- - Global Facal Maaeme Dscou ad Prese alue Techques opyrh 999 by Ers Mau. All rhs reserved. No par of hs lecure may be reproduced whou he permsso of he auhor.. Overvew Las Revso: Sepember 9, 999 I hs lecure we wll roduce dscou echques ad eres rae mahemacs. Ths maeral s mpora several respecs. The echques are fudameal for almos ay facal calculao, ra from smple asks lke calcula he repaymes o a morae or rack a loa balace o more comple applcaos. The maeral s also foudaoal for subseque lecures, parcularly bod valuao, sock valuao ad vesme apprasal echques. Fally, he dscusso below offers some basc shs o he ma asks ad fuco of facal markes, whch we are o o deepe dur he remader of he course.. Objecves A he ed of hs u you should be able o: Track a loa balace. Decde wheher you should re-morae your house. Deerme he requred mohly corbuo o your peso pla. Decde wha lump sum you eed o se asde oday order o fud he collee educao of your chldre.
- - alue a perpeual bod. Dsush aual perceae raes ad effecve aual raes ad use hem correcly.. Fuure alue: The value omorrow of a dollar oday Suppose you jus receved a bous payme of $,, ad you ca pu o a bak accou a a rae of 6% p. a. (p. a. per year). Pla ahead, you wa o deerme he cosumpo you ca afford oe year from ow. How does hs compare o recev a bous payme of $, oe year from ow? Oe of he mos fudameal prcples of face s ha a dollar oday (or $,, hs case) s worh more oday ha a dollar omorrow. Le s look a hese umbers more deal. You ca ves he $, a 6%, so a he ed of oe year he sum your savs accou has row o $,6: you ow $6 more a year from ow f you receve he $, oday raher ha a year hece. How would hs relaoshp chae, f you compare a $, oday wh a payme of $, wo years from ow? We kow already ha we oba $,6 a he ed of he frs year f we ves hem he savs accou for oe year. A he ed of he secod year we wll have: $,6.6 $,36. We call he umber $,36 he fuure value of $, a 6%, years from ow. We wsh o develop he cocep of a fuure value more eerally. Observe ha he fuure value ca be obaed as follows: fuure value $,.6.6 $,.6.
- 3 - We ca erae hs process order o fd ou wha he fuure value of $, oday a 6% s a he ed of hree, fve or e years, ad how would chae f he eres rae were 8%. We use he follow symbols: Fuure value afer years Ial vesme Ieres rae per year The we ca wre dow a eeral epresso for fuure value: ( ). () Suppose $, ad %. The we oba he follow paer of fuure values for ra from o 3. Fuure alues Fuure value $, $8, $6, $4, $, $, $8, $6, $4, $, $ 4 7 3 6 9 5 8 Perod Afer 3 years, our al value of $, has row o a respecable
- 4 - $,. 3 $74,494. osder also he follow eample. Eample : Suppose you oba wo paymes, $5, oday ad $5, eacly oe year from ow. You ca pu hese paymes o a savs accou ad ear eres a a rae of 5%. Wha s he balace your savs accou eacly 5 years from ow? ompue he follow able: Year ash flow Ieres Balace $5,. $. $5,. $5,. $5. $,5. $. $5.5 $,76.5 3 $. $538.3 $,3.63 4 $. $565.3 $,865.66 5 $. $593.8 $,458.94 A he ed of he frs year you receve $5 eres (5% of $5,), v you a oal balace of $,5 oe year from ow. The you ca compue fuure values for each year as $,5.5, where for he balace wo years from ow, ad 4 for he balace 5 years from ow, so fve years you have a balace of: $,5.5 4 $,458.94. You ca easly eed he above eample o oher applcaos order o keep rack of he fuure developme of a vesme ha you make oday. Ths ad he follow calculaos were doe a spreadshee proram. Please refer o he spreadshee. Replcao of he umbers he eamples wh a calculaor may lead o slhly dffere resuls because of roud errors.
