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

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

Contents 1 Interest Rates and Factors 2 1.1 Interest... 2 1.2 SmpleInterest... 3 1.3 CompoundInterest... 3 1.4 AccumulatedValue... 3 1.5 PresentValue... 4 1.6 Rate of Dscount: d... 5 1.7 Constant Force of Interest: δ... 7 1.8 VaryngForceofInterest... 8 1.9 DscreteChangesnInterestRates... 9 ExercsesandSolutons... 10 2 Level Annutes 21 2.1 Annuty-Immedate... 21 2.2 Annuty Due... 25 2.3 DeferredAnnutes... 30 2.4 ContnuouslyPayableAnnutes... 36 2.5 Perpetutes... 40 2.6 EquatonsofValue... 44 ExercsesandSolutons... 48 3 Varyng Annutes 58 3.1 IncreasngAnnuty-Immedate... 58 3.2 IncreasngAnnuty-Due... 61 3.3 DecreasngAnnuty-Immedate... 66 3.4 DecreasngAnnuty-Due... 67 3.5 ContnuouslyPayableVaryngAnnutes... 69 3.6 CompoundIncreasngAnnutes... 71 3.7 ContnuouslyVaryngPaymentStreams... 74 3.8 ContnuouslyIncreasngAnnutes... 75 3.9 ContnuouslyDecreasngAnnutes... 76 ExercsesandSolutons... 77 4 Non-Annual Interest Rate and Annutes 88 4.1 Non-AnnualInterestandDscountRates... 88 4.2 Nomnal p thly Interest Rates: (p)... 88 4.3 Nomnal p thly Dscount Rates: d (p)... 89 4.4 Annutes-Immedate Payable p thly... 91 4.5 Annutes-Due Payable p thly... 94 ExercsesandSolutons... 98 5 Project Apprasal and Loans 108 5.1 DscountedCashFlowAnalyss... 108 5.2 Nomnalvs.RealInterestRates... 114 5.3 Investment Funds......... 115 5.4 AllocatngInvestmentIncome... 117 5.5 Loans:TheAmortzatonMethod... 118 5.6 Loans:TheSnkngFundMethod... 120 1

ExercsesandSolutons... 121 6 Fnancal Instruments 132 6.1 TypesofFnancalInstruments... 132 6.1.1 MoneyMarketInstruments... 132 6.1.2 Bonds... 133 6.1.3 CommonStock... 133 6.1.4 PreferredStock... 134 6.1.5 Mutual Funds....... 134 6.1.6 GuaranteedInvestmentContracts(GIC)... 134 6.1.7 DervatveSercurtes... 135 6.2 BondValuaton... 137 6.3 StockValuaton... 148 ExercsesandSolutons... 149 7 Duraton, Convexty and Immunzaton 156 7.1 PrceasaFunctonofYeld... 156 7.2 ModfedDuraton... 156 7.3 MacaulayDuraton... 157 7.4 EffectveDuraton... 159 7.5 Convexty... 159 7.5.1 MacaulayConvexty... 160 7.5.2 EffectveConvexty... 160 7.6 Duraton,ConvextyandPrces:PuttngtallTogether... 160 7.6.1 RevstngthePercentageChangenPrce... 160 7.6.2 ThePassageofTmeandDuraton... 161 7.6.3 PortfoloDuratonandConvexty... 161 7.7 Immunzaton... 162 7.8 FullImmunzaton... 166 ExercsesandSolutons... 167 8 The Term Structure of Interest Rates 177 8.1 Yeld-to-Maturty... 177 8.2 SpotRates... 177 2

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 one-year 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

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

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

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 end-of-year 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 end-of-year,, and begnnng-of-year 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

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

1.7 Constant Force of Interest: δ Defntons annual effectve rate of nterest s appled over a one-year perod a constant annual force of nterest can be appled over the smallest sub-perod 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

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

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

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 = 615 500 1 =9.2% 2.5 In how many years wll 500 accumulate to 630 at 7.8% smple nterest? Soluton (b) Exercse (c) 500[1 +.078(n)] = 630 = 630 500 1 =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[1 + 3 2n] = 500[1 + (1.5)(.11)] = 582.50 4 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).04 1+.04(n 1) n =16 10

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, 624.73 and B =2, 107.46. We seek A(1.03) 10 + B(1.025) 10 whch equals Exercse (b) 3, 624.73(1.03) 10 +2, 107.46(1.025) 10 =7, 569.07 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 one-half 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) 10 1.05K(1.04) 6 1.05K(1.04) 5 K(1.04) 4 K(1.04) 3 =10, 000 14, 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 = 980 11

1.4 Accumulated Value Exercse (a) 100 s deposted nto an account at the begnnng of every 4-year 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+) 4 +100(1+) 8 +...+100(1+) 40 = X = 5[100(1+) 4 +100(1+) 8 +...+100(1+) 20 ] 100(1+) 4 [1+100(1+) 4 +...+100(1+) 36 ]=5 100(1+) 4 [1+100(1+) 4 +...+100(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+) 20 +4 = 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(4 15 1 4 2 ) = 100(61.9472) = 6194.72 1 (1 + ) 4 1 4 1 5 12

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% 3 +...+(n3 )v0% n ]+vn 0% [(n2 )v25% 1 +(n2 )v25% 2 +(n2 )v25% 3 +...] =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 2 4 1 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) 121 + 121v = 144v 2 + 144v 3 = K 121(1 + v) = 144v 2 (1 + v) v = 11 12 K = 121(1 + 11 12 ) K = 231.92 13

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 24 +... =5 2v 8 [1 + v 8 + v 16 +... =5 [ ] 2v 8 1 1 v 8 =5 2v 8 =5 5v 8 7v 8 =5 v 8 = 5 7 (1 + ) 8 = 7 5 1.6 Rate of Dscount Exercse (a) 1+ = ( ) 1 7 8 =.04296 5 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 one-half 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 1 12 1 2 =$0.97v 1 24 (1 r) =.97(1 d) 1 24 =.97(1.22) 1 24 =.96 r =.04 = 4% 14

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, 000.5 + 2, 000.75 =1, 000(1.10)10 v n +2, 000(1.10) 10 v 2n 4, 666.67 = 2, 593.74v n +5, 187.48v 2n 5, 187.48v 2n +2, 593.74v n 4, 666.67 =0 } {{ } } {{ } } {{ } a b c v n = 2, 593.74 + 2, 593.74 2 4(5, 187.48)( 4, 666.67) 2(5, 187.48) (1.10) n = 1 ln(1.36824) =1.36824 n = =3.29.7306 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 =0.0905 15

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 )+2 10 2 (8)( 1 2 )2 +(2)( 1 = ) + 100 103 =0.09709 2 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 =.10 1+.10t δ B t = Equatng and solve for t d dt AV F B t AV F B t =.05(1.05t) 2 1.05t) 1 =.05(1.05t) 1.10 1+.10t =.05.10.005t =.05 +.005t t =5 1.05t 16

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

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 10 0 10 t [k 2 +8tk 2 +(tk) 2 ](1 + ) 10 t dt =88, 690 δ s ds = e 10 t 8+2s 10 1+8s + s ds f (s) 2 = e t f(s) ds = f(10) f(t) = 1 + 8(10) + (10)2 181 1+8t + t 2 = 1+8t + t 2 10 0 [k 2 +8tk 2 +(tk) 2 181 ] dt =88, 690 1+8t + t2 10 k 2 [1 + 8t + t 2 181 ] 1+8t + t dt =88, 690 2 k2 (181)(10) = 88, 690 k 2 =49 k =7 0 18

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 0.05 1+.05s ds [ ] f(t) + 500e.05t =1, 000 + 500e.05t f(0) [ ] 1+.05(t) AVt C =1, 000 + 500(.05)e.05t = 1000[1 +.05t] + 500e.05t 1+.05(0) d dt AV C t =1, 000(.05) + 500(.05)e.05t =50+25.05t δ t = δ 2 = d dt AV t C AVt C = 50 + 25e.05t 1000[1 +.05t] + 500e.05t 50 + 25e.05(2) =4.697% 1, 000[1 +.05(2)] + 500e.05(2) 19

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 0 1 1+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 = 1 4 20

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 annuty-certan 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 Annuty-Immedate Defnton payments of 1 are made at the end of every year for n years 1 1... 1 1 0 1 2... n - 1 n Annuty-Immedate 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 2 + + 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

Annuty-Immedate 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. 1 +... 0 1 2... 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

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

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

2.2 Annuty Due Defnton payments of 1 are made at the begnnng of every year for n years 1 1 1... 1 0 1 2... n - 1 n Annuty-Due 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 Annuty-Due 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 + ) 2 + + (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

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 1 0 1 2... 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

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 annuty-mmedate and as a result, earns one more perod of nterest, hence t wll be larger. 27

Basc Relatonshp 5:ä n =1+a n 1 ä n =1+[v + v 2 + + 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-1 1 + 1 1... 1 0 1 2... 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

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. 1 +.. 1 1... 1 s n-1 0 1 2... 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

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. 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 8 9 10 11 12 Present Values More than One Perod Before The Frst Payment Date At t = 2, there exsts 7 future end-of-year 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 end-of-year 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 begnnng-of-year 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 begnnng-of-year 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 end-of-year 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

1 1 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 8 9 10 11 12 At t = 0, there now exsts 9 end-of-year 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

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 1. 1 1 1 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 8 9 10 11 12 At t = 0, there now exsts 10 begnnng-of-year 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 end-of-year 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 end-of-year payment) s s 7 (1 + ) 3. Alternatvely, at t = 10, there exsts 7 past begnnng-of-year 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 begnnng-of-year payment) s s 7 (1 + ) 2. Another way to examne ths stuaton s to pretend that there are 10 end-of-year 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

1 1 1 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 8 9 10 11 12 At t = 12, there now exsts 10 end-of-year 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

1 1 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 8 9 10 11 12 At t = 12, there now exsts 9 begnnng-of-year 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

Current Values Between The Frst And Last Payment Dates The 7 payments can be represented by an annuty-mmedate or by an annuty-due dependng on the tme that they are evaluated at. For example, at t= 2, the present value of the 7 end-of-year 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 end-of-year payments whose accumulated value s s 4. The 3 end-of-year 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 begnnng-of-year 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

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

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

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

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 one-year perod, the contnuous payments wll accumulate at the end of the year to a lump sum of s 1. If ths end-of-year lump sum exsts for each year of the n-year annuty-mmedate, then the present value (at t = 0) of these end-of-year 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 end-of-year 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

2.5 Perpetutes Defnton Of A Perpetuty-Immedate payments of 1 are made at the end of every year forever.e. n = 1 1... 1... 0 1 2... n... Perpetuty-Immedate 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 1 0 = 1+ d ( ) ( ) 1 1 = 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

Basc Relatonshp 1:a n = a v n a The present value formula for an annuty-mmedate can be expressed as the dfference between two perpetuty-mmedates: a n = 1 vn = 1 vn = 1 vn 1 = a v n a. In ths case, a perpetuty-mmedate that s payable forever s reduced by perpetuty-mmedate payments that start after n years. The present value of both of these ncome streams, at t = 0, results n end-of-year payments remanng only for the frst n years. 41

Defnton Of A Perpetuty-Due payments of 1 are made at the begnnng of every year forever.e. n = 1 1 1... 1... 0 1 2... n... Perpetuty-Due 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

Basc Relatonshp 1:ä n =ä v n ä The present value formula for an annuty-due can be expressed as the dfference between two perpetuty-dues: ä n = 1 vn d = 1 d vn d = 1 d vn 1 d =ä v n ä. In ths case, a perpetuty-due that s payable forever s reduced by perpetuty-due payments that start after n years. The present value of both of these ncome streams, at t =0,results n begnnng-of-year 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

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 n-1 P n 0 1 2... t... n-1 n W 1 W 2 W t W n-1 W n 44

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. 600 100 200 X 0 5 8 10 compare the values at t =0 600 100 200 X 0 5 8 10 600v 8 8% = 100 + 200v5 8% + Xv10 8% X = 600v8 8% 100 200v5 8% v 10 8% = 190.08 45

compare the values at t =5 600 100 200 X 0 5 8 10 600v 3 = 100(1 + ) 5 + 200 + Xv 5 X = 600v3 100(1 + ) 5 200 v 5 = 190.08 compare the values at t =10 600 100 200 X 0 5 8 10 600(1 + ) 2 = 100(1 + ) 10 + 200(1 + ) 5 + X X = 600(1 + ) 2 100(1 + ) 10 200(1 + ) 5 = 190.08 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

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 2 1 + P 1 P P 1 P 2 ( 2 1 ) 47

Exercses and Solutons 2.1 Annuty-Immedate 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 0. 1+8t + 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+) 4 1 4 4 δ s ds δ s ds δ s ds =1+e 3 + e 2 + e 1 4 4+s 4 =1+e 3 1+8s + s ds 4+s 4 2 + e 2 1+8s + s ds 2 + e 1 4 f (s) 4 1 2 =1+e 3 f(s) ds f (s) 4 1 2 + e 2 f(s) ds f (s) 1 2 + e 1 f(s) ds [ ] 1 [ ] 1 [ ] 1 f(4) 2 f(4) 2 f(4) 2 =1+ + + f(3) f(2) f(1) [ ] 1 [ ] 1 [ 1 + 8(4) + (4) 2 2 1 + 8(4) + (4) 2 2 1 + 8(4) + (4) 2 =1+ + + 1 + 8(3) + (3) 2 1 + 8(2) + (2) 2 1 + 8(1) + (1) 2 =1+1.2005 + 1.5275 + 1.4878 = 5.2158 4 or ] 4+s 1+8s + s 2 ] 1 2 48

Exercse (c) You are gven the followng nformaton: () The present value of a 6n-year annuty-mmedate of 1 at the end of every year s 9.7578. () The present value of a 6n-year annuty-mmedate of 1 at the end of every second year s 4.760. () The present value of a 6n-year annuty-mmedate of 1 at the end of every thrd year s K. Determne K assumng an annual effectve nterest rate of. Soluton (c) 9.758 = a 6n 4.760 = a 3n (1+) 2 1 = a 6n s 2 2.2 Annuty-Due Exercse (a) K = a 2n (1+) 3 1 = a 6n s 3 9.758 4.760 = s 2 =2.05 = 1 + (1 + ) =5% K = 9.758 s 3 5% = 9.758 3.1525 =3.095 Smplfy a 15 (1 + ) 45 ä 3 j to one actuaral symbol, gven that j =(1+) 15 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 end-of-year payments (45 n total) that are beng made and carred forward to t = 45. Therefore, at t=45 you wll have 45 end-of-year payment that have been accumulated and whose value s s 45. 49

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, 840 100[20 + (1 + 2 + 3 +... + 20)] = 2, 840 Exercse (c) 20 + ( 20 21 )=28.40 =.04 and d =.03846 2 100 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 1-14 No deposts at the begnnng of years 15-32; and Y at the begnnng of years 33-42. 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, 000 100 Y (100 + Y )(160.42997) + Y (14.78368) = 100, 000 Y = 479.17 50

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 n-year deferred annuty-mmedate 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

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, 000 + 100a 4 100 =13.53438 1 v n.0526316 =13.52438.288190 = v n 3.47 = (1.0526316) n n = ln(3.47) ln(1.0526316) =24.2553 n =24 1, 000 = 100a 24 + Xv 24 100a 4 X = 1, 000 100(a 24 a 4 ) v 24 X =7.217526(1.0526316) 24 Thus, the total fnal payment s 124.72. X =24.72 52

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 1 + 12δ = 1 1 4δ 1+8δ 48δ 2 =1orδ = 4 48 = 1 6 53

2.5 Perpetutes Exercse (a) A perpetuty-mmedate 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 =.64. 54

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 + 1 1+ 1 d (1+) 2 1 = 1+ + (1 + ) (1 + ) 2 1 Exercse (c) K = 1+ + 1 (1 + ) 1+(1+)2 1 (1 + ) 2 = 1+ + 1 1 (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+2 + 2 ] 1+ =(1+) [(1 + ) 2 1] [1 + 2 + 2 1 [ 2 + 2 ] [ ] + 2+ +1 K =(1+) =(1+) [2 + 2 ] 2 + 2 [ ] 3+ =(1+) = 3+ (2 + ) d(2 + ) A perpetuty-mmedate 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

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 15-year perod. Soluton (a) PV of cash flow n=pv of cash flow out 10, 000 + 1, 000a 10 10% = v 10 X a 5 10% X = 10, 000 + 1, 000a 10 10% v 10 a 5 10% X = 10, 000 + 1, 000(6.144571) (.3855433)(3.7907868) X =11, 046 56

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. () 400.94 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 5 + 500v 10 = 400.94v 5 500v 10 = 200.94v 5 v 5 =.40188 v 5 =.40188 (1 + ) 5 =2.48831 Exercse (c) P = 100(1 + ) 10 + 120(1 + ) 5 = 100(2.48831) 2 + 120(2.48831) = 917.76 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)) = 275.00 57

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 Annuty-Immedate Annuty-Immedate An annuty-mmedate 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 + (n-2)q P + (n-1)q 0 1 2... n - 1 n 58

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 2 + + v n 1 + v n ]+Q[v 2 +2v 3 +(n 2)v n 1 +(n 1)v n ] = P [v + v 2 + + v n 1 + v n ]+Qv 2 [1 + 2v +(n 2)v n 3 +(n 1)v n 2 ] = P [v + v 2 + + 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

Let P =1andQ = 1. In ths case, the payments start at 1 and ncrease by 1 every year untl the fnal payment of n s made at tme n. 1 2... n - 1 n 0 1 2... n - 1 n Increasng Annuty-Immedate Present Value Factor The present value (at t = 0) of ths annual ncreasng annuty mmedate, where the annual effectve rate of nterest s, shall be denoted as (Ia) n and s calculated as follows: [ a n (Ia) n =(1) a n +(1) nv n ] = 1 vn + a n nv n = 1 vn + a n nv n = ä n nv n Increasng Annuty-Immedate Accumulaton Factor The accumulated value (at t = n) of ths annual ncreasng annuty mmedate, where the annual effectve rate of nterest s, shall be denoted as (Is) n and can be calculated usng the same general approach as above, or alternatvely, by smply usng the basc prncple where an accumulated value s equal to ts present value carred forward wth nterest: (Is) n =(Ia) n (1 + ) n (ä n = nv n ) (1 + ) n = ä n (1 + ) n nv n (1 + ) n = s n n 60

Increasng Perpetuty-Immedate Present Value Factor The present value (at t = 0) of ths annual ncreasng perpetuty mmedate, where the annual effectve rate of nterest s, shall be denoted as (Ia) and s calculated as follows: (Ia) = v +2v 2 +3v 3 +... v(ia) = v 2 +2v 3 +3v 4 +... (Ia) v(ia) = v + v 2 + v 3 +... (Ia) (1 v) =a (Ia) = a 1 v = a d = d 1+ = 1+ 2 3.2 Increasng Annuty-Due An annuty-due 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 + (2)Q... P + (n-1)q 0 1 2... n - 1 n 61

