Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities


 Imogen Todd
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1 Table of conens Chaper 1 Ineres raes and facors Ineres Simple ineres Compound ineres Accumulaed value Presen value Rae of discoun Consan force of ineres Varying force of ineres Discree changes in ineres raes 21 Chaper 1 pracice quesions 23 Chaper 2 Level annuiies Annuiyimmediae Annuiydue Deferred annuiies Coninuously payable annuiies Perpeuiies Equaions of value 44 Chaper 2 pracice quesions 49 Chaper 3 Varying annuiies Increasing annuiyimmediae Increasing annuiydue Decreasing annuiyimmediae Decreasing annuiydue Coninuously payable varying annuiies Compound increasing annuiies Coninuously varying paymen sreams Coninuously increasing annuiies Coninuously decreasing annuiies 89 Chaper 3 pracice quesions 92 Chaper 4 Nonannual ineres raes and annuiies Nonannual ineres and discoun raes Nominal phly ineres raes Nominal phly discoun raes Annuiiesimmediae payable phly Annuiiesdue payable phly Increasing phly annuiies Phly increasing phly annuiies 118 Chaper 4 pracice quesions 123 v
2 Chaper 5 Projec appraisal and loans Discouned cash flow analysis Nominal vs. real ineres raes Invesmen funds Allocaing invesmen income Loans: he amorizaion mehod Loans: he sinking fund mehod 153 Chaper 5 pracice quesions 158 Chaper 6 Financial insrumens Types of financial insrumens Bond valuaion Sock valuaion Shor sales Derivaive valuaion 198 Chaper 6 pracice quesions 208 Chaper 7 Duraion, convexiy and immunizaion Price as a funcion of yield Modified duraion Macaulay duraion Effecive duraion Convexiy Duraion, convexiy and prices: puing i all ogeher Immunizaion Full immunizaion Dedicaion 244 Chaper 7 pracice quesions 248 Chaper 8 The erm srucure of ineres raes Yieldomauriy Spo raes Forward raes Arbirage Nonannual compounding Nonannual forward raes Key informaion when using ineres raes 272 Chaper 8 pracice quesions 274 Chaper 9 Sochasic ineres raes Ineres raes as random variables Independen and idenically disribued ineres raes Lognormal ineres rae model Binomial ineres rae rees 293 Chaper 9 pracice quesions 300 Review quesions 305 Appendix A The normal disribuion 319 Appendix B The 30/360 day coun mehod 320 s o pracice quesions 321 Bibliography 329 Index 331 vi
3 Ineres raes Overview In addiion o being a medium of exchange, money can also be hough of as a cash amoun ha is invesed o earn even more cash as ineres. The word ineres can be ranslaed from he Greek language as he birh of money from money. Ineres is defined as he paymen by one pary (he borrower) for he use of an asse ha belongs o anoher pary (he lender) over a period of ime. This asse is also known as capial. In his sense, capial is jus money ha earns ineres. Oher forms of producive capial exis, bu wih ineres heory, we are primarily ineresed in moneary capial, which we also refer o as he principal amoun. When ineres is expressed as a percen of he capial amoun, i is referred o as an ineres rae. Ineres raes are mos ofen compued on an annual basis, bu hey can be deermined for nonannual ime periods as well. We have wo perspecives o consider. The firs is he perspecive of he owner of he capial who would like o be compensaed for lending i. The second is he perspecive of he borrower of he capial who is willing o pay he lender for he righ o borrow i. 1
4 Ineres raes Chaper 1 The borrower pays ineres o he lender for he use of he money. The lender may require addiional compensaion for he risk of defaul, which is he risk ha he borrower will no be able o repay he loan principal. If some risk of defaul exis, he lender will ypically demand a higher ineres rae o compensae for assuming his risk of defaul. Lenders ypically require collaeral, ie somehing of value pledged as securiy agains he risk of defaul, o help guaranee he loan will be repaid. The comedian Bob Hope once observed ha A bank is a place ha will lend you money if you can prove you don need i. In his firs chaper, we consider ineres earned over annual ime periods, bu in laer chapers we consider nonannual ime periods as well. We look a five ways o express ineres: simple ineres, compound ineres, simple discoun, compound discoun, and coninuous ineres. We can break hese mehods down ino wo classes: simplisic and realisic. The simple ineres and simple discoun mehods, as heir names imply, belong o he simplisic class. These mehods are impracical for large, complicaed financial ransacions. They may be used, however, for small, simple ransacions. The compound ineres, compound discoun, and coninuous ineres mehods belong o he realisic class. Compound ineres is generally used for financial ransacions. Compound discoun is less common, bu i isn difficul o grasp once we become comforable wih compound ineres. Coninuous ineres appears frequenly in academic and heoreical discussions of valuaion. All hree of he realisic mehods are consisen wih one anoher. Jus as disance can be measured in miles or kilomeers, ineres can be measured using compound ineres, compound discoun, or coninuous ineres raes. And jus as a European visiing America could insis on convering all disance measuremens ino meric, a finance professor working a a bank could insis on convering compound ineres raes ino coninuous ineres raes. The professor s calculaions would produce he correc values regardless of which ype of rae was used. Compound ineres, which allows ineres o grow on previously earned ineres, has amazing accumulaion powers when compared o simple ineres, which does no allow ineres o