of logarithms. I-FOR A SINGLE SUM OF MONEY . Let P=the principal; or present value of an amount M. - ' A/=the amount oT this principal at the end of a given titfte ; or the sum due at the end o( a g.verv. time, of ^hjch we wish to finflTh^ the present value. ^ - '"'period"' "" ^'' '^'' °'' °"''' """'^ ^^' °"^ y^^^ °^ other given r=( I + /) the sum of $1, £x, or other unity for .one year or ogriod «= the whole number of years or periods. Then i^or the amoifnt' For the ra/ = log. M - n X log. r. (3) IM ^ =■ v-p- or log. r qs log. M - log. /> For the nukber of years or periods log. /»/ - log , p log, r « . Then r - i gives /or the rat^of interest for one year or period. (4) \r^ or « = ai (• JOI / 1 -i— Tf epay it iffe if twtween it "be found e end of a ,and 12%, lid for the e,,an(l the d the rate . o years at. ■j||er given :olumti of divisiqn, [1. )nvertible e alno^nt ' a more ulae are all the on page iritluns. he sum nil the r given iod. ^ - ' . . \ ' USES 6V ri4E TABLES. « , . • 9 i}--^S RELA-^NQ TO ANNUITIES. Lev/ =, the /rue inieresl on i lorsitigle period of the annuity. r -= ( I -h /) the a;^«;// of I for one period. '^ . « = the number of periods or instalments of the annuity. W = the amount 6( the whole of the annuities uf the end of n periods. K (_ \\\c principal ; or \\^g present value of the annuity of,$fl or £a for ^ « P5"uds (each annuity being assumed to be payable at tne«i^ op' Its own period) " ,* <* = ^^%.\^^tin\. being made '' at tn% end oi // + i periods. '. , - d = the /j«w^/l'o find the //-«(r;;/ 7'(r/«^ of any annuity. ~ • • (5) ^= jy--^ ' orlog.A'=log. ji -y}j + log. a - log.// log: -^ = ^ f o — n X Jog; r. ' , , or(6) F,=:^-^^> orlog. ^'=log.(r--i) + log.a-#g./_„xlog... - V rt a a ■ or (7) K = - - — or log- - = log. a - log. /; and ^ , . . - ■ a \ ^ ■ ^ log-77R.= log. rt - log. / - ;; ^ log. r. „ To find {hejimount of any annuity. ' "" . * (8) -^ = — I , - ^^1 or^og. A = loK. |i - ;„| ■+ log. a + « ,c log. r- log A Of (9) // = 'J (,-_ ,j or log. // = log. (r»- ,) + log. a -log. /. orOo) // = J-- - ,orlog. — = log. a + n x \og.,r - log. /; and / . , a . '• -> 'og- 7= log. a,r- log. /^ \ To find the annuity which a given sum ^ will purchase. yt (") « = ,_i or(i2) a = ^v-i es^or Tiod. or log. a = log. y + log. / - log. (-:■)-■■■■: or log. a =iog. f' + log. 7 +»« ^ log. r -log. (r»- i). The period and rat^ bein^ given, tb find what annZTty it would take to amount to a given 3um (^) at the end of a certain number of periods. .At -^^^ ** ~ r» At - -1 ""^ '°^' '^^ '°^' ^"^ "^ '°^- ' ~ ? " '°8' '' - 'og- (i - -) " ^ (I4> tf^ =^^ — lY ^"^ 'og- « = l<«. ^ + Jog. /' - leg. t r*^T) — i I. .. m i5' ifil;; ;,'i III ■ Mi^ nmm^ „ 10 USES OF THE TABLES. T(i find the number of annuities. (15) r' = a- Vt or (16) > = I + dl a then « = log, a - log, (a ~ Vt) ft = then log. r log a - log. ^ log. r „ ^ log. (i + -') or, as 5 = a - Vt, d X log. r (first find log. V by log. r To find present value of a deferred annuity. (17) . -O = ;s or log. D = log. V form 5, 6 or 7). General formulaeapplied to interest compounded, or convertible into principal, half-yearly, quarterly, &c. 10. The following Tables are based upon interest convertible4'. |i + -1 or log. M = log. P + 2n % log. (i + -j (19) For interest convertible quarterly. M = P.