The present value (at t = 0) of ths annual annuty due, where the annual effectve rate of nterest s, shall be calculated as follows: PV 0 =[P ]+[P + Q]v + +[P +(n 2)Q]v n 2 +[P +(n 1)Q]v n 1 = P [1 + v + + v n 2 + v n 1 ]+Q[v +2v 2 +(n 2)v n 2 +(n 1)v n 1 ] = P [1 + v + + v n 2 + v n 1 ]+Qv[1 + 2v +(n 2)v n 3 +(n 1)v n 2 ] = P [1 + v + + v n 2 + v n 1 ]+Qv d dv [1 + v + v2 + + v n 2 + v n 1 ] = P ä n + Qv d dv [ä n ] = P ä n + Qv d [ ] 1 v n dv 1 v [ (1 v) ( nv n 1 ) (1 v n ] ) ( 1) = P ä n + Qv (1 v) 2 = P ä n + Q [ nv n (v 1 1) + (1 v n ] ) (1 + ) (/1+) 2 [ (1 v n ) nv n 1 nv n ] = P ä n + Q 2 (1 + ) [ (1 v n ) nv n (v 1 ] 1) = P ä n + Q 2 (1 + ) [ (1 v n ) nv n ] (1 + 1) = P ä n + Q (1 + ) = P ä n + Q (1 v n ) 2 nvn () [ a n = P ä n + Q nv n ] (1 + ) [ a n = P ä n + Q nv n ] d (1 + ) Ths present value of the annual annuty-due could also have been calculated usng the basc prncple that snce payments under an annuty-due start one year earler than under an annuty-mmedate, the annuty due wll earn one more year of nterest and thus, wll be greater than an annuty-mmedate by (1 + ): PV due 0 = PV0 mmedate (1 + ) ( [ a n = P a n + Q nv n ]) (1 + ) [ a n =(P a n ) (1 + )+Q nv n ] (1 + ) [ a n = P ä n + Q nv n ] d 62

The accumulated value (at t = n) of an annuty due, where the annual effectve rate of nterest s, can be calculated usng the same general approach as above, or alternatvely, 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 ä n + Q nv n ]) (1 + ) n d [ a = P ä n (1 + ) n n + Q (1 + ) n nv n (1 + ) n ] d [ ] s n = P s n + Q n d 63

Let P =1andQ = 1. In ths case, the payments start at 1 and ncrease by 1 every year untl the fnal payment of n s made at tme n 1. 1 2 3... n 0 1 2... n - 1 n Increasng Annuty-Due Present Value Factor The present value (at t = 0) of ths annual ncreasng annuty due, where the annual effectve rate of nterest s, shall be denoted as (Iä) n and s calculated as follows: [ a n (Iä) n =(1) ä n +(1) nv n ] d = 1 vn + a n nv n d d = 1 vn + a n nv n d = ä n nv n d Increasng Annuty-Due Accumulated Value Factor The accumulated value (at t = n) of ths annual ncreasng annuty mmedate, where the annual effectve rate of nterest s, shall be denoted as (I s) n and can be calculated usng the same approach as above or by smply usng the basc prncple where an accumulated value s equal to ts present value carred forward wth nterest: (I s) n =(Iä) n (1 + ) n (ä n = nv n ) (1 + ) n d = ä n (1 + ) n nv n (1 + ) n d = s n n d 64

Increasng Perpetuty-Due Present Value Factor The present value (at t=0) of ths annual ncreasng perpetuty-due, where the annual effectve rate of nterest s, shall be denoted as (Iä) and s calculated as follows: (Iä) =1+2v +3v 2 +4v 3 +... v(iä) = v +2v 2 +3v 3 +... (Iä) v(iä) =1+v + v 2 +... (Iä) (1 v) = ä 1 1 v = d d = 1 d 2 65

3.3 Decreasng Annuty-Immedate Let P = n and Q = 1. In ths case, the payments start at n and decrease by 1 every year untl the fnal payment of 1 s made at tme n. n n - 1... 2 1 0 1 2... n - 1 n Decreasng Annuty-Immedate Present Value Factor The present value (at t = 0) of ths annual decreasng annuty mmedate, where the annual effectve rate of nterest s, shall be denoted as (Da) n and s calculated as follows: [ a n (Da) n =(n) a n +( 1) nv n ] = n 1 vn a n nv n = n nvn a n + nv n = n a n Decreasng Annuty-Immedate Accumulaton Value Factor The accumulated value (at t = n) of ths annual decreasng annuty mmedate, where the annual effectve rate of nterest s, shall be denoted as (Ds) n and can be calculated by usng the same general approach as above, or alternatvely, by smply usng the basc prncple where an accumulated value s equal to ts present value carred forward wth nterest: (Ds) n =(Da) n (1 + ) n ( ) n a n = (1 + ) n = n (1 + )n s n 66

3.4 Decreasng Annuty-Due Let P = n and Q = 1. In ths case, the payments start at n and decrease by 1 every year untl the fnal payment of 1 s made at tme n. n n - 1 n - 2... 1 0 1 2... n - 1 n Decreasng Annuty-Due Present Value Factor The present value (at t = 0) of ths annual decreasng annuty due, where the annual effectve rate of nterest s, shall be denoted as (Dä) n and s calculated as follows: [ a n (Dä) n =(n) ä n +( 1) nv n ] d = n 1 vn a n nv n d d = n nvn a n + nv n d = n a n d Decreasng Annuty-Due Accumulated Value Factor The accumulated value (at t = n) of ths annual decreasng annuty due, where the annual effectve rate of nterest s, shall be denoted as (D s) n and can be calculated usng the same approach as above or by smply usng the basc prncple where an accumulated value s equal to ts present value carred forward wth nterest: (D s) n =( Da) n (1 + ) n ( ) n a n = (1 + ) n d = n (1 + )n s n d 67

Basc Relatonshp 1:ä n = (Ia) n + nv n Consder an n year nvestment where 1 s nvested at the begnnng of each year. The present value of ths multple payment ncome stream at t =0sä n. Alternatvely, consder a n year nvestment where 1 s nvested at the begnnng of each year and produces ncreasng annual nterest payments progressng to n by the end of the last year wth the total payments (n 1) refunded at t = n. The present value of ths multple payment ncome stream at t =0s (Ia) n + nv n. Therefore, the present value of both nvestment opportuntes are equal. Also note that (Ia) n = ä n n v n ä n = (Ia) n + nv n. 68

3.5 Contnuously Payable Varyng Annutes payments are made at a contnuous rate although they ncrease at dscrete tmes. Contnuously Payable Increasng Annuty Present Value Factors The present value (at t = 0) of an ncreasng annuty, where payments are beng made contnuously at an annual rate t, ncreasng at a dscrete tme t and where the annual effectve rate of nterest s, shall be denoted as (Iā) n and s calculated as follows: (Iā) n =ā 1 +2ā 1 v +3ā 1 v 2 +... + nā 1 v n 1 =ā 1 (1 + 2v +3v 2 +... + nv n 1 ) =ā 1 (Iä) n = 1 v (ä n nv n ) δ d = ä n nv n δ Contnuously Payable Increasng Annuty Accumulated Value Factor The accumulated value (at t = n) of ths annual ncreasng annuty where the annual effectve rate of nterest s, shall be denoted as (I s) n and can be calculated usng the basc prncple where an accumulated value s equal to ts present value carred forward wth nterest. (I s) n =(1+) n (Iā) n (ä =(1+) n n nv n ) δ = s n n δ 69

Contnuously Payable Increasng Perpetuty Present Value Factor The present value (at t = 0) of an ncreasng perpetuty, where payments are beng made contnuously at an annual rate of t, ncreasng at a dscrete tme t and where the annual effectve rate of nterest s, shall be denoted as (Iā) and s calculated as follows: (Iā) =ā 1 +2ā 1 v +3ā 1 v 2 +... =ā 1 (1 + 2v +3v 2 +...) =ā 1 (Iä) = 1 v 1 δ d 2 = 1 δ d Contnuously Payable Decreasng Annuty Present Value Factor The present value (at t = 0) of a decreasng annuty where payments are beng made contnuously at an annual rate t, decreasng at a dscrete tme t and where the annual effectve rate of nterest s, shall be denoted as (Dā) n and s calculated as follows: (Dā) n = nā 1 +(n 1)ā 1 v +(n 2)ā 1 v 2 +... +ā 1 v n 1 =ā 1 (n +(n 1)v +(n 2)v 2 +... + v n 1 ) =ā 1 (Dä) n = 1 v n a n δ d = n a n δ Contnuously Payable Decreasng Annuty Accumulated Value Factor The accumulated value (at t = n) of ths annual decreasng annuty, where the effectve rate of nterest s, shall be denoted as (D s) n and can be calculated usng the basc prncple where an accumulated value s equal to ts present value carred forward wth nterest. (D s) n =(1+) n (Dā) n =(1+) n ( n a n δ = n(1 + )n s n δ ) 70

3.6 Compound Increasng Annutes Annuty-Immedate An annuty-mmedate s payable over n years wth the frst payment equal to 1 and each subsequent payment ncreasng by (1 + k). The tme lne dagram below llustrates the above scenaro: n-2 n-1 1 1(1+k)... 1(1+k) 1(1+k) 0 1 2... n - 1 n Compound Increasng Annuty-Immedate Present Value Factor The present value (at t = 0) of ths annual geometrcally ncreasng annuty mmedate, where the annual effectve rate of nterest s, shall be calculated as follows: PV 0 =(1)v +(1+k)v 2 + +(1+k) n 2 v n 1 = v [1 + (1 + k)v + +(1+k) n 2 v n 2 = = ( 1 1+ ( 1 1+ ( 1 = 1+ ( 1 = 1+ ) [ ( 1+k 1+ 1+ ( ) n 1+k ) 1 1+ ( ) 1+k 1 1+ ) [ 1 v n j= 1+ 1+k 1 1 v j= 1+ 1+k 1 ) ä n j= 1+ 1+k 1 ) + + ] ( 1+k 1+ +(1+k) n 1 v n +(1+k) n 1 v n 1 ] ) n 2 ( ) ] n 1 1+k + 1+ 71

Compound Increasng Annuty-Immedate Accumulated Value Factor The accumulated value (at t = n) of an annual geometrc ncreasng 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 ( ) 1 = 1+ ( 1 = 1+ ä n j= 1+ 1+k 1 (1 + ) n ) s n j= 1+ 1+k 1 72

Annuty-Due An annuty-due s payable over n years wth the frst payment equal to 1 and each subsequent payment ncreasng by (1 + k). The tme lne dagram below llustrates the above scenaro: 1 1(1+k) 1(1+k) 2... 1(1+k) n-1 0 1 2... n-1 n Compound Increasng Annuty-Due Present Value Factor The present value (at t = 0) of ths annual geometrcally ncreasng annuty due, where the annual effectve rate of nterest s, shall be calculated as follows: PV 0 =(1)+(1+k)v + +(1+k) n 2 v n 2 +(1+k) n 1 v n 1 ( ) ( ) n 2 ( ) n 1 1+k 1+k 1+k =1+ + + + 1+ 1+ 1+ ( ) n 1+k 1 1+ = ( ) 1+k 1 1+ 1 v n j= 1+ 1+k = 1 1 v j= 1+ 1+k 1 =ä n 1+ j= 1+k 1 Ths present value could ( also ) have been acheved by smply multplyng the annuty-mmedate 1 verson by (1 + ); ä n 1+ j= 1+ (1 + ). 1 1+k Compound Increasng Annuty-Due Accumulated Value Factor The accumulated value (at t = n) of an annual geometrc ncreasng annuty-due, 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 =ä n j= 1+ 1(1 + )n 1+k = s n j= 1+ 1+k 1 73

3.7 Contnuously Varyng Payment Streams Contnuously Varyng Payment Stream Present Value Factor The present value (at t=a) of ths contnuously varyng payment stream, where the payment stream ρ t s from t = a to t = b and the annual force of nterest s δ t : b a ρ t e t a δ s ds dt Contnuously Varyng Payment Stream Accumulated Value Factor The accumulated value (at t=b) of ths contnuously varyng payment stream, where the payment ρ s from t = a to t = b and the annual effectve rate of nterest s δ t : b a ρ t e t a δ s ds dt 74

3.8 Contnuously Increasng Annutes payments are made contnuously at a varyng rate every year for the next n years Contnuously Increasng Annuty Present Value Factor The present value (at t = 0) of an ncreasng annuty, where payments are beng made contnuously at annual rate t at tme t and where the annual effectve rate of nterest s, shall be denoted as ( Īā ) n and s calculated as follows (usng ntegraton by parts): (Īā ) n = n 0 n = 0 = t e δt δ } {{ } u v tv t dt }{{} t e } δt {{ dt } u ] n = n e δn δ 0 =ā n n v n δ dv n 0 + ā n δ e δt δ dt } {{ } v du The present value (at t = 0) of an annuty, where payment at tme t s defned as f(t)dt and where the annual effectve rate of nterest s, shall be calculated as follows: n 0 f(t) v t dt f the force of nterest becomes varable then the above formula becomes: n 0 f(t) e t 0 δsds dt Contnuously Increasng Annuty Accumulated Value Factor The accumulated value (at t = n) of an ncreasng annuty, where payments are beng made contnuously at annual rate t at tme t and where the annual effectve rate of nterest s, shall be denoted as ( Ī s ) and s calculated as follows: n ) (Ī s n =(1 + ) n ( Īā ) n ( ā =(1 + ) n n nv n δ = s n n δ ) 75

Contnuously Payable Contnuously Increasng Perpetuty Present Value Factor The present value (at t = 0) of an ncreasng perpetuty, where payments are beng made contnuously at an annual rate of t, ncreasng at a contnuous tme t and where the annual effectve rate of nterest s, shall be denoted as (Īā) and s calculated as follows: (Īā ) = 0 tv t dt =ā v δ 1 δ = 0 δ = 1 δ 2 3.9 Contnuously Decreasng Annutes payments are made contnuously at a varyng rate every year for the next n years Contnuously Decreasng Annuty Present Value Factor The present value (at t = 0) of a decreasng annuty where payments are beng made contnuously at an annual rate t, decreasng at a contnuous tme t and where the annual effectve rate of nterest s, shall be denoted as ( Dā) n and s calculated as follows: ( Dā) n = n 0 n (n t) v t dt =n v t dt t v t dt 0 0 } {{ } } {{ } ā n n (Īā) n ( ā n nv n ) =n ā n δ = n (1 vn ) ā n + nv n δ = n ā n δ Contnuously Decreasng Annuty Accumulated Value Factor The accumulated value (at t = n) of ths annual decreasng annuty, where the effectve rate of nterest s, shall be denoted as ( D s) n and can be calculated usng the basc prncple where an accumulated value s equal to ts present value carred forward wth nterest. ( D s) n =(1 + ) n ( Dā) n ( ) n ā =(1 + ) n n δ = n(1 + )n s n δ 76

Exercses and Solutons 3.1 Increasng Annuty-Immedate Exercse (a) A depost of 1 s made at the end of each year for 10 years nto a bank account that pays nterest at the end of each year at an annual effectve rate of j. Each nterest payment s renvested nto another account where t earns an annual effectve nterest rate of j 2. The accumulated value of these nterest payments at the end of 10 years s 4.0122. Calculate j. Soluton (a) j (Is) 9 j 2 =4.0122 9 j j ä 2 j 2 9 =4.0122 ä 9 j 2 = 4.0122 2 +9=11.0061 j =4% j =8% 2 77

Exercse (b) Youaregventwoseresofpayments. Seres A s a perpetuty wth payments of 1 at the begnnng of each of the frst three years, 2 at the begnnng of each of the next three years, 3 at the begnnng of each of the next three years, and so on. Seres B s a perpetuty wth payments of K at the begnnng of each of the frst two years, 2K at the begnnng of each of the next two years, 3K at the begnnng of each of the next two years, and so on. The present value of these two seres s equal to each other. Determne K. Soluton (b) ä 3 (Ia) =(1+) 3 1 =(Kä 2 ) (Ia) =(1+) 2 1 ä 3 1 d =(Kä 2 ) 1 d ä 3 1+ =(Kä 2 ) 1+ (1 + ) 3 1 d ä 3 1+(1+)3 1 =(Ka [(1 + ) 3 1] 2 2 ) 1+(1+)2 1 [(1 + ) 2 1] 2 (1 + ) 3 [(1 + ) 3 1] = K (1 + )2 1 (1 + ) 2 2 d [(1 + ) 2 1] 2 (1 + ) (1 + ) 3 1 = K (1 + ) 2 1 K = (1 + )2 1 (1 + ) 3 1 (1 + ) =s 2 (1 + ) = v2 s 2 = a 2 s 3 v 3 s 3 a 3 78

Exercse (c) Danelle borrows 100,000 to be repad over 30 years. You are gven: () Her frst payment s X at the end of year 1. () Her payments ncrease at the rate of 100 per year for the next 19 years and reman level for the followng 10 years. () The effectve rate of nterest s 5% per annum. Calculate X. Soluton (c) 10, 000 = Xa 30 + v100(ia) 19 + v 20 1, 900a 10 [ ] [ä 1 v 30 10, 000 = X + 100v 19 19v 19 ] [ ] 1 v +1, 900v 20 10 3.2 Increasng Annuty-Due Exercse (a) X =5, 504.74 You buy an ncreasng perpetuty-due wth annual payments startng at 5 and ncreasng by 5 each year untl the payment reaches 100. The payments reman at 100 thereafter. The annual effectve nterest rate s 7.5%. Determne the present value of ths perpetuty. Soluton (a) (Iä) = 1 d 2 = 205.4444 5(Iä) 5v 20 (Iä) = 5(205.4444)(1.235413) = 785.40 79

3.3 Decreasng Annuty-Immedate Exercse (a) You can purchase one of two annutes: Annuty 1: An 11-year annuty-mmedate wth payments startng at 1 and then ncreasng by 1 for 5 years, followed by payments startng at 5 and then decreasng by 1 for 4 years. The frst payment s n two years. Annuty 2: A 13-year annuty-mmedate wth payments startng at 5 and then ncreasng by 2 for 5 years, followed by payments startng at 16 and then decreasng by 4 for 3 years and then decreasng by 1 for 3 years. The frst payment s n one year. The present value of Annuty 1 s equal to 36. Determne the present value of Annuty 2. Soluton (a) PV 1 = v[pyramd annuty] = v [ (Ia) 6 + v 6 (Da) 5 ] = v [ä 6 6v 6 + v 6 5 a 5 ] v [ä 6 a 6 ]=a 6 a 6 =[a 6 ] 2 =36 a 6 =6 =0% Exercse (b) PV 2 =5+7+9+11+13+15+16+12+8+4+3+2+1=106 1,000 s deposted nto Fund X, whch earns an annual effectve rate of nterest of. Atthe end of each year, the nterest earned plus an addtonal 100 s wthdrawn from the fund. At the end of the tenth year, the fund s depleted. The annual wthdrawals of nterest and prncpal are deposted nto Fund Y,whch earns an annual effectve rate of 9%. The accumulated value of Fund Y at the end of year 10 s 2,085. Determne. Soluton (b) [ 1, 000 10(1.09) 10 s 10 (Ds) 10 10 9% + 100 s 10 9% = 100 9%.09 ] + 100 (15.19293) 100(94.23) + 1, 519.29 = 2, 085 =6% 80

