Mathematics of Finance

Transcription

1 CATE Mathematcs of ace.. TODUCTO ths chapter we wll dscuss mathematcal methods ad formulae whch are helpful busess ad persoal face. Oe of the fudametal cocepts the mathematcs of face s the tme value of moey,.e., the value of a partcular sum of moey at dfferet pots of tme. or example, f you have s. today, what wll t be worth at the ed of oe year?... Smple terest (S..) Whe we take loa from Bak for a certa perod we pay off the loa ad a certa sum of moey to the Bak for the use of the moey let. The loa amout s called the rcpal ad the addtoal amout pad for the use of the loa s called terest. The sum of the prcpal ad the terest due at the ed of the perod s called the Amout,.e., Amout rcpal + terest. The borrower s called the Debtor ad the leder s called the Credtor. f s. 0 s pad as terest o s. for year, the rate of terest s sad to be 0% p.a. Smple terest (S..) s the terest o the prcpal aloe for the tme for whch t s used. S.. o the prcpal for years at the rate of % p.a. s gve by: S...., where or ; ad the amout (A) s gve by: A A + S ( + ) or + Uless otherwse stated the terest s always calculated yearly. Example.. () d the smple terest o s.,000 for 0 years at 0% p.a. () At what rate percet wll s.,000 amout to s.,500 years? () What prcpal wll amout to s. 3,000 5 years at 0% p.a. S..? (v) what tme wll s.,00 amout to s. 3,600 at 0% p.a. S..? Aswer: () terest o s. for year s. 0 terest o s. for year s. 0 terest o s.,000 for year s. 0, 000

2 acal Mathematcs 0, 000 terest o s.,000 for 0 years s. 0 s.,000 () ere s.,000, A s.,500, years ow, A ( + ) or,,500,000 ( + ) or, +.5 or Aga, () ere, A s. 3,000, 5 years, 0 0. ow, A (+ ), or, 3,000 (+ 5 0.), or, 3, 000 s.,500 (v) ere, s.,00, A s. 3,600, 0% 0. ow, A (+ ), or, 3,600,00 (+ 0.) or, or 0 years COMOUD TEEST (C..) f the terest, as ad whe t becomes due, s added to the prcpal ad the whole amout produces terest for the subsequet perod, the t s called compoud terest. The perod after whch the terest becomes due s kow as terest perod. terest s compouded mothly, quarterly, half-yearly, yearly etc., f t s specfcally metoed. f ths s ot gve the problem we assume that the terest s payable yearly. Example.. () d the dfferece betwee Smple ad Compoud terests o s. 3,000 vested for 3 years at 6% p.a., terest payable aually. () What s the preset value of s.,000 due years at 6% p.a., compoud terest payable half-yearly? () What rate of terest p.a., does a ma get who s 6% compoud terest payable / yearly. (v) d the compoud terest o 4% for the frst year, 5% for the secod year ad 6% for the thrd year. 3, Aswer: () Smple terest s. 540 Compoud terest: rcpal (Orgal) s. 3, terest for st year rcpal for d year 3,80.00 terest for d year rcpal for 3rd year 3, terest for 3rd year 0.5 Amout 3 years s. 3, Compoud terest s Compoud terest Smple terest s. ( ) s

3 Mathematcs of ace 3 () Let, rcpal (orgal) s..00 terest for st / year 3.00 rcpal terest for d / year 3.09 rcpal terest for 3rd / year 3.8 rcpal 09.7 terest for 4th / year 3.8 Amout years s..55 The requred prcpal or preset value s., s () Let, rcpal (orgal) s..00 terest for / year 3.00 rcpal terest for / year 3.09 s Compoud terest s. ( ) s The requred rate of terest 6.09% (v) rcpal (orgal) s., terest for st 4% rcpal for d year, terest for d 5% 5.00 rcpal for 3rd year,09.00 terest for 3rd 6% 65.5 Amout 3 years,57.5 The requred compoud terest s. (,57.5,000) s OMULA O COMOUD TEEST Let, rcpal, o. of years or terest perods, terest o ut sum for terest perod or year, A Amout ad Total terest. ow, rcpal terest the st perod Amout st perod, or rcpal for d perod (+ ) terest for d perod (+ )

4 4 acal Mathematcs Amout the d perod or rcpal for 3rd perod (+ ) + (+ ) (+ ) {+ } (+ ) terest for 3rd perod (+ ) Amout the 3rd perod (+ ) + (+ ) (+ ) {+ } (+ ) 3 ad so o. A Amout perods (+ )...() K f s the rate per cet, the A +,.e., compoud terest the amouts at the ed of dfferet years are G.. formula (), s called the preset value of the sum A due perods. A ( + ) A ( + )...() Compoud terest A (+ ) {(+ ) }...(3) Cor.. f (+ ),.e., Amout of prcpal perod, the A ad { } ece, for the compoud terest, the amout creases geometrc progresso. Cor.. rom equato (), log A log + log (+ ). f we kow ay three of the four ukows A,,,, we ca fd the other. Cor 3. case of uform decrease, we use stead of all the above formulae for compoud terest. ece, for deprecato at compoud rate, we apply the formula, A ( ) Smlarly, f reset populato of a coutry ad % ate of decrease of populato p.a., the populato after years Cor. 4. We use the followg formulae: () A () A () A (v) A +, whe terest s pad half-yearly. K 4 +, whe terest s pad quarterly. 4 K +, whe terest s pad mothly. K K 365 +, whe terest s pad daly. 365 K

5 Mathematcs of ace 5 Cor. 5. f s the ormal rate of terest o e. for year ad the terest s covertble b tmes a year, the the effectve rate of terest s S T + bk b ote : compoud terest calculatos, the prcpal ad terest vary for each ut of tme. The terest for ay ut of tme aga ears terest at the same rate over subsequet uts of tme. ece prcpal for ay ut of tme s the amout due at the ed of prevous ut of tme. Ut of tme s kow as the coverso perod. ote : The rate of terest % p.a. compouded at gve umber of tmes per year s kow as omal rate. The rate of terest % p.a. whch f compouded yearly would yeld the same amout of terest as r% rate compouded m tmes per year, the % s called the effectve rate. or example, f a ma borrows s. at 0 0% p.a. compouded half-yearly, the 0% p.a. s the omal rate ad + 0.5% s M 00K Q the effectve rate (here ). f the omal rate s compouded aually, the the omal rate of terest becomes equal to the effectve rate. Cor. 6. f the rate of terest s dfferet for each year, e.g., r, r, r 3 for the frst, secod ad thrd years respectvely, the the amout after 3 years s gve by: U W r r r A K K K L M 3. ote 3: The compoud terest law A + apples to ay quatty whch creases or decreases so that K amout at the ed of each perod of costat legth bears a costat rato to the amout at the startg of that perod. Ths rato s kow as growth factor f t s more tha, ad decay factor f less tha. or example, f the populato of a cty steadly creases by 3% p.a. of the populato at the begg of each year, the the yearly growth factor wll be.03 ad the populato after years wll be (.03) tmes the populato at the begg of the perod. Aga, f the value of the machery deprecates steadly by % p.a. of ts value at the begg of each year, the the yearly decay factor wll be ( 0.) 0.88 ad the value after years wll be (0.88) tmes ts value whe ew. Example.3. () The smple terest o a sum of moey oe year s s. 50 ad compoud terest two years s s. 0. d the prcpal ad the rate of terest. () A mache s deprecated at the rate of 0% o reducg balace. The orgal cost was s. 0,000 ad the ultmate scrap value was s. 3,750. d the effectve lfe of the mache. [B.Com. (C.U.), 966, B.Com. (B.U.), os. 990] () d what tme a sum of moey wll double tself at 5% terest compouded aually [log 0.300; log 05.0] (v) The machery a factory s valued at s. 4,537 ad t s decded to reduce the estmated value at the ed of each year by 8% of the value at the begg of that year. Whe wll the value be s. 0,000? (v) A ma left s. 8,000 wth the drecto that t should be dvded such a way that hs 3 sos aged 9,, 5 years should each receve the same amout whe they would reach the O