\i+A or log. M = log. /> + 4w X log. (1 + (20) For interest convertible monthly. yI/ = /'. i+ — I or log. M = log. F + im ^ log. (1 + —) To find the true rate of interest for any given period of time. Let r = I + its true interest for one year. By considering/ as i and n as I, M will then represent r. Then raise this formula to the power represented by the number'by which i year would have to be multiplied in order to produce the given time for which we wish to find the rate. The result will show i + its true interest for the given time, and this result minus i will leave the rate of interest. Ex I. — What is the rate of interest per 2 years at 10%, copipounded half- yearly? Here 1 year has to be multiplied by 2 for 2 years .-. we raise the formula to the power of 2. Then r»= j( i +— ) [ or r» = 1.05* = t. 2155 rate = ^2155 per 2 years on i, or 21.55%. Ex. 2. — What is the rate per month at 10% convertible half yearly? Here I year is multiplied by the fraction ^ to make i month .-. we raise the formula to the power of ^. Then ;Vt= |( i -i- -^ H " = i.o5« x t'j = 1.05*. We then raise j.oj-to the power of i = 1.05, and extract the 6th root of the result 6 VidSfr^ i-ooSf&f . . Bate per month lated true rates). =«f^;Sf64 . .% |see fabtp- -'f USES OF THE TABLES. 11 Ex. 3 — What is the true rate for 7 days of 10% convertible half-yearly ? Here I year has to be multiplied by J Jy to make 7 days. •.«?«= ■!/ i + — I [ * = 1.05* ^ sJt= i.osf'sV. Here we have to raise 1.05 to the i4ih power and then extract^e 365th root of the result, or by logarithms, multiply log. r by 14 and divide the result by 365, and then, taking the corresponding number, rslt = 1. 00187 • • R-attJ per 7 days = 0.187%. Single payment Examptes. „ Ex. I (Amount, Form i). — What is the amount of $527.75 put out at com- pound interest for 34 years at 4%% yearly? P=. 527.75, r= 1.045, 'o8- ^ — 01911,62904. « = 34. ' log. 527.75 = 2-7224382 34 X log. r = -6499S89 ) " log. M = 3-3723821 = $2357.12, answer. By Table VH, 4.4664 x 527.75 = $2357.14. Ex. 2 (Present value^ Form 2).— What is the present value and the discount of $3600 due after 7 years, the interest being 6% convertible half-yearly ? M = 3600, r = 1.03, log. r = -01283,72247, « = 14 half-years. log. 3600 = 3-S063020 I4 X log. r = -1797211 leg. P= 3-3766814; /* = $2380.02. Discount = 3600-2380.0.1, = $1219.98. By Table I, .66112 x 3600 = $2380.03. Ex. 3 (Rate, Form 3). — A Borrower returns $5000 for the use of $1335 during 30 years. What is the rate per cent, yearly? M = 5000, P = 1335, 9^ « = 30- log. 5000 = 3 6989700 'Og- ^335 = 3-1264813 By Tables, rate = 4}4 log. yso — S734887 and -r 30, ^rp,- . log. r = -0191163, r— 1.045, '■''ts = 4/^%- See Example § 7. Ex. 4 (Time, Form 4). — How many years must $3000 be put out at 4% interest, compounded yearly, in order to amount to $102,358. Here M = 102,358, P = 3000, r = 1.04, log. r = 0170334. log. Af = 60101218 log. P = 3 47712 13 log. if/- log. P = 1-6330006 and -f -0170334' = 90 years. This example is beyond the limits of the Tables. j Ex. 5 (Amount, Forms 18, 19 and 20). — What will $1 amount to atlthe end of 50 years when put out at compound interest at 8%. t,he interest being con- vertible half-yearly, quarterly and monthly? Here P = I'.oo, / = .08, n =. 