Exercse (c) You are gven an annuty-mmedate payng 10 for 10 years, then decreasng by one per year for nne years and payng one per year thereafter, forever. The annual effectve rate of nterest s 4%. Calculate the present value of ths annuty. Soluton (c) 3.4 Decreasng Annuty-Due Exercse (a) PV =10a 10 + v 10 (Da) 9 + v 19 a [ ] [ ] 1 v 10 9 a =10 + v 10 9 + v 19 1 = 199.40 Debbe receves her frst annual payment of 5 today. Each subsequent payment decreases by 1 per year untl tme 4 years. After year 4, each payment ncreases by 1 untl tme 8 years. The annual nterest rate s 6%. Determne the present value. Soluton (a) The present value at tme 0: 5+4v +3v 2 +2v 3 + v 4 +2v 5 +3v 6 +4v 7 +5v 8 Breakng up the payments from tme 0 to tme 4 s a decreasng annuty-due. 5+4v +3v 2 +2v 3 + v 4 =(Dä) 5 6% (Dä) 5 6% =13.914903 Then from tme 5 to tme 8 s an ncreasng annuty-due whch s mssng t s frst payment. 2v +3v 2 +4v 3 +5v 4 =(Iä) 5 6% (Iä) 5 6% 1=11.875734 PV 0 =[(Iä) 5 6% 1]v 4 6% =23.32 81

3.5 Contnuously Payable Varyng Annutes Exercse (a) Calculate the accumulated value at tme 10 years where payments are receved contnuously over each year. The payment s 100 durng the frst year and subsequent payments ncrease by 10 each year. The annual effectve rate of nterest s 4%. Soluton (b) =90 s 10 4% + 10(I s) 10 4% = 90(12.244660) + 10(63.393834) 3.6 Compoundng Increasng Annutes Exercse (a) = 1165.41 You have just purchased an ncreasng annuty mmedate for 75,000 that makes twenty annual payments as follows: () 5P, 10P,..., 50P durng years 1 through 10, and () 50P (1.05), 50P (1.05) 2,..., 50P (1.05) 10 durng years 11 through 20. The annual effectve nterest rate s 7% for the frst 10 years and 5%, thereafter. Solve for P. Soluton (a) 75, 000 = 5P (Ia) 10 7% +50P (1.05) v 10 7% v 5% ä 10 j=0 75, 000 = 5 P [ä ] 10 10v10 7% 7% +50P v7% 10 (10) P = 75, 000 [ ä ] 10v 10 10 7% 5 7% + 500 v7% 10 = 75, 000 427.8703 = 175.29 82

Exercse (b) You have just purchased an annuty mmedate for 75,000 that makes ten annual payments as follows: () 12.45P, 4P, 6P,8P,10P for years 1 through 5 and () 10P (1.07), 10P (1.07) 2, 10P (1.07) 3, 10P (1.07) 4, 10P (1.07) 5 for years 6 through 10. The annual effectve nterest rate s 5% for the frst 5 years and 7%, thereafter. Solve for P. Soluton (b) 75, 000 = (10.45P )v 5% +(2P ) (Ia) 5 5% + v 5 5% v 7% [10P (1.07)]ä 5 = 1.07 1.07 1=0% [ (ä 5 75, 000 = P 10.45v 5% +2 ) ] 5v5 5% 5% + v5% 5 v 7% [10(1.07)](5) Exercse (c) 75, 000 = P (9.9524 + 25.1328 + 39.1763) = P (74.2615) P =1, 009.94. You have just purchased an ncreasng annuty-mmedate for 50,000 that makes twenty annual payments as follows: () P, 2P,..., 10P n years 1 through 10; and () 10P (1.05), 10P (1.05) 2,..., 10P (1.05) 10 n years 11 through 20. The annual effectve nterest rate s 7% for the frst 10 years and 5% thereafter. Calculate P. Soluton (c) [ 50, 000 = P (Ia) 10 7% + 10P (1.05) (1.05) ] (1.05)2 (1.05)10 +10P +... +10P (1.05) 2 (1.05) 10 v7% 10 [ä 50, 000 = P 10 10v 10 ] +[10P 10]v7% 10 P = [ ä 10 10v 10 50, 000 ] = + 100v7% 10 P = 584.29 50, 000 85.57406 83

3.7 Contnuously Varyng Payment Streams Exercse (a) Fnd the present value of a contnuous ncreasng annuty wth a term of 10 years f the force of nterest s δ =0.04 and f the rate of payment at tme t s t 2 per annum. Soluton (a) Usng ntegraton by parts: n t 2 e δ t dt = 10 0 0 t 2 e 0.04t dt = t2 10 0.04 e 0.04t 0 = 102 0.04 e 0.04(10) 2t + 2 0.04 (0.04) 2 e 0.04t 0 10 10 = 102 0.04 e 0.4 20 (0.04) 2 e 0.04(10) = 102 0.04 e 0.4 20 (0.04) 2 e.4 = 0 + t e 0.04t dt 2 (0.04) 2 2 10 0 (0.04) 3 e 0.04t 0 2 (0.04) 3 e 0.04(10) + [ 2 10 2 (0.04) 3 e 0.4 0.04 + 20 ] (0.04) 2 + 2 (0.04) 3 = 247.6978 e 0.04t dt 10 2 (0.04) 3 84

3.8 Contnuously Increasng Annutes Exercse (a) Fnd the rato of the total payments made under (Īā) 10 of the annuty to those made durng the frst half. Soluton (a) The payment s t dt at tme t. Durng the second half the total payment s: durng the second half of the term 10 Durng the frst half the total payment s: t dt = 1 ] 10 2 t2 =37.50 5 5 The rato s: 5 0 t dt = 1 2 t2 ] 5 0 =12.50 Exercse (b) 37.50 12.50 =3 Evaluate (Īā) f δ =.08. Soluton (b) (Īā) = = 0 = t.08 e.08t 0 0 t dt t e.08t dt + 1.08 0 e.08t dt = t ( ) 2 1.08 e.08t e.08t.08 0 = ( ) 2 1 = 156.25.08 85

Exercse (c) A perpetuty s payable contnuously at the annual rate of 1 + t 2 at tme t. If δ =.05, fnd the present value of the perpetuty. Soluton (c) usng ntegraton by parts PV = 0 (1 + t 2 )e.05t dt = 1 = 1.05 2.05 e.05t t2 0.05 e.05t 0 = 1.05 + 2 [ t.05 (.05) 3 e.05t 0.05 e.05t 0 + 2.05 + 1.05 0 0 t e.05t dt ] e.05t dt = 1.05 + 2 (.05) 3 = 20 + 2(20)3 =16, 020. Exercse (d) A contnuously ncreasng annuty wth a term of n years has payments payable at an annual rate t at tme t. The force of nterest s equal to 1 n. Calculate the present value of ths annuty. Soluton (d) (Īā) n = ā n nv n δ = 1 e δn ne δn δ δ = 1 e 1 n n ne 1 n n 1 n 1 n = n ( n(1 e 1 ) ne 1) = n(n ne 1 ne 1 )=n(n 2ne 1 )=n 2 (1 2e 1 ) 86

3.9 Contnuously Decreasng Annutes Exercse (a) Fnd the rato of the total payments made under ( Dā) 10 durng the second half of the term of the annuty to those made durng the frst half. Soluton (a) The payment s (10 t) dt at tme t. Durng the second half the total payment s: 10 5 (10 t)dt = Durng the frst half the total payment s: (10 t)2 2 10 5 =12.5 The rato s: 5 0 (10 t)dt = (10 t)2 2 5 0 =37.5 = 12.5 37.5 = 1 3 87

4 Non-Annual Interest Rate and Annutes Overvew an effectve rate of nterest,, s pad once per year at the end of the year a nomnal rate of nterest, (p), s pad more frequently durng the year (p tmes) and at the end of the sub-perod (nomnal rates are also quoted as annual rates) nomnal rates are adjusted to reflect the rate to be pad durng the sub perod; for example, (2) s the nomnal rate of nterest convertble sem-annually (2) = 10% (2) 2 = 10% = 5% pad every 6 months 2 4.1 Non-Annual Interest and Dscount Rates Non-Annual pthly Effectve Interest Rates wth effectve nterest, you have nterest, padatp perods per year (1 + ) 1 p 1 Non-Annual pthly Effectve Dscount Rates wth effectve dscount, you have dscount d, padatp perods per year 1 (1 d) 1 p =1 (1 + ) 1 p 4.2 Nomnal p thly Interest Rates: (p) Equvalency to Effectve Rates of Interest:, (p) wth effectve nterest, you have nterest,, pad at the end of the year wth nomnal nterest, you have nterest (p), pad at the end of each sub-perod and ths s p done p tmes over the year (p sub-perods per year) (1 + ) = ) p (1+ (p) f gven an effectve rate of nterest, a nomnal rate of nterest can be determned: (p) = p[(1 + ) 1/p 1] f gven an effectve rate of nterest, the nterest rate per sub-perod can be determned: (p) p p =(1+)1/p 1 88

4.3 Nomnal p thly Dscount Rates: d (p) Equvalency to Effectve Rates of Dscount: d, d (p) wth effectve dscount, you have dscount, d, pad at the begnnng of the year wth nomnal dscount, you have dscount d(p), pad at the begnnng of each sub-perod and p ths s done p tmes over the year (p sub-perods per year) (1 d) = (1 d(p) p f gven an effectve rate of dscount, a nomnal rate of dscount can be determned )p d (p) = p[1 (1 d) 1/p ] the dscount rate per sub-perod can be determned, f gven the effectve dscount rate Relatonshp Between (p) p d (p) p d(p) and p =1 (1 d)1/p when usng effectve rates, you must have (1 + ) or(1 d) 1 by the end of the year (1 + ) = 1 v = 1 =(1 d) 1 (1 d) when replacng the effectve rate formulas wth ther nomnal rate counterparts, you have (1+ (m) ) m = (1 d(p) ) p m p 89

whenp = m (1+ (m) m ) m = (1 d(m) 1+ (m) m = m (1 d(m) m ) m ) 1 1+ (m) m = m m d (m) (m) m = m m m + d(m) 1= m d (m) m d (m) (m) m = d(m) m d (m) (m) m = d (m) m 1 d(m) m the nterest rate over the sub-perod s the rato of the dscount pad to the amount at the begnnng of the sub-perod (prncple of the nterest rate stll holds) d (m) m = (m) m 1+ (m) m the dscount rate over the sub perod s the rato of nterest pad to the amount at the end of the sub-perod (prncple of the dscount rate stll holds) the dfference between nterest pad at the end and at the begnnng of the sub-perod depends on the dfference that s borrowed at the begnnng of the sub-perod and on the nterest earned on that dfference (prncple of the nterest and dscount rates stll holds) (m) m d(m) m = (m) m = (m) m [ 1 )] (1 d(m) d(m) m 0 m 90

4.4 Annutes-Immedate Payable p thly Present Value Factor for a p thly Annuty-Immedate paymentsof 1 p are made at the end of every 1 pth of year for the next n years the present value (at t =0)ofanp th ly annuty mmedate, where the annual effectve rate of nterest s, shall be denoted as a (p) n and s calculated as follows: a (p) n =( 1 p )v 1 p +( 1 p )v 2 p + +( 1 p 1 p )v p +( 1 p )v p p (1st year) +( 1 p+1 p )v p +( 1 p+2 m )v p. +( 1 (n 1)p+1 p )v p =( 1 p )v 1 p +( 1 p. p+1 p )v [ 1+v 1 p +( 1 (n 1)p+1 p )v p [ 1+v 1 p + +( 1 2p 1 p )v p +( 1 (n 1)p+2 p )v p p 2 p + + v [ 1+v 1 p + + v p 2 p + +( 1 p + v p 1 p ] + v p 1 p p 2 p + + v +( 1 p )v 2p p ] + v np 1 p )v p 1 p ] +( 1 p )v np p (2nd year) (last year) (1st year) (2nd year) (last year) ( = ( 1 p )v 1 p +( 1 p+1 p )v p =( 1 p )v 1 p ( 1+v p p + +( 1 p =( 1 ( ) p ) 1 1 v n (1 + ) 1 p 1 v 1 =( 1 p ) 1 1 vn (1 + ) 1 p 1 v 1 p ( ) =( 1 p ) 1 v n (1 + ) 1 p 1 1 v n = ] p [(1 + ) 1 p 1 = 1 ( ) vn 1 = (p) p p 1 vn (p) (n 1)p+1 m ) )v ) + + v (n 1) 1 (v 1 p 1 v 1 p 1 v1 1 v 1 p )p )[ 1+v 1 p p 2 p + + v + v p 1 p ] 91

Accumulated Value Factor for a p thly Annuty-Immedate the accumulated value (at t = n) ofanp th ly annuty mmedate, where the annual effectve rate of nterest s, shall be denoted as s (p) n and s calculated as follows: s (p) n =( 1 p )+(1 p )(1 + ) 1 1 p + +( p +( 1 p )(1 + ) p 1 p+1 p +( )(1 + ) p p. p 2 )(1 + ) p +( 1 (n 1)p )(1 + ) p +( 1 (n 1)p+1 )(1 + ) m p p +( 1 p 1 )(1 + ) p p + +( 1 2p 2 )(1 + ) p +( 1 2p 1 )(1 + ) p p p + +( 1 np 2 )(1 + ) p 1 m +( p np 1 )(1 + ) p (last year) (2nd last year) (frst year) =( 1 [ p ) 1+(1+) 1 p 2 p + +(1+) p +( 1 p )(1 + ) p p ] +(1+) p 1 p [1+(1+) 1 p + +(1+) p 2 p ] +(1+) p 1 p (last year) (2nd last year). +( 1 (n 1)p )(1 + ) p p [1+(1+) 1 p + +(1+) p 2 p ] +(1+) p 1 p (frst year) ( = ( 1 p )+(1 p )(1 + ) p p + +( 1 p ) (n 1)p [ )(1 + ) p ) 1+(1+) 1 p 2 p + +(1+) p =( 1 ( ) [ p ) 1+(1+) p p + +(1+) (n 1) 1 ((1 + ) 1 p ) p 1 (1 + ) 1 p =( 1 ( ) [ ] 1 (1 + ) n p ) 1 (1 + ) 1 1 (1 + ) 1 1 (1 + ) 1 p ( ) =( 1 p ) 1 (1 + ) n 1 (1 + ) 1 p = (1 + )n 1 ] p [(1 + ) 1 p 1 = (1 + )n 1 (p) = ( ) 1 p p (1 + )n 1 (p) ] ] +(1+) p 1 p 92

Basc Relatonshp 1:1= (p) a (p) n + vn Basc Relatonshp 2:PV(1 + ) n = FV and PV = FV v n f the future value at tme n, s (p) n, s dscounted back to tme 0, then you wll have ts present value, a (p) n s (p) n vn = [ (1 + ) n 1 (p) ] v n = (1 + )n v n v n (p) = 1 vn (p) = a (p) n f the present value at tme 0, a (p) n, s accumulated forward to tme n, then you wll have ts future value, s (p) n a (p) n (1 + )n = [ 1 v n (p) ] (1 + ) n = (1 + )n v n (1 + ) n (p) = (1 + )n 1 (p) = s (p) n Basc Relatonshp 3:a (p) n = a (p) n, s (p) n = s (p) n Consderpaymentsof 1 p made at the end of every 1 th of year for the next n years. Over a p one-year perod, payments of 1 p made at the end of each pth perod wll accumulate at the ( ) 1 end of the year to a lump sum of p p s (p). If ths end-of-year lump sum exsts for each 1 year of the n-year annuty-mmedate, ( ) then the present value (at t = 0) of these end-of-year 1 lump sums s the same as p p a (p) n : ( ) ( ) 1 p p a (p) 1 n = p p s (p) a 1 n a (p) n = (p) a n Therefore, ( ) the accumulated value (at t = n) of these end-of-year lump sums s the same as 1 p p s (p) n : ( ) ( ) 1 p p s (p) 1 n = m p s (p) s 1 n s (p) n = (p) s n 93

4.5 Annutes-Due Payable p thly Present Value Factor of a p thly Annuty-Due paymentsof 1 p are made at the begnnng of every 1 pth of year for the next n years the present value (at t =0)ofanp th ly annuty due, where the annual effectve rate of nterest s, shall be denoted as ä (p) n and s calculated as follows: ä (p) n =( 1 p )+(1 p )v 1 p + +( 1 p 2 p )v p +( 1 p 1 p )v p (1st year) +( 1 p p )v p +( 1 p+1 p )v p. +( 1 (n 1)p p )v p =( 1 p ) [ 1+v 1 p +( 1 p )v p p. +( 1 (n 1)p p )v p [ 1+v 1 p + +( 1 2p 2 p )v p +( 1 (n 1)p+1 p )v p p 2 p + + v [ 1+v 1 p + +( 1 p + v + + v p 2 p p 1 p ] + v p 1 p p 2 p + + v +( 1 2p 1 p )v p ] + v np 2 p )v p 1 p ] +( 1 p np 1 p )v (2nd year) (last year) (1st year) (2nd year) (last year) = (( 1p p )+(1p )v p =( 1 p ) ( 1+v p p =( 1 ( ) 1 v n p ) 1 v 1 =( 1 p ) 1 vn 1 v 1 p =( 1 p ) ( = [ p 1 v n + +( 1 p )v (n 1)p p ) )[ ) + + v (n 1) 1 (v 1 p 1 v1 1 v 1 p ) 1 (1 d) 1 p 1 v n ] 1 (1 d) 1 p ( ) 1 p p = 1 vn d (p) = 1 vn d (p) 1+v 1 p )p 1 v 1 p p 2 p + + v + v p 1 p ] 94

the accumulated value (at t = n) ofanp th ly annuty due, where the annual effectve rate of nterest s, shall be denoted as s (p) n and s calculated as follows: s (p) n =( 1 p )(1 + ) 1 p +( 1 p )(1 + ) 2 1 p + +( p )(1 + ) p 1 p +( 1 p )(1 + ) p p (last year) +( 1 p+1 )(1 + ) p +( 1 p+2 )(1 + ) p + +( 1 2p 1 )(1 + ) p +( 1 p p p p )(1 + ) 2p p (2nd last year). +( 1 (n 1)p+1 )(1 + ) p p +( 1 (n 1)p+2 )(1 + ) p + +( 1 np 1 )(1 + ) p +( 1 np )(1 + ) p p p p (frst year) =( 1 p )(1 + ) 1 p [1+(1+) 1 p + +(1+) p 2 p +( 1 p+1 )(1 + ) p [1+(1+) 1 p 2 p + +(1+) p p ] +(1+) p 1 p ] +(1+) p 1 p (last year) (2nd last year). +( 1 (n 1)p+1 )(1 + ) p p [1+(1+) 1 p + +(1+) p 2 p ] +(1+) p 1 p (frst year) ( = ( 1 p )(1 + ) 1 p +( 1 p =( 1 p )(1 + ) 1 p p+1 )(1 + ) p + +( 1 ) [ (n 1)p+1 )(1 + ) p ) 1 ((1 + ) 1 p ) p p 1 (1 + ) 1 p ] (1+(1+) p p + +(1+) (n 1) ) [ 1 ((1 + ) 1 p ) p =( 1 ( ) [ ] p )(1 + ) 1 p 1 (1 + ) n 1 (1 + ) 1 1 (1 + ) 1 1 (1 + ) 1 p ( ) =( 1 p )(1 + ) 1 1 (1 + ) n p 1 (1 + ) 1 p (1 + ) n 1 = [ ] p v 1 p (1 + ) 1 p 1 = (1 + )n 1 [ ] p 1 v 1 p = (1 + )n 1 [ ] p 1 (1 d) 1 p = (1 + )n 1 d (p) = ( ) 1 p p (1 + )n 1 d (p) 1 (1 + ) 1 p ] 95