6 6 acal Mathematcs age of 5 years. f the rate of compoud terest s 3.5% p.a., what should each so receve whe he s 5 years old? (v) Whe a boy s bor, s. 500 s placed to hs credt a accout that 6% compouded aually, (b) 6% compouded quarterly, (c) 6% compouded mothly. f the accout s ot dstrbuted, what amout wll there be to hs credt o hs tweteth brthday? (CWA, Dec. 979) (v) A truck purchased by a trasport compay at s. 60,000 deprecates at the rate of 0% p.a. ad ts mateace cost for the frst year s s.,000 whch creases by % every year. f the scrap value realsed whe sold s s. 35,49.40, fd the mmum average aual retur from the truck the compay should get so as ot to susta ay loss. [CWA, Ja. 978] (v) A ma borrowed s. 0,000 from a moey-leder but he could ot repay ay amout for a perod of 4 years. Accordgly, the moey leder s demad showed s. 6,500 due from hm. At what rate per cet p.a. C.. dd the moey leder led hs moey? [Gve: log , log 0.300, log ] [C.A. (Et), ov. 99] Aswer: () S.., or,, where rcpal, umber of Years, terest o e. for oe year Compoud terest for years {(+ ) } { + } ( + ) ow, 50...() ad ( + ) 0...() () rom (), 50 ( + ) 0, or, rom (), 50 s., The requred prcpal ad the rate of terest are s.,50 ad 4% respectvely. 4% Let the effectve lfe of the mache be years. or deprecato at costat rate, A ( )...() ere, s. 0,000, A s. 3,750, 0 0 rom (), 3,750 0,000 or, (0.9) or, log log 0.9 or, or, ( ) ( ) or, or, years K

7 Mathematcs of ace 7 () f s., the A s. 00, 5 ow, A ( + ) or, 00 ( ) or, (.05), or, log log.05 or, log (log 05 log ) or, (.0 ), or, years. 00. (v) ere, s. 4,537, A s. 0,000, 0.8 ow, A ( ) or, 0,000 4,537 ( 0.8) or, , or, log 0.85 log 0.8 or, or, ( ) ( ) or, years (v) Let the sos aged 9, ad 5 years receve s., s. ad s. 3 respectvely. Aga, each of them receve s. at the age of 5 years ,000...() or the so aged 9 years, Smlarly, " " years, (.035) 3 " " 5 years, 3 (.035) (. 035) K rom (), (.035) 6 + (.035) 3 + (.035) 0 8,000...() ow, (.035) 6 x (say) or, 6 log.035 log x or, log x or, log x or,. 766 log x or, x Smlarly, usg logarthm we get, (.035) (.035) rom () ( ) 8,000 or, 8, 000 s. 9, (v) (a) ere s. 500, 0.06, 0 ow A ( + ) or, A 500 (.06) 0 or, log A log log A Atlog s.,603. (b) ow, A + 4 K ( ) 80 or, log A log log A Atlog 3.0 s.,66.

8 8 acal Mathematcs (c) ow A K ( ) 40 or, log A log log A Atlog s.,596. (v) ere, s. 60,000, 0., A s. 35,49.40, the truck s sold after years. ow, A ( ) or, or, 35, ,000 ( 0.) or, log log 0.9 or, or, ( ) ( ) or, 0.87 ( ) or, years The mateace cost for 5 years: s. [,000 +,040 +, ,.4 +,64.86] s. 0, Mmum retur requred for o loss: s. (60, ,408.08) s. 35,49.40 s. 34, s. 34, Average mmum yearly retur requred for o loss s. 6, 995, (v) f r% p.a. s the rate at whch the moey-leder leds hs moey, the 6,500 0,000 ( + r/) 4 ; or log 6,500 r log 0, log ( + r/) or, log ( + r/); or, 4 log + 0.; or, log K ( + r/) ; or ( + r/).073; or r/ 0.073, or r 7.3%. Example.4. () A sum of moey put out at smple terest amouts to s. 690 two years ad to s ½ years. d the sum vested ad the rate of smple terest. [CWA (rel.), Jue 99, Dec. 993] () Two equal sums are let at 6.75% ad 4.5% smple terest per aum respectvely. f the former s recovered two years earler tha the later ad the amout each case s s.,905. d the sum let each case. [B.Com. (Bagalore), ov. 99] () A pressure cooker s avalable for s. 50 cash or s. cash dow paymet followed by s. 65 after sx moths. d the rate of terest charged uder the stalmet pla. [CWA (rel.), Dec. 99, Dec. 993] (v) am deposted a sum of s. 0,000 a bak. After years, he wthdrew s. 4,000 ad at the ed of 5 years he receved a amout of s. 7,50. d the rate of smple terest. [CWA (rel.), Jue 990] (v) f ask you for a loa ad agree to repay you s. 300 after e moths from to-day, how much should you loa me f you are wllg to make the loa at the rate of 6% p.a.? [CWA (rel.), Jue 986]