grow on previously earned ineres. Alber Einsein is said o have noed ha he mos powerful force in he universe is compound ineres. In his chaper, we also develop mehods for accumulaing paymens o a fuure poin in ime and discouning paymens o a previous poin in ime. When a paymen is accumulaed o a fuure poin in ime, is fuure value is calculaed, aking ino accoun any ineres ha will be earned during he invesmen period. When a paymen is discouned o a previous poin in ime, is presen value is calculaed, aking ino accoun any ineres ha will be earned during he invesmen period. In order o value a cash flow, we need o know he exac amoun of he cash flow and he exac iming of he cash flow. This chaper illusraes he ime value of money, ie a $1 paymen now is worh more han $1 payable in one year s ime. 1.1 Ineres Ineres on savings accouns When money is deposied ino a bank accoun, i ypically earns ineres. The deposior can be hough of as he lender and he bank can be hough of as he borrower. The borrower pays ineres o he lender o compensae for he use of he money and any risk of defaul. Le s say a person deposis $1,000 ino a bank accoun. One year laer, he accoun has accumulaed o $1,050. This amoun consiss of $1,000, represening he iniial deposi or capial, and $50, represening he ineres earned on he deposi over he year. The amoun of ineres 2
5 Chaper 1 Ineres raes earned over a period of ime is simply he difference beween he accumulaed accoun value a he end of he period and he accumulaed accoun value a he beginning of he period. For any amoun of ineres earned over a given period, we can also calculae he associaed ineres rae. The ineres rae in effec for a oneyear period is he amoun of ineres earned over he year divided by he iniial accumulaed value. In his example, he ineres rae for he year is: 1,050 1, = = 0.05 or 5% 1,000 1,000 A his poin, le s keep hings simple by considering annual ime periods only, bu in laer chapers we will consider nonannual ime periods. Ineres The amoun of ineres earned from ime o ime + s is: where Ineres rae AV+ s AV AV is he accumulaed value a ime. The annual ineres rae i in effec from ime o ime + 1 is: AV+ 1 AV i = AV where is measured in years. For example, Bob invess $3,200 in a savings accoun on January 1, On December 31, 2004, he accoun balance has grown o $3, The oal ineres earned during 2004 is: 3, , = $94.08 The annual ineres rae during 2004 is: 3, , = = or 2.94% 3, , Ineres on loans The same heory and definiions can be applied o a loan. A person may borrow money from a bank, eg o buy a car. In his case, he bank is he lender, and he individual is he borrower. As before, he borrower pays ineres o he lender o compensae for he use of he money and any risk of defaul. Le s say a person borrows $12,000 from a bank. The loan is o be repaid in full in one year s ime wih a paymen o he bank of $12,780. This amoun consiss of $12,000, represening he iniial amoun of he loan, and $780, represening he ineres paid on he loan over he year. The annual ineres rae in his example is: 12, , = = or 6.5% 12, , 000 3
6 Ineres raes Chaper 1 When we use ineres raes in he conex of a loan, wo imporan poins should be kep in mind: 1. The principal amoun of he loan is he amoun provided o he borrower when he loan is originaed. (In he example above, he principal amoun is $12,000.) 2. Ineres begins o accrue when he loan is originaed and coninues o accrue unil he loan is repaid in full. The second poin is analogous o a savings accoun: ineres begins o accrue when he money is deposied and coninues o accrue unil he deposi is repaid (wihdrawn) in full. 1.2 Simple ineres When money is invesed in an accoun paying simple ineres, ineres is only earned on he iniial deposi. Ineres is no earned on he ineres ha has previously accrued. Simple ineres If $X is invesed in an accoun ha pays simple ineres a a rae of i per year, hen he accumulaed value of he invesmen afer years is: AV = X(1 + i) The accumulaed value is composed of he iniial invesmen of $X and years of simple ineres a he rae of i per year, $Xi. The ineres earned each year is $Xi. For example, le s assume a bank accoun is opened wih a deposi of $100. The accoun pays simple ineres of 8% per year. We have i = 0.08 and X = 100. The accumulaed value of he accoun a he end of one year is: AV 1 = 100( ) = $ The accumulaed value of he accoun a he end of en years is: AV 10 = 100( ) = $ The accumulaed value of he accoun a he end of weny years is: AV 20 = 100( ) = $ Wih simple ineres, he accoun value increases linearly over ime. This is shown in he following graph. Simple Ineres Value Time 4
7 Chaper 1 Ineres raes As he coninuous naure of he graph implies, he formula for he accumulaed value of a deposi under simple ineres sill applies if is no an ineger. When is no an ineger, ineres is paid on a proraa (proporional) basis. For example, a bank accoun ha pays simple ineres of 6% per year is opened wih a deposi of $100. Wha is he accumulaed value of he accoun a he end of 9 monhs? Since we re working wih an annual simple ineres rae, should be expressed in erms of he number of years on deposi. So we have X = 100, = 9/12 = 0.75 years, and i = Hence, he accumulaed value afer 9 monhs is: Example 1.1 AV 0.75 = 100( ) = $ A bank accouns pays 3.6% simple ineres. Anna deposis $10,000 on January 1, 2004 and leaves her funds o earn ineres. Calculae he accumulaed value of Anna s accoun on April 1, 2006 and January 1, On April 1, 2006, we have = 2 years 3 monhs = 2.25 years, so he accumulaed value is: AV 2.25 = 10, 000( ) = $10, On January 1, 2007, we