Basc Relatonshp 1:1=d (p) ä (p) n + vn Basc Relatonshp 2:PV(1 + ) n = FV and PV = FV v n f the future value at tme n, s (p) n, s dscounted back to tme 0, then you wll have ts present value, ä (p) n [ (1 + ) n 1 s (p) n vn = d (p) ] v n = (1 + )n v n v n d (p) = 1 vn d (p) =ä (p) n f the present value at tme 0, ä (p) n, s accumulated forward to tme n, then you wll have ts future value, s (p) n [ 1 v n ä (p) n (1 + )n = d (p) ] (1 + ) n = (1 + )n v n (1 + ) n d (p) = (1 + )n 1 d (p) = s (p) n Basc Relatonshp 3:ä (p) n = d d ä (p) n, s (p) n = d d s (p) n Consderpaymentsof 1 p made at the begnnng of every 1 p th of year for the next n years. Over a one-year perod, payments of 1 p made at the begnnng of each pth perod wll ( ) 1 accumulate at the end of the year to a lump sum of p p s (p). If ths end-of-year 1 lump sum exsts for each year of the n-year annuty-mmedate, ( ) then the present value 1 (at t = 0) of these end-of-year lump sums s the same as p p ä (p) n : ( ) ( ) 1 p p ä (p) 1 n = p p s (p) a 1 n ä (p) n = d (p) a n Therefore, ( ) the accumulated value (at t = n) of these end-of-year lump sums s the same 1 as p p s (p) n : ( ) ( ) 1 p p s (p) 1 n = p p s (p) s 1 n s (p) n = (p) s n 96

Basc Relatonshp 5: Due= Immedate (1 + ) 1 p ä (p) n = 1 vn = ( 1 vn ) = a (p) d (p) (p) 1+ (p) p s (p) n = (1 + )n 1 = (1 + )n 1 ( d (p) (p) 1+ (p) p n (1+ (p) ) = s (p) n p ) = a (p) n (1 + ) 1 p ) (1+ (p) = s (p) n p (1 + ) 1 p An p th ly annuty due starts one p th of a year earler than an p th ly annuty-mmedate and as a result, earns one p th of a year more nterest, hence t wll be larger. 97

Exercses and Solutons 4.2 Nomnal p th ly Interest Rates Exercse (a) 100 s deposted nto a bank account. The account s credted at a nomnal rate of nterest convertble semannually. At the same tme, 100 s deposted nto another account where nterest s credted at a force of nterest, δ. After 8.25 years, the value of each account s 300. Calculate ( δ). Soluton (a) 100 ) 2 (8.25) ( ) 2 (1+ (2) = 300 1+ (2) 2 2 =3 1 16.5 =1.06885 = 1+ (2) 1=.142439 2 OR 100(1 + ) 8.25 = 300 1+ =3 1 8.25 =3 1 8.25 1=.142439 100e 8.25δ = 300 e 8.25δ =3 8.25δ = ln[3] δ = ln[3] 8.25 =.133165 δ =.142439.133165 =.009274 =.93% 98

Exercse (b) X s deposted nto a savngs account at tme 0, whch pays nterest at a nomnal rate of j convertble semannually. 2X s deposted nto a savngs account at tme 0, whch pays smple nterest at an effectve rate of j. The same amount of nterest s earned durng the last 6 months of the 8th year under both savngs accounts. Calculate j. Soluton (b) X A(8) A(7.5) = X [ ( 1+ j ) 2 (8) ( 1+ j ) ] 2 (7.5) 2 2 [ ( 1+ j ) 16 ( 1+ j ) ] 15 ( = X 1+ j ) 15 [1+ j2 ] ( 2 2 2 1 = X 1+ j ) 15 [ ] j 2 2 A(8) A(7.5) = 2X[(1 + j(8)) (1 + j(7.5))] = 2X(j(.5)) = X(j) ( 1+ j ) 15 [ ] j = j 2 2 ( 1+ j ) 15 =2 j =.094588 = 9.46% 2 99

Exercse (c) At tme 0, K s deposted nto Fund X, whch accumulates at a force of nterest of δ t = t +3 t 2 +6t +9. At tme m, 2K s deposted nto Fund Y, whch accumulates at an annual nomnal rate of 10.25%, convertble quarterly. At tme n, wheren>m, the accumulated value of each fund s 3K. Calculatem. Soluton (c) K e n 0 δ t dt ( =2K 1+.1025 ) 4(n m) =3K e 4 n 0 δ t dt ( =2 1+.1025 ) 4(n m) =3 4 e n 0 t +3 t 2 +6t +9 dt =3 e 1 2 n 0 2t +6 n f (t) t 2 1 2 +6t +9 =3 e 0 f(t) dt =3 ( f(n) f(0) ) 1 2 =3 ( n 2 +6n +9 9 ) 1 2 =3 n 2 +6n +9=81 (n 6)(n + 12) = 0 n =6 ( 2 1+.1025 ) 4(6 m) =3 4(6 m) = 4 ln [ ] 3 2 ln [ 1+.1025 4 ] 6 m =4 m =2 Exercse (d) You make a depost nto a savngs account, whch pays nterest at an annual nomnal rate of (2). At the same tme, your professor deposts 1,000 nto a dfferent savngs account, whch pays smple nterest. At the end of fve years, the forces of nterest, δ 5, on the two accounts are equal, and your professor s account has accumulated to 2,000. Calculate (2). Soluton (d) 1, 000(1 + 5) =2, 000 = 20% δ t = ( 1+ (2) 2 1+t δ 5 =.2 1+(.2)(5) = 10% ) 2 ) = e δ(1) (1+ (2) = e δ 2 = e.05 2 (2) =2(e.05 1) = 10.2542% 100

Exercse (e) If an nvestment wll be trpled n 8 years at a force of nterest δ, n how many years wll an nvestment be doubled at a nomnal rate of nterest numercally equal to δ and convertble once every three years? Soluton (e) e 8δ =3 δ = ln[3] 8 (1 + 3δ) n 3 =2 ( 1+ 3ln[3] 8 ) n 3 =2 n 3 = ln[2] [ ln 1+ 3ln[3] 8 ] n 3 =2 n =6 Exercse (f) Calculate the nomnal rate of nterest convertble once every four years that s equvalent to a nomnal rate of dscount convertble quarterly. Soluton (f) 1 1 [ 4 = (1 + ) 4 1 ] = 1 [ (1 d) 4 1 ] 4 4 [ ( ) 4 4 1 d(4) 1] [ ( ) 16 1 d(4) 1] = 1 4 4 = 1 4 4 101

4.3 Nomnal p th ly Dscount Rates Exercse (a) You are gven a loan for whch nterest s to be charged over a four-year perod, as follows: () At an annual effectve rate of nterest of 6%, for the frst year. () At an annual nomnal rate of nterest of 5%, convertble every two years, for the second year. () At an annual nomnal rate of dscount of 5%, convertble semannually, for the thrd year. (v) At a force of nterest of 5%, for the fourth year. Calculate the annual effectve rate of dscount over thew four-year perod. Soluton (a) ( (1 + ) 1+2 ( 1 )) 1 2 12 (1.06) (1 + 2(.05)) 1 2 ) 2 (1 d(2) e δ =(1+) 4 =(1 d) 4 ( 2 1.05 2 ) 2 e.05 =(1 d) 4 1.22944 = (1 d) 4 d =1 1.22944 1 4 =5.03% 102

Exercse (b) Express d (4) as a functon f (3). Soluton (b) Exercse (c) Express (6) as a functon of d (2). Soluton (c) (1+ (3) 3 d (4) =4 1 ) 3 = (1 d(4) (1+ (3) 3 4 ) 4 ) 3 4 Exercse (d) (1+ (6) 6 ) 6 ) 2 = (1 d(2) 2 (6) =6 (1 d(2) 2 ) 1 3 1 Gven a nomnal nterest rate of 7.5% convertble semannually, determne the sum of the: () force of nterest; and () nomnal dscount rate compounded monthly. Soluton (d) (1+ (2) 2 ) 2 = e δ (1) ( δ =2ln 1+.075 ) =.073627946 2 ) 12 ) 2 (1 d(12) = (1+ (12) 12 2 d (12) =12 1 (1+ (2) 2 ) 2 12 =.073402529.073627946 +.073402529 =.147030475 103

4.4 Annutes-Immedate Payable p th ly Exercse (a) On January 1, 1985, Mchael has the followng two optons for repayng a loan: () Sxty monthly payments of 100 begnnng February 1, 1985. () A sngle payment of 6,000 at the end of K months. Interest s at a nomnal annual rate of 12% compounded monthly. The two optons have the same present value. Determne K. Soluton (a) (12) 12 =.01/month 100a 60.01 = 6000v K 4495.5 = 6000v K 6000 4495.50 =(1.01)K K = ln ( ) 6000 4495.50 =29months ln(1.01) Exercse (b) Steve elects to receve hs retrement beneft over 20 years at the rate of 2,000 per month begnnng one month from now. The monthly beneft ncreases by 5% each year. At a nomnal nterest rate of 6% convertble monthly, calculate the present value of the retrement beneft. Soluton (b) R =2, 000s 12.06.12 =24, 671.12 PV = Rv + R(1.05)v 2 +... + R(1.05) 19 v 20 ( 1 1.05 1+ PV =24, 671.12.05 ) 20, where =(1.005) 12 1=.0616778 PV = 419, 253.14 104

Exercse (c) A pensoner elects to receve her retrement beneft over 20 years at a rate of 2,000 per month begnnng one month from now. The monthly beneft ncreases by 5% each year. At a nomnal nterest rate of 6% convertble monthly, calculate the present value of the retrement beneft. Soluton (c) 2, 000a 12 12 % =23, 237.86 PV =23, 237.86(1 + v(1.05) + v 2 (1.05) 2 +... + v 19 (1.05) 19 ) =23, 237.86 ( where = =23, 237.86 1+ 1.05 1+ + ( ) 2 1.05 +... + 1+ ( 1+.06 ) 12 1=.061678 12 ( ( 1 1.05 1.061678.061678.05 ) 20 ( ) ) 19 1.05 1+ ) (1.061678) PV = 419, 252.46 105

Exercse (d) If 3a (2) n Soluton (d) =2a(2) 2n =45s(2), fnd. 1 ( ) ( ) 1 v n 1 v 2n 3 =2 (2) (2) 3 3v n =2 2v 2n ( ) =45 (2) 2v 2n 3v n +1=0 Then the equaton: gves us v n = 3 9 4(2) 4 ( ) 1 v n 3 (2) = 1 2 ( ) =45 (2) ( 3 1 1 ) =45 2 4.5 Annutes-Due Payable p th ly Exercse (a) = 1.5 45 = 1 30 At tme t =0youdepostP nto a fund whch credts nterest at an effectve annual nterest rate of 8%. At the end of each year n years 6 through 20, you wthdraw an amount suffcent to purchase an annuty due of 100 per month for 10 years at a nomnal nterest rate of 12% compounded monthly. Immedately after the wthdrawl at the end of year 20, the fund value s zero. Calculate P. Soluton (a) PMT = 100ä 120 1% =7, 039.75 P (1.08) 6 =7, 039.75ä 15 8% P =41, 009.64 106

Exercse (b) At tme t = 0, Paul deposts P nto a fund credtng nterest at an effectve annual nterest rate of 8%. At the end of each year n year 6 through 20, Paul wthdraws an amount suffcent to purchase an annuty-due of 100 per month for 10 years at a nomnal nterest rate of 12% compounded monthly. Immedately after the wthdrawl at the end of year 20, the fund value s zero. Calculate P. Soluton (b) PMT = (100 12)ä (12) 10 = 100ä 120.01 =7, 039.75 0=P (1.08) 20 v[(100 12)ä (12) } {{ 10 ] s 15 }.08 =41, 009 100ä 120.01 107

5 Project Apprasal and Loans Overvew ths chapter extends the concepts and technques covered to date and apples them to common fnancal stuatons smple fnancal transactons such as borrowng and lendng are now replaced by a broader range of busness/fnancal transactons taxes and nvestment expenses are to be gnored unless stated 5.1 Dscounted Cash Flow Analyss by takng the present value of any pattern of future payments, you are performng a dscounted cash flow analyss letcft n represent the returns made back to the nvestor that are made at tme t letcft out represent the contrbutons by an nvestor that are made at tme t Net Present Value the net present value can be calculated by takng the present value of the cash-flows-n and reducng them by the present value of the cash-flows-out NPV = = n t=0 n t=0 ( CF n t CF n t ) CFt out v t v t n t=0 CF out t v t 108

Example A 10 year nvestment project requres an ntal nvestment of 1, 000, 000 and subsequent begnnng-of-year payments of 100, 000 for the followng 9 years. The project s expected to produce 5 annual nvestment returns of 600, 000 commencng 6 years after the ntal nvestment. From a cash flow perspectve, we have The net present value can be calculated as: CF0 out =1, 000, 000 CF1 out = CF2 out = = CF9 out = 100, 000 CF6 n = CF7 n = = CF10 n = 600, 000 NPV = NPV = n t=0 n t=0 ( CF n t CFt n v t CFt out ) v t n t=0 CF out t = 600, 000v 6 + 600, 000v 7 + + 600, 000v 10 [ 1, 000, 000 + 100, 000v 1 + 100, 000v2 + + 100, ] 000v9 NPV = 600, 000v 6 ä 5 1, 000, 000 100, 000v 1 ä 9 = 1, 000, 000 100, 000v 1 ä 5 + 500, 000v 6 ä 4 + 600, 000v 10 The net present value depends on the annual effectve rate of nterest,, that s adopted. Under ether approach, f the net present value s negatve, then the nvestment project would not be a desrable pursut. v t 109

Internal Rate of Return There exsts a certan nterest rate where the net present value s equal to 0. Under the cash flow approach, PV 0 = NPV = n t=0 n t=0 CF n t n t=0 netcf t (1 + IRR) t =0 v t CF n t v t = n t=0 n t=0 CF out t v t =0 CF out t In other words, there s an effectve rate of nterest that exsts such that the present value of cash-flows-n wll yeld the same present value of cash-flows out. Ths nterest rate s called a yeld rate or an nternal rate of return (IRR) as t ndcates the rate of return that the nvestor can expect to earn on ther nvestment (.e. on ther contrbutons or cash-flows-n). v t 110

Ths yeld rate for the above nvestment project can be determned as follows: n t=0 CF n t v t = n t=0 CF out t v t 600, 000v 6 ä 5 =1, 000, 000 + 100, 000v 1 ä 9 600, 000v 6 ä 5 100, 000v 1 ä 9 1, 000, 000 = 0 600, 000v 6 1 v5 1 v 1 100, 000v 1 1 v9 1 v 1 1, 000, 000 = 0 600, 000v 6 600, 000v 11 100, 000v 1 + 100, 000v 10 1, 000, 000(1 v 1 )=0 600, 000v 6 600, 000v 11 + 900, 000v 1 + 100, 000v 10 1, 000, 000 = 0 The yeld rate (solved by usng a pocket calculator wth advanced fnancal functons or by usng Excel wth ts Goal Seek functon) s 8.062%. Usng ths yeld rate, nvestment projects wth a 10 year lfe can be compared to the above project. In general, those nvestment opportuntes that have hgher(lower) expected returns than the 8.062% may be more(less) desrable. 111

Renvestment Rates the yeld rate that s calculated assumes that the postve returns (or cash-flows-out) wll be renvested at the same yeld rate the actual rate of return can be hgher or lower than the calculated yeld rate dependng on the renvestment rates Example an nvestment of 1 s nvested for n years and earns an annual effectve rate of. The nterest payments are renvested n an account that credts an annual effectve rate of nterest of j. f the nterest s payable at the end of every year and earns a rate of j, then the accumulated value at tme n of the nterest payments and the orgnal nvestment s: FV n =( 1)s n j +1 f = j, then the accumulated value at tme n s: FV n = (1 + )n 1 +1=(1+) n 112

Example an nvestment of 1 s made at the end of every year for n years and earns an annual effectve rate of nterest of. The nterest payments are renvested n an account that credts an annual effectve rate of nterest of j. note that nterest payments ncrease every year by 1 as each extra dollar s deposted each year nto the orgnal account payments nterest 1 1 1... 1 1 2... (n-2) (n-1) 0 1 2 3... n-1 n f the nterest s payable at the end of every year and earns a rate of j, then the accumulated value at tme n of the nterest payments and the orgnal nvestment s: FV n = (Is) n 1 j + n 1 f = j, then the accumulated value at tme n s: FV n = s n 1 (n 1) + n = s n 1 n +1+n = s n 1 +1 = s n 113

5.2 Nomnal vs. Real Interest Rates Interest Rates and Inflaton nterest rates and nflaton are assumed to move n the same drecton over tme snce lenders wll charge hgher nterest rates to make up for the loss of purchasng power due to hgher nflaton. the relatonshp s actually between the current rate of nterest and the expected (not current) rate of nflaton. a nomnal nterest rate s one that has not been adjusted for nflaton Nomnal Interest Rates vs. Real Interest Rates let real represent the real rate of nterest, represent the nomnal nterest rate and π represent the rate of nflaton where real = 1+ 1+π 1 Present Value Formulas Usng Nomnal and Real Interest Rates PV = = = n nomnalcf t v t t=0 n t=0 n t=0 nomnalcf t v t (1 + real ) t realcf t (1 + real ) t 114