- 5 -.3. Prese alue: The value oday of a dollar omorrow Ofe you wll eed o do he reverse operao of he oe we performed above, ad ask he queso: How much do you have o pu o your bak accou oday, so ha oe year from ow, he balace s eacly $,, f I accrue eres a a rae of 6% o he balace. Hece, you wsh o deerme he amou P ha solves: P.6 $, whch ves you mmedaely: $, P.6 $9,433.96 Hece, f you pu $9,433.96 o your bak accou oday, he hs amou wll row o eacly $, oe year from ow. We call he amou P (here $9,433.96) he prese value of $, oe year from ow a a eres rae of 6%. I s easy o see from equao () above ha f s he fuure value (. e., he ed of perod balace our savs accou), he s he prese value. We ca herefore solve () for he prese value o fd: () ( ) The prese value formula () preses he flp sde of he prcple ha a dollar omorrow s worh less ha a dollar oday. I our case, oe dollar a year from ow s
- 6 - worh oly $.9433. Prese values are mpora f you wsh o compare dffere lables. osder he follow eample: Eample : Suppose your dauher Jae jus raduaed from collee ad wshes o ake a posraduae course. Jae has he choce bewee wo uverses of comparable qualy ha offer he wo-year course of her choce. Uversy A chares $8, of uo fees for he frs year ad s epeced o crease hese fees o $, he secod year, whereas uversy B chares $9, he frs year ad $9,5 he secod year. You ve Jae a sum suffce o cover all her uo fees, ad she ca ves hs a a rae of 5.5% p. a. How much do you have o ve o Jae f she aeds he course a eher uversy A or B? Assume all uo fees are always due a he ed of he year. We use a sep-by-sep approach o calculae he amou Jae has o borrow f she aeds he course a A. The prese value of a payme of $8, a 5.5% a he ed of oe year s: $8,/.55$7,58.94. The prese value of a payme of $, a he ed of he secod year s: $,/.55 $8984.5. Hece, f Jae chooses A, you have o ve her $6,567.46 oday o fully cover her uo fees. Smlarly, for B we fd: $9, $9,5 $7,66..55.55. Le s check ha hs acually works. If Jae chooses B ad you had over he sum of $7,66., she pus o her savs accou ad accumulaes a balace of
- 7 - $8,4.74$7,66..55 a he ed of he year. The she pays $9, o he uversy ad reas $9,4.74 her accou. A he ed of he secod year she has accumulaed $9,4.74.55$9,5. her savs accou, jus eouh o mee he bll for uo fees a he ed of he secod year. The follow able racks hepaymes ad balaces of jae s accou. Dae Ial Balace Payme Rema Balace $7,66. $. $7,66. $8,4.74 $9,. $9,4.74 $9,5. $9,5. $. The las eample roduces aoher aspec of prese values ha s worh emphasz. We could smply add he prese values of he wo paymes for uo fees. Ths propery s called value addvy ad makes work wh prese values very srahforward. To eeralze hs uo, we use he symbol for he cash flow a he ed of year. I he prevous eample, we have $9, ad $9,5 for uversy B. The we defe prese value of a sream of paymes as:... ( ) ( ) ( ) ( ) ( ) (3) whch shows ha we ca always calculae he prese value of a seres of paymes by add he prese values of each dvdual payme. Noe ha each eleme of he The Greek leer Σ deoes he so-called summao operaor ad reads as follows: Sum a seres of elemes dsplayed o he rh had sde of he leer Σ for all
- 8 - seres (3) has a very smple srucure. We mulply he payme perod by he facor ( ) whch depeds oly o he dae of he payme ad he eres rae. Ths facor s called he dscou facor, ad f we eed o be more precse, we refer also as he -perod dscou facor. The follow able ad raph ve he dscou facor for a eres rae of 6% for up o fve perods: Perod Dscou facor /.6.943.89 3.6 4.47 5.33 49.58 5.54 bewee ad. Hece, he rh had sde of equao (3) s a shorhad for he lef had sde of he equao.