50, li-H-l = i.o4>''», I -I- I = i.oz'oo, li +— I = i.pod'oo. 8% half-yearly. (18) log. /'^o- 100 X log. 1.04 = 1(^038339 log. M= 17083389 8% quarterly. 8%. monthly. (19) log. ^=0- (20) log. /' = o 200 X log. 1.02 = 1 7200344 600 x log. I o} = 1-7814139 i/-: $50. 5049 log. M =17 200844 iW'=$5»4849 log. M= 1781 4189 A/= $53.8782 ■'^•"'X'r i * ii I. I *Y 12 USES OF THi TABLES Annuity Examples. Ex I (Present v;,{ue, Forms 5, 6 and 7;.-What is the present value of 20 h.lf-year^y mstnlments of $, at ,0% interest convertible half-yearlyT Here i7\ ''~\ '■°^' ^■°^' ^^; := 021X8,92991. log. /= 2 6989700. « = 20 periods. Kl) log. 1=0- (6) 20x|og.>= -4237860 (7) • log. r» •> = . .4337860 '"'" = 2.65330 log. /= 2 6989700 r'^"- 1 = 1.65330 log. {r-i» - i) log.- a = •Clog. r-"' = 18762140 Clog. / = 1-3010300 log. r= 10968960 ^=$12.4622 log. //■*«= 11227860 2183610 log. a = o. - log./r * o = 8772440 . = $7 53779 Perpetuity or M I , = = $20. t .01; * 35 $12.46221 (By Table IV, 12.462 = p. v. 20 insulmenls). 20 X log. r= 4237860 '°8- ;nrn = 1 6762140 ^■!o =-3^68895 I I -^ = -6231105 '"g- (^-7;^) =1-7946660 log. <7 = 0- = r7946660 log. / = 2^89700 log. F= 10988960 ^=$124622 1 1. From Form 5 it will he seen that the presait value of an annuity of $1 mav be found by d.v.d.ng the discount of a single payment of $, due at the end of the required number pf periods by the intere.st on $1 for a single peiid of such annmty, and the present value of any similar annuity can ttn be found b5 multiplying the result by the annuity. ^ °' ^^ Koi. ^'^-2 -Required the present v.lue of an annuity of $1, payable yearly !;alf:j:;';ly'>'""''^^'^ '"' """'"^' '^^ ^° ^^^^-^ ^^ 6%'perannumfco'7ertible By Table I the present value of $1 due 20 years hence = .^o6s6 .- the dis- i v"ear - o'6iV: • 1? "" '" '"'^"''^^'^^ "°'"'"^' ^"'^ ^^"^ '^'^^ '^e interest f^i .69344"^ ^ ' "'^'^■>'^"^=°3 ; quarter = .0,4889.. ; month =.0049386.. Then .0609 =$"-3866 = p. v. 20 yearly instalments of $1 (see Table V). •69344 ,. - .03 -*23ii5 = P- V. 40 half-yearly instalments of $1 (see Table IV). ■ .69344 .01^889 = ^46.574 = ?. V. 80 quarterly instalments of $1 (see Table III). •69344 ^ .0049386 -* '40.41 i = p. V. 240 monthly instalments of $1 (see Table II). As shown by Form 9 the amount of an annuity may also be found in a s.mila*nanner by dividing the interest on $, to the end of the requ red number theTnuit^" ''' '"'^""' of $1 - by the interest on $, for a s?ngle periS of * C is a contraction for arithmetical complement. It is the algebmic remainder after subtracnns the number from zero or o. By using the O we save subtLions.ThcZd tionTf U to a number hav.ng the same efrectas subtracting the < yr i^inal^mb.r.4h.^:-;-;;';^"^ P •■•■wi ii'^'iii niiM USES OF THE TABLES. 13 tfi> ^$i^:Sf ITid^ . - '^^ ^"^"""' °^ ^' ^' 3% yearly for ,o years nuit eit^-^U. ^f :°/ ^'^i^ '■! '"'^'"''- '^° fi"'^ the amount of lo yearly an- nu.t.es o. $1 at 3^ yearly, divide .3439 by the interest on $1 for r Jenodi" year or .03 ; ^;^J^-= $. 1.463. (See Table VIII). Ex. 4.— (^//a;/////, Forms 8, 9 and 10). convemi:;: u-al^v V^ °' ' ^"^"'^ ^""""^^ °' ^'° '^^ - ^ears, at^% interest, •I a =10; //=2o; / = i.o3; ^=.03; jog. r = 01383,72247 ; log. / = 2-4771213 (8) l.'g..