5.3 Investment Funds Dollar-weghted Interest Rate nvestment funds typcally experence multple contrbutons and wthdrawals durng ts lfe nterest payments are often made at rregular perods rather than only at the end of the year here the yeld rate s nfluenced by the dollar amount of the contrbuton, t s often referred to as a dollar-weghted rate of nterest the dollar-weghted rate of nterest s the actual return that the nvestor experences over the year n F 0 (1 + ) T + c s (1 + ) T ts = F T where F 0 s the value of the fund at tme 0 where F T s the value of the fund at tme T where c s s the deposts or wthdrawls of the fund where t s s the tme and s the nterest rate Smple Interest Approxmaton for Compound Interest to approxmate the dollar-weghted rate of nterest assume that (1 + ) n =(1+n) ths approach can produce results that are farly close to the exact approach s=1 115

Tme-Weghted Interest Rate to measure annual fund performance wthout the nfluence of the contrbutons, one must look at the fund s performance over a varety of sub-perods. These sub-perods are trggered whenever a contrbuton takes place and end just before the next contrbuton (or when the end of the year s fnally met). n general, the nterest rate earned for the sub-perod s determned by takng the rato of the fund at the begnnng of the sub-perod versus the fund at the end of the sub-perod. The yeld rate derved from ths method s called the tme-weghted rate of nterest and s determned usng the followng formula: where F 0 s the value of the fund at tme 0 where F T s the value of the fund at tme T where c n s the external cash flows (1 + ) T = F 1 F 2 F 3 F T... F 0 F 1 + c 1 F 2 + c 2 F n + c n where F n s the value of the fund before each of the cash flows 116

5.4 Allocatng Investment Income you are gven a fund for whch there are many nvestors. Each nvestor holds a share of the fund expressed as a percentage. For example, nvestor k mght hold 5% of the fund at the begnnng of the year over a one-year perod, the fund wll earn nvestment ncome, I there are two ways n whch the fund s nvestment ncome can be dstrbuted to the nvestors at the end of the year () Portfolo Method () Investment Year Method () Portfolo Method fnvestork owns 5% of the fund at the begnnng of the year, then nvestor k gets 5% of the nvestment ncome (5% I). ths s the same approach as f the fund s dollar-weghted rate of return was calculated and all the nvestors were credted wth that same yeld rate the dsadvantage of the portfolo method s that t doesn t reward those nvestors who make good decsons For example, nvestor k may have contrbuted large amounts of money durng the last sx months of the year when the fund was earnng, say 100%. Investor c may have wthdrawn money durng that same perod, and yet both would be credted wth the same rate of return. If the fund s overall yeld rate was say, 0%, obvously, nvestor k would rather have ther contrbutons credted wth the actual 100% as opposed to (1 + 0%) 1 2. () Investment Year Method an nvestors s contrbuton wll be credted durng the year wth the nterest rate that was n effect at the tme of the contrbuton. ths nterest rate s often referred to as the new-money rate. Renvestment rates can be handled n one of two ways: (a) Declnng Index System - only prncpal s credted at new money rates (b) Fxed Index System - prncpal and nterest s credted at new money rates ths earmarkng of money for new money rates wll only go on for a specfed perod before the portfolo method commences 117

5.5 Loans: The Amortzaton Method there are two methods for payng off a loan () Amortzaton Method - borrower makes nstallment payments at perodc ntervals () Snkng Fund Method - borrower makes nstallment payments as the annual nterest comes due and pays back the orgnal loan as a lump-sum at the end. The lump-sum s bult up wth perodc payments gong nto a fund called a snkng fund. ths chapter also dscusses how to calculate: (a) the outstandng loan balance once the repayment schedule has begun, and (b) what porton of an annual payment s made up of the nterest payment and the prncpal repayment Fndng The Outstandng Loan There are two methods for determnng the outstandng loan once the payment process commences () Prospectve Method () Retrospectve Method () Prospectve Method (see the future) the orgnal loan at tme 0 represents the present value of future repayments. If the repayments, X, are to be level and payable at the end of each year, then the orgnal loan can be represented as follows: Loan = X a n the outstandng loan at tme t, O/S Loan t, represents the present value of the remanng future repayments O/S Loan t = P a n t ths also assumes that the repayment schedule determned at tme 0 has been adhered to; otherwse, the prospectve method wll not work () Retrospectve Method (see the past) If the repayments, X, are to be level and payable at the end of each year, then the outstandng loan at tme t s equal to the accumulated value of the loan less the accumulated value of the payments made to date O/S Loan t = Loan (1 + ) t X s t ths also assumes that the repayment schedule determned at tme 0 has been adhered to; otherwse, the accumulated value of past payments wll need to be adjusted to reflect what the actual payments were, wth nterest 118

Basc Relatonshp 1: Prospectve Method = Retrospectve Method let a loan be repad wth end-of-year payments of 1 over the next n years: Present Value of Payments = Present Value of Loan (1)a n = Loan Accumulated Value of Payments = Accumulated Value of Loan (1)a n (1 + ) t = Loan (1 + ) t Accumulated Value of Past Payments +Present Value Future Payments = Accumulated Value of Loan (1)s t +(1)a n t = Loan (1 + ) t = a n (1 + ) t Present Value Future Payments = Accumulated Value of Loan Accumulated Value of Past Payments (1)a n t = a n (1 + ) t (1)s t Prospectve Method = Retrospectve Method the prospectve method s preferable when the sze of each level payment and the number of remanng payments s known the retrospectve method s preferable when the number of remanng payments or a fnal rregular payment s unknown. Amortzaton Schedules an orgnal loan of (1)a n s to be repad wth end-of-year payments of 1 over n years an annual end-of-year payment of 1 usng the amortzaton method wll contan an nterest payment, I t, and a prncpal repayment, P t notherwords,1=i t + P t Interest Payment I t s ntended to cover the nterest oblgaton that s payable at the end of year t. The nterest s based on the outstandng loan balance at the begnnng of year t. B t 1 s the prncpal balance outstandng after a prevous payment. usng the prospectve method for evaluatng the outstandng loan balance, the nterest payment s: Prncpal Repayment I t = B t 1 once the nterest owed for the year s pad off, then the remanng porton of the amortzaton payment goes towards payng back the prncpal: P t = L a n I t 119

5.6 Loans: The Snkng Fund Method let a loan of (1)a n be repad wth sngle lump-sum payment at tme n. If annual end-of-year nterest payments of a n are beng met each year, then the lump-sum requred at t = n, s the orgnal loan amount (.e. the nterest on the loan never gets to grow wth nterest). servce payments are the nterest payments that are pad to the lender let the lump-sum that s to be bult up n a snkng fund be credted wth nterest rate j and the rate of nterest on the loan be credted wth nterest rate Snkng Fund Loan: Equaton of Value (when servce payments equal nterest due) L = Accumulated Value of snkng fund payments (SFP) at tme n = SFP s n j Snkng Fund Loan: Payments (when servce payments equal nterest due) SFP = L s n j Net Amount of Snkng Fund Loan (when servce payments equal nterest due) we defne the loan amount that s not covered by the balance n the snkng fund as the NetAmountofLoanoutstandng and s equal to: Net Amount of Loan t = L SFP s t j note that the NetAmountofLoanoutstandng under the snkng fund method s the same value as the outstandng loan balance under the amortzaton method Snkng Fund Loan: General Equaton of Value usually, the nterest rate on borrowng,, s greater than the nterest rate offered by nvestng n a fund, j the total payment under the snkng fund approach s then L(1 + ) n = SP s n + SFP s n j ths s where SP s servce payments and SPF s snkng fund payments Net Amount of Snkng Fund Loan NetAmountofLoan t = L(1 + ) t SP s t SFP s t j 120

Exercses and Solutons 5.1 Dscounted Cash Flow Analyss Exercse (a) You nvest 300 nto a bank account at the begnnng of each year for 20 years. The account pays out nterest at the end of every year at an annual effectve nterest rate of. Thenterest s renvested at an annual effectve rate of ( ) 2. The annual effectve yeld rate on the entre nvestment over the 20-year perod s 10%. Determne. Soluton (a) 300ä 20 10% =20 300 + (300)(Is) 20 2 18, 900.75 = 6, 000 + (300) ( a 20 2 2 ) 20 ( ) 18, 900.75 = 6, 000 + (600) s 21 2 21 18, 900.75 = 6, 000 + 600s 21 2 12, 600 s 21 2 =42.50 use a calculator to fnd: =6.531% =2(6.531%) = 13.062% 2 Exercse (b) The nternal rate of return for an nvestment n whch C 0 =4, 800, C 1 = X, R 1 = X +4, 000 and R 2 =2, 500 can be expressed as 1 n.fndn. Soluton (b) PV of CF n =PVofCF out C 0 + C 1 v = R 1 v + R 2 v 2 4, 800 + Xv =(X +4, 000)v +2, 500v 2 2, 500v 2 +4, 000v 4, 800 = 0 v = 4, 000 + 4, 000 2 4(2, 500)( 4, 800) 2(2, 500) = 4 5 1+ = 5 4 = 1 4 121

Exercse (c) An nvestor deposts 5,000 at the begnnng of each year for fve years n a fund earnng an annual effectve nterest rate of 5%. The nterest rate of 5%. The nterest from ths fund can be renvested at an annual effectve nterest rate of 4%. Prove that the future value of ths nvestment at tme t =10sequalto6, 250 ( s 11 4% s 6 4% 1 ). Soluton (c) Interest n 1st year = 5, 000(5%) = 250 Interest n 2nd year = 10, 000(5%) = 500 Interest n 3rd year = 15, 000(5%) = 750 Interest n 4th year = 20, 000(5%) = 1, 000 Interest n 5th year = 25, 000(5%) = 1, 250 Interest n 6th year = 25, 000(5%) = 1, 250 Interest n 7th year = 25, 000(5%) = 1, 250 Interest n 8th year = 25, 000(5%) = 1, 250 Interest n 9th year = 25, 000(5%) = 1, 250 Interest n 10th year = 25, 000(5%) = 1, 250 FV 10 =5 5, 000 + 250(Is) 5 4% (1.04) 5 +1, 250s 5 4% (ä ) ( 5 FV 10 =25, 000 + 250 4% 5 (1.04) (1.04) 5 5 ) 1 +1, 250.04.04 ( ) FV 10 =25, 000 + 6, 250 s 6 4% 6 (1.04) 5 +31, 250 ( (1.04) 5 1 ) ( ) FV 10 =25, 000 + 6, 250 s 6 4% 6 (1.04) 5 +31, 250(1.04) 5 31, 250 FV 10 =6, 250s 6 4% (1.04) 5 6, 250(1.04) 5 6, 250 ( ) FV 10 =6, 250 s 6 4% (1.04) 5 (1.04) 5 1 ( ) FV 10 =6, 250 s 11 4% s 6 4% 1 122

Exercse (d) You have 10,000 and are lookng for a sold nvestment. Your professor suggests that you lend hm the 10,000 and agrees to repay you wth 10 annual end-of-year payments that decrease arthmetcally. The annual effectve nterest rate that you wll charge hm s 25% and you are able to renvest the repayments at 10%. After 5 years, your professor flees the country and leaves you wth nothng. What s your yeld on ths foolsh nvestment? Soluton (d) Payments= 10, 000 (Da) 10 25% = 10, 000 ( 10 a 10 25% 25% ) = ] 10, 000(1 + ) 5 = 388.83 [(Ds) 5 10% +5s 5 10% 10, 000(1 + ) 5 = 388.83[19.475 + 30.526] 10, 000(1 + ) 5 = 388.83[50.00] 10, 000(1 + ) 5 =19, 441.65 (1 + ) 5 =1.944165 =14.22% 10, 000 25.719 = 388.83 5.2 Nomnal vs. Real Interest Rates Exercse (a) The real nterest rate s 6% and the nflaton rate s 4%. Alan receves a payment of 1,000 at tme 1, and subsequent payments ncrease by 50 for 5 more years. Determne the accumulated value of these payments at tme 6 years. Soluton (b) The nomnal nterest rate s: The accumulated value of the cash flows s: =(1.06)(1.04) 1=10.24% =1, 000(1.1024) 5 +1, 050(1.1024) 4 +1, 100(1.1024) 3 +1, 150(1.1024) 2 +1, 200(1.1024) 1 +1, 250 = 950s 6 10.24% + 50(Is) 6 10.24% = 8623.08 123

5.3 Investment Funds Exercse (a) On January 1, an nvestment account s worth 100. On May 1, the value has ncreased to 120 and W s wthdrawn. On November 1, the value s 100 and W s deposted. On January 1 of the followng year, the nvestment account s worth 100. The tme-weghted rate of nterest s 0%. Calculate the dollar-weghted rate of nterest. Soluton (a) ( )( )( ) 120 100 100 1=0% 100 120 W 100 + W ( 100 )( 100 ) 120 W 100 + W = 1 120 100 10, 000 12, 000 + 20W W 2 =0.8333 0=16.67W.8333W 2 0=W (16.67.8333) 20 = W Usng smple nterest approxmaton: 100(1 + ) 20(1 + ) 8 12 + 20(1 + ) 2 12 = 100 100(1 + ) 20(1 + 8 2 ) + 20(1 + ) = 100 12 12 100 + 90 = 100 90 =0 =0 124

Exercse (b) On January 1, an nvestment s worth 100. On May 1, the value has ncreased to 120 and D s deposted. On November 1, the value s 100 and 40 s wthdrawn. On January 1 of the followng year, the nvestment account s worth 65. The tme-weghted rate of return s 0%. Calculate the dollar-weghted rate of return. Soluton (b) ( )( )( ) 120 100 65 1=0% 100 120 + D 60 ( )( )( ) 120 100 65 =1 100 120 + D 60 100 120 + D =0.76923 100 = 92.3076 +.76923D 10 = D Usng smple nterest approxmaton: 100(1 + ) + 10(1 + ) 8 12 40(1 + ) 2 12 =65 100(1 + ) + 10(1 + 8 12 ) 40(1 + 2 12 )=65 70 + 100 =65 100 = 5 = 5% 125

Exercse (c) You are gven the followng nformaton about two funds: Value of Value of Fund X before Fund Y before Fund X Fund Y Fund X Fund Y Deposts and Deposts and Date Deposts Deposts Wthdrawls Wthdrawls Wthdrawls Wthdrawls 01-Jan-03 50,000 100,000 01-Mar-03 55,000 01-May-03 24,000 50,000 01-Jul-03 15,000 105,000 01-Nov-03 36,000 77,310 01-Dec-03 10,000 F 31-Dec-03 31,500 94,250 Fund Y s dollar-weghted rate of return n 2003 s equal to Fund X s tme-weghted rate of return n 2003. Calculate F. Soluton (c) Calculate the dollar-weghted rate of return for Fund Y. Under smple nterest for compound nterest: 100, 000(1 + ) 15, 000(1 + ) 6 12 =94, 250 100, 000(1 + ) 15, 000(1 + 6 )=94, 250 12 100, 000 + 100, 000 15, 000 7, 500 =94, 250 92, 500 =9, 250 = 10% Calculate the tme-weghted rate of return for Fund X. 1.10 = (1 + j 1 )(1 + j 2 )(1 + j 3 )(1 + j 4 )=1+ 1.10 = (1 + j 1 )(1 + j 2 )(1 + j 3 )(1 + j 4 ) 1.10 = 50, 000 50, 000 77, 310 50, 000 + 24, 000 F 77, 310 36, 000 31, 500 F =36, 260 F 10, 000 126

5.4 Allocatng Investment Income Exercse (a) The followng table shows the annual effectve nterest rates beng credted n an nvestment account, by the calendar year of the nvestment. The nvestment year method s applcable for the frst three years, after whch a portfolo rate s used: Calendar Year Investment Year Rates Calendar Year Portfolo of Investment 1 2 3 of Portfolo Rate Rate 1996 12% 12% t% 1999 8% 1997 12% 5% 10% 2000 (t 1)% 1998 8% (t.02)% 12% 2001 6% 1999 9% 11% 6% 2002 9% 2000 7% 7% 10% 2003 10% An nvestment of 100 s made at the begnnng of years 1996, 1997 and 1998. The total amount of nterest credted by the fund durng 1999 s equal to 28.49. Calculate t. Soluton (a) Interest earned n 1999 s equal to the 1999 nterest rate multpled by the balance at January 01, 1999. 1996 Money 1997 Money 100(1.12)(1.12)(1 + t)(8%) 1998 Money 100(1.12)(1.05)(10%) = 11.76 100(1.08)(t.02) = 108(t.02) Total Interest = 10.0353(1 + t)+11.76 + 108(t.02) = 28.49 t =7.5% 127

5.5 Loans: The Amortzaton Method Exercse (a) You take out a loan for 2,000,000 that wll be dsbursed to you n three payments. The frst payment of 1,000,000 s made mmedately and s followed sx months later by a payment of 500,000 and then sx months after that by another payment of 500,000. The nterest on the payments s calculated at a nomnal rate of nterest of 26.66%, convertble sem-annually. After two years, you replace the outstandng loan wth a 30-year loan at a nomnal rate of nterest of 12%, convertble monthly. The amount of the monthly payments for the frst fve years on ths loan wll be one-half of the monthly payment requred after 5 years. Payments are to be made at the begnnng of each month. Calculate the amount of the 12th repayment. Soluton (a) FV =1, 000, 000(1.13333) 4 + 500, 000(1.13333) 3 + 500, 000(1.13333) 2 =3, 019, 493.52 3, 019, 493.52 = P ä 60 1% + v 60 1% 2P ä 300 1% Exercse (b) P =20, 000 A loan s amortzed over fve years wth monthly payments at a nomnal rate of nterest of 6%, convertble monthly. The frst payment s 1,000 and s to be pad one month after the date of the loan. Each succeedng monthly payment wll be 5% lower than the pror payment. Calculate the outstandng loan balance mmedately after the 40th payment s made. Soluton (b) Payment 41 =1, 000(.95) 40 = 128.51 j = 6% 12 =.005 ] O/SLoan 40 = v.005 [128.51 a 20 = 1+.005 1.95 1=.05789 = v.005 [128.51 (12.344)] = v.005 [1, 586.33] =1, 578.44 128