- 9 - Dscou facor.9.8.7.6.5.4.3.. 3 4 5 Perods Eample 3: Recosder he uo fee eample, bu assume ha uversy A requres ha half of he uo fee for a parcular year s pad before ad he oher half a he ed of he academc year. Hece, Jae has o make he follow paymes: Year Payme $4, $9,$4,$5, $5, Noe ha he payme a he ed of he frs year covers half of he uo for he frs ad half of he uo for he secod year. Hece, ow he amou of moey she has o pu asde o cover her uo a uversy A s: $9, $5, $ 4, $7,3.7.55.55 almos as much as he prese value of uo fees for B.
- - The ma advaae of prese values s ha hey make payme sreams wh dffere ms comparable. I our eample, he paymes a uversy A are due a mes ad, hose for B a, ad. I order o compare lke wh lke, we eed o epress hese paymes oe commo u, here oday s dollars. However, he prcple s oly o epress paymes erms of dollars of he same year, we could equally well choose he las year. Suppose we apply formula () o he prese value epresso equao (3). We oba: ( ). (4) We derve he seps lead o (4) he apped. Epresso (4) has a uve erpreao as a loa balace. 3 Suppose you eed o make a sequece of paymes order o mee a oblao (lke collee uo fees), ad you have o borrow he moey you eed for hese paymes from he bak because you have o come. The he value s he balace o your loa o he day afer you made he las payme. To see hs, oe ha you compoud eres o he frs payme over years, so afer years you owe ( ) for borrow. O payme you compoud eres oly over - years, so you owe ( ) a he ed of year. ou hs way, afer he -h payme of (o whch you have o accrued ay eres), your loa balace s ve by (4). 3 Effecvely, here s o dfferece wheher you aalyze hs from he po of vew of a borrower as a loa balace, or from he po of vew of a leder as a fud o whch you make perod paymes.
- - Eample 4: For our eample we ca epress he payme sreams o A ad B erms of her ed of year values by us he fuure value formula: $7,3.7.55 $8,947. Suppose you dd o ve ay moey o Jae, ad she had o ake ou a loa o cover her uo. The she would make he requred paymes as descrbed eample 3. O he frs payme of $4, a he be of he frs year she owes $4,.55$4, a he ed of he frs year. The she makes aoher payme of $9,, add hs amou o her loa balace Over her secod year she has o pay eres o he $4,, so her lably from hs payme creases o $4,.55$4,45.$4,.55. Add o hs he prcpal ad eres from he secod payme, whch amous o $9,.55$9,495. o ve a oal of $3,947. Add he las payme afer year of $5, ves a oal loa balace of $8,947., whch s eacly he fuure value we jus compued. The follow able ves he balaces ad paymes: Year Ial Balace Payme Ieres Balace $. $4,. $. $4,. $4,. $9,. $. $3,. $3,. $5,. $77. $8,947. Hece, we ca summarze ha prese values ad fuure values are useful order o compare payme sreams. The mpora prcple s value addvy: we ca add prese values ad fuure values, provded hey are epressed erms of dollars of he same year. Prese values have he erpreao of moey o be se asde (for a lably), or wealh erms of curre dollars (for a asse). If I have o make paymes
- - over he e years, ad I ca accumulae eres a a rae % per year, he he lump sum I have o se asde oday order o mee hese oblaos s he prese value of hese paymes. Smlarly, fuure values keep rack of accou or loa balaces. If I have o make paymes over he e years, ad I ca borrow he moey ad accumulae eres a a rae % per year, he he loa balace afer mak he las payme s he fuure value from hese paymes. Smlarly, f I make paymes o a savs accou ad accumulae eres, he he balace of my savs accou s ve by he fuure value of all paymes o hs accou..4. ompoud Iervals So far we have made oe lm assumpo by maa ha eres s compouded aually. Ths s ulkely, ad dffere facal coracs come wh dffere compoud ervals: morae ad cred card loas ypcally compoud eres mohly, savs accous quarerly, ad bods sem-aually. Hece, we modfy our symbols as follows: m m R R/m r Number of years Number of compoud perods per year Number of compoud perods Nomal or saed eres rae, also called he APR (aual perceae rae) perodc eres rae effecve aual eres rae Eample 5: Suppose you have a 5-year morae wh a saed APR of 9%, where eres s compouded mohly. The 5, m ad he umber of compoud perods
- 3 - s 53 mohs. The APR s R9%, ad.75% s he mohly eres rae. Eample 6: You ake ou a loa o face your car a a eres rae of %, wh quarerly paymes. The, m4 ad /45 years. The APR s R%, 3% s he quarerly eres rae. I order o roduce he cocep of a effecve aual rae, we eed o sudy he mpac of dffere compoud ervals a lle more closely. The mos srahforward case here s fuure value. Suppose you ake ou a loa of dollars oday wh a APR of R ad compoud ervals. The formula () above s sll vald, bu s ow he umber of compoud ervals, whch s eerally o he umber of years, ad s he eres rae per compoud erval, o per year. I order o epress hs erms of years ad aual perceae raes, we ca subsue o oba: R ( ) m m (5) Eample 7: Suppose you borrow $, a a APR of % ad repay oe lump sum a he ed of he year. If eres s compouded aually, he you owe $, a he ed of he year. However, f eres s compouded sem-aually, he your eres rae for half a year s 6%, so your loa balace afer s mohs s $,6. Therefore, a he ed of he year you eed o repay $,6.6$,36..