have = 3 years, so he accumulaed value is: AV 3 = 10, 000( ) = $11, Banks don ypically use simple ineres because i is jus oo, well, simplisic. If banks acually paid simple ineres, hen deposiors could earn more ineres by pursuing he following simple sraegy: as soon as ineres is credied o he accoun, wihdraw he oal accoun value and immediaely redeposi i, using he ineres paidodae o increase he size of he deposi. This is illusraed in he following example. Example 1.2 A bank accouns pays 6% simple ineres. Randy deposis $100 and leaves his funds o earn ineres for 2 years. Leonard also deposis $100, bu Leonard wihdraws his accumulaed value a he end of 1 year, and he hen immediaely reurns he money o he bank, deposiing i in a new accoun. Who has he greaer accumulaed value a he end of 2 years: Randy or Leonard? A he end of wo years, Randy has $112: AV 2 = 100( ) = $ A he end of 1 year, Leonard has $106: AV 1 = 100( ) = $ Bu Leonard hen wihdraws he $106 a he end of 1 year and deposis he $106 in a new accoun, which also earns a simple ineres rae of 6% per year. A he end of he second year, Leonard s accumulaed value is $112.36: AV 2 = 106( ) = $ Leonard has he greaer accumulaed value a he end of wo years. 5
8 Ineres raes Chaper 1 In he example above, Leonard found a way o earn ineres on he ineres he earned in he firs year. As we ll see in he nex secion, when ineres is earned on ineres, he ineres is compounding. If banks used simple ineres, hen deposiors who wihdrew and redeposied heir funds would have higher accoun values han he deposiors who simply lef heir funds in heir accouns. Since i doesn make sense o reward deposiors for wihdrawing and redeposiing heir funds, banks and oher financial insiuions don acually use simple ineres when calculaing accumulaed values. However, simple ineres is someimes used in circumsances where accuracy is no very imporan. For example, if he ime period is shor or if he amoun of money involved is small, hen simple ineres migh be considered sufficienly accurae. 1.3 Compound ineres When money is deposied ino an accoun paying compound ineres, ineres is earned on he iniial deposi and he ineres ha has previously accrued. This is analogous o using simple ineres, bu periodically he ineres is credied o he accoun and he ineres rae hen applies o he new, larger balance. We examine annual compounding in his chaper, which means ha ineres is credied o he accoun annually. I is also possible o compound more or less frequenly han annually, and we will learn more abou nonannual compounding in Chaper 4. Compound ineres If $X is invesed in an accoun ha pays compound ineres a a rae of i per year, hen he accumulaed value of he invesmen afer years is: AV = X(1 + i) Consider anoher bank accoun ha is opened wih a deposi of $100. This accoun pays compound ineres of 8% per year. Le s compare his compound ineres accoun o he simple ineres accoun ha we examined earlier ha paid simple ineres of 8% per year. Under boh compound and simple ineres, he accumulaed value a he end of one year is $108: Compound ineres accoun: = 100( ) = $ Simple ineres accoun: = 100( ) = $ AV1 AV1 Wih compound ineres, he enire accumulaed value, no jus he original principal, earns ineres. This causes he accumulaed value in he second year o grow more quickly under compound ineres han under simple ineres. The accumulaed value of he accoun under compound ineres is $0.64 greaer a he end of wo years han he accumulaed value under simple ineres: Compound ineres accoun: AV = 100( )( ) = $ Simple ineres accoun: AV = 100( ) = $ The $0.64 difference is due o he fac ha under compound ineres, he $8 of ineres ha was earned in he firs year earns $0.64 in ineres over he second year: 8(0.08) = $
9 Chaper 1 Ineres raes Insead of calculaing he accumulaed value year by year, we can use he formula from he box above o obain he same accumulaed value a he end of wo years under compound ineres: 2 AV 2 = 100( ) = $ The accumulaed value of he compound ineres accoun a he end of en years is: 10 AV 10 = 100( ) = $ Noice ha a he end of he enh year, he accumulaed value under compound ineres exceeds he $180 accumulaed value ha we calculaed under simple ineres. The accumulaed value of he compound ineres accoun a he end of he wenieh year is: 20 AV 20 = 100( ) = $ A he end of he wenieh year, he accumulaed value under compound ineres is much greaer han he $260 accumulaed value under simple ineres. This illusraes he power of compound ineres. As ime passes, an accoun paying compound ineres exhibis geomeric (or exponenial) growh, while an accoun paying simple ineres exhibis linear growh. The following graph shows how he value of he accoun using compound ineres increases over ime. The doed line shows he linear progression of he deposi earning simple ineres. Compound vs. Simple Ineres Value Compound Simple Time As he graph implies, he formula for he accumulaed value of a deposi under compound ineres sill applies if is no an ineger, assuming ha ineres is paid on a proraa basis. For example, afer 2.5 years he accumulaed value of he compound ineres accoun in he curren example is: 2.5 AV 2.5 = 100( ) = $ Example 1.3 Drew invess $100 on January 1, 2004 in a bank accoun ha pays compound ineres of 5% per year. Wha is he accumulaed value of he accoun on Ocober 1, 2005? We have X = 100, i = 0.05, and = 1 year 9 monhs = 1.75 years. So, he accumulaed value of he accoun on Ocober 1, 2005 is: 1.75 AV 1.75 = 100( ) = $