9 Mathematcs of ace 9 (v) A sum of s.,00 becomes s.,33 years at compoud terest compouded aually. d the rate per cet. [CWA (rel.), Dec. 990, Jue 993] (v) f the populato of a tow creases every year by per cet of the populato at the begg of that year, how may years wll the total crease of populato be 40%? [B.Com. (C.U.), os. 990] (v) A sum of s.,000 s vested for 5 years at % terest per year. What s the smple terest? f the same amout had bee vested for the same perod at 0% p.a. compoud terest compouded per year, how much more terest would he get? [CWA (rel.), Jue 987] (x) O what sum the dfferece betwee smple ad compoud terest for 3 years at the rate of 0% s s.,600? [CWA (rel.), Dec. 993] (x) A ma deposts s. 5,000 a Savgs Bak whch pays compoud terest at the rate of 4½ % for frst two years ad the at the rate of 5% p.a. for ext three years. d hs amout after 5 years. [B.Com. (C.U.), 98] (x) A mache deprecates at the rate of 7% of ts value at the begg of a year. f the mache was purchased for s. 8,500, what s the mmum umber of complete years at the ed of whch the worth of the mache wll be less tha or equal to half of ts orgal cost prce? [CWA, Dec. 976] (x) A ma wshes to have s.,500 avalable a bak accout whe hs daughter s frst year college expeses beg. ow much must he depost ow at 3.5% compouded aually, f the grl s to start college sx years from ow? [CWA, Dec. 98] (x) A mache, the lfe of whch s estmated to be 0 years, costs s. 0,000. Calculate the scrap value at the ed of ts lfe, deprecato o the reducg stalmet system beg charged at 0% p.a. [Gve: log ad log ] [CA (Et.), May 99] Aswer: () Let, rcpal ad ate of smple terest s. 690 ( + )...() s ( + 3.5)...() rom () ad (), or, 690 +, ,55 or, , or, 0.075,.e., 7.5%. rom (), 690 ( ) or, 690 s () Let s. Two equal sums, ad sums at 6.75% ad 4.5% S.. p.a. be recovered after years ad ( + ) years respectvely L M K b g. O Q s.,905...() s. 905,...() rom () ad (), + + or, or, years

10 0 acal Mathematcs rom (), K, 905, or, s. 500,. 7. () terest o s. 50 [.e., s. (50 )] for 6 moths s. (65 50) s. 5. S.. o s. 50 for year s. (5 ) s. 30. s. 30 % of terest 0%. s. 50 (v) Total terest receved by am s. (7,50 6,000) s.,50. f % p.a. ate of terest, the S.. o 0, 000 s. 0,000 for years s. 00 Aga, prcpal after years s. (0,000 4,000) s. 6,000 6, S.. o s. 6,000 for 3 years s. 80 Total terest s. ( ) s.,50 or, 4%. (v) Let s. be the loa. 3 6 Amout s. + s K or, s s year (v) ere, s.,00, A s.,33,,?,33,00 (+ ) 33, or, ( + ) 05., 00 Takg logarthms of both sdes, we get log ( + ) log.05 or, log ( + ) or, log ( + ) 0.03 or, ( + ) Atlog or, 0.05 ad %. (v) Let Orgal populato A ere,,? 40 + or, +.40 or, log.0 log.40 or, K 046. or, years K L M O Q 0, 5 (v) S.. o s.,000 for 5 % p.a. s. Compoud terest o s.,000 for 5 0% p.a. S 5 U T 0 K W ( ) 5, 0,. s s. 600.

11 [Let, x,000 (.) 5 or, log x log, log. or, log x 3 log or, x s.,6] Compoud terest s. {,000 (.) 5,000} s. (,6,000) s. 6. Dfferece s. (6 600) s.. (x) Let, s. Sum of moey S.. s. 3 0 K s. 60 s. S T ST 0 Compoud terest (C..) ( + ) s + K U 3 6 s. S s. T 6 5K W 5 UW 9, s. s C.. S.. s. ( ) s..80 Dfferece s s..80 whe sum of moey s s. " e. " ". 80 " s.,600 " " s., (x) Amout after 5 years s. 5000, + L M G K J O Q G + 5 s.,500. K J 3 3 U W s. [5,000 (.045) ] (.05) 3. Mathematcs of ace s. [5, ].5765 s. (5, ) s. 6,30. (x) or deprecato at compoud rate: A ( ).,500,? ere s. 8,500, 7% 0.07, A s K 8 A 8,500 ( 0.07) 8,500 (0.93). Aga, 8,500 ( 093. ) 8,500 or, ( 093. ) or, log 093. log 05. or, or, ( ) ( ) (x) or, ( 0 035) or, years Mmum umber of complete years 0. Let be the moey deposted ow. ere, A s.,500, 3.5% 0.035, 6,?,500 ( ) 6 or, log,500 log + 6 log.035 or, log 6 log log,500 or, log 6 (0.049) or, log or, Atlog s.,034.

12 acal Mathematcs (x) A Scrap value ( ) 0,000 ( 0.) 0 0,000 (0.9) 0 Let x ; log x 0 [log 9 log 0] 0 [ log 3 ] 0 [ ] 0 [0.954 ] T.54 log x ; A s. (0, ) s. 3, SOME ELATED TEMS () Exact Tme, Exact terest, Ordary terest: may trasactos, the tme may be gve moths, weeks or days. But the smple terest formula, must be expressed years. Thus, moths, weeks or days are to be coverted to years, e.g., moths year, 3 weeks year Whe a aual smple terest rate ad the tme d days are gve, the followg methods are used to covert days to years: () f ( years) d( days), the the terest s sad to be exact. 365 () f ( years) d( days), the the terest s sad to be ordary. 360 The exact umber of days betwee the date of depost ad date of terest calculato s referred to as exact tme. or example, the exact tme days from ebruary, 00 to March 0, 00 s the exact umber of days betwee the two dates. Sce 00 s ot a leap year, the exact tme wll be as follows: o. of remag days ebruary 8 7 o. of days March 0 Exact tme 37 days llustrato.. A borrows s.,000 o Jue, 00, for 60 days. The smple terest rate s 5½%. () Calculate the exact smple terest. () Calculate the ordary smple terest. Aswer: () ere, s.,000, 0.055, year. Exact S.. s., s () ere year Ordary S.. s., s () Equatos of alue, Tme alue of Moey ad S..: Cosder a case whe A borrows s. from B at 5% S.. ad agrees to pay s. 50 o the loa 6 moths. What paymet year from ow wll settle the debt?

13 Mathematcs of ace 3 Set up the formato o a tme dagram as follows: s. (Moey borrowed) 0 6 moths moths s. 50 x (aymet moths) or ths type of problem where paymets are made at dfferet dates, we eed the followg fudametal prcple of mathematcs of face: Equato of alue;.e., alue of loa at focal date alue of paymets at focal date ocal date s the partcular date at whch amouts of moey payable at dfferet tmes ca oly be compared. ocal date s fxed by the leder ad the borrower. (a) ths problem, f the focal date s chose year from ow, the the value of each sum of moey must be calculated at the focal date as follows: alue of loa at focal date alue of paymets at focal date. alue of s. at focal date alue of s. 50 at focal date + alue of x at focal date L s s f tme) s G M K J Q + xo ( shft o s or, x s [ocal date at moths]. (b) ths problem, f the focal date s chose 6 moths from ow, the the equato of value wll be as follows: alue of s. at focal date alue of s. 50 at focal date + alue of x at focal date L M A (+), s G K JO O s. 50 (o shft of tme) Q x s A x e.. or, 5. 5 or, x s ( + ) 05. The tme dagram wll be as follows: ocal date (moths) (moths) 50 x x G K J x,.. e. 05 Thus, dfferet focal dates gve dfferet values of x. Ths dfferece wll exst smple terest trasactos ad hece t s mportat for the partes cocered to agree o the focal date.