- 4 - The addoal $36. represes he compoud eres (6% of he $6 eres added o your loa balace afer 6 mohs). Smlarly, wh quarerly compoud your loa balace would accumulae o $,55.9. The follow able ves your lably for dffere compoud ervals f you repay afer oe or afer wo years oe lump sum: ompoud Perod ompoud Iervals Year $,. $,544. 6 mohs $,36. $,64.77 4 mohs 3 $,48.64 $,653.9 Quarer 4 $,55.9 $,667.7 mohs 6 $,6.6 $,68.4 Mohs $,68.5 $,697.35 Days 365 $,74.75 $,7.99 Hours 876 $,74.96 $,7.47 Secods 556 $,74.97 $,7.49 We ca see mmedaely ha creas he umber of compoud perods also creases he effecve coss of a loa. Hece, f he compoud erval s o oe year, he he APR does o ve us he eres coss of he loa for oe year ay more. Ths leads us o he defo of a effecve eres rae whch we deoe by r, ad whch s dffere from he saed or omal eres rae or APR. To see he dfferece, cosder he case of a mohly compoud perod eample 7 ad he prevous able. The saed eres rae (APR) s %. The eres ha accumulaes o he loa s he same as f we had aual compoud ad a eres rae of.685%, subsaally hher ha he saed rae of %. (see he shaded row he able) Ths eres rae of.685% s our effecve aual rae. I s defed as he rae ha we eed o apply o he oral loa order o oba he oal eres ha accumulaes o he loa oe year. Noe ha hs rae also works for wo or ay umber of years. I our eample, afer wo years we have
- 5 - accumulaed $,.685 $,697.35. The effecve aual rae s wha we eed order o compue he effecve eres coss of he loa. Hece, mus sasfy: R ( r) m m (6) whch ves us mmedaely ha: r R m m (7) whch s derved he apped. Eample 8: For he umbers eample 7 we oba for he effecve aual rae: ompoud Perod ompoud Iervals Effecve aual eres rae Year.% 6 mohs.36% 4 mohs 3.4864% Quarer 4.559% mohs 6.66% Mohs.685% Days 365.7475% Hours 876.7496% Secods 556.7497% Oe observao s mmedae from hs able: he effecve eres rae creases wh he umber of compoud ervals, bu does so a a ever smaller rae. As he umber of compoud ervals becomes fely lare (or, equvalely, as he leh of oe compoud erval becomes fesmally small), we ca fd a very covee epresso for he effecve aual rae as:
- 6 - r e R (8) where e represes Euler s umber (e.788). 4 Noe ha you have o epress he eres rae decmal form here, e.., % as.. I our eample we oba: r e..7497 Ths s he same umber we oba wh compoud every secod up o 6 decmal places. I markes where we have o work wh daly ervals (e.. fuures ad opos markes) couous compoud s mos of he me easer ha compu effecve aual raes from (8). Eample 9: You have o make a payme o a loa wh a curre balace of $, ha maures 5 days from ow. Ieres accumulaes daly o hs loa a a rae of 6% p. a. Wha s he effecve aual rae o hs loa? Wha error do you make your calculao f you assume ha eres s compouded couously? If eres accumulaes daly, he he effecve aual rae s:.6 r 365 365.683 or 6.83%. Ths ves a loa balace 5 days from ow of 5 5 $ $, 365.6, 365 ( r ) $,98. 3 4 Noe ha e s also he base of he aural loarhm, commoly deoed by l: l(e).