10 Ineres raes Chaper 1 Example 1.4 Helen borrows $1,000 for 3 years a a compound ineres rae of 11.65%. Wha will he oal paymen be in 3 years o repay boh he loan principal and ineres due on he loan? We have X = 1,000, i = , and = 3. So, he oal paymen required in 3 years is: 3 AV 3 = 1, 000(1.1165) = $1, Based on he discussion so far, one migh hink ha he accumulaed value of an accoun earning compound ineres always exceeds he accumulaed value of an accoun earning he same simple ineres rae. Bu his is only rue when he period is greaer han one year. When he invesmen period is less han one year, he accumulaed accoun value under simple ineres acually exceeds he accumulaed accoun value under he same compound ineres rae. Since he difference beween he accumulaed values is small when he elapsed ime is less han one year, his effec is no easily observed in he previous char. So, le s magnify he porion of he char from ime 0 o 1 year. The graph below illusraes ha he accumulaed value under compound ineres is less han he accumulaed value under simple ineres during he firs year. Compound vs. Simple Ineres Value Compound Simple Time Noe: his graph is no drawn o scale. For example, if he simple ineres rae and he compound ineres rae are each 5%, le s deermine he accumulaed amoun of a $100 deposi afer 6 monhs. Under simple ineres and compound ineres, he deposi accumulaes o: 8 Compound ineres accoun: AV0.5 = 100( ) = $ Simple ineres accoun: AV = 100( ) = $ This may no seem like a big difference, bu consider a large pension fund deposi of $100,000,000. Insead of a $0.03 difference beween accoun values, we have a $30,000 difference! We can also sae he relaionship beween simple and compound ineres mahemaically. For i > 0, we have: 1 + i > (1 + i) for 0 < < 1 ie simple ineres greaer before one year 1 + i = (1 + i) for = 1 ie simple and compound ineres equal over one year 1 + i < (1 + i) for > 1 ie compound ineres greaer afer more han one year 0.5
11 Chaper 1 Ineres raes Le s ake anoher look a Example 1.2. This ime, he bank credis ineres o Leonard and Randy a a compound ineres rae, and he resuling reamen is more equiable. Example 1.5 A bank accouns pays 6% compound ineres. Randy deposis $100 and leaves his funds o earn ineres for 2 years. Leonard also deposis $100, bu Leonard wihdraws his accumulaed value a he end of 1 year, and he hen immediaely reurns he money o he bank, deposiing i in a new accoun. Who has he greaer accumulaed value a he end of 2 years: Randy or Leonard? A he end of wo years, Randy has $112.36: 2 AV 2 = 100( ) = $ A he end of 1 year, Leonard has $106: AV 1 = 100( ) = $ Leonard wihdraws he $106 a he end of 1 year and hen deposis he $106 in a new accoun, which also earns a compound ineres rae of 6% per year. A he end of he second year, Leonard s accumulaed value is $112.36: AV 2 = 106( ) = $ Leonard and Randy each have $ a he end of wo years. If he ype of ineres is no specified, he convenion is o use compound ineres, especially if a period longer han one year is being considered. From now on, a compound ineres rae of x % per year compounded annually will be referred o as an annual effecive ineres rae of x %. We discuss his in more deail laer, bu for now we should undersand ha he effecive ineres rae is effecive over he ime period in quesion, which is one year in he case of an annual effecive ineres rae. Example 1.6 Carmen borrows $1,000 for 90 days a an annual effecive ineres rae of 8.25%. Wha will he oal paymen be in 90 days o repay boh he loan principal and ineres due on he loan? The erm of he loan is 3 monhs, so = 3/12 = The oal paymen is: 0.25 AV 0.25 = 1, 000(1.0825) = $1, Example 1.7 Sam borrows $20,000. He repays he loan 4 years laer wih a paymen of $26, Wha is he annual effecive ineres rae on he loan? We have X = 20,000, = 4, and AV 4 = 26, Solving for i we have: 4 26, = 20, 000(1 + i) 1/4 26, i = 20, 000 i = 7.5% 9
12 Ineres raes Chaper Accumulaed value We have already learned how o accumulae a single paymen ino he fuure. The same can be done for several paymens. In laer chapers, we ll develop useful noaion and formulas o do his, bu for now we ll jus use he basic principles ha we ve learned so far. Compound ineres accumulaed value facor Under compound ineres, he accumulaed value afer years of a deposi of $1 is he compound ineres accumulaed value facor: AVF = (1 + i) Simple ineres accumulaed value facor Under simple ineres, he accumulaed value afer years of a deposi of $1 is he simple ineres accumulaed value facor: AVF = (1 + i ) Le s consider he siuaion when here is more han one deposi. For example, a deposi of $100 is invesed oday and anoher $100 deposi is invesed a he end of 5 years. Using an annual effecive ineres rae of 6%, how much is his invesmen worh a he end of 10 years? The iniial deposi of $100 is invesed for 10 years. The accumulaed value a ime 10 years is: (1.06) The second deposi is invesed for 10 5 = 5 years. The accumulaed value a ime 10 years is: 5 100(1.06) So, he oal accumulaed value a ime 10 years is: (1.06) + 100(1.06) = = $ These cash flows can be illusraed on a imeline Cash flow: Time: Timelines such as his are paricularly useful when he cash flows are complicaed. Example 1.8 A deposi of $100 is invesed oday. Anoher $100 is invesed a he end of 5 years. Using an annual simple ineres rae of 6%, how much is his invesmen worh a he end of 10 years? Under simple ineres, he oal invesmen a ime 10 years is worh: 100( ) + 100( ) = $
