FINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals


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1 FINANCIAL MATHEMATICS A Practcal Gude for Actuares and other Busness Professonals Second Edton CHRIS RUCKMAN, FSA, MAAA JOE FRANCIS, FSA, MAAA, CFA Study Notes Prepared by Kevn Shand, FSA, FCIA Assstant Professor Warren Centre for Actuaral Studes and Research
2 Contents 1 Interest Rates and Factors Interest SmpleInterest CompoundInterest AccumulatedValue PresentValue Rate of Dscount: d Constant Force of Interest: δ VaryngForceofInterest DscreteChangesnInterestRates... 9 ExercsesandSolutons Level Annutes AnnutyImmedate Annuty Due DeferredAnnutes ContnuouslyPayableAnnutes Perpetutes EquatonsofValue ExercsesandSolutons Varyng Annutes IncreasngAnnutyImmedate IncreasngAnnutyDue DecreasngAnnutyImmedate DecreasngAnnutyDue ContnuouslyPayableVaryngAnnutes CompoundIncreasngAnnutes ContnuouslyVaryngPaymentStreams ContnuouslyIncreasngAnnutes ContnuouslyDecreasngAnnutes ExercsesandSolutons NonAnnual Interest Rate and Annutes NonAnnualInterestandDscountRates Nomnal p thly Interest Rates: (p) Nomnal p thly Dscount Rates: d (p) AnnutesImmedate Payable p thly AnnutesDue Payable p thly ExercsesandSolutons Project Apprasal and Loans DscountedCashFlowAnalyss Nomnalvs.RealInterestRates Investment Funds AllocatngInvestmentIncome Loans:TheAmortzatonMethod Loans:TheSnkngFundMethod
3 ExercsesandSolutons Fnancal Instruments TypesofFnancalInstruments MoneyMarketInstruments Bonds CommonStock PreferredStock Mutual Funds GuaranteedInvestmentContracts(GIC) DervatveSercurtes BondValuaton StockValuaton ExercsesandSolutons Duraton, Convexty and Immunzaton PrceasaFunctonofYeld ModfedDuraton MacaulayDuraton EffectveDuraton Convexty MacaulayConvexty EffectveConvexty Duraton,ConvextyandPrces:PuttngtallTogether RevstngthePercentageChangenPrce ThePassageofTmeandDuraton PortfoloDuratonandConvexty Immunzaton FullImmunzaton ExercsesandSolutons The Term Structure of Interest Rates YeldtoMaturty SpotRates
4 1 Interest Rates and Factors Overvew nterest s the payment made by a borrower (.e. the cost of dong busness) for usng a lender s captal assets (usually money); an example s a loan transacton nterest rate s the percentage of nterest to the captal asset n queston nterest takes nto account the rsk of default (rsk that the borrower can t pay back the loan) the rsk of default can be reduced f the borrower promses to release an asset of thers n the event of ther default (the asset s called collateral) 1.1 Interest Interest on Savngs Accounts a bank borrows a depostor s money and pays them nterest for the use of ther money the greater the need for money, the greater the nterest rate offered Interest Earned Durng the Perod t to t + s: AV t+s AV t the Accumulated Value at tme n s denoted as AV n nterest earned durng a perod of tme s the dfference between the Accumulated Value at the end of the perod and the Accumulated Value at the begnnng of the perod TheEffectveRateofInterest: s the amount of nterest earned over a oneyear perod when 1 s nvested s also defned as the rato of the amount of Interest Earned durng the perod to the Accumulated Value at the begnnng of the perod Interest on Loans = AV t+1 AV t AV t compensaton a borrower of captal pays to a lender of captal lender has to be compensated snce they have temporarly lost use of ther captal nterest and captal are almost always expressed n terms of money 2
5 1.2 Smple Interest let the nterest amount earned each year on an nvestment of X be constant where the annual rate of nterest s : AV t = X(1 + t), where (1 + t) s a lnear functon smple nterest has the property that nterest s NOT renvested to earn addtonal nterest amount of Interest Earned to tme t s 1.3 Compound Interest I = AV t AV 0 = X(1 + t) X = X t let the nterest amount earned each year on an nvestment of X also allow the nterest earned to earn nterest where the annual rate of nterest s : where (1 + ) t s an exponental functon AV t = X(1 + ) t, compound nterest produces larger accumulatons than smple nterest when t>1 note that a constant rate of compound nterest mples a constant effectve rate of nterest 1.4 Accumulated Value Accumulated Value Factor: AV F t assume that AV t s contnuously ncreasng letx be the ntal Prncpal nvested (X >0) where AV 0 = X AV t defnes the Accumulated Value that amount X grows to n t years the Accumulated Value at tme t s the product of the ntal captal nvestment of X (Prncpal) made at tme zero and the Accumulaton Value Factor: AV t = X AV F t, where AV F t =(1+t) f smple nterest s beng appled and AV F t =(1+) t f compound nterest s beng appled 3
6 1.5 Present Value Dscountng Accumulated Value s a future value pertanng to payment(s) made n the past Dscounted Value s a present value pertanng to payment(s) to be made n the future dscountng determnes how much must be nvested ntally (Z) sothatx wll be accumulated after t years Z (1 + ) t = X Z = X = X(1 + ) t (1 + ) t Z represents the present value of X to be pad n t years letv = 1, v s called a dscount factor or present value factor 1+ Z = X v t Dscount Functon (Present Value Factor): PVF t smple nterest: PVF t = 1 1+t compound nterest: PVF t = 1 (1 + ) t = vt compound nterest produces smaller Dscount Values than smple nterest when t>1 4