- 7 - Wh couous compoud we oba: r e.6 $, e.6837 5.6 365 $,98.39 a dfferece of 6 ces o a loa balace of $,. Eample : Suppose he effecve aual eres rae s 9%. Whch APR do you have o use f eres o hs accou accumulaes mohly? ouously? Ths s a more advaced queso. We eed a ukow eres rae R such ha: R.9 R (.9 ). 865 or 8.65%. The operao for couous compoud volves ak los: e R (.9). 86.9 R l or 8.6%. Noe ha you have o be careful wh epress he eres rae ad caledar me here. R s a aual rae, hece me has o be epressed fracos of oe year. 5 Eample : Suppose you accrue eres o your cred card couously a a rae of.5% per moh. Wha s he effecve mohly rae? Wha s he effecve aual rae? 5 If we had epressed he eres rae for aoher compoud erval, e.., as a mohly rae, he we would have o epress me erms of mohs. Ths s smply a requreme o kep he us of measureme cosse.
- 8 - Wha would be f eres compouded mohly? Hece, how much eres do you accrue o a balace of $, f you repay afer 6 weeks (4 days)? ompue.5 he effecve mohly rae frs as e. 5. ompoud over mohs ves.5 -.97-.97 or 9.7%. If eres compouded mohly he effecve aual rae would be oly.5 -.956, or 9.56%. To compue your loa balace afer 4 days wh couous compoud, use: 4 365.97 $, $,.93..5. Dscou wh a fe me horzo Equaos (3) ad (4) sum up all he coceps reard prese values ad fuure values. However, some cases s possble o smplfy hese epressos f he sream of paymes has a cera paer. Surprsly, he eases formula obas he case where paymes () are cosa, ad () coue defely o he fuure. Ths paer s kow from so-called osols. These are perpeual bods ("cosoldao bods") ssued by he Brsh overme he 9 h ceury ha have a cosa coupo ad are ever repad. 6 Aoher applcao s compay valuao, where we cao assume ha dvded paymes sop a a defe po me. 7 6 7 We wll have more o say abou hese he lecure o bod valuao. We wll dscuss hs more deal he lecure o sock valuao.
- 9 - Therefore, assume ha for all perods sar a (. e., sar a he ed of he curre perod) for all perods o he fuure, ad also assume ha s he approprae dscou rae. The urs ou ha he prese value of hs sream of perpeual paymes s: ( ) ( ). (9) We derve hs epresso he apped us a sadard formula for he summao of eomerc seres. Eample : Suppose you are offered a perpeual bod ha ves you oe aual payme of $5 a he ed of each year, ad he e payme s eacly oe year from oday. The approprae dscou rae s 4% p. a. How much are you wll o pay for hs bod? Apply equao (9) ves: $5 $,5..4 You may fd equao (9) more uve by epress as:. Ths ca be ve he follow erpreao. Suppose you borrow, ad you pay he eres due o hs loa a he ed of each perod, bu you ever make a repayme of prcpal. The s your eres payme a he ed of each year. Of course, f you ever repay ay prcpal, he you have o keep mak eres paymes a he ed of each year defely.