Exercse (c) You borrow 1,000 at an annual effectve nterest rate of 4% and agree to repay t wth 3 annual nstallments. The amount of each payment n the last 2 years s set at twce that n the frst year. At the end of 1 year, you have the opton to repay the entre loan wth a fnal payment of X, n addton to the regular payment. Ths wll yeld the lender an annual effectve rate of 4.5% over the 1-year perod. Soluton (c) 1, 000 = P v 1 4% +2P (v2 4% + v3 4% ) P = 1, 000 v 1 4% +2(v2 4% + v3 4% ) = 217.93 1, 000 = (217.93) v 1 4.5% + Xv1 4.5% X = 1, 000 (217.93) v1 4.5% v 1 4.5% = 827.07 Exercse (d) A loan s to be repad by annual nstallments of P at the end of each year for 10 years. You are gven: () The total prncpal repad n the frst 3 years s 290.35. () The total prncpal repad n the last 3 years s 408.55. Determne the total amount of nterest pad durng the lfe of the loan. Soluton (d) 290.35 = Pv 10 + Pv 9 + Pv 8 = Pv 7 (v 3 + v 4 + v 5 ) Dvdng the 2 equatons 408.55 = Pv 3 + Pv 4 + Pv 5 = P (v 3 + v 4 + v 5 ) 290.35 408.55 = v7 =5% P = 408.55 v 3 5% + v4 5% + v5 5% = 150.03 L = P a 10 5% =1, 158.44 Total Interest Pad=Total Payments-Loan = 10(150.03) 1, 158.44 = 341.76 129

5.6 Loans: The Snkng Fund Method Exercse (a) An nvestor wshes to take out a loan at an annual effectve nterest rate of 9% and then buy a 10-year annuty whose present value s 1,000 calculated at an annual effectve nterest rate of 8%. The loan s to be repad over the next 10 years wth annual end-of-year nterest payments. Annual end-of-year snkng fund deposts are also to be made that are credted at an annual effectve nterest rate of 7%. Soluton (a) Annual annuty payment s 1, 000 a 10 8% = 149.03 The annuty payment must cover the nterest payment and the snkng fund depost. Exercse (b) 149.03 = Prce s 10 7% + Prce 9% Prce= 149.03 1 s 10 7% +9% = 917.80 A 5% 10-year loan of 10,000 s to be pad by the snkng fund method, wth nterest and snkng fund payments made at the end of each year. The effectve rate of nterest earned n the snkng fund s 3% per annum. Immedately before the ffth payment s due, the lender requests that the loan be repad mmedately. Calculate the net amount of the loan at that tme. Soluton (b) Snkng fund depost s 10, 000 s 10 3% = 872.30 Snkng fund balance just before the ffth depost s to be made s 892.30s 4 3% (1.03) = 3, 758.86 Value of the loan just before the nterest payment s due s 10, 000(1.05) = 10, 500 Net amount of the loan at tme 5 s the value of the loan less the snkng fund balance s 10, 500 3, 786.86 = 6, 741.14 130

Exercse (c) You borrow 10,000 for 10 years at an annual effectve nterest rate of and accumulate the amount necessary to repay the loan by usng a snkng fund. Ten payments of X are made at the end of each year, whch ncludes nterest on the loan and the payment nto the snkng fund, whch earns an annual effectve rate of 8%. If the annual effectve rate of the loan had been 2, your total annual payment would have been 1.5X. Calculate. Soluton (c) X = 10, 000 + 1.5X =2 10, 000 + 10, 000 s 10 8% 10, 000 s 10 8% 0.5X = 10, 000 X =2 10, 000 2 10, 000 = 10, 000 + = 690.29 10, 000 =6.9% 10, 000 s 10 8% 131

6 Fnancal Instruments Overvew nterest theory s used to evaluate the prces and values of: 1. bonds 2. equty (common stock, preferred stock) ths chapter wll show how to: 1. calculate the prce of a securty, gven a yeld rate 2. calculate the yeld rate of a securty, gve the prce 6.1 Types of Fnancal Instruments There are seven types of common securtes avalable n the fnancal markets that are dscussed n ths chapter: 6.1.1 Money Market Instruments provde hgh lqudty and attractve yelds; some allow cheque wrtng contans a varety of short-term, fxed-ncome securtes ssued by governments and prvate frms credted rates fluctuate frequently wth movements n short-term nterest rates nvestors wll park ther money n an MMF whle contemplatng ther nvestment optons Treasury Blls (T-blls) Government ssued bonds are called: - Treasury Blls (f less than one year) - Treasury Notes (f n between one year and long-term) - Treasury Bonds (f long-term) Treasury Blls are valued on a dscount yeld bass usng actual/360 Queston: Fnd the prce of a 13-week T-bll that matures for 10, 000 and s bought at dscount to yeld 7.5%. Soluton: [ 10, 000 1 91 ] 360 (7.5%) =9, 810.42 Certfcate of Deposts (CD) rates are guaranteed for a fxed perod of tme rangng from 30 days to 6 months hgher denomnatons wll usually credt hgher rates of nterest yeld rates are usually more stable than MMF s but less lqud wthdrawal penaltes tend to encourage a secondary tradng market rather than cashng out 132

Commercal Paper 6.1.2 Bonds s an unsecured debt note that usually lasts one to two months, up to 270 days usually ssued by a large corporaton the face amount s most lkely n multples of $100,000 promse to pay nterest over a specfc term at stated future dates and then pay lump sum at the end of the term (smlar qualtes to an amortzed loan approach). ssued by corporatons and governments as a way to rase money (.e. borrowng). the end of the term s called a maturty date; some bonds can be repad at the dscreton of the bond ssuer at any redempton date (callable bonds). nterest payments from bonds are called coupon payments. bonds wthout coupons and that pay out a lump sum n the future are called accumulaton bonds or zero-coupon bonds. a bond that has a fxed nterest rate over the term of the bond s called fxed-rate bonds. a bond that has a fluctuated nterest rate over the term of the bond s called a floatng-rate bond. the rsk that a bond ssuer does not pay the coupon or prncpal payments s called default rsk. a mortgage bond s a bond backed by collateral; n ths case, by a mortgage on real property (more secure) an ncome bond pays coupons f company had suffcent funds; no threat of bankruptcy for mssedcouponpayment junk bonds have a hgh rsk of defaultng on payments and therefore need to offer hgher nterest rates to encourage nvestment a convertble bond can be converted nto the common stock of the company at the opton of the bond owner borrowers n need of a large amount money may choose to ssue seral bonds that have staggered maturty dates 6.1.3 Common Stock s an ownershp securty, lke preferred stock, but does not have fxed dvdends level of dvdend s determned by company s drectors common stock dvdends are pad after nterest payments for bonds and preferred stock are pad out varable dvdend rates means prces are more volatle than bonds and preferred stock 133

6.1.4 Preferred Stock provdes a fxed rate of return (smlar to bonds); called a dvdend ownershp of stock means ownershp of company (not borrowng) no maturty date for credtor purpose, preferred stock s second n lne, behnd bond owners (common stock s thrd) falure to pay dvdend does not result n default cumulatve preferred stock wll make up for any mssed dvdends; regular preferred stock does not have to convertble preferred stock gves the owner the opton of convertng to common stock under certan condtons 6.1.5 Mutual Funds pooled nvestment accounts; an nvestor buys shares n the fund offers more dversfcaton than what an ndvdual can acheve on ther own a money manager controls the nvestment; f the money manager trys to outbeat the market t s sad that the nvestment strategy s actve, f the money manager matches the ndex fund then the nvestment strategy s passve. 6.1.6 Guaranteed Investment Contracts (GIC) ssued by nsurance companes to large nvestors smlar to CD s; market value does not change wth nterest rate movements GIC mght allow for addtonal deposts and can offer nsurance contract features.e. annuty purchase optons nterest rates hgher than CD s; closer to Treasury securtes banks compete wth ther bank nvestment contracts (BIC) 134

6.1.7 Dervatve Sercurtes the value of a dervatve securty depends on the value of the underlyng asset n the marketplace Optons a contract that allows the owner to buy or sell a securty at a fxed prce at a future date call opton gves the owner the rght to buy; put opton gves the owner the rght to sell European opton can be used on a fxed date; Amercan opton can be used any tme untl ts expry date nvestors wll buy(sell) call optons or sell(buy) put optons f they thnk a securty s prce s gong to rse (fall) one motvaton for buyng or sellng optons s speculaton; opton prces depend on the value of the underlyng asset (leverage) another motvaton (and qute opposte to speculaton) s developng hedgng strateges to reduce nvestment rsk (see Secton 8.7 Short Sales) a warrant s smlar to a call opton, but has more dstnct expry dates; the ssung frm also has to own the underlyng securty a convertble bond may be consdered the combnaton of a regular bond and a warrant Futures ths s a contract where the nvestor agrees, at ssue, to buy or sell an asset at a fxed date (delvery date) at a fxed prce (futures prce) the current prce of the asset s called the spot prce an nvestor has two nvestment alternatves: () buy the asset mmedately and pay the spot prce now, or () buy a futures contract and pay the futures prce at the delvery date; earn nterest on the money deferred, but lose the opportunty to receve dvdends/nterest payments Forwards smlar to futures except that forwards are talored made between two partes (no actve market to trade n) banks wll buy and sell forwards wth nvestors who want protecton for currency rate fluctuatons for one year or longer banks also sell futures to nvestors who wsh to lock-n now a borrowng nterest rate that wll be appled to a future loan rsk s that nterest rates drop n the future and the nvestor s stuck wth the hgher nterest rate 135

Swaps a swap s an exchange of two smlar fnancal quanttes for example, a change n loan repayments from Canadan dollars to Amercan dollars s called a currency swap (rsk depends on the exchange rate) an nterest rate swap s where you agree to make nterest payments based on a varable loan rate (floatng rate) nstead of on a predetermned loan rate (fxed rate) Caps an nterest rate cap places an upper lmt on an nterest rate to protect the nvestor aganst ncreasng nterest rates. [ ] ndex rate-strke rate Cap payment = Max, 0 notatonal amount p where p s the amount of tmes per year nterest s pad Floors an nterest rate floor a lower lmt on an nterest rate to protect the nvestor aganst decreasng nterest rates. [ ] strke rate-ndex rate Floor payment = Max, 0 notatonal amount p where p s the amount of tmes per year nterest s pad 136

6.2 Bond Valuaton lke any loan, the prce(value) of a bond can be determned by takng the present value of ts future payments prces wll be calculated mmedately after a coupon payment has been made, or alternatvely, at ssue date f the bond s brand new letp represent the prce of a bond that offers coupon payments of (Fr) and a fnal lump sum payment of C. The present value s calculated as follows: P =(Fr) a n + Cv n F represents the face amount(par value) of a bond. It s used to defne the coupon payments that are to be made by the bond. C represents the redempton value of a bond. Ths s the amount that s returned to the bond-holder(lender) at the end of the bond s term (.e. at the maturty date). r represents the coupon rate of a bond. Ths s used wth F to defne the bond s coupon payments. Ths nterest rate s usually quoted frst on a nomnal bass, convertble semannually snce the coupon payments are often pad on a sem-annual bass. Ths nterest rate wll need to be converted before t can be appled. (Fr) represents the sem-annual coupon payment of a bond. n s the number of coupon payments remanng or the tme untl maturty. s the bond s yeld rate or yeld-to-maturty. It s the IRR to the bond-holder for acqurng ths nvestment. Recall that the yeld rate for an nvestment s determned by settng the present value of cash-flows-n (the purchase prce, P ) equal to the present value of the cashflows-out (the coupon payments, (Fr), and the redempton value, C). there are three formulas that can be used n order to determne the prce of a bond: () Basc Formula () Premum/Dscount Formula () Base Amount Formula () Basc Formula P =(Fr) a n + Cv n As stated before, a bond s prce s equal to the present value of ts future payments. () Premum/Dscount Formula P =(Fr) a n + Cv n =(Fr) a n + C (1 a n ) = C +(Fr C) a n If we let C loosely represent the loan amount that the bond-holder gets back, then (Fr C) represents how much better the actual payments, Fr, are relatve to the expected nterest 137

payments, C. When (F r C) > 0 the bond wll pay out a superor nterest payment than what the yeld rate says to expect. As a result, the bond-buyer s wllng to pay(lend) a bt more, P C, than what wll be returned at maturty. Ths extra amount s referred to as a premum. On the other hand, f the coupon payments are less than what s expected accordng to the yeld rate, (Fr C) < 0, then the bond-buyer won t buy the bond unless t s offered at a prce less than C or, n other words, at a dscount. () Base Amount Formula Let G represent the base amount of the bond such that f multpled by the yeld rate, t would produce the same coupon payments that the bond s provdng: G = Fr G = Fr. The prce of the bond s then calculated as follows: P =(Fr) a n + Cv n =(G) a n + C (1 a n ) = G (1 v n )+Cv n = G +(C G) v n If we now let G loosely represent the loan, then the amount that the bond-holder receves at maturty, n excess of the loan, would be a bonus. As a result, the bond becomes more valuable and a bond-buyer would be wllng to pay a hgher prce than G. On the other hand, f the payout at maturty s perceved to be less than the loan G, then the bond-buyer wll not purchase the bond unless the prce s less than the loan amount. Determnaton Of Yeld Rates gven a purchase prce of a securty, the yeld rate can be determned under a number of methods. Problem: What s the yeld rate convertble sem-annually for a 100 par value 10-year bond wth 8% sem-annual coupons that s currently sellng for 90? Soluton: 1. Use an Socety of Actuares recommended calculator wth bult n fnancal functons: PV =90, N =2 10 = 20, FV = 100, PMT = 8% 2 100 = 4, CPT % 4.788% 2=9.5676% 2. Do a lnear nterpolaton wth bond tables (not a very popular method anymore). 138

Premum And Dscount the redempton value, C, loosely represents a loan that s returned back to the lender after a certan perod of tme the coupon payments, (Fr), loosely represent the nterest payments that the borrower makes so that the outstandng loan does not grow let the prce of a bond, P, loosely represent the orgnal value of the loan that the bondbuyer(lender) gves to the bond-ssuer(borrower) the dfference between what s lent and what s eventually returned, P C, wll represent the extra value (f P C>0) or the shortfall n value (f P C<0) that the bond offers the bond-buyer s wllng to pay(lend) more than C f he/she perceves that the coupon payments, (Fr), are better than what the yeld rate says to expect, whch s (C). the bond-buyer wll pay less than C f the coupon payments are perceved to be nferor to the expected nterest returns, Fr< C. the bond s prced at a premum f P > C (or Fr > C)oratadscountfP < C (or Fr < C). recall the Premum/Dscount Formula from the pror secton: P = C +(Fr C) a n P C =(Fr C) a n =(Cg C) a n = C (g ) a n n other words, f the modfed coupon rate, g, s better than the yeld rate,, the bond sells at a premum. Otherwse, f g<, then the bond wll have to be prced at a dscount. g = Fr C represents the true nterest rate that the bond-holder enjoys and s based on what the lump sum wll be returned at maturty represents the expected nterest rate (the yeld rate) and depends on the prce of the bond the yeld rate,, s nversely related to the prce of the bond. f the yeld rate,, s low, then the modfed coupon rate, g, looks better and one s wllng to pay a hgher prce. f the yeld rate,, s hgh, then the modfed coupon rate, g, s not as attractve and one s not wllng to pay a hgher prce (the prce wll have to come down). 139

Example LetthereexstabondsuchthatC = 1. g(fr = Cg = g). The coupon payments are therefore equal to Let the prce of the bond be denoted as 1 + p where p s the premum (f p>0) or the dscount (f p<0). P =(Fr) a n + Cv n 1+p = g a n +(1)v n = g a n +1 a n =1+(g ) a n Amortzaton of Premum and Accrual of Dscount Interest Earned: I t usng the prospectve method for evaluatng the value (current prce) of the bond, the nterest payment s derved as follows: I t = BV t 1 [ ] = g a n (t 1) +(1)v n (t 1) ( = g 1 v n (t 1) ) = g (g ) v n (t 1) + v n (t 1) therefore, each coupon payment, g, s ntended to represent a perodc nterest payment plus(less) a return of porton of the premum(dscount) that was made at purchase. I t = g (g ) v n (t 1) g = I t +(g ) v n (t 1) Premum Amortzaton: PA t the premum amortzaton n ths case s equal to the coupon payment less the nterest payment, g I t : PA t = g I t Other Premum Amortzaton Formulas = g [g (g ) v n (t 1) ] PA t =(g ) v n (t 1) PA t = BV t 1 BV t Amortzaton of Premum over the t-th perod=pa t Accumulaton of Dscount over the t-th perod= PA t 140

Book Value Of The Bond the bond s value starts at prce P (or 1 + p, n our example) and eventually, t wll become value C (or 1, n our example) at maturty. ths value, or current prce of the bond, at any tme between ssue date and maturty date can be determned usng the prospectve approach: BV t =(Fr) a n t + Cv n t 1+p = g a n t +(1)v n t = g a n t +1 a n t =1+(g ) a n t or t can be determned usng the retrospectve approach: BV t = P (1 + ) t Frs t = P (1 + ) t Cgs t when ths asset s frst acqured by the bond-buyer, ts value s recorded nto the accountng records, or the books, at ts purchase prce. Snce t s assumed that the bond wll now be held untl maturty, the bond s future value wll contnue to be calculated usng the orgnal expected rate of return (the yeld rate that was used at purchase). the current value of the bond s often referred to as a book value. the value of the bond wll eventually drop or rse to value C dependng f t was orgnally purchased at a premum or at a dscount. f Book Value Formulas BV t = Cga n t + Cv n t t = P (1 + t) t + Cgs t = BV t 1 (1 + ) Cg = BV t 1 + I t Cg = BV t 1 PA t = BV t 1 Amortzaton of Premum = BV t 1 + Accumulaton of dscount 141

Bond Amortzaton Schedule an amortzaton schedule for ths type of loan can also be developed that wll show how the bond s beng wrtten down, f t was purchased at a premum, or how t s beng wrtten up, f t was purchased at a dscount. the followng bond amortzaton schedule llustrates the progresson of the coupon payments when C = 1 and when the orgnal prce of the bond was 1 + p =1+(g ) a n : Perod (t) Coupon Interest Earned I t Amorzaton of Premum PA t Book ValueBV t 1 g g (g ) v n (g ) v n 1+(g ) a n 1 2 g g (g ) v n 1 (g ) v n 1 1+(g ) a n 2..... t g g (g ) v n (t 1) (g ) v n (t 1) 1+(g ) a n t..... n 1 g g (g ) v 2 (g ) v 2 1+(g ) a 1 n g g (g ) v 1 (g ) v 1 Total n g n g (g )a n (g ) a n = p 142

note that the total of all the nterest payments s represented by the total of all coupon payments less what s beng returned as the premum (or plus what s beng removed as dscount snce the loan s apprecatng to C). n I k = k=1 = n k=1 g (g ) v n (t 1) n g k=1 n (g ) v n (k 1) k=1 = ng (g )a n = ng p note that the total of all the prncpal payments must equal the premum/dscount n n P k = (g )v n (k 1) =(g )a n = p k=1 k=1 note that the book value at t = n s equal to 1, the last and fnal payment back to the bond-holder. ( ) note that the prncpal (premum) repayments ncrease geometrcally by 1+ (2) 2 ( ) n..e. PA t+n = PA t 1+ (2) 2 remember that the above example s based on a redempton value of C =1. Theabove formulas need to be multpled by the actual redempton value f C s not equal to 1 143