=,0- (9) 2o.log.U .2„„4« (,o) lo-,a=.i. 20 X log. r= 2M7446 r^-rH^^ .' '^-I.aoOn 2oxlog. r= 206744S ^-t= ^06 1 1 C l(jg. / = 18338787 log. ( ;^«> - I ) = 1.9063960 , a r"" ^3 7796333 'ogp!0 = i-74325BB C' log. / = 1-6238787 « ^ ^ ^ = -553676 I '<»g- ('-/«») =16496800 log. a = i C' log. / = 16338787 log. y»= ^3867446 log. ^ = 34392737 ^ = $268,704 Ex. 5.~(Annuify, Forms u and 12). ' poun^e^^^Sy^'"""^ '" '' ^"" ^•'" ^^'°3«-7S purchase at 5^%. corn- log. /=Vi»m3" ''" '•°'^'^^' '=-°'36S7; « = 6a; log r = = '•-4,38.. ; ,=.„„„38.. ,„g. . =^.„, log. fl - 0- log. (a- Vt)orS = U^^n^ lOfr "3;= 4343209 (% Table ir, 203 monthly instalments of $, = J.^S.oor). .Ex. 7.— (Z>^^^^^^^,^,,^^.^^ p^^^ ^^^ further term of 27 years in reversion aflprThf ^ ^""' "^ 9 years ; C has a .n perpetuity after "the 54 years What is thf^n^''"'';""? ^ ""'' '^^ '^'^^'^on each .nthe estate, at 4%. compounded Jearly^ ^ ' '"'"' °^ ^^ •"'^••^«' °f a-IOO; r_x.04; /^ 04; log-- ^1703.33393; log. / = ,oo20600; for A;« = n • forB,«=x8and^=9;forC.« = 27and^=27forD r -^ ' 9, petuity and d= 54 ' 27 , for D. r= - or p. v. of per- (6) 9 X log. r= 1633001 (6) '■"= 1-42331 log. /"- I =l^6266604 log. a = 2- C* log. / = 1 3979400 C log. r» = 18466999 log. F = 28713003 ^=$743-533 B 18 X log. r= 3066001 '^'^^ 2.02582 '-"- 1 = 1.02582 log. ;'»-!= ^^Hom log. a = 2 Clog. /■ = 13979400 Clog, a" = 16933999 log. r = 31024096 (6) 27 X log. r= 4599002 ^"=2.88337 '■"-1=^^88337 log. t^-i= 27mB* log. tf = 2 C' log. / = 13979400 Clog. ^" = rB400998 log. f^-Vmma (By 1 able V). ^I7^nxl«„ '"S- »- =3 2129782 ^ = $889,425 Z? = $566.337 P.V. o, $1 tor 9 years = 7.4353 - lOO = 743.53 = A's ah.re H " ^7 .. i^:^^9^ D = P-n.etu.t,- =2500.00 - 9 .. r43S3 -(A + B + C)= 2199.30 l!9^3JLioo=: 889.43 =B's .. D's share = $^300. 70 " 54 " = 21.9930 - 27 II = 16 3296 u __ A^3i!L'° ° = ^56634 = c' s 11 2199.30 loan of $i2» • A- = 0031896; >g. r) = 203 > has a tercn s ; C has a le reversion interest of 6rA,« = 9; V. of per- '= '4S99002 = 2.88337 = 27493S4 = 2 = 13979400 = 10400998 = 3 21397S2 = -4099002 = 2 7030700 $566,337 = 2500.00 _2299^3o = $^300. 70 , , USES OF THE TABLKS. . iK n .' '■ I?";'"^*''^ annuitiesmay be Considered as being composed of two distinct parts-the Interest and Sinking Fund. The annuity, in order to pay off the pr.nc.pal, must e;^peed the interest. The difference between the annuity and TJTT'^- v""t'''' Sinking Fund. If 5 denote, the yearly sum appropri- ated to the Smking-Fund. the amount of which will produce the principal, we ' have a=yf + S and from Form 5 we deduce directly a = T/ + 4 so that S=^ But a denotes the annuity and~the present value of the «"'^o'r last annuit*);,' therefore the Sinkine; Fund is equal to the present value of the jast annuity. half ^^-^—^ •"ail of $500,000 is contracted by a corporation at <7 nav ible to h'rn^^' '° be re.mbuhed by means of 90 halfyearly annuities. VVhat^sum is L to^n? " ' ^'"'''"' '^""'' '"''^^ '''' '"^^'«^'' for the purp :,se of redeeming r/ (in":::str=;2'so°oTth^e; = '■'''' ' ' = ■'"''-'" = ^° ■' '°^- ^ = ""^^^^ (12) 90 X log. r = 01^01479 log. .-= ^51479 a (annuity) = $14,01904 ' r -922886 log. F/r. 4-0969100 F/ (interest) = 12,50000 r"- I =8.22886 C'log.