7 1.6 Rate of Dscount: d Defnton an effectve rate of nterest s taken as a percentage of the balance at the begnnng of the year, whle an effectve rate of dscount s at the end of the year. eg. f 1 s nvested and 6% nterest s pad at the end of the year, then the Accumulated Value s 1.06 eg. f 0.94 s nvested after a 6% dscount s pad at the begnnng of the year, then the Accumulated Value at the end of the year s 1.00 d s also defned as the rato of the amount of nterest (amount of dscount) earned durng the perod to the amount nvested at the end of the perod d = AV t+1 AV t AV t+1 f nterest s constant, then dscount s constant the amount of dscount earned from tme t to t + s s AV t+s AV t Relatonshps Between and d f 1 s borrowed and nterest s pad at the begnnng of the year, then 1 d remans the accumulated value of 1 d at the end of the year s 1: (1 d)(1 + ) =1 nterest rate s the rato of the dscount pad to the amount at the begnnng of the perod: = d 1 d dscount rate s the rato of the nterest pad to the amount at the end of the perod: d = 1+ the present value of endofyear nterest s the dscount pad at the begnnng of the year v = d the present value of 1 to be pad at the end of the year s the same as borrowng 1 d and repayng 1 at the end of the year (f both have the same value at the end of the year, then they have to have the same value at the begnnng of the year) 1 v =1 d the dfference between endofyear,, and begnnngofyear nterest, d, depends on the dfference that s borrowed at the begnnng of the year and the nterest earned on that dfference d = [1 (1 d)] = d 0 5
8 Dscount Factors: PVF t and AV F t under the smple dscount model the Dscount Present Value Factor s: PVF t =1 dt for 0 t<1/d under the smple dscount model the Dscount Accumulated Value Factor s: AV F t =(1 dt) 1 for 0 t<1/d under the compound dscount model, the Dscount Present Value Factor s: PVF t =(1 d) t = v t for t 0 under the compound dscount model, the Dscount Accumulated Value Factor s: AV F t =(1 d) t for t 0 a constant rate of smple dscount mples an ncreasng effectve rate of dscount a constant rate of compound dscount mples a constant effectve rate of dscount 6
9 1.7 Constant Force of Interest: δ Defntons annual effectve rate of nterest s appled over a oneyear perod a constant annual force of nterest can be appled over the smallest subperod magnable (at a moment n tme) and s denoted as δ an nstantaneous change at tme t, due to nterest rate δ, where the accumulated value at tme t s X, can be defned as follows: δ = d dt AV t AV t = d dt ln(av t) = d X(1 + )t dt X(1 + ) t = (1 + )t ln(1 + ) (1 + ) t δ = ln(1 + ) takng the exponental functon of δ results n e δ =1+ takng the nverse of the above formula results n e δ = 1 1+ = v Accumulated Value Factor (AV F t ) usng constant force of nterest s AV F t = e δt PresentValueFactor(PVF t ) usng constant force of nterest s PVF t = e δt 7
10 1.8 Varyng Force of Interest let the constant force of nterest δ now vary at each nftesmal pont n tme and be denoted as δ t a change from tme t 1 to t 2, due to nterest rate δ t, where the accumulated value at tme t 1 s X, can be defned as follows: t2 δ t = d dt AV t AV t = d dt ln(av t) t2 d δ t dt = t 1 t 1 dt ln(av t) dt = ln(av t2 ) ln(av t1 ) ( ) AVt2 δ t dt =ln t2 t 1 t2 AV t1 e t 1 δ t dt = AV t 2 AV t1 Varyng Force of Interest Accumulaton Factor  AV F t1,t 2 let AV F t1,t 2 = e t2 t 1 δ t dt represent an accumulaton factor over the perod t 1 to t 2, where the force of nterest s varyng ft 1 = 0, then the notaton smplfes from AV F 0,t2 to AV F t.e. AV F t = e fδ t s readly ntegrable, then AV F t1,t 2 can be derved easly t 0 δ t dt fδ t s not readly ntegrable, then approxmate methods of ntegraton are requred Varyng Force of Interest Present Value Factor  PVF t1,t 2 let 1 1 PVF t1,t 2 = = AV F t2 = AV δ t dt t 1 = e t 1 t1,t 2 AV t2 δ t dt e t 1 represent a present value factor over the perod t 1 to t 2, where the force of nterest s varyng ft 1 = 0, then the notaton smplfes from PVF 0,t2 t2 to PVF t.e. PVF t = e t 0 δ t dt 8
11 1.9 Dscrete Changes n Interest Rates the most common applcaton of the accumulaton and present value factors over a perod of t years s t AV F t = (1 + k ) and PFV t = k=1 t k=1 1 (1 + k ) where k s the constant rate of nterest between tme k 1andtmek 9
12 Exercses and Solutons 1.2 Smple Interest Exercse (a) At what rate of smple nterest wll 500 accumulate to 615 n 2.5 years? Soluton (a) Exercse (b) 500[1 + (2.5)] = 615 = =9.2% 2.5 In how many years wll 500 accumulate to 630 at 7.8% smple nterest? Soluton (b) Exercse (c) 500[ (n)] = 630 = =3.33 years.078 At a certan rate of smple nterest 1,000 wll accumulate to 1,100 after a certan perod of tme. Fnd the accumulated value of 500 at a rate of smple nterest three fourths as great over twce as long a perod of tme. Soluton (c) 1, 000[1 + n] =1, 100 n =.11 Exercse (d) 500[ n] = 500[1 + (1.5)(.11)] = Smple nterest of = 4% s beng credted to a fund. In whch perod s ths equvalent to an effectve rate of 2.5%? Soluton (d) 0.25 = n = 1+(n 1) (n 1) n =16 10
13 1.3 Compound Interest Exercse (a) Fund A s nvested at an effectve annual nterest rate of 3%. Fund B s nvested at an effectve annual nterest rate of 2.5%. At the end of 20 years, the total n the two funds s 10,000. At the end of 31 years, the amount n Fund A s twce the amount n Fund B. Calculate the total n the two funds at the end of 10 years. Soluton (a) Let the ntal funds be A and B. Therefore, we have two equatons and two unknowns: A(1.03) 20 + B(1.025) 20 =10, 000 A(1.03) 31 =2B(1.025) 31 Solvng for B n the second equaton and pluggng t nto the frst equaton gves us A = 3, and B =2, We seek A(1.03) 10 + B(1.025) 10 whch equals Exercse (b) 3, (1.03) 10 +2, (1.025) 10 =7, Carl puts 10,000 nto a bank account that pays an annual effectve nterest rate of 4% for ten years. If a wthdrawal s made durng the frst fve and onehalf years, a penalty of 5% of the wthdrawal amount s made. Carl wthdrawals K at the end of each of years 4, 5, 6, 7. The balance n the account at the end of year 10 s 10,000. Calculate K. Soluton (b) 10, 000(1.04) K(1.04) K(1.04) 5 K(1.04) 4 K(1.04) 3 =10, , 802 K[(1.05)(1.04) 6 +(1.05)(1.04) 5 +(1.04) 4 +(1.04) 3 ]=10, 000 4, 802 = K 4.9 K =
14 1.4 Accumulated Value Exercse (a) 100 s deposted nto an account at the begnnng of every 4year perod for 40 years. The account credts nterest at an annual effectve rate of. The accumulated value n the account at the end of 40 years s X, whch s 5 tmes the accumulated amount at the end of 20 years. Calculate X. Soluton (a) 100(1+) (1+) (1+) 40 = X = 5[100(1+) (1+) (1+) 20 ] 100(1+) 4 [1+100(1+) (1+) 36 ]=5 100(1+) 4 [1+100(1+) (1+) 16 ] 100(1 + ) 4 [ 1 [(1 + ) 4 ] 10 1 (1 + ) 4 ] [ 1 [(1 + ) =5 100(1 + ) 4 4 ] 5 ] 1 (1 + ) 40 =5[1 (1 + ) 20 ] 1 (1 + ) 4 (1 +) 40 5(1+) = 0 [(1+) 20 1][(1+) 20 4] = 0 (1 +) 20 =1or(1+) 20 =4 (1 + ) 20 =1 =0% AV 40 = 1000 and AV 20 = 500, mpossble snce AV 40 =5AV 20 therefore, (1 + ) 20 =4 [ ] [ ] 1 (1 + ) X = 100(1 + ) 4 40 = 100( ) = 100( ) = (1 + )