14 4 acal Mathematcs llustrato.. A borrows s. 00 ow ad agrees to pay s. 50 after moths ad s. 70 after 6 moths. What fal paymet should A make 8 moths from ow to settle ths debt f the S.. rate s 0% ad the focal date s ow? Aswer: Let s. x be the fal paymet the followg tme dagram. The values of the loa ad paymets at the focal date are show Table.. alue of moey at focal date ocal date 00 s (moths) 49.0 s. 50 s. 70 s. x x 5. (Tme dagram) Table. s. ormula to use alue at focal date (s.) 00 o shft tme 00 A G K J 70 A G K J x A x x G K J.. The equato of value at the focal date s x or, x or, x s () Equatos of alue ad C..: Let us aga cosder trasactos whch oe or more debts are repad wth oe or more paymets due at varous pots of tme. llustrato.3. A borrows s. 00 ad agrees to pay s. 50 after moths ad s. 70 after 6 moths. What fal paymet should A make 8 moths from ow to settle the debt f terest s 6% compouded mothly? Aswer: Ths llustrato s smlar to llustrato.. The oly dfferece s that C.. s used here. Wth C.., the focal date may be ay date at whch terest s compouded, ad the resultg equatos of value wll gve detcal result for the quatty to be determed. (a) Let 8 moths be the focal date ad x be the amout of fal paymet.

15 Mathematcs of ace 5 alue of moey at focal date ocal date s (.005) (moths) s. 50 s. 70 x x 70 (.005) (Tme Dagram) 50 (.005) 6 Table. shows the value of each amout of moey at the focal date. Table. s. ormula used alue at focal date (s.) 00 A + K K " K " K x o shft tme x alue of loa at focal date alue of paymets at focal date 00 (.005) 8 50 (.005) (.005) + x or, 00 ( ) 50 ( ) + 70 ( ) + x or, x or, x s [Usg Calculator] (b) Let us solve ths llustrato by selectg focal date ow. Table.3 shows the values of each amout of moey at focal date (.e., t 0). The tme dagram s show below: alue of moey at focal date ocal date (moths) 50 (.005) x 70 (.005) 6 x (.005) 8 a a a f f f

16 6 acal Mathematcs Table.3 s. ormula used alue at focal date 00 o shft tme A A x A + K K K (.005) 70 (.005) 6 x (.005) 8 The equato of value s: alue of loa at focal date alue of paymets at focal date (.005) + 70 (.005) 6 + x (.005) ( ) + 70( ) + x( ) or, x or, x 8.56 or, x s [usg calculator] Thus values of x (a) ad (b) are equal. Also f we multply both sdes of equato of value of (b) by (.005) 8, we get the equato of value of (a), whch shows that the two equatos (for two dfferet focal dates) are algebracally equvalet. (v) Cotuous Compoudg: The compoud amout A for a depost of s. at terest rate per year compouded cotuously for years s gve by: A. e. ( decmal form) To get s. A at the ed of years, a tal vestmet of A. e s requred. The value of e s The values of e x ad e x ca be obtaed from tables or by usg calculators. or a costat prcpal, tme perod ad aual terest rate, the more frequet the compoudg, the larger s the retur o the vestmet. But there s a theoretcal upper lmt o the retur that ca be obtaed ths way. f we mage the umber of yearly coversos to crease deftely, we arrve at a stuato whe terest s compouded cotuously,.e., at each stat of tme the vestmet grows proporto to ts curret value. Ths s kow as cotuous compoudg. EXECSE (a). Mr. ama vested equal amouts oe at 6% smple terest ad the other at 5% compoud terest. f the former ears s more as terest at the ed of two years, fd the total amout vested. [CWA (rel.), Dec. 985]. Mr. am borrowed s. 5,000 from a moeyleder but he could ot repay ay amout a perod of 5 years. Accordgly the moeyleder demads ow s. 35,880 from hm. At what rate per cet per aum compoud terest dd the latter led hs moey? [CWA, Jue 987]

17 Mathematcs of ace 7 3. how may years wll the populato of a vllage chage from,500 to,60, f the rate of crease s % per aum? [CWA (rel.), Dec. 99] 4. Two parters A ad B together led s.,65 at 5% compouded aually. The amout A gets years s the same as B gets at the ed of 4 years. Determe the share of each the prcpal. [CWA (rel.), Jue 99] [ts: f A leds s., the B leds s. (,65 ). (+ 0.05) (,65 ) ( ) 4 ] 5. A sum of moey vested at compoud terest amouts to s. 0,86 at the ed of secod year ad to s.,48.64 at the ed of the thrd year. d the rate of terest ad the sum vested. [B.Com. (C.U.), 983] 6. The populato of a tow s,5,000. f the aual brth rate s 3.3% ad aual death rate s.3%, calculate the populato of the tow after 3 years. [CWA, Jue 99] [ts: opulato creases each year by (3.3.3) % 3 A,5,000 + K ] 7. A ma left for hs three sos aged 0, ad 4 years s. 0,000, s. 8,000 ad s. 6,000 respectvely. The moey s vested 3%, 6% ad 0% compoud terest respectvely. They wll receve the amout whe each of them attas the age of years. d, usg a fve-fgure log-table, how much each would receve. [B.Com. (C.U.), 967] 8. A borrower pays terest o hs loa at the rate of 4% quarterly stalmets. e wshes to pay mothly the future. What should be the ew omal rate so that the leder wll receve a equvalet amout? [CWA (rel.), Dec. 989] 9. A mache deprecates 0% p.a. for frst two years ad the 7% p.a. for the ext three years, deprecato beg calculated o the dmshg value. f the value of the mache be s. 0,000 tally, fd the average rate of deprecato ad the deprecated value of the mache at the ed of the ffth year. [B.Com. (C.U.), 974] 0. The dfferece betwee the smple ad the compoud terests at the same rate for years o a certa amout s /400 of amout. d the rate of terest. [CWA (rel.), Jue 986]. The compoud terest o a certa sum of moey vested for two years at 5% s s. 38. What wll be the smple terest o t at the same rate ad for the same perod? [CWA (rel.), Dec. 986]. A sum of moey vested ow at x% per aum compoud terest quadruples 8 years. d x. [CWA (rel.), Jue 984] 3. how may years wll the populato of a vllage chage from 5,65 to 7,576 f the rate of crease s 4% per year? [Gve: ad ] [CWA (rel.), Dec. 984] 4. A mache deprecates at the rate of 0 p.c. of ts value at the begg of a year. The mache was purchased for s. 44,000 ad the scrap value realsed whe sold was s. 5, d the umber of years the mache was used. [CWA, Dec. 983]