(/-««- I) = 10846603 ■S(sinkingfund) = $"77^7^ log. (r*'-l)= 9103397 log. a = 41487183 T o == *> «i»»7i8a To prove S=p. v. of a = $14,019.04 ' last annuity. ' log. a = 4-1467182 log. /*»= -9601479 (By-; .„ , , ,„ 500,000 \ ^^^'^^^'" 1^:666 =$'4,oi8.97J log. 5=3-1810708 5=$I, 519.04 The sinking fund is a little over }4 of 1% per annum, consequently an addition annual rate of about 6 mills on the $ would extinguish the debt in the period ot 45 years.. Ex. 9.— How many yearly annuities would be required to repay a loan of siSTnind ? '"'^''^''' ^^^^^'^ ^^"^^' '^ ^^*^'''°°^^ b^'"g added to form a c- L-^"i7^°°i°/°oi ''"'■°S5' '=055; log. '■=•02320,24096; interest ( f7)= 1 1,000 • Sinking Fund (5 ) = 2,000 ; a or (K/ + 5) =13,000 ' (14) log. = 41139434 log. 5=8 3010300 ' ' log. (a - 5)=n -8129184 and -J- -02820 (log. r) = 34.96 or 35 annuities (nearly). (200 000 13,000 = ^5-3846. In Table V, 35 years= 15.3905 and 34 years = 15.2370) Ex. 10.— A Corporation issues 2,000 Debentures of $500 each, beating nV interest, and to be liquidated by means of 30 annuities. What sum is to be ^id annually, and whafr number of Debentures will be redeemed in ^eh of ttiB== first 3 years, and also the 20th year? r= 1,000,000 ; /• = 1.05 ; / = .05 ; « = 30 ; log. r = -0311898 : log. / - 3-6989T00 • *'» = 50,000 , ' r J, ■^ 1 fi USES OF THE TABLES. (I2) 30xIog. r= -6356790 log I^-a I • *^ r» = 4.2iQ^ 1 . 4nnuuy = $65,05, 44 '^ 4.32194 log. /= 6989700 interest= 50000 ,„ /T ;~^-3^'94 log. r»= .6356790 sinking\fund= Ts,o7r74 log. (;^*'-I = 6213918 C'log.(r«'-,) = 1.4786082 > ,A 5.o5«-44 6V I) 14786082 3b debtnlures, as a,whole a (mean annui ';)t|67o5r4" """1— be uscd." and for redemption of 30 Debentures " ' " " \' " ^^-^ °°° \. 15 000 Total,...), $61; 000 At the end. of the 2nd year, mean annuity \ tr Interest on 1,970 Debentures = $g8s 000 T^^S-osi 44 , ^ ^^ ■• =^ 49.250 Left for Sinking Fund ^«, 5 go, ^^ This would redeem 3, more Debenture-s, which brings their number \^ ■ down to ,,939. and the Corporation would have pnd Tnihl \ 2nd year, interest on ,.970 Debentures ... =$LVro \ and for redemption of ^i more ■' ^ ^>49.25o \ at the end of the 3rd year, mean annuity . . = fB67o'cTT.*^'*'^^° interest on 1,939 I>ebentures = $969,500 ...;;'.;: = 48;475 Left for Sinking Fund $^7576^ u-v would redeem ^^ Debentures And er. r,r. r^, fU i 1 —7— ■ whifch number of Dentures Redeemed in the ,o/h ^^"'.' ""^ ''" "™^- ^«^ ^h\ last instalment at that date anTdi!^! by 500.' ' '''' ''' P'"^"' ^^'"« "^ ''^^^ (2) log. <7 = 4-8132672 log. »-"'= 211 8930 log S=^^i^; 5= $39,935.96 or nearly 80 Debentures. " 13 It may be mentioned that the sums paid for the Sinking F„nH so as to pay the buyer the liven rate of iZZ^ ^""""les, and discounting these aremade^obearag1ve7ra^rof nte^^^^^ "k?"^'^' "'^^" "'^ '^^"ds might bear an agree'ment tSll the"S;"Fund ^e^L? Jt:""; '^^'/'r annual assessment for redemption of the who"e issue sho^H h T''' °i ^^^ bonds to the Amount of the sum wi,ich remah^s Sr n u •"'^'^ '" '^^«^^'" year on the balances remainrng unpar Tl^I ' ^ I ;_ J ■ -^ .'° /° ^""'Q "ot amount to one-tenth of thp inco ...... n incurred m remvesting the Sinking Fund. * usually. )X\