13 Chaper 1 Ineres raes Example 1.9 A deposi of $X is invesed a ime 6 years a an annual effecive ineres rae of 8%. A second deposi of $X is invesed a ime 8 years a he same ineres rae. A ime 11 years, he accumulaed amoun of he invesmen is $976. Calculae X. The firs deposi is invesed for 11 6 = 5 years, and he second deposi is invesed for 11 8 = 3 years. The imeline is as follows: Cash flow: X X 976 Time: The accumulaed amoun a ime 11 years is: 5 3 X(1.08) + X(1.08) = Solving his, we have: Example X = X = $ Jim invess $500 a he beginning of 2002, 2003, and 2004 in a bank accoun ha pays simple ineres. A he end of 2004, he accumulaed value of he accoun is $1,635. Calculae he rae of ineres paid by he bank. The firs deposi is invesed for 3 full years; he second deposi for 2 full years; and he final deposi for jus one year. Under simple ineres, he oal accumulaed value a he end of 2004 is: 500(1 + 3 i) + 500(1 + 2 i) + 500(1 + i) = 500(3 + 6 i ) Solving for i, we have: 500(3 + 6 i) = 1, 635 i = 4.5% 1.5 Presen value No only can we deermine he accumulaed value of invesmens a a fuure poin in ime, bu we can also find he value now, a ime 0, of a paymen o be made in he fuure, aking ino accoun any ineres ha will be earned during he invesmen period. This is known as he presen value of a fuure paymen. The process of allowing for fuure ineres in deermining a presen value is also known as discouning a paymen. The presen value of $X payable in years is he amoun ha, if invesed now a an annual effecive ineres rae i, will accumulae o $X a ime years. 11
14 Ineres raes Chaper 1 For example, suppose ha he annual effecive ineres rae is 5%, and we need o make a paymen of $100 in one year s ime. Wha is he presen value of he $100 payable in one year? Clearly he answer mus be less han $100. If we have $100 now, we can inves i a 5% o obain $105 in one year. So, how much do we need o inves now a 5% in order o have $100 in 1 year? If we denoe his unknown quaniy as PV, hen PV can be found by solving: PV 1.05 = PV = = Since $95.24 would accumulae o $100 in 1 year, he presen value of $100 in 1 year is $ Presen value Assuming an annual compound ineres rae of i, he presen value of a paymen of $X o be made in years is: X PV = = X(1 + i) (1 + i) In ineres heory applicaions, many presen values are calculaed. Noaion has been developed o assis wih expressing presen values. General oneyear presen value facor The oneyear presen value facor, which is also known as he oneyear discoun facor, is: 1 1 v = = (1 + i) 1 + i Compound ineres presen value facor Under compound ineres, he presen value of a paymen of $1 o be made in years is he compound ineres presen value facor: PVF = v = (1 + i) Simple ineres presen value facor Under simple ineres, he presen value of a paymen of $1 o be made in years is he simple ineres presen value facor: PVF = (1 + i ) 1 A paymen s presen value can be deermined as of any poin in ime. If he valuaion dae is no specified, we usually assume ha we are a ime 0 when we calculae presen values. However, we can also imagine ourselves o be a a laer poin in ime. So i is also valid o refer o a presen value a ime n. Noice ha we have inroduced a new variable, v, which is he presen value facor for one year. 1 When we sudy annuiies in Chaper 2, we ll find i convenien o wrie v insead of (1 + i). The presen value facor is simply he inverse of he accumulaed value facor. We should also noice ha he presen value facors for boh compound and simple ineres are equivalen only when = 1. 12
15 Chaper 1 Ineres raes Le s derive a formula for i in erms of v. We have: 1 v = 1 + i Rearranging, we have: i = i 1 v = v or 1 v i = v Example 1.11 A paymen of $10 is o be made a ime 7 years. Deermine he presen value of his paymen a ime 0 and a ime 4 years. The annual effecive ineres rae is 6%. The invesmen is discouned for 7 0 = 7 years o deermine he presen value a ime 0: 7 10 PV7 = 10 v = = $ (1.06) The presen value of an invesmen a ime 4 years of $10 made a ime 7 years is he amoun of money ha would need o be se aside a ime 4 years so ha he accumulaed value a ime 7 is $10. The invesmen is discouned for 7 4 = 3years o deermine he presen value a ime 4: 3 10 PV3 = 10 v = = $ (1.06) Example 1.12 A paymen of $10 is made a ime 7 years. Deermine he presen value of his paymen a ime 0 and a ime 4 years. The annual simple ineres rae is 6%. The presen value a ime 0 under simple ineres is: 10 PV 7 = = $7.04 ( ) The presen value a ime 4 under simple ineres is: 10 PV 3 = = $8.47 ( ) 1.6 Rae of discoun So far we have considered simple and compound ineres raes. Given an annual ineres rae i, if an invesor deposis or loans $1 a ime 0, a paymen of (1 + i) is reurned a ime 1 year. The ineres of i is paid a he end of he ime period. Anoher valid approach is o view he ineres as being paid a he beginning of he ime period. When ineres is paid a he beginning of he ime period, ineres is paid in advance, and i is known as discoun. Jus as wih ineres, he amoun of discoun earned is simply he difference beween he accumulaed accoun value a he end of he period and he accumulaed accoun value a he beginning of he period. 13