18 8 acal Mathematcs 5. A mache deprecates each year by 0% of ts value at the begg of the year. At the ed of fourth year ts value s s.,3,00. d ts orgal value. [Gve: log , log , Atlog ,000] [B.Com. (C.U.), 98] 6. Mr. eedy borrowed moey from the short-term loa compay ad promsed to pay s. 00 at the ed of the year. The terest rate s ½% per moth o the frst s. ad % o the secod s.. ow much does he receve? [CWA, Dec. 979] 7. What s the effectve rate of terest correspodg to a omal rate of 5% p.a. f terest s compouded quarterly? [CWA, Jue 99] 8. A leds B s. 30. B s to pay terest o whatever amout he has ot pad back at the rate of 5% per aum for the frst year, 6% for the secod year ad 7% for the thrd year. B pays s. at the ed of frst year, s. at the ed of the secod year ad eough to pay off completely the debt ad the terest at the ed of the thrd year. ow much s the last paymet? [CWA, 97] 9. The populato of a coutry creases every year by.4% of the populato at the begg of that year. what tme wll the populato double tself? Aswer to the earest year. [CWA, Jue 977] 0. The dfferece betwee smple ad compoud terest o a sum of moey put out for 4 years at 5% p.a. s s. 50. d the sum. [CWA, Jue 989]. A mache deprecates value each year at the rate of 0% of ts value at the begg of a year. The mache was purchased for s. 0,000. Obta, to the earest rupee, ts value at the ed of the teth year. [CWA, Jue 975]. A sum of moey vested at compoud terest, payable yearly, amouts to s.,704 at the ed of the secod year ad to s.,8.6 at the ed of the thrd year. d the rate of terest ad the sum. [B.Com. (C.U.), 983] 3. a certa populato the aual brth ad death rates per,000 are 39.4 ad 9.4 respectvely. d the umber of years whch the populato wll be doubled assumg that there s o mmgrato. [CWA, Dec. 974] 4. d the amout that s. wll become after 0 years at compoud 5% calculated aually. [B.Com. (C.U.), 966] 5. A ma wats to vest s. 5,000 for four years. e may vest the amout at 0% p.a. compoud terest, terest accrug at the ed of each quarter of the year, or, he may vest t at 0½% p.a. compoud terest, terest accrug at the ed of each year. Whch vestmet wll gve hm slghtly better retur? [CWA, Jue 976] 6. The terest o a sum of moey vested at compoud terest are s. 83 for the secod year ad s for the thrd year. d the rate of terest ad the sum vested. [CWA, Jue 986 (old)] 7. Mr. Brow was gve the choce of two paymet plas o a pece of property. e may pay s. 0,000 at the ed of 4 years, or, s.,000 at the ed of 9 years. Assumg moey ca be vested aually at 4% p.a. coverted aually, what pla should Mr. Brow choose? [CWA, Jue 983]

19 ASWES Mathematcs of ace 9 () s. 7,800; () 7.5%; (3) years; (4) s. 6,65, s. 6,000; (5) 4%, s. 0,000; (6),3,65; (7) s. 3,843, s. 3,57; s.,69; (8) 3.6%; (9) 8.%, s. 6,55; (0) 5% p.a.; () s. 3.0; () 8%; (3) 3 years; (4) 5 years; (5) s.,00,000; (6) s ; (7) %; (8) s ; (9) 9 years; (0) s. 9,673.5; () s. 3,483; () 4% p.a., s.,500; (3) 35 years; (4) s ; (5) Secod vestmet wll gve hm better retur; (6) 4%, s. 0,000; (7) Secod pla..5. AUTES A auty s a fxed amout pad at regular tervals e.g., mothly, quarterly, yearly, etc., uder certa codtos. Whe the terval s ot gve, we take t as oe year. A auty payable for a fxed umber of perods, or, years s defed as Auty Certa. Whe a auty s to cotue for ever, t s sad to be a erpetual Auty or erpetuty. f the paymet of a auty commeces or ceases at the occurrece of a cotget evet, t s sad to be a Auty Cotget. f the paymets are made at the ed of each perod, the auty s called mmedate or Ordary Auty. Whe the paymets are made at the begg of each perod, the auty s defed as Auty Due. A auty s take as mmedate uless otherwse stated. f the paymets of a auty are deferred or delayed for certa perods, or years, t s called a Deferred Auty. f a auty s deferred for years, ts frst stalmet wll be pad at the ed of ( + ) years. or a Deferred erpetuty, the paymets commece after the deferred perod ad thereafter cotue for ever. f we add the preset values of all the paymets of a auty, we get ts reset alue. A ree-hold Estate s that whch geerates a perpetual auty,.e., ret. A Lease-hold Estate geerates ret for the fxed lease perod. The value of a freehold estate s equal to the preset value of the perpetuty (or ret). f a auty remas upad for a certa perod, t s sad to be Upad Auty for that perod. The amout of the upad auty s obtaed by addg the stalmets ad the compoud terest o each for the perod durg whch t remaed upad..6. OMULAE () Amout of a auty: A ( +) where, A Amout of a auty or (mmedate auty) Auty; Upad years terest o ut sum for year or perod roof: The frst stalmet due at the ed of the frst year wll ear compoud terest for ( ) years ad the amout wll be (+ ). Smlarly, the secod stalmet wll ear compoud terest for ( ) years ad amout to (+ ) ad so o. The last stalmet wll ot ear ay terest.

20 0 acal Mathematcs b g b g... A [Ths s a G.. wth C.. /( + )] ( + ) ( + ) S T K + + S T U W U ( + ) S T ( + ) ( + ) W ( + ) Cor. f t Tmes of paymets made per year. U ( + ) ( + ) W ( + ) + s s. et per year, s. t et per perod terest per ut sum per year, the A Amout of the auty for years S T t U S T t + K W + t t tk () reset value of a auty: ( + ) s where, Auty to cotue for years. reset value of the auty (mmedate). terest o ut sum for year. roof: The preset values of the frst, secod, thrd,..., th paymets are: +, +, + 3,..., + respectvely. a f a f a f a f Sum of the preset values of all the paymets. t U W a f a f a f a f b g { } + + b gs T G + + K J U W

21 Mathematcs of ace Cor. f t Tmes of paymets made per year s. et per year s. t et per perod terest o ut sum for year, the t t t S + U t T tk W S T + tk (3) reset value of a perpetuty: as zero. roof: or perpetuty, s deftely large. ece the value of S T rom ( ), ( + ) U W (4) reset value of a deferred auty: S T ( + ) m+ ( + ) U W S T U W ( 0 ) ( + ) m+ U S W T ( + ) ( + ) () may be take roof: Let Auty, reset value, terest o ut sum for year. aymet starts at the ed of m years ad thereafter cotues for years. ece, the frst, secod, thrd,..., last paymets are due at the ed of (m + ), (m + ), (m + 3),..., (m + ) years respectvely. The preset values of the successve paymets are: m+ m m +, + +,..., + + respectvely. a f a f a f m+ m+ m a f a f a f ( + ) S U m+ T ( + ) W L O + M L m m a + f a + f Q M ( + ) m+ O L M m U W Q ( + ) [fte geometrc seres] [reset value of a mmedate auty to cotue for (m + ) years] [reset value of a mmedate auty to cotue for m years.] m O Q