15 1.5 Present Value Exercse (a) Annual payments are made at the end of each year, forever. The payments at tme n s defned as n 3 for the frst n years. After year n, the payments reman constant at n 2. The present value of these payments at tme 0 s 20n 2 when the annual effectve rate of nterest s 0% for the frst n years and 25% thereafter. Calculate n. Soluton (a) [(1 3 )v 0% +(2 3 )v0% 2 +(33 )v0% (n3 )v0% n ]+vn 0% [(n2 )v25% 1 +(n2 )v25% 2 +(n2 )v25% ] =20n2 [(1 3 )+(2 3 )+(3 3 )+... +(n 3 )] + (n 2 )v 25% [1 + v 1 25% + v2 25% +...]=20n2 [ n 2 (n +1) 2 ] [ ] +(n 2 1 )v 25% =20n v 25% [ (n +1) 2 4 ] [ ] 1 +(.8) 1.8 =20 [ ] (n +1) 2 +4=20 4 (n +1)2 4 =16 (n +1) 2 =4 n =7 Exercse (b) At an effectve annual nterest rate of, >0, each of the followng two sets of payments has present value K: () A payment of 121 mmedately and another payment of 121 at the end of one year. () A payment of 144 at the end of two years and another payment of 144 at the end of three years. Calculate K. Soluton (b) v = 144v v 3 = K 121(1 + v) = 144v 2 (1 + v) v = K = 121( ) K =
16 Exercse (c) The present value of a seres of payments of 2 at the end of every eght years, forever, s equal to 5. Calculate the effectve rate of nterest. Soluton (c) 2v 8 +2v 16 +2v =5 2v 8 [1 + v 8 + v =5 [ ] 2v v 8 =5 2v 8 =5 5v 8 7v 8 =5 v 8 = 5 7 (1 + ) 8 = Rate of Dscount Exercse (a) 1+ = ( ) = A busness permts ts customers to pay wth a credt card or to receve a percentage dscount of r for payng cash. For credt card purchases, the busness receves 97% of the purchase prce onehalf month later. At an annual effectve rate of dscount of 22%, the two payments are equvalent. Fnd r. Soluton (a) $1(1 r) =$0.97v =$0.97v 1 24 (1 r) =.97(1 d) 1 24 =.97(1.22) 1 24 =.96 r =.04 = 4% 14
17 Exercse (b) You depost 1,000 today and another 2,000 n fve years nto a fund that pays smple dscountng at 5% per year. Your frend makes the same deposts nto another fund, but at tme n and 2n, respectvely. Ths fund credts nterest at an annual effectve rate of 10%. At the end of 10 years, the accumulated value of your deposts s exactly the same as the accumulated value of your frend s deposts. Calculate n. Soluton (b) 1, 000[1 5%(10)] 1 +2, 000[1 5%(5)] 1 =1, 000(1.10) 10 n +2, 000(1.10) 10 2n 1, , =1, 000(1.10)10 v n +2, 000(1.10) 10 v 2n 4, = 2, v n +5, v 2n 5, v 2n +2, v n 4, =0 } {{ } } {{ } } {{ } a b c v n = 2, , (5, )( 4, ) 2(5, ) (1.10) n = 1 ln( ) = n = = ln(1.10) =.7306 Exercse (c) A depost of X s made nto a fund whch pays an annual effectve nterest rate of 6% for 10 years. At the same tme, X/2 s deposted nto another fund whch pays an annual effectve rate of dscount of d for 10 years. The amounts of nterest earned over the 10 years are equal for both funds. Calculate d. Soluton (c) X(1.06) 10 X = X 2 (1 d) 10 X 2 d =
18 1.7 Constant Force of Interest Exercse (a) You are gven that AV t = Kt 2 + Lt + M, for0 t 2, and that AV 0 = 100, AV 1 = 110, and AV 2 = 136. Determne the force of nterest at tme t = 1 2. Soluton (a) AV 0 = M = 100 AV 1 = K + L + M = 110 K + L =10 AV 2 =4K +2L + M = 136 4K +2L =36 These equatons solve for K =8,L =2,M = 100. We know that δ t = d dt AV t AV t = 2Kt + L Kt 2 + Lt + M δ 1 = (2)(8)( 1 2 ) (8)( 1 2 )2 +(2)( 1 = ) = Exercse (b) Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces of nterest on the two funds are equal. Soluton (b) AV F A t =1+.10t and AV F B t =(1+.05t) 1 δ A t = d dt AV F A t AV F A t = t δ B t = Equatng and solve for t d dt AV F B t AV F B t =.05(1.05t) t) 1 =.05(1.05t) t = t = t t =5 1.05t 16
19 1.8 Varyng Force of Interest Exercse (a) On 15 March 2003, a student deposts X nto a bank account. The account s credted wth smple nterest where =7.5% On the same date, the student s professor deposts X nto a dfferent bank account where nterest s credted at a force of nterest δ t = 2t,t 0. t 2 + k From the end of the fourth year untl the end of the eghth year, both accounts earn the same dollar amount of nterest. Calculate k. Soluton (a) Smple Interest Earned = AV 8 AV 4 = X[1+.075(8)] X[1+.075(4)] = X(0.075)(4) = X(.3) Compound Interest Earned = AV 8 AV 4 = Xe 8 0 δ t dt Xe 4 0 δ t dt Xe 8 0 2t 4 t 2 + k dt Xe 0 2t 8 t 2 + k dt f (t) 4 = Xe 0 f(t) dt f (t) Xe 0 X (8)2 + k (0) 2 + k X (4)2 + k (0) 2 + k = X 48, compound nterest k X(.3) = X 48 k.3 =48 k.30k =48 k = 160 f(t) dt = X f(8) f(4) X f(0) f(0) 17
20 Exercse (b) Payments are made to an account at a contnuous rate of, k 2 +8tk 2 +(tk) 2,where0 t 10 and k>0. Interest s credted at a force of nterest where After 10 years, the account s worth 88, 690. Calculate k. Soluton (b) δ t = 8+2t 1+8t + t 2 (1 + ) 10 t = e t [k 2 +8tk 2 +(tk) 2 ](1 + ) 10 t dt =88, 690 δ s ds = e 10 t 8+2s s + s ds f (s) 2 = e t f(s) ds = f(10) f(t) = 1 + 8(10) + (10) t + t 2 = 1+8t + t [k 2 +8tk 2 +(tk) ] dt =88, t + t2 10 k 2 [1 + 8t + t ] 1+8t + t dt =88, k2 (181)(10) = 88, 690 k 2 =49 k =7 0 18
21 Exercse (c) Fund A accumulates at a constant force of nterest of δt A =.05 at tme t, fort 0, and 1+.05t Fund B accumulates at a constant force of nterest of δ B =5%. Youaregven: () The amount n Fund A at tme zero s 1,000. () The amount n Fund B at tme zero s 500. () The amount n Fund C at any tme t, t 0, s equal to the sum of the amounts n Fund A and Fund B. Fund C accumulates at a force of nterest of δt C,fort0. Calculate δ C 2. Soluton (c) AV C t = AV A t + AV B t =1, 000e t 0 δ s dt + 500e.05t AV C t =1, 000e t s ds [ ] f(t) + 500e.05t =1, e.05t f(0) [ ] 1+.05(t) AVt C =1, (.05)e.05t = 1000[1 +.05t] + 500e.05t 1+.05(0) d dt AV C t =1, 000(.05) + 500(.05)e.05t = t δ t = δ 2 = d dt AV t C AVt C = e.05t 1000[1 +.05t] + 500e.05t e.05(2) =4.697% 1, 000[1 +.05(2)] + 500e.05(2) 19