16 Ineres raes Chaper 1 For any amoun of discoun earned over a period, we can also calculae an associaed discoun rae. The discoun rae in effec for a oneyear period is he amoun of discoun earned over he year divided by he ending accumulaed value. Discoun The amoun of discoun earned from ime o ime + s is: where AV+ s AV AV is he accumulaed value a ime. Discoun rae The annual discoun rae d in effec for he year from ime o ime + 1 is: AV+ 1 AV d = AV + 1 where is measured in years. Le s work hrough a simple example o explain his. Consider he case of a oneyear loan of $1. The lender loans $1 a ime 0. The borrower pays discoun (ie ineres) o he lender a ime 0, and reurns a paymen of $1 a ime 1 year. The ineres is paid a he beginning of he ime period. If ineres were payable on he loan a he end of year, he amoun of ineres payable a ha ime would be i. Bu he discoun is payable a he beginning of he year, so we mus find he presen value of he paymen of i. Hence, he discoun payable a ime 0 is: i iv = 1 + i The ne amoun ha he borrower receives a ime 0 from he loan is he amoun of he loan ($1) less he discoun ( iv ): 1 Loan amoun discoun = 1 i iv = 1 v 1 + i = 1 + i = So, he borrower receives a ne paymen of v now in exchange for a promise o repay $1 in one year. In oher words, he presen value a he beginning of he year is v, and he accumulaed value a he end of he year is $1. This is consisen wih he fac ha he presen value of $1 payable one year from now is v. The rae of discoun, which is denoed by d, is defined as he amoun of he discoun ( iv ) divided by he accumulaed value a he end of he year ($1). Hence, he discoun rae is: d = iv We can derive an alernaive expression for d : i 1 d = iv = = 1 = 1 v 1+ i 1+ i This is consisen wih he fac ha he borrower effecively pays ineres of (1 v) a he beginning of he year. 14
17 Chaper 1 Ineres raes Rae of discoun The annual rae of discoun, d, is he amoun of ineres payable a he sar of he year, on a loan of $1 for one year. i d = (1 v) = = iv 1 + i I is very useful o become comforable convering beween hese variables. Given ha i represens he ineres paid a he end of a year, and d represens he discoun paid a he sar of he year, his relaionship makes sense. We can hink of d as he presen value of a paymen of i payable a he end of he year. We can rearrange he definiion o obain a relaionship beween he variables i and d : d i = 1 d As we ve seen in he previous secions, ineres raes can be used o accumulae and discoun cash flows. We can also accumulae and discoun paymens using he rae of discoun. (Don be confused by he fac ha discoun raes bear he adjecive discoun in his respec, he erminology can be a lile confusing.) Since d = (1 v), we have v = 1 d. We already know ha v is he oneyear discoun facor. We also know ha he accumulaion facor is he inverse of he discoun facor. Using his, we can deermine he presen value and accumulaed value facors for compound raes of discoun. Compound rae of discoun presen value facor Assuming an annual compound rae of discoun of d, he presen value of a paymen of $1 o be made in years is he compound rae of discoun presen value facor: PVF = (1 d) Compound rae of discoun accumulaed value facor Assuming an annual compound rae of discoun of d, he accumulaed value afer years of a deposi of $1 is he compound rae of discoun accumulaed value facor: AVF = (1 d) Example 1.13 An invesor would like o have $5,000 a he end of 20 years. The annual compound rae of discoun is 5%. How much should he invesor deposi oday o reach ha goal? The presen value facor for compound discoun is (1 d). The invesor should se aside: , 000(1 d ) = 5, 000(1 0.05) = 5, 000(0.95) = $1,
18 Ineres raes Chaper 1 Example 1.14 An invesor deposis $1,000 oday. The annual compound rae of discoun is 6%. Wha is he accumulaed value of he invesmen a he end of 10 years? The accumulaed value facor for compound discoun is (1 d ). The accumulaed value a he end of 10 years is: = 1, 000(1 ) = 1, 000(1 0.06) = $1, AV d We can also have a simple rae of discoun insead of a compound rae of discoun. Recall ha wih simple ineres, he ineres iself does no earn ineres. Simple rae of discoun presen value facor Assuming an annual simple rae of discoun of d, he presen value of a paymen of $1 o be made in years is he simple rae of discoun presen value facor: PVF = (1 d) Simple rae of discoun accumulaed value facor Assuming an annual simple rae of discoun of d, he accumulaed value afer years of a deposi of $1 is he simple rae of discoun accumulaed value facor: AVF 1 = (1 d) Example 1.15 An invesor would like o have $10,000 a he end of 5 years. The annual simple rae of discoun is 3%. How much should he invesor deposi oday o reach ha goal? The presen value facor for simple discoun is The invesor should se aside: Example 1.16 (1 d ). 10, 000(1 d) = 10, 000( ) = $8, An invesor deposis $5,000 oday. The annual simple rae of discoun is 5%. Wha is he accumulaed value of he invesmen a he end of 7 monhs? The accumulaed value facor for simple discoun is (1 d). Since we re working wih an annual discoun rae, he variable should be expressed in erms of years. So, = 7/12 years. The accumulaed value a he end of 7 monhs is: , 000(1 ) 5, 000(1 0.05) $5, AV = d = =
19 Chaper 1 Ineres raes Simple raes of ineres and discoun are no used very ofen, and when hey are used, i is generally over shor periods of ime. The nex example illusraes why simple discoun is suiable only for shor periods of ime. Example 1.17 An invesor would like o have $5,000 a he end of 20 years. The simple rae of discoun is 5%. How much should he invesor deposi oday o reach ha goal? The presen value facor for simple discoun is (1 d). The invesor should se aside: PV20 = 5, 000(1 d ) = 5, 000( ) = 5, 000(0) = $0 This seems a lile silly, bu i is he correc answer if we use a simple rae of discoun. The lender would lend $5,000 now. Each year s ineres is: $5, 000(0.05) = $250 Therefore, 20 years of ineres is $5,000: $250(20) = $5, 000 Subracing he ineres received of $5,000 from he principal of $5,000, he invesor s ne paymen is zero. Since all of he ineres is paid up fron, he borrower is lef wih nohing a ime 0, bu has an obligaion o pay $5,000 a ime 20 