22 acal Mathematcs (5) The preset value of a deferred perpetuty: L M ( + ) m where, reset value of a deferred perpetuty to beg after m years. roof: (4), puttg O Q 0, as ( m + ) m+ s deftely large, we get ( + ) + ( + ) m ( + ) (6) Skg fud: t s a fud whch s created by vestg aually a fxed amout at compoud terest to pay off a loa or a debeture stock or bod o a gve date or to redeem certa labltes or to provde for replacemet of assets (.e., captal expedture), e.g., plat ad machery, etc. f A s the amout of loa to be pad off at the ed of years ad s the accumulatos of the aual sum, the A ( +) where, terest o ut sum for year [Same as formula ()] (7) reset value of ueve cash flows: f,,..., be the uequal cash flows receved at the ed of year,,... respectvely, the the preset value of these sums at terest rate s gve by: ( + ) ( + ) ( + ) Example.5. () A mache costs s. 97,000 ad ts effectve lfe s estmated to be years. A fud s created for replacg the mache at the ed of ts effectve lfe tme. f the scrap realses s.,000, what amout should be retaed out of profts at the ed of each year to accumulate at compoud terest at 5% p.a.? [Gve (.05).797]. () S. oy borrows s. 0,000 at 4% compoud terest ad agrees to pay both the prcpal ad the terest 0 equal aual stalmets at the ed of each year. d the amout of these stalmets. [C.U., 97] () The aual ret of a free-hold estate s s.,000. What s ts preset value, f the compoud terest rate s 4% p.a.? (v) Whch s better, a auty of s. to last for years, or, the reverso of a free-hold estate of s. 80 p.a. to commece 6 years hece, the rate of terest beg 6%? (v) A ma buys a old pao for s. 500, agreeg to pay s. dow ad the balace equal mothly stalmet of s. 0 wth terest at 6%. ow log wll t take hm to complete paymet? [CWA, Dec. 979] m.

23 Mathematcs of ace 3 (v) The accumulatos a rovdet ud are vested at the ed of every year to ear 0% p.a. A perso cotrbutes ½% of hs salary to whch hs employer adds 0% every moth. d how much the accumulatos wll amout to at the ed of 30 years of hs servce, for every rupees of hs mothly salary [Gve the aswer to the earest rupee]. [CWA, Jue 975] (v) A wago s purchased o stalmet bass such that s. 5,000 s to be pad o the sgg of the cotract ad four-yearly stalmets of s. 3,000 each payable at the ed of the frst, secod, thrd ad fourth years. f terest s charged at 5% p.a., what would be the cash dow prce? [B.Com. (C.U.), C.A. (Et.), ov. 99] (v) or edowg a aual scholarshp of s.,000 a ma wshes to make three equal cotrbutos. The frst award of the scholarshp s to be made 3 years after the last of hs three cotrbutos. What would be the value of each cotrbuto, assumg terest at.5% p.a. compouded aually? (Assume that the frst cotrbuto s to be made ow ad the other two at a terval of oe year thereafter.) [CWA, Dec. 980] (x) A skg fud s created for redempto of debetures of s.,00,000 at the ed of 0 years. ow much moey should be provded out of proft each year for the skg fud f the vestmet ca ear 4% p.a.? Aswer: () ere, A Cost prce of the mache Scrap value s. (97,000,000) s. 95,000 years, 0.05 ow, Amout of a auty A ( +) s where, Amout to be retaed out of profts at the ed of each year Auty s 95, [Let x.05 log x log x Atlog ] or, 4750, 4,750 (.797 ) or, s. 5, () ere, s. 0,000.. of a auty of s. to cotue for 0 years at 4% p.a., 0, 0.04, Amout of each stalmet or auty. ow, reset value of auty s or, 0, 000 ( + ) L M 004. ( 04. ) 0 [Let, x (.04) 0 log x 0 log , x.479] L M O or, Q (..479 ) s., s.,470 () A freehold estate yelds a perpetual auty. f.. of the freehold estate, the, 000 s. 5, O Q

24 4 acal Mathematcs (v) rst case: ( + ).. of a auty to cotue for years. - ere,, 0.06, or, [Let x.06, log x log x Atlog ] or,, ( ) s Secod case: t s a deferred perpetuty whch commeces after 6 years. ow, m ( + ).. of a deferred perpetuty to commece after m years ere, s. 80, 0.06, m 6 (. 06), (. 06) L M b [Let, x (.06) 6, log x 6 log x Atlog ] or,, s Sce.. of the secod case,.e., s s greater tha the.. of the frst case,.e., s , the secod case s better. (v) Amout pad cash s.. s. (500 ) s. 400 The balace amout. f be the umber of years, the + K 40 ere, s. 0 s. 40, or, 400 4,000 { (.005) } or, 0 L M ( 5. ) 9 0 or, ( 5. ) or, (. 005) or, log.005 log. or, or,.76 ere, the requred tme years moths.76 moths (v) Total mothly cotrbutos to s..5; f we assume the mothly salary of the perso as s.. Total aual cotrbuto to...5 s. 70. f A Total accumulato at the ed of 30 years, the A 70 ( +) s. ere, s. 70, 0., 30 (. ) 30 s, 700d. 30 h 0. [Let x. 30, or log x 30 log x Atlog ] A, s. 44,44. (v) f.. of the auty of s. 3,000 for 4 years at 5% compoud terest, the cash dow prce of the wago wll be s. ( + 5,000). ow, , { b g } ( ) s ere s. 3,000, 4, g O Q L M O Q K O Q

25 Mathematcs of ace 5 [Let, x.05 4, log x 4 log x Atlog ] or 60,000 ( 0.86) s. 0,644 The requred cash dow prce s. (0, ,000) s. 5,644. (v) The frst scholarshp s to be pad at the ed of the 5th year ad thereafter t wll cotue for ever. ece we have a perpetuty of s.,000 deferred by 4 years. x.. of the edowmets x + ( + ) s where, x The aual cotrbuto ere,, as the frst cotrbuto s made ow, x x s. x x x K 94. x...() Aga, for the.. of deferred perpetuty ( + ) m, ere, s.,000, 0.05, m 4, 000 ( ) [Let, x.05 4, or log x 4 log x Atlog ] 4,80, s. 4,34,98...() rom () ad (),.94x s. 4,34,98 x s.,48,744.8 (x) Amout of auty, A 0 oa+ f t; or 00000,, oa04. f t , 8, 000 or, s ( 04 ) ( 88 ). 6, s [Let x (.04) 0 ; log x 0 log , x.88] Example.6. () A govermet costructed housg flats costg s.,36,000; 40% to be pad at the tme of possesso ad the balace reckog 9% p.a. s to be pad equal aual stalmets. d the amout of each such stalmet. Gve: L M ( 09. ) O Q [B.Com. (C.U.), 984] () A loa of s. 0,000 s to be repad 30 equal aual stalmets of s.. d f the compoud terest charged s at the rate of 4% p.a. (Auty s a mmedate auty,.e., frst paymet s made at the ed of frst year). [Gve: (.04) ] [B.Com. (C.U.), 98] () A mache costs a compay s. 5,000 ad ts effectve lfe s estmated to be 5 years. A skg fud s created for replacg the mache by a ew model at the ed of ts lfe tme, whe ts scrap realses a sum of s.,500 oly. The prce of the ew model s estmated to