- - Eample 3: Suppose you ake ou a loa of $, a a rae of 7.5% wh aual compoud, ad do o repay ay prcpal ad make aual eres paymes a he ed of he year. The you pay he leder $5.75$, a he ed of each year, ad you always owe he prcpal..6 Aues ad Moraes osa perpeual paymes are easy o aalyze, bu hey are o very commo. A more commo payme paer s a so-called auy, where paymes are also cosa, bu eed over a fe perod of me. The mos freque eample for hs s a morae loa, where he borrower repays he leder a loa a specfed umber of equal salmes. Eample 4: You ake ou a loa of $5, o your house a a morae rae (APR) of 6% over 3 years. Ths meas ha you repay he bak by mak 36 mohly paymes. Wha s he mohly repayme o hs loa? Noe ha.6/.5%, so, clearly, he aswer s more ha $75 (.5% of $5,), sce $75 would oly repay he eres, bu o repay ay prcpal. We eed o deerme how much more. Our objecve s ow o value a sream of cosa paymes. Apply (3) oce more ves: ( ). ()
- - Aa, we ve a rorous demosrao of () he apped. However, () ca also be demosraed very uvely. I s easy o see ha a auy s smply a dfferece bewee wo perpeues. To see hs, cosder he follow able: Tme - Perpeuy P Perpeuy P Auy A P-P We ca ow epress he paymes o he auy as: ash Flow ( A) ash Flow( P) ash Flow( P). Hece, apply he prcple of value addvy, ( A) ( P ) ( ) P. We have already esablshed he prevous seco ha he value of P a me zero s /. Moreover, by he same prcple he value of P a me s also /. Hece, he prese value of P s: ( P) ( ). Noe ha he paymes for P sar a me. However, he prese value of P a me s /, so we eed o dscou / over perods, o perods. Hece:
( A) ( P ) ( P) ( ). - - ( ) By remember how o epress a auy as a dfferece bewee wo perpeues, all you eed o remember s he prese value formula (), ad he formula for perpeues (9), ad you wll always kow how o derve () ad wha umber o use he epoe for. Eample 5: Sar Moraes cosders buy a morae from Moo Bak. The morae was orally a hry-year fed rae morae ad sll has eacly wey years of mohly paymes. The morae rae areed o he morae s 9%, ad he mohly paymes are $,5 per moh. How much s Sar Moraes wll o pay whe hey purchase he morae f he curre -year morae rae s 6%? The frs mpora observao here s ha he oral morae rae of 9% s compleely rreleva here. Sce eres raes have falle, Moo s dscou fuure paymes a 6%. Hece, he perodc eres rae s.6/.5 or.5%, ad he umber of perods s 4 mohs. Hece, we use () as follows: $,5.5.5 4 $9,37 Hece, Sar s wll o pay $9,37 for hs morae. If you ake ou a morae you are probably more eresed a dffere queso: ve he eres rae ad he amou you wsh o borrow, wha s your mohly repayme? Noe ha () has a very smply srucure. You smply mulply he cosa perodc
- 3 - payme by a facor ha oly depeds o he eres rae ad he umber of perods. Ths facor s called he auy facor. We use he symbol A () for hs facor, ha s defed as: A () ( ) () We ca herefore epress () as: A () Now s smple o solve for : A () () Equao () has a mpora erpreao. Suppose we wsh o ake ou a loa wh a amou, he s he cosa perodc payme we eed o make order o repay he loa over perods f he perodc eres rae s. Eample 6: You wa o ake ou a morae of $, o your house, ad you are offered a eres rae (APR) of 6% o a 5-year morae. Ieres s compouded mohly. Wha s your mohly repayme? Smply apply () ad () above wh 6%/.5% ad 58 o e: $,.5.5 8 $,687.7