years. This is an excellen deal for he lender, bu he realiy is ha he lender will no find anyone willing o agree o hese erms. Simple discoun and simple ineres are simply unrealisic for use over longer periods of ime, so i is usually bes o assume ha any rae of ineres or rae of discoun is a compound rae unless oherwise saed. 1.7 Consan force of ineres We have considered he discree cases where ineres is payable a he sar or a he end of he year. An annual compound ineres rae is he change in he accoun value over one year, expressed as a percenage of he beginningofyear value. An annual compound discoun rae is he change in he accoun value over one year, expressed as a percenage of he endofyear value. We now consider he case of ineres ha is compounded coninuously. A coninuously compounded ineres rae is called he force of ineres. The force of ineres a ime is denoed δ. Nonannual compounding is covered in more deail in Chaper 4. The force of ineres is he insananeous change in he accoun value, expressed as an annualized percenage of he curren value. If he annual effecive ineres rae is consan, hen he force of ineres is also consan. When he force of ineres is consan, he subscrip is omied from δ since he force of ineres does no vary over ime. We can derive he value of δ as follows. In erms of calculus, he force of ineres is he derivaive of he accumulaed value wih respec o ime expressed as a percenage of he accumulaed value a ime : AV δ = AV where AV is he firs derivaive of AV wih respec o. 17
20 Ineres raes Chaper 1 Using he sandard calculus resul: d a x x = a ln a dx We have: d AV = X(1 + i) = X(1 + i) ln(1 + i) d Hence we can solve for he force of ineres: AV X(1 + i) ln(1 + i) δ = = = ln(1 + i) AV X(1 + i) Consan force of ineres rae When he force of ineres is consan, he force of ineres can be expressed in erms of he annual effecive ineres rae i : δ = ln(1 + i ) For example, if he annual effecive ineres rae is 5%, hen he force of ineres is: δ = ln(1 + i ) = ln(1.05) = Le s derive expressions for i and v in erms of δ. Rearranging δ = ln(1 + i), we have: We also have: δ δ 1+ i = e i = e 1 1 δ v = (1 + i) = e We recognize (1 + i ) as he oneyear accumulaed value facor and v as he oneyear presen value facor, so we can now express accumulaed values and presen values in erms of he consan force of ineres. Consan force of ineres accumulaed value facor Assuming a consan force of ineres of δ, he accumulaed value afer years of a paymen of $1 is he consan force of ineres accumulaed value facor: δ AVF = e Consan force of ineres presen value facor Assuming a consan force of ineres of δ, he presen value of a paymen of $1 o be made in years is he consan force of ineres presen value facor: δ PVF = e Example 1.18 If he consan force of ineres is 6%, wha is he corresponding annual effecive rae of ineres? 18
21 Chaper 1 Ineres raes The corresponding annual effecive rae of ineres is calculaed as follows: Example 1.19 δ 0.06 i = e 1 = e 1 = or 6.184% Using a consan force of ineres of 4.2%, calculae he presen value of a paymen of $1,000 o be made in 8 years ime. The presen value is: 8δ , 000e = 1, 000 e = $ Example 1.20 A deposi of $500 is invesed a ime 5 years. The consan force of ineres is 6% per year. Deermine he accumulaed value of he invesmen a he end of 10 years. The money is invesed for 10 5 = 5 years. The accumulaed amoun is: 5δ e = 500 e = $ Varying force of ineres In he previous secion, he force of ineres was consan over ime. The force of ineres can vary over ime, in which case we wrie i as δ. I is defined as: AV δ = AV Using he sandard calculus resul: d f ( x) ln ( f( x) ) = dx f ( x) We can wrie: d AV ln( AV ) = d AV Hence we have: d δ = ln( AV) d Inegraing boh sides of he equaion from 1 o 2, where 1 < 2 : 2 δ = 2 d AV d ln[ AV] d = ln[ AV ] ln[ AV ] = ln d AV 1 19
22 Ineres raes Chaper 1 Hence: exp AV δd = 1 AV 2 2 Le s consider he meaning of his imporan resul. 1 The accumulaed value a ime 2 divided by he accumulaed value a ime 1 is equal o 1 plus he percenage increase in he accumulaed value from ime 1 o ime 2. Hence he accumulaed value a ime 2 of $1 invesed a ime 1 is given by: exp δ 2 d 1 Before, we only used one subscrip of o simplify he noaion when we have a deposi a ime 0 ha is accumulaed o a laer ime. When he deposi is made a a ime oher han ime 0, he following noaion is more useful. Varying force of ineres accumulaed value facor The varying force of ineres accumulaed value facor a ime 2 of an invesmen of $1 made a ime 1 is: Example 1.21 AVF = δ 2 1, exp 2 d 1 A deposi of $10 is invesed a ime 2 years. Using a force of ineres of δ = , find he accumulaed value of his paymen a he end of 5 years. The accumulaed value is: ( ) AV2,5 = 10 exp = ( ) d 10 exp = 10e = 10 e = $14.77 In a similar manner, we can deermine he presen value of a paymen using a varying force of ineres. We recall ha he presen value is he inverse of he accumulaed value. Varying force of ineres presen value facor The varying force of ineres presen value facor a ime 1 of a paymen of $1 o be made a ime 2 is: PVF 2 1, = exp 2 δ 1 d Noice ha he only difference beween he presen value and he accumulaed value using a nonconsan force of ineres is he negaive sign in he exponen of he presen value equaion. 20