26 6 acal Mathematcs be 5% hgher tha the prce of the preset oe. d what amout be set asde each year out of the profts for the skg fud, f t accumulates at 3½% per aum compoud? [Gve: log , log ] [C.A. (Et.), May 99] (v) A vestor has a captal of s. 0,000 o whch he ears 5% p.a. f he speds s.,800 per year, show that he wll be rued of hs captal before the ed of 7th year. (v) Determe the preset value of a perpetual auty of s. payable at the ed of st year, s. 00 at the ed of d year, ad s. 300 at the ed of 3rd year, ad so o, creasg s. payable at the ed of each subsequet year. Assume a tme preferece rate of 5% p.a., compouded aually. Aswer: () Amout to be pad at the tme of possesso s. (,36, ) s. 54,400 Balace to be pad stalmets alog wth terest s. (,36,000 54,400) s. 8,600. f s. s the aual stalmet, the ,. ( 09. ) [Let x (.09) ; log x log x Atlog ] or, , or, s., L M O Q () ,. ( ) 30 [Let x ; log x 30 log x Atlog ] or, 0, ( ) or, s () A Cost of the mache Scrap value s. (.5 5,000,500) s. (65,000,500) s. 6,500 Amout of a auty A ( + ) s where, A Amout to be retaed out of profts at the ed of each year 5 s 6,500 ( ) [Let x ; log x 5 log log.363 x.363], or, (.363 ),87.5 or, s. s., (v) ( + ) s or, 0 000, 800, 005. ( 05. ) or, 0, ( 05. ) or, (. ) or, log.05 log 4 log 9, L M s O Q

27 Mathematcs of ace or, or, thyear 0. 0 (v) The preset value of the perpetual auty ca be obtaed as follows: ( + ) ( + ) ( + ) 05 ( ) 05 ( 05 ) or, ( 05. ) ( 05. ) or, A x+ x + 3x ut x ad A. 05 Multplyg both sdes by x, we get Ax x + x 3 + 3x ow, A Ax x+ x + x or, A ( x) x (+ x+ x +...) x ( x) or, A x ( x) or, x ( x) L M O Q (. 05) ( 05. ) s K s. 4, 000. ( 005. ) AMOTZATO A loa s amortzed f both the prcpal ad terest are pad by a sequece of equal perodc paymets. Amortzato meas removal of loa. Each paymet cossts of: () The terest o the loa outstadg at the begg of the paymet perod. () A part repaymet of the loa or prcpal. To facltate accoutg, t s ofte requred to splt up each stalmet pad to repaymet of loa ad paymet of terest o the outstadg balaces. Wth each paymet, the actual loa or prcpal decreases, terest part each paymet successvely decreases ad loa repaymet (.e., amortzato) creases..7.. To calculate the amortzatos for the st, d, 3rd, etc., years, of a loa repad by fxed aual paymets years, compoud terest beg allowed Let A be the amout of each aual paymet pad at the ed of each year for years. ece the loa s the preset value of the aual paymets. Let be the rate of terest per e. p.a. The loa s gve by: a f a f a f a f a f { a f a f... a f a f} 3 A + + A + + A A + A

28 8 acal Mathematcs ( + ) A ( + ) ( + ) ( + ) ( + ) A ( + ) A + a f a a f f + + a a f f S a T a f f + + A + + U W ece, A ( + ) ( + ) ow, the aual paymet A pad at the ed of the st year cotas the terest o for year, whch s ad the rest of A s amortzato. Amortzato at the ed of the st year...() A ( + ) ( + )...[rom ()] ( + ) ( + ) + ( + ) ( + ) After amortzato the loa amout at the ed of the st year ( + )...() terest o () year ( + ) ece amortzato at the ed of the d year S U T + W + A + + ( ) ( ) ( + ) ( + ) ( + ) + + ( + ) ( + ) a f [rom ()] s ( + ) ( + ). ( + ) Smlarly, amortzato at the ed of the 3rd year ( + ) + Amortzato at the ed of the th year + Total amortzato years a f a f a f ( ) ( ) ( ) ( + ) a a f f ad so o

29 Mathematcs of ace 9 + ( + ) + ( + ) ( + ) ( + ) + ( + ) Loa amout. ( ) ( + ) Example.7. A loa of s.,000 s to be pad 5 equal aual paymets, terest beg at 6% p.a. compoud terest ad frst paymet beg made after a year. Aalyse the paymets to those o accout of terest ad o accout of amortzato of the prcpal. [CWA, Ja. 970] Aswer: reset value of a auty, ( + ), where s the auty (.e., yearly paymet) rupees. s ere,,000, 0.06, 5,? 0, (. 06 ) 5,. or, 60 (. 06) 5 s o t o t...() [Let x (.06) 5, log x 5 log x Atlog ( ) ] 60 rom (), 60 { } or, s Amortzato Table Ed of Yearly terest Amortzato rcpal year paymet due (s) (s) earg terest (s) (s) st , d rd th th Total,87 87, ESET ALUE (..) CATAL EXEDTUE The expedtures made for buyg fxed assets lke lad, buldg, plat ad machery projects, etc. are called captal expedture. We have already see how preset values of such assets ca be calculated... cocept captal expedture helps us to select the best out of alteratves. or example, suppose a ppe le s due for repars. t wll cost s. 0,000 ad lasts for 3 years. Alterately, a ew ppe le ca be lad at a cost of s. 30,000 whch wll last for 0 years. Assume cost of captal s 0% ad gore salvage value.

30 30 acal Mathematcs The preset values of both the cases are to be calculated ad compared. The project havg lesser.. s to be selected..8.. ree-hold ad Lease-hold Estates. A free-hold estate yelds perpetual auty,.e., ret whereas a lease-hold estate held for a fxed perod yelds ret for that fxed perod oly.. The value of free-hold estate ca be take as the.. of the perpetuty havg each paymet equal to ret. The value of certa lease hold property s the.. of a auty certa havg each paymet equal to ret, the status beg the o. of years to the tme the lease termates. 3. alue of a free-hold estate.. of the perpetuty of (Aual ret) x ( say), x, s the o. of year s purchase. 4. f a ma havg purchased a lease yeldg a ret of s. yearly ad lastg for years, wats, after x years ( > x), to reew t for aother perod of y years, he must pay fe for the reewal. The fe s the.. of a auty of for y years deferred ( x) years. 5. To obta the.. of a lease hold property wth a ret, we are to fd the.. of the auty of for the remag perod of the lease. 6. A lease-hold property reverts to the orgal holder after termato of the lease. A estate X worth X to be reverted after years has the reverso value.. of X. ( + ) f the yearly ret s, the from (3), X. The value of reverso ( + ) 7. A compay creases ts captal by ssug loas, whch are called Debetures. The debeture-holders get ther fxed rate of terest whether the compay ears proft or ot. t s also ssued by State or Cetral Govts., or, other publc bodes, e.g., mprovemet Trusts, Corporatos, etc. Lke debeture bod t s also a wrtte promse to pay or do somethg..9. ESET ALUE AD DSCOUT The.. of a gve sum A due after a gve perod s that sum of moey [ (say)] whch becomes the sum A after the gve perod. A (a).. (Smple terest) + A True Dscout (T.D.) A A A + + A (b).. ( + ) A T.D. A ( + ) A ( + ) s K A + (Compoud terest).