Straght Lne Method If an approxmaton for wrtng up or wrtng down the bond s acceptable, then the prncpal payments can be defned as follows: or PA t = 1+p 1 n = p,fc =1. n PA t = BV 0 BV n. n I t wll then be the coupon payment less the premum amortzaton: or I t = g p,fc =1. n I t =(Cg) PA t =(Cg) BV 0 BV n. n The book value of the bond, BV t, wll then be equal the orgnal prce less the sum of the premum repayments made to date: t ( ) BV0 BV n BV t = BV 0 PA k = BV 0 t = BV 0 tp A t. n k=1 or Prce t =1+p t p ( ) n t n =1+ p, fc =1. n 144

The bond amortzaton table would then be developed as follows: Perod (t) Payment I t P t Book Value t 1 g g p n 2 g g p n p n p n 1+ ( ) n 1 n p 1+ ( ) n 2 n p..... t g g p n p n 1+ ( ) n t n p..... n 1 g g p n n g g p n Total ng ng p p n 1+ ( 1 n ) p p n 1 p n n = p Callable Bonds ths s a bond where the ssuer can redeem the bond pror to the maturty date f they so chooseto;thsscalledacall date. the challenge n prcng callable bonds s tryng to determne the most lkely call date assumng that the redempton date s the same at any call date, then () the call date wll most lkely be at the earlest date possble f the bond was sold at a premum (ssuer would lke to stop repayng the premum va the coupon payments as soon as possble). () the call date wll most lkely be at the latest date possble f the bond was sold at a dscount (ssuer s n no rush to pay out the redepton value). when the redempton date s not the same at every call date, then one needs to examne all possble call dates. 145

Example A 100 par value 4% bond wth sem-annual coupons s callable at the followng tmes: 109.00, 5 to 9 years after ssue 104.50, 10 to 14 years after ssue 100.00, 15 years after ssue. Queston: What prce should an nvestor pay for the callable bond f they wsh to realze a yeld rate of (1) 5% payable sem-annually and (2) 3% payable sem-annually? Soluton: (1) Snce the market rate s better than the coupon rate, the bond would have to be sold at a dscount and as a result, the ssuer wll wat untl the last possble date to redeem the bond: P =2.00 a 30 2.5% + $100.00 v 30 2.5% =89.53 (2) Snce the coupon rate s better than the market rate, the bond would sell at a premum and as a result, the ssuer wll redeem at the earlest possble date for each of the three dfferent redempton values: P =2.00 a 10 1.5% + 109.00 v1.5% 10 = 112.37 P =2.00 a 20 1.5% + 104.50 v1.5% 20 = 111.93 P =2.00 a 30 1.5% + 100.00 v1.5% 30 = 112.01 In ths case, the nvestor would only be wllng to pay 111.93. Note that the excess of the redempton value over the par value s referred to as a call premum and starts at 9.00, before droppng to 4.50, before droppng to 0.00. 146

Bond Prce Between Coupon Payment Dates up to now, bond prces and book values have been calculated assumng the coupon has just been pad. letp t be the bond prce (book value) just after a coupon payment has been made. P t = Fr.a n t + Cv n t = B t 1 (1 + ) Fr when buyng an exstng bond between ts coupon dates, one must decde how to splt up the coupon between the pror owner and the new owner. letfr k represent the amount of coupon (accrued coupon) that the pror owner s due where 0 <k<1 the prce of the bond (full prce) to be pad to the pror owner at tme t + k should be based on the clean prce of the bond and the accrued nterest (accrued coupon) P t+k = Clean Prce + Accrued Interest Accrued Interest notce that the clean prce of the bond, wll only recognze future coupon payments; hence, the reason for a separate calculaton to account for the accrued coupon, Fr k. k maybecalculatedonanactual/actual or 30/360bassfdaysaretobeused. AI = coupon k = Fr k the number of days between the last coupon payment date and settlement date where k= total number days between coupon payment dates Full Prce between Coupon Payments Dates full prce s equal to the bond value as at the last coupon payment date, carred forward wth compound nterest. or the full prce s equal to the sum of the next coupon Fr, and the prce of the next coupon P t+1, all dscounted back to the settlement date. P t+k = P t (1 + ) k = P t+k =(P t+1 + Fr)v (1 k) Clean Prce of Bond between Coupon Payment Dates Clean prce = P t+k AI = P t (1 + ) k (coupon k) = P t (1 + ) k (Fr k) 147

6.3 Stock Valuaton Common Stock ssued by corporatons.e. borrowng, but not payng back the prncpal. pays out annually a dvdend, Dv, rather than nterest wth no requrement to guarantee payments. Also the dvdend can be n any amount (not fxed ncome). an assumpton s requred wth respect to the annual growth rate of dvdends, k. Prce s equal to the present value of future dvdends at a gven yeld rate r and as gven growth rate, g. General DCF Stock Valuaton Formula Level Dvdend Stock Valuaton Formula PVstock = t=1 dv t (1 + r) t PVstock = dv 1 r Constant Dvdend Growth the technques to be used are exactly the same as those methods presented n Secton 3.6, Compound Increasng Annutes. ) PV = v r (Dv ä r = 1+r = Dv 1+g 1 r g Example Assumng an annual effectve yeld rate of 10%, calculate the prce of a common stock that pays a 2 annual dvdend at the end of every year and grows at 5% for the frst 5 years, 2.5% for the next 5 years and 0%, thereafter: ( ) P = v 10% 2 ä 5 r = 1+10% Prce to Earnng (P/E) rato [v 10% ( 2(1 + 5%) 5 ä 5 r = 1+10% 1+5% 1 )] + v10% 5 1+2.5% ( 1 + v10% [v 10% 2(1 + 5%) 5 (1 + 2.5%) 5 ä r = 1+10% 1+0% 1 P =8.30 + 6.29 + 11.13 = 25.72 )] P/Erato = stock prce per share earnngs per share = P 0 EPS where earnngs per share = net ncome number of outstandng shares 148

Exercses and Solutons 6.2 Bond Valuaton Exercse (a) Youaregventwon year 1000 par value bonds. Bond X has 14% semannual coupons and a prce of 1407.70, to yeld, compounded semannually. Bond Y has 12% semannual coupons and a prce of 1271.80, to yeld the same rate, compounded semannually. Calculate the prce of Bond X to yeld 1%. Soluton (a) ( 1, 407.70 = 1, 000 + 1, 000 7% ) ( a 2 2n 2.40770 = 7% ) a 2 2n 2 ( 1, 271.80 = 1, 000 + 1, 000 6% ) ( a 2 2n 2.27180 = 6% ) a 2 2n 2.40770.27180 = 7% 2 6% 2 2 =.04 =.08.40770 = (7% 4%)a 2n 4% 2n =20 ( P =1, 000 + 1, 000 7% ) 8% 1% a 2 20 8% 1% 2 =1, 497.43 Exercse (b) A bond wth a par value of 1,000 and 6% semannual coupons s redeemable for 1,100. You are gven the followng nformaton. () The bond s purchased a P to yeld 8%, convertble semannually. () The amount of the dscount adjustment from the 16th coupon payment s 5. Calculate P. Soluton (b) Fr BV 15 = 5 30 (4%)BV 15 = 5 BV 15 = 875 BV 15 = P (1 + ) 15 Fr s 15 875 = P (1.04) 15 30s 15 4% P = 819.41 149

Exercse (c) A 700 par value fve-year bond wth 10% semannual coupons s purchased for 670.60. The present value of the redempton value s 372.05. Calculate the redempton value. Soluton (c) P = K + Fra n P K = Fr a n 670.60 372.05 = 35a 10 = 298.55 =3% Exercse (d) 372.05 = C v 10 3% C = 500 You buy an n year 1,000 par value bond wth 6.5% annual coupons at a prce of 825.44, assumng an annual yeld rate of, >0. After the frst two years, the bond s book value has changed by 23.76. Calculate. Soluton (d) BV 2 BV 0 = BV 0 (1 + ) 2 Fr(1 + ) Fr BV 0 23.76 = 825.44(1 + ) 2 65(1 + ) 65 825.44 0 = } 825.44 {{ } (1 + ) 2 }{{} 65 (1 + ) 914.20 } {{ } a b c 1+ = ( 65) + (65) 2 4(825.44)( 914.20) 2(825.44) =1.0925 =9.25% 150

Exercse (e) A 30-year 10,000 bond that pays 3% annual coupons matures at par. It s purchased to yeld 5% for the frst 15 years and 4% thereafter. Calculate the premum or dscount adjustment for year 8. Soluton (e) ( ) Fr BV 7 = 300 (5%) 300a 8 5% + v5% 8 300a 15 4% +10, 000v5% 8 v15 4% = 300 (5%)(7954.82) = 300 397.74 Exercse (f) = 97.74 Among a company s asset and accountng records, an actuary fnds an n year annual coupon-payng bond that was purchased at a dscount. From the records, the actuary has determned the followng. () the amount for amortzaton of the dscount n the (n 12th) coupon payment and (n 9th) coupon payment were 850 and 984, respectvely () the dscount s equal to 18,117. What s the value of n. Soluton (f) P 1 = P n 9 P n 12 = 984 850 =(1+)3 =5% P n 12 (1 + ) = 850 n 12 1 (1 + ) = 850 n 12 1 (1.05) n 13 dscount = P 1 + P 2 + P 3 +... + P n 18, 117 = P 1 +P 1 (1.05)+P 1 (1.05) 2 +...+P 1 (1.05) n 1 = P 1 [1+(1.05)+(1.05)5 2 +...+(1.05) n 1 ] 18, 117 = 850 (1.05 n 13[s n 5% ]= 850vn 5% (1.05) 13[s n 5% ] = 850(1.05) 13 [a n 5% ] a n 5% =11.3033 n =17.07 151

Exercse (g) A par value 10-year bond wth 10% annual coupons s bought at a premum to yeld an annual effectve rate of 8% for the frst 5 years and 6%, thereafter. The nterest porton of the 9th coupon s 12,880. Calculate the par value. Soluton (g) I 9 =6% BV 8 [ ] 12, 880 = 6% F (10%)a 2 6% + Fv6% 2 12, 880 = 6% F [1.07335707] Exercse (h) F = 200, 000 A machne costs P. At the end of 10 years, t wll have a salvage value of 0.28P. The book value at the end of 5 years usng the snkng fund method at an annual effectve nterest rate of 6% s X. The book value at the end of 5 years usng the straght lne method s Y. You are gven X Y =650.92. Calculate P. Soluton (h) (P.28P ) BV 5 = P s 10 6% s 5 6% = X BV 5 = P 5(P.28P ) 10 (P.28P ) P ( s 10 6% s 5 6% P 5(P.28P ) 10 = Y ( 1 (P.28P ) 2 s ) 5 = 650.92 s 10 P =12, 500 ) = 650.92 152

6.3 Stock Valuaton Exercse (a) You and a frend each sell a dfferent stock short for a prce of 1,000. The margn requrement s 60% of the sellng prce and the nterest on the margn account s credted at an annual effectve rate of 6%. You buy back your stock one year later at a prce of P. At the end of the year, the stock pad a dvdend of X. Your frend also buys back ther stock one year later but at a prce of (P 25). At the end of the year, ther stock pad a dvdend of 2X. Both you and your frend earn an annual effectve yeld rate of 20% on your sales. Calculate P. Soluton (a) Margn (I) = 1, 000(60%) = 600 Interest on Margn = 600(6%) = 36 Dvdend = X Gan on Sale = 1, 000 P Yeld (I) = 1, 000 P +36 X 600 = 20% X = 916 P Margn (II) = 1, 000(60%) = 600 Interest on Margn = 600(6%) = 36 Dvdend = 2X Gan on Sale = 1, 000 (P 25) Yeld (II) = 1, 000 P +25+36 2X 600 = 1061 P 2(916 P ) 600 = 20% P = 891 153

Exercse (b) You and a frend each sell a dfferent stock short for a prce of 1,000. The margn requrement s 50% of the sellng prce and the nterest on the margn account s credted at an annual effectve rate of 6%. You buy back your stock one year later at a prce of P. At the end of the year, the stock pad a dvdend of X. Your frend also buys back ther stock one year later but at a prce of (P 25). At the end of the year, ther stock pad a dvdend of 2X. Both you and your frend earn an annual effectve yeld rate of 20% on your sales. Calculate X. Soluton (b) Margn (I) = 1, 000(50%) = 500 Interest on Margn = 500(6%) = 30 Dvdend = X Gan on Sale = 1, 000 P Yeld (I) = 1, 000 P +30 X 500 = 20% X = 930 P Margn (II) = 1, 000(50%) = 500 Interest on Margn = 500(6%) = 30 Dvdend = 2X Gan on Sale = 1, 000 (P 25) Yeld (II) = 1, 000 P +25+30 2X 500 = 1055 P 2(930 P ) 500 = 20% X =25 154

Exercse (c) A common stock s purchased on January 1, 1992. It s expected to pay a dvdend of 15 per share at the end of each year through December 31, 2001. Startng n 2002 dvdends are expected to ncrease K% per year ndefntely, K<8%. The theoretcal prce to yeld an annual effectve rate of 8% s 200.90. Calculate K. Soluton (c) Exercse (d) ( ) 15 200.90 = 15a 9 8% +(1.08) 9.08 K K =.01 Steven buys a stock for 200 whch pays a dvdend of 12 at the end of every 6 months. Steven deposts the dvdend payments nto a bank account earnng a nomnal nterest rate of 10% convertble semannually. At the end of 10 years, mmedately after recevng the 20th dvdend payment of 12, Steven sells the stock. the sale prce assumes a nomnal yeld of 8% convertble semannually and that the semannual dvdend of 12 wll contnue forever. Steven s annual effectve yeld over the 10-year perod s. Calculate. Soluton (d) AV of dvdends =12s 20 5% = 396.79 P = 12.04 = 300 200(1 + ) 10 = 396.79 + 300 (1 + ) 10 =3.4840 =13.29% 155

7 Duraton, Convexty and Immunzaton Overvew there are two types of bond senstvty, they are duraton and convexty. modfed, Macaulay, and effectve are three types of convexty and duraton. the present value of a company s assets mnus the present value of a companes labltes s called a surplus. when a company s assets and labltes are protected from nterest rate fluctuatons, ths s called mmunzaton. when matchng asset cash flows to lablty cash flows for the protecton of the surplus, ths s called dedcaton. 7.1 Prce as a Functon of Yeld examnng the prce curve, prce falls as the yeld ncreases prce s calculated as follows: P = t>0 ( CF t 1+ y ) mt m where P =asset prce, y=yeld (e. y = (m) )andcf t =cash flow at tme t years the estmated percentage change n a bond s prce s approxmately: 7.2 Modfed Duraton Modfed duraton s calculated as follows %ΔP (Δy) P (y) P (y) wherep s: ModD = P (y) P (y) P = ( CF t 1+ y ) mt m dp dy = tcf t (1+ y ) ( mt 1 tcft 1+ y = ( ) m m 1+ y m ) mt ModD = dp ( dy P = tcft 1+ y m P ( 1+ y ) m ) mt 156

Modfed Duraton ModD = t=0 tcf ( t 1+ y m P ( 1+ y ) m ) mt Relatonshp between prce and modfed duraton where one hundred bass ponts equals 1.0% 7.3 Macaulay Duraton Macaulay Duraton s calculated as follows: Solvng for Macaulay Duraton: %ΔP (Δy)(ModD) MacD = P ( ) (δ) dp 1 P (δ) = dδ P P = CF t e δt MacD = dp dδ = t CF t e δt ( ) dp 1 t dδ P = CFt e δt CFt e δt ( ) t CFt e δt t CFt 1+ y mt m t CFt v t MacD = = = P P P Relatonshp between Macaulay Duraton and Modfed Duraton ModD = ( MacD ) 1+ y m 157

Example of Duraton: Assumng an nterest rate of 8%, calculate the duraton of: (1) An n-year zero coupon bond: MacD = n t v t CF t t=n = n v t CF t t=n (2) An n-year bond wth 8% coupons: n v n CF n v n CF n = n MacD = n t v t CF t t=1 = n v t CF t t=1 8% (Ia) n +10v n 8% a n + v n (3) An n-year mortgage repad wth level payments of prncpal and nterest: MacD = n t v t CF t t=1 = n v t CF t t=1 (Ia) n a n (4) A preferred stock payng level dvdends nto perpetuty: MacD = t v t CF t t=1 = v t CF t t=1 (Ia) a Macaulay Duraton for Bonds Prced at Par MacD =ä (m) n 158

7.4 Effectve Duraton for bonds that do not have fxed cash flows, such as callable bonds, one can use effectve duraton Effectve Duraton EffD = P P + P 0 (2Δy) where P 0 s the current prce of a bond, P + s the bond prce f nterest rates shft up by Δy, and P s the bond prce f nterest rates shft down by Δy Relatonshp between Prce and Effectve Duraton 7.5 Convexty %ΔP (Δy)(EffD) convexty s descrbed as the rate of change n nterest senstvty. It s desrable to have postve (negatve) changes n the asset values to be greater (less) than postve (negatve) changes n lablty values. If the changes were plotted on a curve aganst nterest changes, you d lke the curve to be convex. convexty s calculated as follows: solvng for convexty: Convexty = P (y) P (y) P = ( CF t 1+ y ) mt m d 2 P dy 2 Convexty = d2 P dy 2 dp dy = t CF t (1+ y ) mt 1 m = ( t t + 1 ) ( CF t 1+ y ) mt 2 m m = ( t t + 1 ) ( CF t 1+ y ) mt 2 m m Relatonshp between Prce, Duraton, and Convexty %ΔP (Δy)(Duraton) + (Δy)2 (Convexty) 2 159