23 Chaper 1 Ineres raes Example 1.22 A deposi of $200 is invesed a ime 8 years. Using a force of ineres of δ = , find he presen value of his paymen a he end of 3 years. The presen value is: ( ) PV3,8 = 200exp = ( ) d 200exp ( ) = 200e = 200 e = $ Discree changes in ineres raes The resuls and echniques used so far in his chaper can be applied o oher siuaions. This secion looks a some examples where he ineres rae changes a specific poins in ime. For example, he annual effecive ineres rae migh be x % for he firs m years and hen y % for he nex n years. Example 1.23 A deposi of $100 is invesed a ime 0. The annual effecive ineres rae is 5% from ime 0 o ime 7 years and hereafer is 6%. Calculae he accumulaed value of he invesmen a ime 10 years. A ime 7 years, he accumulaed value of he invesmen is: 7 AV 7 = 100(1.05) This amoun is hen accumulaed for a furher 3 years a a rae of 6% per year. The accumulaed value a ime 10 is: Example AV 10 = 100(1.05) (1.06) = $ A deposi of $2,500 is invesed a ime 0. The annual effecive rae of ineres is 2.5% from ime 0 o ime 6 years. The annual effecive rae of discoun is 2.5% from ime 6 o ime 10 years and he annual force of ineres is 2.5% hereafer. Find he accumulaed value of he invesmen a ime 13 years. The accumulaion facor from ime 0 o ime 6 years is: 6 AVF 0,6 = The accumulaion facor from ime 6 o ime 10 years is: 4 4 AVF 6,10 = ( ) =
24 Ineres raes Chaper 1 The accumulaion facor from ime 10 o ime 13 years is: AVF = e 10, The accumulaed value of he invesmen from ime 0 o ime 13 years is herefore: AV = 2, 500 AVF AVF AVF 0,13 0,6 6,10 10, = 2, e = $3,
25 Chaper 1 Ineres raes Chaper 1 Pracice Quesions Quesion guide Quesions es maerial from Secions Quesions es maerial from Secions Quesions are from he SOA/CAS Course 2 exam or he IOA/FOA 102 exam. Quesion 1.1 $300 is deposied in a bank accoun, which pays simple ineres of 3.5% a year. Calculae he accumulaed value of he deposi afer 6 years. Quesion 1.2 Fund P earns ineres a a simple rae of 4% a year. Fund Q earns ineres a a simple rae of i % a year. $100 is invesed in fund P and $ is invesed in fund Q. The accumulaed amoun in he wo funds will be equal afer 5 years. Deermine he simple rae of ineres i. Quesion 1.3 $800 is deposied in a bank accoun, which pays compound ineres of 5.25% a year. Calculae he accumulaed value of he deposi afer 15 years. Quesion 1.4 Fund P earns ineres a a compound rae of 4% a year. Fund Q earns ineres a a compound rae of i % a year. $100 is invesed in fund P and $ is invesed in fund Q. The accumulaed amoun in he wo funds will be equal afer 5 years. Deermine he compound rae of ineres i. Quesion 1.5 Using an annual effecive rae of ineres of 5%, find he presen value of an invesmen of $5,000 a ime 20 years. Quesion 1.6 Your greagreagrea grandfaher se aside $10 on July 1, 1876 in an accoun paying 5% annual effecive ineres. Assuming no addiional deposis or wihdrawals were made, wha is he accoun balance on January 1, 2004? Wha would he accoun balance have been if he accoun had been paying an annual effecive ineres rae of 10% insead of 5% all of hese years? Quesion 1.7 Larry has a credi card balance of $10,000. The annual effecive ineres rae on he credi card is 15%. Larry akes ou a home equiy loan o pay off his credi card balance. The ineres rae on he home equiy loan is 5%. Ignoring axes, how much does his sraegy save Larry, assuming he pays off he loan in full 18 monhs from now? 23
26 Ineres raes Chaper 1 Quesion 1.8 Using an annual effecive rae of discoun of 5% per year, find he accumulaed value a ime 20 years of an invesmen of $5,000 a ime 0. Quesion 1.9 Using a simple rae of discoun of 4% per year, find he presen value of a paymen of $5,000 a ime 3 monhs. Quesion 1.10 A $15,000 car loan is repaid wih one paymen of $18, afer 36 monhs. Wha is he annual effecive discoun rae? Quesion 1.11 $500 is paid a ime 8 years a a consan force of ineres of 10%. Deermine he presen value of he invesmen a ime 0. Quesion 1.12 $1,000 is paid a ime 6 years. Find he presen value a ime 4 years using a force of ineres of δ = Quesion 1.13 Find he accumulaed value a ime 5 years of $30 ha is invesed a ime 0. Use a force of ineres of δ = Quesion 1.14 A paymen of $1,000 is due a ime 15 years. Beween imes 0 o 5 years, he annual effecive rae of ineres is 7%. Beween imes 5 and 10 years i is 9% and beween imes 10 and 15 years i is 4%. Calculae he presen value of he paymen a ime 0. Quesion 1.15 $100 is invesed a ime 0. The consan force of ineres is 7% from ime 0 o ime 5 years and 5% from ime 5 o ime 8 years. Deermine he accumulaed value of he invesmen a ime 8 years. Quesion 1.16 The effecive annual rae of discoun has been 4% for he las 5 years. Prior o ha, i was 5%. A bank accoun has a balance of $457 oday. A single deposi of $X was placed in he accoun 8 years ago. Calculae he value of X. 24
27 Chaper 1 Ineres raes Quesion 1.17 SOA/CAS Bruce and Robbie each open up new bank accouns a ime 0. Bruce deposis $100 ino his bank accoun, and Robbie deposis $50 ino his. Each accoun earns an annual effecive discoun rae of d. The amoun of ineres earned in Bruce s accoun during he 11h year is equal o $X. The amoun of ineres earned in Robbie s accoun during he 17h year is also equal o $X. Calculae X. Quesion 1.18 SOA/CAS Ernie makes deposis of $100 a ime 0, and $X a ime 3. The fund grows a a force of ineres 2 δ = 0.01, > 0. The amoun of ineres earned from ime 3 o ime 6 is $X. Calculae X. Quesion 1.19 David can receive one of he following wo paymen sreams: (i) $100 a ime 0, $200 a ime n, and $300 a ime 2n (ii) $600 a ime 10. A an annual effecive ineres rae of i, he presen values of he wo sreams are equal. n Given v = , deermine i. SOA/CAS Quesion 1.20 The force of ineres, δ is: IOA/FOA < 5 δ = ( ) > 5 Calculae he presen value of $100 payable a ime
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