22 Exercse (d) Fund F accumulates at the rate δ t = 1 1+t. Fund G accumulates at the rate δ t = 4t 1+2t. 2 F (t) s the amount n Fund F at tme t, and G(t) s the amount n fund G at tme t, wth F (0) = G(0). Let H(t) =F (t) G(t). Calculate T, the value of tme t when H(t) sa maxmum. Soluton (d) Snce F (0) = G(0), we can assume an ntal depost of 1. Then we have, F (t) =e t r dr = e ln(1+t) ln(1) =1+t G(t) =e t 0 4r 1+2r 2 dr = e ln(1+2t2 ) ln(1) =1+2t 2 H(t) =t 2t 2 and d dt H(t) =1 4t =0 t =
23 2 Level Annutes Overvew Defnton of An Annuty a seres of payments made at equal ntervals of tme (annually or otherwse) payments made for certan over a fxed perod of tme are called an annutycertan payments made for an uncertan perod of tme are called a contngent annuty the payment frequency and the nterest converson perod are equal the payments are level 2.1 AnnutyImmedate Defnton payments of 1 are made at the end of every year for n years n  1 n AnnutyImmedate Present Value Factor the present value (at t = 0) of an annuty mmedate, where the annual effectve rate of nterest s, shall be denoted as a n and s calculated as follows: a n =(1)v +(1)v 2 + +(1)v n 1 +(1)v n = v(1 + v + v v n 2 + v n 1 ) ( )( ) 1 1 v n = 1+ 1 v ( )( ) 1 1 v n = 1+ d ( ) ( ) 1 1 v n = 1+ = 1 vn 1+ 21
24 AnnutyImmedate Accumulated Value Factor the accumulated value (at t = n) of an annuty mmedate, where the annual effectve rate of nterest s, shall be denoted as s n and s calculated as follows: s n = 1 + (1)(1 + )+ + (1)(1 + ) n 2 + (1)(1 + ) n 1 1 (1 + )n = 1 (1 + ) 1 (1 + )n = = (1 + )n 1 Basc Relatonshp 1:1= a n + v n Consder an n year nvestment where 1 s nvested at tme 0. The present value of ths sngle payment ncome stream at t =0s1. Alternatvely, consder a n year nvestment where 1 s nvested at tme 0 and produces annual nterest payments of (1) at the end of each year and then the 1 s refunded at t = n n  1 n The present value of ths multple payment ncome stream at t =0s a n +(1)v n. Therefore, the present value of both nvestment opportuntes are equal. Also note that a n = 1 vn 1= a n + v n. 22
25 Basc Relatonshp 2:PV(1 + ) n = FV and PV = FV v n f the future value at tme n, s n, s dscounted back to tme 0, then you wll have ts present value, a n [ (1 + ) s n v n n ] 1 = v n = (1 + )n v n v n = 1 vn = a n f the present value at tme 0, a n, s accumulated forward to tme n, then you wll have ts future value, s n [ ] 1 v a n (1 + ) n n = (1 + ) n = (1 + )n v n (1 + ) n = (1 + )n 1 = s n 23
26 1 Basc Relatonshp 3: = 1 + a n s n Consder a loan of 1, to be pad back over n years wth equal annual payments of P made at the end of each year. An annual effectve rate of nterest,, s used. The present value of ths sngle payment loan must be equal to the present value of the multple payment ncome stream. P a n =1 P = 1 a n Alternatvely, consder a loan of 1, where the annual nterest due on the loan, (1), spadat the end of each year for n years and the loan amount s pad back at tme n. In order to produce the loan amount at tme n, annual payments at the end of each year, for n years, wll be made nto an account that credts nterest at an annual effectve rate of nterest. The future value of the multple depost ncome stream must equal the future value of the sngle payment, whch s the loan of 1. D s n =1 D = 1 s n The total annual payment wll be the nterest payment and account payment: + 1 s n Therefore, a level annual annuty payment on a loan s the same as makng an annual nterest payment each year plus makng annual deposts n order to save for the loan repayment. Also note that 1 (1 + )n (1 + )n = = a n 1 vn (1 + ) n (1 + ) n 1 = (1 + )n + (1 + ) n 1 = [(1 + )n 1] + (1 + ) n 1 = + (1 + ) n 1 = + 1 s n 24
27 2.2 Annuty Due Defnton payments of 1 are made at the begnnng of every year for n years n  1 n AnnutyDue Present Value Factor the present value (at t = 0) of an annuty due, where the annual effectve rate of nterest s, shall be denoted as ä n and s calculated as follows: ä n = 1 vn 1 v = 1 vn d AnnutyDue Accumulated Value Factor =1+(1)v +(1)v 2 + +(1)v n 2 +(1)v n 1 the accumulated value (at t = n) of an annuty due, where the annual effectve rate of nterest s, shall be denoted as s n and s calculated as follows: s n = (1)(1 + ) + (1)(1 + ) (1)(1 + ) n 1 + (1)(1 + ) n =(1+)[1 + (1 + )+ +(1+) n 2 +(1+) n 1 ] [ ] 1 (1 + ) n =(1+) 1 (1 + ) [ ] 1 (1 + ) n =(1+) [ (1 + ) n ] 1 =(1+) = (1 + )n 1 d 25
28 Basc Relatonshp 1:1=d ä n + v n Consder an n year nvestment where 1 s nvested at tme 0. The present value of ths sngle payment ncome stream at t =0s1. Alternatvely, consder a n year nvestment where 1 s nvested at tme 0 and produces annual nterest payments of (1) d at the begnnng of each year and then have the 1 refunded at t = n. d d d... d n  1 n The present value of ths multple payment ncome stream at t =0sd ä n +(1)v n. Therefore, the present value of both nvestment opportuntes are equal. Also note that ä n = 1 vn d 1=d ä n + v n. Basc Relatonshp 2:PV(1 + ) n = FV and PV = FV v n f the future value at tme n, s n, s dscounted back to tme 0, then you wll have ts present value, ä n [ (1 + ) s n v n n ] 1 = v n = (1 + )n v n v n = 1 vn =ä n d d d f the present value at tme 0, ä n, s accumulated forward to tme n, then you wll have ts future value, s n [ ] 1 v ä n (1 + ) n n = (1 + ) n = (1 + )n v n (1 + ) n d d = (1 + )n 1 d = s n 26
29 1 Basc Relatonshp 3: = 1 + d ä n s n Consder a loan of 1, to be pad back over n years wth equal annual payments of P made at the begnnng of each year. An annual effectve rate of nterest,, s used. The present value of the sngle payment loan must be equal to the present value of the multple payment stream. P ä n =1 P = 1 ä n Alternatvely, consder a loan of 1, where the annual nterest due on the loan, (1) d, spad at the begnnng of each year for n years and the loan amount s pad back at tme n. In order to produce the loan amount at tme n, annual payments at the begnnng of each year, for n years, wll be made nto an account that credts nterest at an annual effectve rate of nterest. The future value of the multple depost ncome stream must equal the future value of the sngle payment, whch s the loan of 1. D s n =1 D = 1 s n The total annual payment wll be the nterest payment and account payment: d + 1 s n Therefore, a level annual annuty payment s the same as makng an annual nterest payment each year and makng annual deposts n order to save for the loan repayment. Also note that 1 = d (1 + )n d(1 + )n = ä n 1 vn (1 + ) n (1 + ) n 1 = d(1 + )n + d d (1 + ) n 1 = d[(1 + )n 1] + d (1 + ) n 1 = d + d (1 + ) n 1 = d + 1 s n Basc Relatonshp 4: AnnutyDue= Annuty Immedate (1 + ) ä n = 1 vn d s n = (1 + )n 1 d = 1 vn (1 + ) =a n (1 + ) [ (1 + ) n 1 = ] (1 + ) =s n (1 + ) An annuty due starts one perod earler than an annutymmedate and as a result, earns one more perod of nterest, hence t wll be larger. 27
