Stat 6863-Handout 5 Fundamentals of Interest July 2010, Maurice A. Geraghty

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1 S 6863-Hou 5 Fuels of Ieres July 00, Murce A. Gerghy The pror hous resse beef cl occurreces, ous, ol cls e-ulero s ro rbles. The fl copoe of he curl oel oles he ecooc ssupos such s re of reur o sses flo. Ths bref oerew wll rouce he fuel prcples of he eersc pproch o he oel, whch s sll he os wely use pproch curl scece. Ieres Theory Defos The os bsc oel s he ssupo of cos re of eres oer e. Ths s specl cse of he eersc oel. The prcpl s he l ou of oey ese he ou recee fer pero of e s clle he ccuule lue. The fferece of hese wo ous s clle eres. A ccuulo fuco, () ges he lue of l prcpl of soe fuure e. If he l prcpl ws soe lue k, he he ccuule lue e s wre s k () he eres ere s k [ ( ) ]. The effece re of eres,, s he ou h u wll er urg oe yer of ese (0) or () +. There re wo bsc ehos of creg eres oer ffere peros of e Sple eres s rhec ccuulo of eres oer e + whle Copou eres s geoerc ccuulo of eres oer e ( + ). I os cses, copou eres s he preferre eho of clculo. Exple A l ese of $000 ers 6% eres for 5 yers. F he ccuule lue uer boh ehos of creg eres. Sple 000[ + (.06)(5)] Copou 000(.06) The er + s frequely referre o s ccuulo fcor s clcules lue he e of yer of l ese he begg of he yer. Howeer, we que ofe o he reerse, o f he curre lue of pye he fuure. The scou fcor, /( + ) s he lue he begg of he yer of u he e of he yer. The recprocl of he ccuulo fuco s clle he scou fuco c be wre for boh sple copou eres /( ) /( Sple + Copou + )

2 I prculr s clle he prese lue of u o be p fer yers. Exple A zero coupo bo wh fce lue of $00,000 wll be pyble 0.5 yers. F he prese lue of he bo ssug 7% re of eres. PV (/.07) A effece re of scou, or scou re (represee by he leer ), s he esure of eres p he begg of he yer, s oppose o he effece re of eres whch s cree he e of he yer. There re seerl reloshps bewee, + ( ) Exple A bk les borrower $000 eely collecs $80 of eres, leg he borrower wh $90. The 80/000 8% s he scou re. The effece re of eres s.08/(-.08) The effece res of eres scou ssue pye of eres he e begg of he yer respecely. Howeer, eres y be p ore frequely urg yer, for exple ohly eres o cre crs or ly eres o sgs ccous. A ul re of eres h s p us of / oer equl e seges urg he yer s clle he ol re of eres s eoe by he sybol (). Assug copou eres, he followg foruls pply [( + ) ] + Exple A cre cr chrges ol re of eres of % copoue ohly. Deere he effece re of eres. () % The force of eres, δ, s ul re of eres h s p couously oer he yer. I oher wors, s he seous re of eres δ l lso < () δ l( + ) e < (3) <... < δ <... < δ (3) < () < for > 0 Exple F he force of eres whe he effece re of eres s 8%. F he effece re of eres whe he force of eres s %.

3 δ l(.08).0769 e.. 75 A seres of pyes equl slles s clle uy. A uy-ee s whe he pyes re e ully he e of he pero. The prese lue of - yer uy-ee of ul slles of oe u s wre s. A uy-ue s whe he pyes re e ully he begg of he pero. The prese lue of -yer uy-ue of ul slles of oe u s wre s & &. && + + L L + + ( + ) Exple F he prese lue of 0 equl pyes of $000 he e of he yer ssug eres re of 0% PV (/.) $000 $ Exple Peso lw requres h 40(k) Pl srbuos e o prcp before ge 59.5 re subjec o erly srbuo pely of 0%. Oe opo o o hs pely s o ke subslly equle ul pyes oer he lfe expeccy of he prcp. A prcp currely ge 54 hs ccou of $00,000 lfe expeccy of 5 yers. Deere he ul ou pyble he begg of he yer h woul ssfy hs requree ssug he eres re s 9% per yer. P && 5 (.09) 0.09 ( /.09) The ccuule lue of uy-ee whch pys oe u he e of he yer for yers s clle he fuure lue s represee by he sybol s. The fuure lue for uy-ue s represee by & s& ( + ) s + ( + ) + ( + ) + L+ ( + ) ( + ) ( + ) && s ( + ) + ( + ) + L+ ( + ) + ( + ) ( + ) ( + ) && Exple A esor pus $000 o sgs he e of ech of 0 yers. F he lue of he ccou fer 0 yers ssug 7% ul re of reur.

4 FV 000s 0.07 (000) I prcce, ues re usully p ore frequely h ully. The prese lue of uy whch pys / he e of ech h of he yer for ol of yers s eoe by. The prese lue of uy whch pys / he begg of ech h of yer for ol of yers s & & && L + + L / [( + ) ] Exple A peso pl s pyg 0 yer cer beef of $000 ohy for 0 cosecue ohs. F he prese lue of hs beef ssug beefs re p he begg of he oh eres of 8% per yer. 0 () (/.08) & & (000)(6.9974) PVB.08 A slr juse c be e for fuure lue whch wll be oe fro hs hou. Whe we le he frequecy of he pye becoes fe, we ge he heorecl couous uy, whch c lso be use o pproxe ues of gre frequecy (e.g. ly) l & l 0 δ δ Exple Aul pyes of $000 re p couously oer he ex 5 yers. Assug effece eres re of 6% per yer, eere he prese lue. ( 000) ( 000) δ (.06) ( 000) l.06 $ () (3) (3) () Also < < < L < < L < && < && < && for 0 >

5 There re y oher fcl fucos h c be ere, bu wll be oe here. I prcce, hese clculos c be esly perfore usg he fcl fucos Excel. Oe pplco s he reloshp bewee he Prese (or Fuure) lue, he eres re, he uber of pyes he ou of pye. Oce hree of hese es re kow, he oher c be ere. These fucos Excel re clle PV (or FV), RATE, NPER, PMT. I ler hou, we wll scuss eres s ro rble whch reflecs he ersy of oer ese heory. Hoework. For effece eres re of 6.5% ere he followg ers 6 δ & 8 s 4 () () 5 7. Ech yer, corporo kes 4 qurerly corbuos of $,000,000 o s peso pl rus. These pyes occur o Mrch 3, Jue 30, Sepeber 30 Deceber 3. Assug o l lue eres re of 8.5%, wh s he lue of he peso rus fer 0 yers of corbuos. 3. You purchse hoe for $500,000 wh 0% ow pye 30 yer fxere orgge. The ers of hs orgge re pyes he e of he oh ol eres re of 8% per yer chrge ohly. Clcule he ohly orgge pye. Wh s he effece re of ul eres for hs rsco? 4. Afer 5 yers, you refce he orgge escrbe Queso 3 swch o ew 5 yer fxe-re orgge ol re of 5.5% chrge ohly. Deere he ew orgge pye. 5. A bo wh fce lue of $00,000 wll py se-ul coupos of $000 he e of ech sx oh pero. A he e of 8 yers, he bo wll ure reur $00,000 o he holer, o o he fl $000 coupo. Deere he prese lue of he bo uer he followg ul eres re sceros 3%, 5%, 7%. 6. A uy pyble he begg of ech oh pys $000 per oh for he frs 5 yers he $000 per oh for he ex 5 yers. Clcule he prese lue of hs uy usg 8% effece re of eres. 7. Repe queso 5 ssug he pyes re couous hrough he yer.

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