7.5.1 Macaulay Convexty solvng for Macaulay convexty: P = CF t e δt dp dδ = t CF t e δt d 2 P dδ 2 = t 2 CF t e δt Dsperson MacC = t2 CF t e δt P MacC = Dsperson + MacD 2 Dsperson = (t MacD)2 CF t e δt CFt e δt 7.5.2 Effectve Convexty EffC = (P + + P 2P 0 ) (Δy) 2 P 0 7.6 Duraton, Convexty and Prces: Puttng t all Together 7.6.1 Revstng the Percentage Change n Prce The general formula: %ΔP (Δy)(Duraton) + (Δy)2 (Convexty) 2 one can calculate for %ΔP usng modfed duraton and convexty when they have the same frequency as the yeld that s shfted modfed duraton and convexty can be used only when the bond s cash flows are fxed one can calculate for %ΔP usng Macaulay duraton and Macaulay convexty when the yeld s contnuously compoundng Macaulay duraton and Macaulay convexty can be used only when the bond s cash flows are fxed 160

one can calculate for %ΔP usng effectve duraton and effectve convexty when they and Δy have the same frequency as the yeld that s shfted and when the yeld s contnuously compoundng effectve duraton and effectve convexty can be used when the bond s cash flows are fxed and when they are not fxed 7.6.2 The Passage of Tme and Duraton There are two opposte effects that occur as a bond ages: duraton decreases wth tme snce the tmng for each payment decreases duraton ncreases wth tme as more weght s gven to the more dstant cash flows (e. t wll spke as you geet closer to a cash flow) The overall effect s that duraton wll decrease. 7.6.3 Portfolo Duraton and Convexty The duraton of the portfolo s: = P 1 D 1 + P 2 D 2 +... + P n D n MV port MV port MV port where n s how many bonds are n the portfolo, P k s the prce, D k s the duraton and MV port s the market value of the portfolo The convexty of the portfolo: where C k s the bond s convexty = P 1 C 1 + P 2 C 2 +... + P n C n MV port MV port MV port 161

7.7 Immunzaton t s very dffcult for a fnancal enterprse to match the cash flows of ther assets to the cash flows of ther labltes. especally when the cash flows can change due to changes n nterest rates. Surplus S(y) =PV A PV L where y s the nterest rate, PV A s the present value of the assets and PV L s the present value of the labltes A Problem wth Interest Rates a bank ssues a one-year depost and guarantees a certan rate of return. f nterest rates have gone up by the end of the year, then the depost holder wll not renew f the guaranteed rate s too low versus the new nterest rate. the bank wll need to pay out to the depost holder and f the orgnal proceeds were nvested n long-duraton assets ( gong long ), then the bank needs to sell off ts own assets (that have declned n value) n order to pay. f nterest rates have gone down by the end of the year, then t s possble that the backng assets may not be able to meet the guaranteed rate; ths becomes a greater possblty wth short-duraton assets ( gong short ). the bank may have to sell off some ts assets to meet the guarantee. A Soluton to the Interest Rate Problem structure the assets so that ther cash flows move at least the same amount as the labltes cash flows move when nterest rates change. leta t and L t represent the cash flows at tme t from an nsttuton s assets and labltes, respectvely. letr t represent the nsttuton s net cash flows at tme t such that R t = A t L t. n n fp () = v t R t = v t (A t L t ), then we would lke the present value of asset cash t=1 t=1 flows to equal the present value of lablty cash flows.e. P () =0: n n v t A t = v t L t t=1 we d also lke the nterest senstvty (modfed duraton) of the asset cash flows to be equal to the nterest senstvty of the labltes.e. P () P () =0 P () =0. MacD A = n t v t A t t=1 = n v t A t t=1 t=1 n t v t L t t=1 n v t L t t=1 = MacD L 162

n addton, we d also lke the convexty of the asset cash flows to be equal to the convexty of the labltes.e. P () P () =0 P () > 0. an mmunzaton strategy strves to meet these three condtons. recall that n Secton 7.2, modfed duraton was determned by takng the 1 st dervatve of the present value of the payments: ( n ) v t R t ModD = d d t=1 = n v t R t t=1 MacD 1+ we are now also nterested n how senstve the volatlty tself s called convexty. We determne convexty by takng the 1 st dervatve of v: ( d 2 n ) Convexty = d ( ) MacD d 2 v t R t t=1 = d 1+ n v t R t note that the forces that control lablty cash flows are often out of the control of the fnancal nsttuton. as a result, mmunzaton wll tend to focus more on the structure of the assets and how to match ts volatlty and convexty to that of the labltes. Immunzaton s a three-step process: (1) the present value of cash nflows (assets) should be equal to the present value of cash outflows (labltes). (2) the nterest rate senstvty of the present value of cash nflows (assets) should be equal to the nterest rate senstvty of the present value of cash outflows (labltes). (3) the convexty of the present value of cash nflows (assets) should be greater than the convexty of the present value of cash outflows(labltes). In other words, asset growth (declne) should be greater (less) than lablty growth(declne). t=1 Dffcultes/Lmtatons of Immunzaton (a) choce of s not always clear. (b) doesn t work well for large changes n. (c) yeld curve s assumed to change wth Δ; actually, short-term rates are more volatle than long-term rates. (d) frequent rebalancng s requred n order to keep the modfed duraton of the assets and labltes equal. 163

(e) exact cash flows may not be known and may have to be estmated. (f) convexty suggests that proft can be acheved or that arbtrage s possble. (g) assets may not have long enough maturtes of duraton to match labltes. 164

Example: Abanksrequredtopay1, 100 n one year. There are two nvestment optons avalable wth respect to how mones can be nvested now n order to provde for the 1, 100 payback: () a non-nterest bearng cash fund, for whch x wll be nvested, and () a two-year zero-coupon bond earnng 10% per year, for whch y wll be nvested. Queston: based on mmunzaton theory, develop an asset portfolo that wll mnmze the rsk that lablty cash flows wll exceed asset cash flows. Soluton: t s desrable to have the present value of the asset cash flows equal to that of the lablty cash flows: x + y(1.10) 2 v 2 = 1100v 1 t s desrable to have the modfed duraton of the asset cash flows equal to that of the lablty cash flows so that they are equally senstve to nterest rate changes: ( ) x 0 + y ( ) 2 = 1 x + y 1+ x + y 1+ 1+ 2y x + y =1 t s desrable for the convexty of the asset cash flows to be greater than that of the labltes: ( x + y(1.10)2 v 2 ) ( ) 1100v 1 d 2 d 2 x + y(1.10) 2 v 2 y(1.10) 2 ( 2)( 3) v 4 x + y(1.10) 2 v 2 > d 2 d 2 1100v 1 > 1100( 2)v3 1100v 1 f an effectve rate of nterest of 10% s assumed, then: } x + y(1.10)2 = 1100 1.10 2 1.1 x + y = 1000 x = 500,y= 500 2y x+y =1 and the convexty of the assets s greater than convexty of the labltes: 500(1.10) 2 ( 2)( 3) v 4 10% 500 + 500(1.10) 2 v 2 10% > 1100( 2)v3 10% 1100v 1 10% 2.479 > 1.653 the nterest volatlty (modfed duraton) of the assets and lablty are: - ModD A = ( ) x x + y ( ) ( ) 0 y + 1+ x + y ModD L = ( ) ( ) 2 500 = 1+ 1000 ( ) ( ) 1 1 = =.90909 1+ 1.1 ( ) 2 =.90909 1.1 165

7.8 Full Immunzaton Redngton mmunzaton can be used when there are small shfts n a flat yeld curve n order to protect the surplus full mmunzaton can be used when there are small or large shfts n a flat yeld curve n order to protect the surplus a fully mmunzed poston fulfls all condtons of Redngton mmunzaton Condtons For Full Immunzaton of a Sngle Lablty Cash Flow (1) Present value of assets=present value of lablty (2) Duraton of assets=duraton of lablty (3) The asset cash flows occur before and after the lablty cash flow. That s: (T q) < T<(T + r) For a fully mmunzaton poston: MacD L = MacD A = T Dsperson L = [T T ]2 L e δt L e δt =0 Dsperson A = [(T q) T ]2 Q e δ(t q) +[(T + r) T ] 2 δ(t +r) R e Q δ(t q) + R e δ(t r) = q2 Qe δ(t q) + r 2 δ(t +r) R e Q e δ(t q) + R e δ(t +r) Dsperson A > 0 MacC L =0+T 2 = T 2 MacC A = Dspeson A + T 2 MacC A >MacC L 166

Exercses and Solutons 7.2 Modfed Duraton Exercse (a) The annual effectve yeld on a bond s 6%. A 5 year bond pays annual coupons of 7%. Calculate the modfed duraton of the bond. Soluton (a) The prce of the bond s: CF t =(0.07)(100) = 7 P = ( CF t 1+ y ) mt m =7v +7v 2 +7v 3 +7v 4 +7v 5 + 100v 5 =7(v + v 2 + v 3 + v 4 + v 5 ) + 100v 5 =7a 5 6% + 100v 5 The prce of modfed duraton: = 104.212 ModD = ( t CFt 1+ y m P ( 1+ y ) m ) mt = 1 7v 6% +2 7v 2 6% +3 7v3 6% +4 7v4 6% +5 7v5 6% +5 100v5 6% 104.212(1.06) = 7(Ia) 5 6% + 5(100)v 5 6% 104.212(1.06) =4.152 167

Exercse (b) The current prce of a bond s 116.73 and the current yeld s 5%. The modfed duraton of the bond s 8.14. Use the modfed duraton to estmate the prce of the bond f the yeld ncreases to 6.30%. Soluton (b) %ΔP ( Δy)(ModD) ( 0.013)(8.14) The bond prce s: (.10582) 116.73(1 0.10582) = 104.378 Exercse (c) A zero coupon bond matures n 10 years for 3,000. The bonds yeld s 3% compounded sem-annually. Calculate the modfed duraton of the bond. Soluton (c) ModD = t t CF ( t 1+ y m P ( 1+ y ) m ) mt 10(3, 000)(1.015) 20 3, 000(1.015) 20 (1.015) = 10 1.015 =9.852 168

Exercse (d) A two year bond has 6% annual coupons pad sem-annually. The bond s yeld s 8% semannually. Calculate the modfed duraton of the bond. Soluton (d) ModD = ( t CFt 1+ y m P ( 1+ y ) m ) mt The sem-annual coupons are 0.03(100) = 3 The sem-annual yeld s 0.08 2 =0.04 Prce= ( CF t 1+ y ) mt m =3v +3v 2 +3v 3 +3v 4 + 100v 4 =3a 4 4% + 100v 4 P =96.3701 ModD = 3(0.5) 1.04 + 3(1) (1.04) 2 + 3(1.5) (1.04) 3 3(2) (1.04) 4 + 100(2) (1.04) 4 96.3701 (1.04) = 1.5(Ia) 4 4% + 2(100) (1.04) 4 (96.3701)(1.04) = 13.34528 + 170.96 100.2249 =1.8389 169

7.3 Macaulay Duraton Exercse (a) The current prce of a bond s 200. The dervatve wth respect to the yeld to maturty s -800. The yeld to maturty s 7%. Calculate the Macaulay duraton. Soluton (a) ModD = P (y) P (y) ModD = = ( 800) 200 ( MacD ) 1+ y m =4 4= MacD 1.07 MacD =4.28 170

Exercse (b) Calculate the Macaulay duraton of a common stock that pays dvdends at the end of each year nto perpetuty. Assume that the dvdend s constant, and that the effectve rate of nterest s 8%. Soluton (b) MacD = t CF t v t t=1 CF t v t t=1 = t v t t=1 t=1 v t Solvng for the numerator: tv t = v +2v 2 +3v 3 +4v 4 +... + v t=1 Solvng for the denomnator: = 1+ 2 v t = v + v 2 + v 3 + v 4 +...v t=1 = 1 = t v t t=1 = v t t=1 1+ 2 1 = 1+ = 1.08 0.08 =13.5 171

Exercse (c) Calculate the Macaulay Duraton of a common stock that pays dvdends at the end of each year nto perpetuty. Assume that the dvdend ncreases by 3% each year and the effectve rate s 5%. Soluton (c) MacD = t CF t v t t=1 CF t v t t=1 MacD = t v t (1.03) t 1 t=1 v t (1.03) (t 1) t=1 Solvng for the numerator: N = v(1.03) + 2v 2 (1.03) + 3v 3 (1.03) 2 +... + nv n (1.03) n 1 (1.03v)N =(1.03)v 2 +2v 3 (1.03) 2 +3v 4 (1.03) 3 +... +(n 1)v n (1.03) n 1 +... N N(1.03)v = v + v 2 (1.03) + v 3 (1.03) 2 +... +(1.03) n 1 v n +... N(1 (1.03)v) =v(1 + v(1.03) + v 2 (1.03) 2 +... +(1.03) n v n ( N(1 (1.03)v) =v 1 ( 1 ) 1.03 1+ 1.03 1+ [ ] 1 N(1 (1.03)v) =v 1 (1.03)v ) [ ] N(1 (1.03)v) = 1 1 1+ 1 1.03 1+ N(1 (1.03)v) = N = 1 1+ 1+ 1.03 1+ 1+ ( 0.03) 2 = 1 0.03 172

Now solvng for the denomnator: = v + v 2 (1.03) + v 3 (1.03) 2 +... + v n (1.03) n 1 Exercse (d) = v(1 + v(1.03) + v 2 (1.03) 2 +... + v n 1 (1.03) n 1 MacD = 1+ (.03) 2 1.03 = v ( 1 1.03 1+ 1 1.03 1+ ) [ ] = v 1 1+ 1 1.03 1+ = 1 1+.03 1+ = 1.03 The duraton of a bond at nterest rate s defned as: = 1+.03 = 1.05 0.05 0.03 = 1.05 0.02 =52.5 t CF t v t t=1 CF t=1 v t t=1 where CF t represents the net cash flow from the coupons and the maturty value of the bond at tme t. You are gven a 1,000 par value 20-year bond wth 4% annual coupons and a maturty value of 1,000. Calculate the duraton of ths bond at 5% nterest. Soluton (d) Duraton: = 40(Ia) 20 5% + 20(1, 000)v5% 20 40a 20 5% +1, 000v5% 20 = 11, 975.81 875.38 ( ä 20v 20 20 5% 40 5% =13.678 ) + 20(1, 000)v 20 5% 40a 20 5% +1, 000v 20 5% 173

7.4 Effectve Duraton Exercse (a) a 10-year bond yeldng 8% has a prce of 112.76. If the bond yeld falls to 7.75% then the prce of the bond wll ncrease to 115.32. If the bond s yeld ncreases to 8.25% then the prce of the bond wll fall to 110.25. Calculate the effectve duraton of the bond. Soluton (a) = EffD = P P + P 0 (2Δy) 115.32 110.25 112.76(2)(0.0025) = 5.07 0.5638 =8.9926 Exercse (b) A sx year bond wth coupons of 5% pays coupons sem-annually. The effectve yeld s 4%. It s current prce s 98.71. If the bond ncreases by 10 bass ponts then t s prce wll fall to 97.84. If the bond s yeld falls 10 bass ponts, then t s prce rses to 99.13. Caculate the bond s effectve duraton. Soluton (b) = EffD = P P + P 0 (2Δy) 99.13 97.84 (98.71)(2)(0.001) = 1.29.19742 =6.5346 174

7.5 Convexty Exercse (a) The effectve duraton s 5.4. The effectve convexty s calculated by a 20 year bond that has an effectve yeld of 7% and a prce of 103.10. If the bond s yeld falls to 6.5% then the prce ncreases to 103.51. If the bond s yeld falls to 7.5% then the prce decreases to 102.94. Calculate the estmated new prce of the bond f t ncreases by 75 bass ponts. Soluton (a) EffC = P + + P 2P 0 P 0 (Δy) 2 = 103.51 + 102.94 (2)(1.03.10) (103.10)(.005) 2 = 0.25.0025775 =96.9932 %ΔP (Δy)(Duraton) + (Δy)2 (Convexty) 2 = (.0075)(5.4) + (.0075)2 (96.9932) 2 = 0.0378 175

7.6 Duraton, Convexty, and Prces: Puttng t all Together Exercse (a) Leah purchases three bonds to form a portfolo as follows: Bond A has sem-annual coupons at 5%, a duraton of 20.57 years and was purchased for 880. Bond B s an 11-year bond wth a duraton of 11.75 years and was purchased for 1,124. Bond C has a duraton of 17.31 years and was purchased for 1,000. Calculate the duraton of the portfolo at the tme of purchase. Soluton (a) The duraton of the portfolo s: = P 1 D 1 + P 2 D 2 +... + P n D n MV port MV port MV port MV port = P 1 + P 2 +...P n MV port = 880 + 1, 124 + 1, 000 = 3, 004 Duraton port = (880)(20.57) 3, 004 + (1, 124)(11.75) 3, 004 + (1, 000)(17.31) 3, 004 = 4, 8618.6 3, 004 =16.18 176

8 The Term Structure of Interest Rates Overvew ths chapter wll show how to: 1. calculate a spot rate 2. calculate a forward rate 8.1 Yeld-to-Maturty an nterest rate s called a yeld rate or an nternal rate of return (IRR) as t ndcates the rate of return that the nvestor can expect to earn on ther nvestment. fndng prce usng a yeld rate: P = t>0 CF t (1 + y) t where P s prce, t s tme, CF t s the cash flow and y s the yeld rate Yeld Curves usually long-term market nterest rates are hgher than short-term market nterest rates yeld curves are usually upslopng, although they can be flat, downslopng, a peak or a valley when the yeld s constructed by government securtes, t s called an on-the-run yeld curve when each bond s coupon rate s assumed to equal a bond s yeld rate, t s called aparyeld curve 8.2 Spot Rates are the annual nterest rates that make up the yeld curve. lets t represent the spot rate for perod t. let P represent the net present value of a seres of future payments (postve or negatve) dscounted usng spot rates: P = n (1 + s t ) t CF t The present value of annutes can also be found usng spot rates: t=0 a n = 1 1 + 1+s 1 (1 + s 2 ) +... + 1 2 (1 + s n ) n ä n =1+ 1 1 + 1+s 1 (1 + s 2 ) +... + 1 2 (1 + s n 1 ) n 1 177

Bootstrappng determnng future spot rates usng the prce of a coupon bond s used assumng arbtrage s mpossble arbtrage s rsk-free proft Prce usng yeld=prce usng spot rates Forward Rates are consdered to be future renvestment rates. the prce of a securty s calculated as follows: CFt (1 + y) t = CF t (1 + s t ) t P = CF t (1 + f t>0 0 )(1 + f 1 ) (1 + f t 1 ) where f t s the annual effectve forward rate from tme t to t +1 smlar to the bootstrappng concept, forward rates can be calculated usng the yeld curve Prce of yeld curve=prce usng forward rates Relatonshps between Forward Rates and Spot Rates (1 + s t ) t =(1+f 0 )(1 + f 1 ) (1 + f t 1 ) s t = t (1 + f0 )(1 + f 1 ) (1 + f t 1 ) 1 Example: f t 1 = (1 + s t) t (1 + s t 1 ) t 1 1 A frm wshes to borrow money repayable n two years, where the one-year and two-year spot rates are 8% and 7%, respectvely. The estmated one-year deferred one-year spot rate s called the forward rate, f, and s calculated by equatng the two nterest rates such that (1.08) 2 =(1.07)(1 + f) f =9.01% If the borrower thnks that the spot rate for the 2 nd year wll be greater(less) than the forward rate, then they wll select (reject) the 2-year borrowng rate. 178