30 Basc Relatonshp 5:ä n =1+a n 1 ä n =1+[v + v v n 2 + v n 1 ] =1+v[1 + v + + v n 3 + v n 2 ] ( ) 1 v n 1 =1+v 1 v ( )( ) 1 1 v n 1 =1+ 1+ d ( )( ) 1 1 v n 1 =1+ 1+ /1+ =1+ 1 vn 1 =1+a n 1 Ths relatonshp can be vsualzed wth a tme lne dagram. a n n  1 n becomng n payments that now com An addtonal payment of 1 at tme 0 results n a n 1 mence at the begnnng of each year whch s ä n. 28
31 Basc Relatonshp 6:s n =1+ s n 1 s n =1+[(1+)+(1+) 2 + +(1+) n 2 +(1+) n 1 ] =1+(1+)[1 + (1 + )+ +(1+) n 3 +(1+) n 2 ] [ ] 1 (1 + ) n 1 =1+(1+) 1 (1 + ) [ ] 1 (1 + ) n 1 =1+(1+) [ (1 + ) n 1 ] 1 =1+(1+) =1+ (1 + )n 1 1 d =1+ s n 1 Ths relatonshp can also the vsualzed wth a tme lne dagram s n n  1 n An addtonal payment of 1 at tme n results n s n 1 commerce at the end of each year whch s s n becomng n payments that now 29
32 2.3 Deferred Annutes There are three alternatve dates to valung annutes rather than at the begnnng of the term (t =0)orattheendoftheterm(t = n) () present values more than one perod before the frst payment date () accumulated values more than one perod after the last payment date () current value between the frst and last payment dates The followng example wll be used to llustrate the above cases. Consder a seres of payments of 1 that are made at tme t =3tot =9,nclusve Present Values More than One Perod Before The Frst Payment Date At t = 2, there exsts 7 future endofyear payments whose present value s represented by a 7. If ths value s dscounted back to tme t = 0, then the value of ths seres of payments (2 perods before the frst endofyear payment) s The general form s: 2 a 7 = v 2 a 7. m a n = v m a n. Alternatvely, at t = 3, there exsts 7 future begnnngofyear payments whose present value s represented by ä 7. If ths value s dscounted back to tme t = 0, then the value of ths seres of payments (3 perods before the frst begnnngofyear payment) s The general form s: 3 ä 7 = v 3 ä 7. m ä n = v m ä n. Another way to examne ths stuaton s to pretend that there are 9 endofyear payments. Ths can be done by addng 2 more payments to the exstng 7. In ths case, let the 2 addtonal payments be made at t = 1 and 2 and be denoted as 1. 30
33 At t = 0, there now exsts 9 endofyear payments whose present value s a 9. Ths present value of 9 payments would then be reduced by the present value of the two magnary payments, represented by a 2. Therefore, the present value at t =0s and ths results n a 9 a 2, The general form s v 2 a 7 = a 9 a 2. v m a n = a m + n a m. 31
34 Wth the annuty due verson, one can pretend that there are 10 payments beng made. Ths can be done by addng 3 payments to the exstng 7 payments. In ths case, let the 3 addtonal payments be made at t = 0, 1 and 2 and be denoted as At t = 0, there now exsts 10 begnnngofyear payments whose present value s ä 10.Ths present value of 10 payments would then be reduced by the present value of the three magnary payments, represented by ä 3. Therefore, the present value at t =0s and ths results n ä 10 ä 3, The general form s v 3 ä 7 =ä 10 ä 3. v m ä n =ä m + n ä m. Accumulated Values More Than One Perod After The Last Payment Date At t = 9, there exsts 7 past endofyear payments whose accumulated value s represented by s 7. If ths value s accumulated forward to tme t = 12, then the value of ths seres of payments (3 perods after the last endofyear payment) s s 7 (1 + ) 3. Alternatvely, at t = 10, there exsts 7 past begnnngofyear payments whose accumulated value s represented by s 7. If ths value s accumulated forward to tme t = 12, then the value of ths seres of payments (2 perods after the last begnnngofyear payment) s s 7 (1 + ) 2. Another way to examne ths stuaton s to pretend that there are 10 endofyear payments. Ths can be done by addng 3 more payments to the exstng 7. In ths case, let the 3 addtonal payments be made at t = 10, 11 and 12 and be denoted as 1. 32
35 At t = 12, there now exsts 10 endofyear payments whose present value s s 10. Ths future value of 10 payments would then be reduced by the future value of the three magnary payments, represented by s 3. Therefore, the accumulated value at t =12s and ths results n s 10 s 3, The general form s s 7 (1 + ) 3 = s 10 s 3. s n (1 + ) m = s m + n s m. Wth the annuty due verson, one can pretend that there are 9 payments beng made. Ths can be done by addng 2 payments to the exstng 7 payments. In ths case, let the 2 addtonal payments be made at t = 10 and 11 and be denoted as 1. 33
36 At t = 12, there now exsts 9 begnnngofyear payments whose accumulated value s s 9. Ths future value of 9 payments would then be reduced by the future value of the two magnary payments, represented by s 2. Therefore, the accumulated value at t =12s and ths results n s 9 s 2, The general form s s 7 (1 + ) 2 = s 9 s 2. s n (1 + ) m = s m + n s m. 34