- 4 - Noe ha hs s a he ross payme for eres ad prcpal, ad does o ake o accou a deducos or escrow paymes coeced wh hs loa. I s srucve o rewre () by subsu for he auy facor: The umeraor of hs epresso s already famlar from our dscusso of perpeues. If we do o make ay repaymes of prcpal, he we mus make mohly eres paymes equal o. The we would owe he full amou (or par amou) of he loa a he ed of he perod. The epresso he deomaor s clearly smaller ha oe, hece creases he payme o accou for he fac ha we make repaymes of prcpal as well as eres paymes. Eample 4 (co.) I eample 4 we already esablshed ha he mohly payme has o eceed $75. We ca ow calculae as: $5,.5.5 36 $899.33 Noe, however, ha he composo of he mohly payme bewee eres ad prcpal chaes over he lfeme of he loa. I he early saes of repay a loa, eres accous for mos of he mohly paymes. However, as you repay he prcpal, he loa balace decreases ad so does he eres compoe, ad a he ed he mohly
- 5 - paymes are almos erely repaymes of prcpal. The mechacs of hs are he subjec of a repayme schedule, whch s bes demosraed by way of a eample. Eample 7: Recosder eample 6 ad cosder he frs mohly payme of $,687.7. Afer oe moh, he eres owed o $, s eacly $,, or.5% of $,. Hece, he rema $687.7 s a repayme of prcpal, ad a he ed of he frs moh, you owe he bak he rema $,-$687.7$99,3.9. Hece, for he secod moh you have o pay eres oly o hs slhly reduced amou, whch s $996.56, so ha your prcpal payme he secod moh mus be $687.7-$996.56$69.5. The follow able shows he frs ad las few mohs of he repayme schedule: 8 Moh Ial Balace Payme Ieres Prcpal Ed Balace $,. $,687.7,. $687.7 $99,3.9 $99,3.9 $,687.7 996.56 $69.5 $98,6.3 3 $98,6.3 $,687.7 993. $694.6 $97,96.53 4 $97,96.53 $,687.7 989.63 $698.8 $97,8.45 78 $5,.93 $,687.7 5.6 $,66.65 $3,35.8 79 $3,35.8 $,687.7 6.75 $,67.96 $,679.3 8 $,679.3 $,687.7 8.4 $,679.3 $. The fure dsplayes he me-seres paers of he oal payme, ad s decomposo o prcpal ad eres.
- 6 - $, Repayme of a Morae Payme $,5 $, $5 $ Payme Ieres Prcpal 39 58 77 96 Moh 5 34 53 7 So far we have aalyzed he prese value of aues. Ofe we are also eresed he fuure value of a auy, for eample, f you wa o deerme he value you accumulae a peso pla f you make cosa paymes over a cera perod of me. Us () ad apply () ves: ( ) ) ( ) ( ). (3) Eample 8: Today s your 35 h brhday, ad you recko you ca pu asde $,4 a quarer o a peso pla where your moey accumulaes a a rae of 5% p. a., 8 See he spreadshee for he calculao of he complee repayme schedule.
- 7 - compouded quarerly. How much wll you have accumulaed he pla afer you made he las payme o your 65 h brhday? Effecvely, you make 34 quarerly paymes of $,4 over 3 years ha are compouded a.5% per quarer. Us (3): (.5 ) $66,5. 94 $,4..5.7. Grow Perpeues ad Grow Aues Our aalyss of perpeues ad aues above was lmed by he assumpo ha paymes say cosa over me. I s acually srahforward o aalyze a more eeral case, where paymes are allowed o row a a cosa rae over me. Hece, we posulae:... 3 ( ) ( ) ( ) ( )... ( ) Eample 9: The US overme wshes o ssue perpeual bods. I order o provde vesors wh a more aracve vesme, he reasury deparme decdes ha he aual coupo o oe bod s $ he frs year, ad row a a rae of 3% per year afer ha. The he aual coupo s $3 he secod year of he bod, $6.9 he hrd year, ad by he eh year has become a remarkable $3.48. Evdely, he mechacs are he same as he fuure value calculao from () above, where he rowh rae 3% akes he place of he eres rae. I urs ou ha such a row perpeuy s jus as srahforward o value as a cosa perpeuy. We ca adap (9) o ve:
( ) - 8 - ( ) ( ) (4) Equao (4) s derved he apped. Noe ha hs resul s oly vald f >: ca ever become eave f s posve. Eample : Recosder eample 9, ad suppose ha he aual rowh rae of coupos s 3%, ad he eres rae s 5%. Apply (4) o ve: $.5.3 $5, Noe ha (9) s a specal case of (4), hece we eed o memorze oly (4): If we se (4) we oba (9) aa. We ca apply he same loc o row aues, where he umber of paymes s fe. Smlarly, a fe sequece of row paymes s row auy. The parallel epresso for () s: (5) Eample : Recosder he eample from your peso pla eample 8. However, suppose you epec your corbuos o row a a rae of.5% per quarer, so your frs quarer s corbuos are sll o o be $,4 as before, bu he secod quarer you ow corbue $,4. Wha s he ed balace your fud afer you made he las payme o your 65 h brhday ow? Wha corbuo would you have o sar wh f you waed o accumulae $,,
- 9 - by your 65 h brhday? We compue hs wo seps. The frs sep compues he prese value as of your 35 h brhday. Ths s from (5): $,4.5.5.5.5 $88,878.6 The secod sep s o cover he prese value o a fuure value us (), whch ves us $88,878.6.5 $838,66.8. I order o accumulae $,, you eed o crease your paymes by a facor of $,,/$838.66.8.9, whch raslaes o a corbuo he al quarer of $,86.7. Noe, however, ha you are commed o crease hs by.5% per year, so he fal quarer s corbuo s o o be $5,8.67, almos double of wha you corbue oday! ocluso The ma purpose of hs oe s o llusrae dscou echques ad her applcaos. These echques are fudameal for all facal calculaos subseque lecures. I s easy o e los he maze of dffere formulas. I s ofe easer o memorze hese by udersad he relaoshps bewee he ma coceps. Prese values ad fuure values are bascally flp sdes of he same co, smply reverse he dreco me. You oly eed o remember he eerc formula () ad how relaes o (). All oher fuure value formulas are specal applcaos (so (4) follows from (3), (3) follows from ()). The eerc prese value formula s (3). I ecompasses all subseque formulas as specal cases. The ma specal cases are perpeues ad aues.
- 3 - Thk abou dscou raes as apply o perods. Perods are ypcally shorer ha oe year. osa aues ad cosa perpeues are specal cases of row aues ad row perpeues respecvely: Oly lear (4) ad (5) whch are more eeral. (9) s a specal case of (4), () s a specal case of (5). Hece, lear (), (), (3), (4) ad (5) s suffce, ad everyh else falls o place.
- 3 - Apped Dervao of equao (4): ombe () ad (3) as follows: from () ( ) ( ) ( ) ( ) ( )... ( ) ( ) ( ) ( )... ( ) ves (4) ( ) from (3) ( ) Dervao of equao (7): Dvd boh sdes of equao (6) by we e: R m ( r) m Now ake he -h roo o boh sdes. (Apply he epoe / o boh sdes, remember / ha ( ) ): ( r) R m m Subrac o boh sdes ves (7). Dervao of equao (9): We make he follow subsuo:
- 3 - ad observe ha < <. The we ca rewre he frs equaly (9) as: ( )... 3 Mulply boh sdes by - o oba: ( ) ( )( ) { } ( ) { } ( ) { } ( )......... 3 4 3 where he las equaly obas from cacel equvale epressos ad observ ha for <, coveres o zero as becomes lare. Fally, solv for ad subsu for ves: whch proves (9). Dervao of equao (): We use eacly he same sraey as he dervao of equao (9). Frsly: ( )... 3 he, afer mulply wh -:
- 33 - ( ) ( )( ) { } ( ) { } ( ) { } ( ) { } ( ) ( ) ( )...... 3 4 3 The we solve for as before: ( ) afer subsu for, whch ves (). Dervao of equao (4): We proceed as he dervao of equao (9) above, ecep ha we ow defe: The we ca rewre he lef had sde of (4) as: The, us parallel seps o he dervao of (9): Noe ha he sum coveres oly f <, whch s equvale o <. Oherwse
- 34 - we would have ha he sum dveres o fy. Dervao of equao (5): Us he same procedure as for equaos () ad (4), we oba: ( ) ( )
- 35 - Impora Termoloy auy auy facor 3 APR 4 compoud eres 4 osols 8 dscou facor 8 effecve eres rae 4 fuure value row auy 8 row perpeuy 7 omal eres rae 4 par amou 4 prese value 5 value addvy 7
- 36 - Impora Formulae Fuure value: Prese value: ( ) () () ( )... ( ) ( ) ( ) ( ) ( ) (3) Dscou facor: ( ) Auy: ( ) ( ) A () Grow Perpeuy: (4) Grow auy: (5)