31 Mathematcs of ace Baker s Dscout ad reset alue f a sum of moey s due after a gve perod, the the terest (smple or compoud) o the sum of moey for that perod s called the Baker s Dscout. t s the terest o the face value of the bll. [ace value.. + T.D.]. True dscout (T.D.) s the terest (smple or compoud) o the preset value of the sum due or preset value of the amout of the bll. Baker s.. s the dfferece betwee the sum of moey due ad the Baker s Dscout. Baker s Ga Baker s Dscout True Dscout terest o Bll value terest o reset value terest o (Bll value reset value) terest o True Dscout Legal due date omal due date + Three days of grace. Example.8. () The dfferece betwee true dscout ad baker s dscout o a bll due after 6 moths at 4% p.a. s s. 4. d out true dscout, baker s dscout ad bll amout. [B.Com. (Bagalore), 99] () f the dfferece betwee true dscout ad baker s dscout o a sum due 6 moths at 5% p.a. s s., fd the amout of the bll. [B.Com. (Bagalore), 990] () The baker s ga for a sum due 0 moths hece at 6% p.a. s s. 5. d the sum due. [CWA, Jue 993] (v) A bll for s.,900 was draw o 3rd ebruary, 988 at sx moths date ad dscouted o 3th March at the rate of 8% p.a. or what sum was the bll dscouted ad how much dd the baker ga o ths? [B. Com. (Bagalore), 99] Aswer: () (Baker s dscout True dscout) for 6 moths terest o true dscout for 6 moths 4 True dscout or, True dscout s., 00. f s. x s the.. of the bll, the x 004., x s. 60,000 Amout of the bll s. (60,000 +,00) s. 6,00. True dscout s.,00; Baker s dscout s.,4. () Let the amout of the bll be s. x. A x 00 reset value of the bll x x T.D. x Baker s dscout 05 4 x x Baker s dscout True dscout Amout of bll s.,64,000. x x K x s. (gve)

32 3 acal Mathematcs () Let s. x be the.. of the sum due 0 Sum due s x + x K s x + x K x x T.D. x + x 0 0 x Baker s ga terest o T.D (gve) x s. 0,000 0 (v) Due date of the bll 6th August (cludes 3 days of grace) o. of days for whch Baker s dscout s pad days. 46 Baker s dscout s., s reset value of the bll o the day of dscoutg: reset value + reset value 46 L , 900 or, reset value + O Q,, or, reset value s.,500. T.D. s. (,900,500) s Baker s ga s. ( ) s TYCAL EXAMLES Example.9: () Mache A costs s. 0,000 ad has useful lfe of 8 years. Mache B costs s. 8,000 ad has useful lfe of 6 years. Suppose mache A geerates a aual savgs of s.,000 whle mache B geerates a aual savg of s.,800. Assumg the tme value of moey s 0% p.a., whch mache s preferable? [D.U., B.Com. (os.), 997] () ow much s eeded to be saved each year a savgs accout payg 6% p.a. compouded cotuously order to accumulate s. 6,000 three years? [D.U., B.Com. (os.), 99] () A ma retres at the age of 60 years ad hs employer gves hm a peso of s.,00 a year for the rest of hs lfe. eckog hs expectatos of lfe to be 3 years ad that terest s at 4% p.a., what sgle sum s equvalet to hs peso? [Gve: log ad log ]. [C.A. (Et.) May, 99] (v) A fxed royalty of s. 0,000 p.a. for 5 years s grated to a author by a publshg compay. The rght of recevg the royalty s sold after 0 years. d to the earest rupee the prce at whch t s sold, assumg moey s worth % p.a. compouded aually. (v) d the preset value of s. 500 due 0 years hece whe terest of 0% s compouded () half-yearly, () cotuously. [D.U., Eco. (os.), 989] (v) A moey-leder charges terest at the rate of 0 pase per rupee per moth, payable advace. What effectve rate of terest does he charge p.a.? [D.U., B.Com. (os.), 996] (v) s. 3,000 s vested at aual rate of terest of 0%. What s the amout after 3 years f the compoudg s doe: (a) Aually? (b) Sem-aually? (c) Mothly? (d) Daly? M

33 Mathematcs of ace 33 (v) A vestor wats to buy a 3-year s.,000 per value bod havg omal terest rate of 0%. At what prce the bod be purchased ow f t matures at par ad the vestor requres a rate of retur of 4%? (x) What aual rate of terest compouded aually doubles a vestmet 5 years? (x) A ma opeed a accout o Aprl, 004 wth a depost of s The accout pad 5% terest compouded quarterly. O October, 004, he closed the accout ad added eough addtoal moey to vest a 6-moth Tme Depost for s.,00 earg 5% compouded mothly. (a) ow much extra amout dd the ma vest o October, 004? (b) What was the maturty value of hs Tme Depost o Aprl, 005? (c) ow much total terest was eared? (x) s. 0,000 s vested a Term Depost Scheme whch yelds terest 5% p.a. compouded quarterly. What wll be the terest after year? Work-out ab to. What s the effectve rate of terest? (x) Mr. Y has made real estate vestmet for s. 5,000 whch he expects wll have a maturty value equvalet to terest at % compouded mothly for 5 years. f most of the savgs sttutos curretly pay 6% compouded quarterly o a 5-year term, what s the least amout for whch Mr. Y should sell hs property? (x) Mr. X plas to receve a auty of s. 8,000 sem-aually for 0 years after he retres 5 years. Moey s worth 0% compouded sem-aually. (a) ow much amout s eeded to face the auty? (b) What amout of sgle depost made ow would provde the fuds for the auty? (c) What amout wll Mr. Y receve from the auty? (xv) d the effectve rate equvalet to omal rate of 0% compouded (a) half-yearly; (b) quarterly; (c) mothly; (d) cotuously. (xv) A.S.C costs s. 0 ad realses s. 5 after 0 years. d the rate of terest volved whe t s added (a) yearly; (b) cotuously. (xv) A bak ssues e-vestmet certfcates for a perod of years. f s. 5,000 are vested these certfcates, ther maturty value s s. 6,500. Assumg that the terest s compouded every year, what s the rate of terest? Aswer: () Mache A:.. of a sequece of aual savgs of s.,000 for 8 0% p.a. s gve by: 000 o a+ f t o + 00 a 00. f 8 t 0, 000b g 0, s. 0,670. et savg from Mache A s. 0,670 s. 0,000 (cost of mache) s. 670 Mache B:.. of a sequece of aual savgs of s.,800 for 6 0% p.a. s gve by: o a f t k p s. 7, et savgs from Mache B s. (7, ,000) s ece, Mache A s preferable.