37 Current Values Between The Frst And Last Payment Dates The 7 payments can be represented by an annutymmedate or by an annutydue dependng on the tme that they are evaluated at. For example, at t= 2, the present value of the 7 endofyear payments s a 7. At t= 9, the future value of those same payments s s 7. Theresapontbetweentme2and9where the present value and the future value can be accumulated to and dscounted back, respectvely. At t = 6, for example, the present value would need to be accumulated forward 4 years, whle the accumulated value would need to be dscounted back 3 years. a 7 (1 + ) 4 = v 3 s 7 The general form s a n (1 + ) m = v (n m) s n Alternatvely, at t = 3, one can vew the 7 payments as beng pad at the begnnng of the year where the present value of the payments s ä 7.Thefuturevalueatt =10wouldthen be s 7. At t = 6, for example, the present value would need to be accumulated forward 3 years, whle the accumulated value would need to be dscounted back 4 years. ä 7 (1 + ) 3 = v 4 s 7. The general form s ä n (1 + ) m = v (n m) s n. At any tme durng the payments, there wll exsts a seres of past payments and a seres of future payments. For example, at t= 6, one can defne the past payments as 4 endofyear payments whose accumulated value s s 4. The 3 endofyear future payments at t= 6 would then have a present value (at t= 6)equaltoa 3. Therefore, the current value as at t= 6ofthe7 payments s s 4 + a 3. Alternatvely, f the payments are vewed as begnnngofyear payments at t= 6, then there are 3 past payments and 4 future payments whose accumulated value and present value are respectvely, s 3 and ä 4. Therefore, the current value as at t= 6 of the 7 payments can also be calculated as s 3 +ä 4. Ths results n s 4 + a 3 = s 3 +ä 4. The general form s s m + a n = s n +ä m. 35
38 2.4 Contnuously Payable Annutes payments are made contnuously every year for the next n years (.e. m = ) Contnuously Payable Annuty Present Value Factor the present value (at t = 0) of a contnuous annuty, where the annual effectve rate of nterest s, shall be denoted as ā n and s calculated as follows: ā n = = n 0 n 0 v t dt e δt dt = 1 ] n δ e δt 0 = 1 [ e δn e δ0] δ = 1 [ ] 1 e δn δ = 1 vn δ Contnuously Payable Annuty Accumulated Value Factor the accumulated value (at t = n) of a contnuous annuty, where the annual effectve rate of nterest s, shall be denoted as s n and s calculated as follows: s n = = = n 0 n 0 n 0 = 1 δ eδt dt = 1 δ (1 + ) n t dt (1 + ) t dt e δt dt ] n 0 [ e δn e δ0] = (1 + )n 1 δ 36
39 Basc Relatonshp 1:1=δ ā n + v n Basc Relatonshp 2:PV(1 + ) n = FV and PV = FV v n f the future value at tme n, s n, s dscounted back to tme 0, then you wll have ts present value, ā n [ (1 + ) s n v n n ] 1 = v n δ = (1 + )n v n v n δ = 1 vn δ =ā n f the present value at tme 0, ā n, s accumulated forward to tme n, then you wll have ts future value, s n [ ] 1 v ā n (1 + ) n n = (1 + ) n δ = (1 + )n v n (1 + ) n δ = (1 + )n 1 δ = s n 37
40 1 Basc Relatonshp 3: = 1 + δ ā n s n Consder a loan of 1, to be pad back over n years wth annual payments of P that are pad contnuously each year, for the next n years. An annual effectve rate of nterest,, and annual force of nterest, δ, s used. The present value of ths sngle payment loan must be equal to the present value of the multple payment ncome stream. P ā n =1 P = 1 ā n Alternatvely, consder a loan of 1, where the annual nterest due on the loan, (1) δ, s pad contnuously durng the year for n years and the loan amount s pad back at tme n. In order to produce the loan amount at tme n, annual payments of D are pad contnuously each year, for the next n years, nto an account that credts nterest at an annual force of nterest, δ. The future value of the multple depost ncome stream must equal the future value of the sngle payment, whch s the loan of 1. D s n =1 D = 1 s n The total annual payment wll be the nterest payment and account payment: δ + 1 s n Notethat 1 ā n = δ 1 v (1 + )n δ(1 + )n = n (1 + ) n (1 + ) n 1 = δ(1 + )n + δ δ (1 + ) n 1 = δ[(1 + )n 1] + δ (1 + ) n 1 = δ + δ (1 + ) n 1 = δ + 1 s n Therefore, a level contnuous annual annuty payment on a loan s the same as makng an annual contnuous nterest payment each year plus makng level annual contnuous deposts n order to save for the loan repayment. 38
41 Basc Relatonshp 4:ā n = δ a n, s n = δ s n Consder annual payments of 1 made contnuously each year for the next n years. Over a oneyear perod, the contnuous payments wll accumulate at the end of the year to a lump sum of s 1. If ths endofyear lump sum exsts for each year of the nyear annutymmedate, then the present value (at t = 0) of these endofyear lump sums s the same as ā n : ā n = s 1 a n = δ a n Therefore, the accumulated value (at t = n) of these endofyear lump sums s the same as s n : s n = s 1 s n = δ s n Basc Relatonshp 5:ā n = d δ ä n, s n = d δ s n The mathematcal development of ths relatonshp s derved as follows: ä n d δ = 1 vn d d δ = 1 vn =ā n δ s n d δ = (1 + )n 1 d d δ = (1 + )n 1 = s n δ 39
42 2.5 Perpetutes Defnton Of A PerpetutyImmedate payments of 1 are made at the end of every year forever.e. n = n... PerpetutyImmedate Present Value Factor the present value (at t = 0) of a perpetuty mmedate, where the annual effectve rate of nterest s, shall be denoted as a and s calculated as follows: a =(1)v +(1)v 2 +(1)v 3 + = v(1 + v + v 2 + ) ( )( ) 1 1 v = 1+ 1 v ( )( ) = 1+ d ( ) ( ) 1 1 = = 1 one could also derve the above formula by smply substtutng n = nto the orgnal present value formula: a = 1 v = 1 0 = 1 Notethat 1 represents an ntal amount that can be nvested at t = 0. The annual nterest ( ) 1 payments, payable at the end of the year, produced by ths nvestment s =1. s s not defned snce t would equal 40
43 Basc Relatonshp 1:a n = a v n a The present value formula for an annutymmedate can be expressed as the dfference between two perpetutymmedates: a n = 1 vn = 1 vn = 1 vn 1 = a v n a. In ths case, a perpetutymmedate that s payable forever s reduced by perpetutymmedate payments that start after n years. The present value of both of these ncome streams, at t = 0, results n endofyear payments remanng only for the frst n years. 41
44 Defnton Of A PerpetutyDue payments of 1 are made at the begnnng of every year forever.e. n = n... PerpetutyDue Present Value Factor the present value (at t = 0) of a perpetuty due, where the annual effectve rate of nterest s, shall be denoted as ä and s calculated as follows: ä = (1) + (1)v 1 +(1)v 2 + ( ) 1 v = 1 v ( ) 1 0 = d = 1 d one could also derve the above formula by smply substtutng n = nto the orgnal present value formula: ä d = 1 v d = 1 0 d = 1 d Notethat 1 represents an ntal amount that can be nvested at t = 0. The annual nterest d ( ) 1 payments, payable at the begnnng of the year, produced by ths nvestment s d =1. d s s not defned snce t would equal 42
45 Basc Relatonshp 1:ä n =ä v n ä The present value formula for an annutydue can be expressed as the dfference between two perpetutydues: ä n = 1 vn d = 1 d vn d = 1 d vn 1 d =ä v n ä. In ths case, a perpetutydue that s payable forever s reduced by perpetutydue payments that start after n years. The present value of both of these ncome streams, at t =0,results n begnnngofyear payments remanng only for the frst n years. Defnton Of A Contnuously Payable Perpetuty Present Value Factor payments of 1 are made contnuously every year forever the present value (at t=0) of a contnuously payable perpetuty, where the annual effectve rate of nterest s, shall be denoted as ā and s calculated as follows: ā = = 0 0 v t dt e δt dt = 1 δ e δt ] 0 = 1 δ Basc Relatonshps: ā = δ a and ā = d δ ä The mathematcal development of these relatonshps are derved as follows: a δ = 1 δ = 1 δ =ā ä d δ = 1 d d δ = 1 δ =ā 43
46 2.6 Equatons of Value the value at any gven pont n tme, t, wll be ether a present value or a future value (sometmes referred to as the tme value of money) the tme value of money depends on the calculaton date from whch payment(s) are ether accumulated or dscounted to Tme Lne Dagrams t helps to draw out a tme lne and plot the payments and wthdrawals accordngly P 1 P 2... P t... P n1 P n t... n1 n W 1 W 2 W t W n1 W n 44
47 Example a payment of 600 s due n 8 years; the alternatve s to receve 100 now,200 n 5 years and X n 10 years. If = 8%, fnd $X, such that the value of both optons s equal X compare the values at t = X v 8 8% = v5 8% + Xv10 8% X = 600v8 8% v5 8% v 10 8% =
48 compare the values at t = X v 3 = 100(1 + ) Xv 5 X = 600v3 100(1 + ) v 5 = compare the values at t = X (1 + ) 2 = 100(1 + ) (1 + ) 5 + X X = 600(1 + ) 2 100(1 + ) (1 + ) 5 = all 3 equatons gave the same answer because all 3 equatons treated the value of the payments consstently at a gven pont of tme. 46
49 Unknown Rate of Interest Assumng that you do not have a fnancal calculator Lnear Interpolaton need to fnd the value of a n at two dfferent nterest rates where a n 1 = P 1 <P and a n 2 = P 2 >P. } a n 1 = P 1 a n = P a n 2 = P P 1 P P 1 P 2 ( 2 1 ) 47
50 Exercses and Solutons 2.1 AnnutyImmedate Exercse (a) Fence posts set n sol last 9 years and cost Y each whle fence posts set n concrete last 15 years and cost Y + X. The posts wll be needed for 35 years. What s the value of X such that a fence bulder would be ndfferent between the two types of posts? Soluton (a) The sol posts must be set 4 tmes (at t=0,9,18,27). The PV of the cost per post s PV = Y a 36 a 9. The concrete posts must be set 3 tmes (at t=0,15,30). PV =(Y + X) a 45 a. 15 The breakeven value of X s the value for whch PV s = PV c.thus The PV of the cost per post s Exercse (b) You are gven δ t = Calculate s 4. Soluton (b) Y a 36 a 9 =(Y + X) a 45 X a 45 a 15 X = Y 4+t for t t + t2 a 15 [ a = Y 36 a 45 a 9 a 15 [ ] a 36 a 15 1 a 45 a 9 s 4 =1+(1+) 4 3 +(1+) 4 2 +(1+) δ s ds δ s ds δ s ds =1+e 3 + e 2 + e s 4 =1+e 3 1+8s + s ds 4+s e 2 1+8s + s ds 2 + e 1 4 f (s) =1+e 3 f(s) ds f (s) e 2 f(s) ds f (s) e 1 f(s) ds [ ] 1 [ ] 1 [ ] 1 f(4) 2 f(4) 2 f(4) 2 = f(3) f(2) f(1) [ ] 1 [ ] 1 [ 1 + 8(4) + (4) (4) + (4) (4) + (4) 2 = (3) + (3) (2) + (2) (1) + (1) 2 = = or ] 4+s 1+8s + s 2 ]
51 Exercse (c) You are gven the followng nformaton: () The present value of a 6nyear annutymmedate of 1 at the end of every year s () The present value of a 6nyear annutymmedate of 1 at the end of every second year s () The present value of a 6nyear annutymmedate of 1 at the end of every thrd year s K. Determne K assumng an annual effectve nterest rate of. Soluton (c) = a 6n = a 3n (1+) 2 1 = a 6n s AnnutyDue Exercse (a) K = a 2n (1+) 3 1 = a 6n s = s 2 =2.05 = 1 + (1 + ) =5% K = s 3 5% = =3.095 Smplfy a 15 (1 + ) 45 ä 3 j to one actuaral symbol, gven that j =(1+) Soluton (a) a 15 (1 + ) 45 ä 3 j = s 15 (1 + ) 30 (1 + v 1 j + v 2 j ) = s 15 (1 + ) 30 (1 + v 15 + v 30 )=s 15 [(1 + ) 30 +(1+) 15 +1] There are 3 sets of 15 endofyear payments (45 n total) that are beng made and carred forward to t = 45. Therefore, at t=45 you wll have 45 endofyear payment that have been accumulated and whose value s s
52 Exercse (b) A person deposts 100 at the begnnng of each year for 20 years. Smple nterest at an annual rate of s credted to each depost from the date of depost to the end of the twenty year perod. The total amount thus accumulated s 2,840. If nstead, compound nterest had been credted at an effectve annual rate of, what would the accumulated value of these deposts have been at the end of twenty years? Soluton (b) 100[(1 + )+(1+2)+(1+3)+... +(1+20)] = 2, [20 + ( )] = 2, 840 Exercse (c) 20 + ( )=28.40 =.04 and d = s 20 4% =3, 097 You plan to accumulate 100,000 at the end of 42 years by makng the followng deposts: X at the begnnng of years 114 No deposts at the begnnng of years 1532; and Y at the begnnng of years The annual effectve nterest rate s 7%. X Y = 100. Calculate Y. Soluton (c) }{{} X s 14 7% (1.07) 28 + Y s 10 7% = 100, Y (100 + Y )( ) + Y ( ) = 100, 000 Y =
53 2.3 Deferred Annutes Exercse (a) Usng an annual effectve nterest rate j 0, you are gven: () The present value of 2 at the end of each year for 2n years, plus an addtonal 1 at the end of each of the frst n years, s 36. () The present value of an nyear deferred annutymmedate payng 2 per year for n years s 6. Calculate j. Soluton (a) () 36 = 2a 2n + a n then subtractng () from () gves us: () 6 = v n 2a n =2a 2n 2a n 30 = 3a n 10 = a n 6=v n 2(10).3 =v n 10 = a n = 1 vn = 1.3 =7% 51
54 Exercse (b) A loan of 1,000 s to be repad by annual payments of 100 to commence at the end of the ffth year and to contnue thereafter for as long as necessary. The effectve rate of dscount s 5%. Fnd the amount of the fnal payment f t s to be larger than the regular payments. Soluton (b) PV 0 =1, 000 = 100a n 100a 4 a n = 1, a = v n = = v n 3.47 = ( ) n n = ln(3.47) ln( ) = n =24 1, 000 = 100a 24 + Xv a 4 X = 1, (a 24 a 4 ) v 24 X = ( ) 24 Thus, the total fnal payment s X =
55 2.4 Contnuously Payable Annutes Exercse (a) There s 40,000 n a fund whch s accumulatng at 4% per annum convertble contnuously. If money s wthdrawn contnuously at the rate of 2,400 per annum, how long wll the fund last? Soluton (a) If the fund s exhausted at t = n, then the accumulated value of the fund at that tme must equal the accumulated value of the wthdrawals. Thus we have: 40, 000e.04n =2, 400 s n = 2400( e.04n 1 ).04 40, 000 (.04) = 1 e.04n 2, 400 n = 40,000 ln[1 (.04) 2,400 ] =27.47 years.04 Exercse (b) If ā n =4and s n = 12 fnd δ. Soluton (b) ā n = 1 vn δ =4 v n =1 4δ s n = (1 + )n 1 δ (1 + ) n =1+12δ =12 (1 + ) n = v n then δ = 1 1 4δ 1+8δ 48δ 2 =1orδ = 4 48 =
56 2.5 Perpetutes Exercse (a) A perpetutymmedate pays X per year. Kevn receves the frst n payments, Jeffrey receves the next n payments and Hal receves the remanng payments. The present value of Kevn s payments s 20% of the present value of the orgnal perpetuty. The present value of Hal s payments s K of the present value of the orgnal perpetuty. Calculate the present value of Jeffrey s payments as a percentage of the orgnal perpetuty. Soluton (a) The present value of the perpetuty s: The present value of Kevn s payments s: Xa = X Ths leads to: Xa n =.2Xa =.2 X The present value of Hal s payments s: a n =.2 1 vn =.2 v n =.8 Therefore, Xv 2n a = K X a = K X. Therefore, Jeffrey owns.16(1.2.64). X (.8) 2 a K =(.8) 2 =
57 Exercse (b) A perpetuty pays 1 at the begnnng of every year plus an addtonal 1 at the begnnng of every second year. Determne the present value of ths annuty. Soluton (b) K =ä + v ä (1+) 2 1 = 1 d d (1+) 2 1 = 1+ + (1 + ) (1 + ) 2 1 Exercse (c) K = (1 + ) 1+(1+)2 1 (1 + ) 2 = (1 + ) (1 + ) 2 (1 + )(1 + ) 2 1 K = 1+ K =(1+) + [ ] (1 + ) 1 (1 + ) 2 1 =(1+) + 1 (1 + ) 2 1 [ (1 + ) 2 ] [ ] 1+ =(1+) [(1 + ) 2 1] [ [ ] [ ] K =(1+) =(1+) [2 + 2 ] [ ] 3+ =(1+) = 3+ (2 + ) d(2 + ) A perpetutymmedate pays X per year. Ncole receves the frst n payments, Mark receves the next n payments and Cheryl receves the remanng payments. The present value of Ncole s payments s 30% of the present value of the orgnal perpetuty. The present value of Cheryl s payments s K% of the present value of the orgnal perpetuty. Calculate the present value of Cheryl s payments as a percentage of the orgnal perpetuty. Soluton (c) The present value of Ncole s payments s: Ths leads to X a n =.3 X a =.3 X The present value of Cheryl s payments s: a n =.3 1 vn =.3 v n =.7 X v 2n a = K X a = K X Therefore X (.7) 2 a = K X a K =(.7) 2 =.49 55
58 2.6 Equaton of Value Exercse (a) An nvestment requres an ntal payment of 10,000 and annual payments of 1,000 at the end of each of the frst 10 years. Startng at the end of the eleventh year, the nvestment returns fve equal annual payments of X. Determne X to yeld an annual effectve rate of 10% over the 15year perod. Soluton (a) PV of cash flow n=pv of cash flow out 10, , 000a 10 10% = v 10 X a 5 10% X = 10, , 000a 10 10% v 10 a 5 10% X = 10, , 000( ) ( )( ) X =11,
59 Exercse(b) At a certan nterest rate the present value of the followng two payment patterns are equal: () 200 at the end of 5 years plus 500 at the end of 10 years. () at the end of 5 years. At the same nterest rate, 100 nvested now plus 120 nvested at the end of 5 years wll accumulate to P at the end of 10 years. Calculate P. Soluton (b) 200v v 10 = v 5 500v 10 = v 5 v 5 = v 5 = (1 + ) 5 = Exercse (c) P = 100(1 + ) (1 + ) 5 = 100( ) ( ) = Whereas the choce of a comparson date has no effect on the answer obtaned wth compound nterest, the same cannot be sad of smple nterest. Fnd the amount to be pad at the end of 10 years whch s equvalent to two payments of 100 each, the frst to be pad mmedately and the second to be pad at the end of 5 years. Assume 5% smple nterest s earned from the date each payment s made and use a comparson date of the end of 10 years. Soluton (c) Equatng at t =10 X = 100(1 + 10) + 100(1 + 5) = [100(1 + 10(.05)) + 100(1 + 5(.05)) =
60 3 Varyng Annutes Overvew n ths secton, payments wll now vary; but the nterest converson perod wll contnue to concde wth the payment frequency annutes can vary n 3 dfferent ways () where the payments ncrease or decrease by a fxed amount (sectons 3.1, 3.2, 3.3, 3.4 and 3.5) () where the payments ncrease or decrease by a fxed rate (secton 3.6) () where the payments ncrease or decrease by a varable amount or rate (secton 3.7) 3.1 Increasng AnnutyImmedate AnnutyImmedate An annutymmedate s payable over n years wth the frst payment equal to P and each subsequent payment ncreasng by Q. The tme lne dagram below llustrates the above scenaro: P P + (1)Q... P + (n2)q P + (n1)q n  1 n 58
61 The present value (at t = 0) of ths annual annuty mmedate, where the annual effectve rate of nterest s, shall be calculated as follows: PV 0 =[P ]v +[P + Q]v 2 + +[P +(n 2)Q]v n 1 +[P +(n 1)Q]v n = P [v + v v n 1 + v n ]+Q[v 2 +2v 3 +(n 2)v n 1 +(n 1)v n ] = P [v + v v n 1 + v n ]+Qv 2 [1 + 2v +(n 2)v n 3 +(n 1)v n 2 ] = P [v + v v n 1 + v n ]+Qv 2 d dv [1 + v + v2 + + v n 2 + v n 1 ] = P a n + Qv 2 d dv [ä n ] = P a n + Qv 2 d [ ] 1 v n dv 1 v [ (1 v) ( nv = P a n + Qv 2 n 1 ) (1 v n ] ) ( 1) (1 v) 2 [ Q nv n (v 1 1) + (1 v n ] ) = P a n + (1 + ) 2 (/1+) 2 [ (1 v n ) nv n 1 nv n ] = P a n + Q 2 [ (1 v n ) nv n (v 1 ] 1) = P a n + Q 2 [ (1 v n ) nv n ] (1 + 1) = P a n + Q = P a n + Q (1 v n ) [ a n = P a n + Q nv n ] 2 nvn () The accumulated value (at t = n) of an annuty mmedate, where the annual effectve rate of nterest s, can be calculated usng the same approach as above or calculated by usng the basc prncple where an accumulated value s equal to ts present value carred forward wth nterest: FV n = PV 0 (1 + ) n ( [ a n = P a n + Q nv n ]) (1 + ) n [ a = P a n (1 + ) n n + Q (1 + ) n nv n (1 + ) n ] [ ] s n = P s n + Q n 59
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