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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 FINANCIAL MATHEMATICS A Practcal Gude for Actuares and other Busness Professonals Second Edton CHRIS RUCKMAN, FSA, MAAA JOE FRANCIS, FSA, MAAA, CFA Study Notes Prepared by Kevn Shand, FSA, FCIA Assstant Professor Warren Centre for Actuaral Studes and Research

2 Contents 1 Interest Rates and Factors Interest SmpleInterest CompoundInterest AccumulatedValue PresentValue Rate of Dscount: d Constant Force of Interest: δ VaryngForceofInterest DscreteChangesnInterestRates... 9 ExercsesandSolutons Level Annutes Annuty-Immedate Annuty Due DeferredAnnutes ContnuouslyPayableAnnutes Perpetutes EquatonsofValue ExercsesandSolutons Varyng Annutes IncreasngAnnuty-Immedate IncreasngAnnuty-Due DecreasngAnnuty-Immedate DecreasngAnnuty-Due ContnuouslyPayableVaryngAnnutes CompoundIncreasngAnnutes ContnuouslyVaryngPaymentStreams ContnuouslyIncreasngAnnutes ContnuouslyDecreasngAnnutes ExercsesandSolutons Non-Annual Interest Rate and Annutes Non-AnnualInterestandDscountRates Nomnal p thly Interest Rates: (p) Nomnal p thly Dscount Rates: d (p) Annutes-Immedate Payable p thly Annutes-Due Payable p thly ExercsesandSolutons Project Apprasal and Loans DscountedCashFlowAnalyss Nomnalvs.RealInterestRates Investment Funds AllocatngInvestmentIncome Loans:TheAmortzatonMethod Loans:TheSnkngFundMethod

3 ExercsesandSolutons Fnancal Instruments TypesofFnancalInstruments MoneyMarketInstruments Bonds CommonStock PreferredStock Mutual Funds GuaranteedInvestmentContracts(GIC) DervatveSercurtes BondValuaton StockValuaton ExercsesandSolutons Duraton, Convexty and Immunzaton PrceasaFunctonofYeld ModfedDuraton MacaulayDuraton EffectveDuraton Convexty MacaulayConvexty EffectveConvexty Duraton,ConvextyandPrces:PuttngtallTogether RevstngthePercentageChangenPrce ThePassageofTmeandDuraton PortfoloDuratonandConvexty Immunzaton FullImmunzaton ExercsesandSolutons The Term Structure of Interest Rates Yeld-to-Maturty SpotRates

4 1 Interest Rates and Factors Overvew nterest s the payment made by a borrower (.e. the cost of dong busness) for usng a lender s captal assets (usually money); an example s a loan transacton nterest rate s the percentage of nterest to the captal asset n queston nterest takes nto account the rsk of default (rsk that the borrower can t pay back the loan) the rsk of default can be reduced f the borrower promses to release an asset of thers n the event of ther default (the asset s called collateral) 1.1 Interest Interest on Savngs Accounts a bank borrows a depostor s money and pays them nterest for the use of ther money the greater the need for money, the greater the nterest rate offered Interest Earned Durng the Perod t to t + s: AV t+s AV t the Accumulated Value at tme n s denoted as AV n nterest earned durng a perod of tme s the dfference between the Accumulated Value at the end of the perod and the Accumulated Value at the begnnng of the perod TheEffectveRateofInterest: s the amount of nterest earned over a 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

5 1.2 Smple Interest let the nterest amount earned each year on an nvestment of X be constant where the annual rate of nterest s : AV t = X(1 + t), where (1 + t) s a lnear functon smple nterest has the property that nterest s NOT renvested to earn addtonal nterest amount of Interest Earned to tme t s 1.3 Compound Interest I = AV t AV 0 = X(1 + t) X = X t let the nterest amount earned each year on an nvestment of X also allow the nterest earned to earn nterest where the annual rate of nterest s : where (1 + ) t s an exponental functon AV t = X(1 + ) t, compound nterest produces larger accumulatons than smple nterest when t>1 note that a constant rate of compound nterest mples a constant effectve rate of nterest 1.4 Accumulated Value Accumulated Value Factor: AV F t assume that AV t s contnuously ncreasng letx be the ntal Prncpal nvested (X >0) where AV 0 = X AV t defnes the Accumulated Value that amount X grows to n t years the Accumulated Value at tme t s the product of the ntal captal nvestment of X (Prncpal) made at tme zero and the Accumulaton Value Factor: AV t = X AV F t, where AV F t =(1+t) f smple nterest s beng appled and AV F t =(1+) t f compound nterest s beng appled 3

6 1.5 Present Value Dscountng Accumulated Value s a future value pertanng to payment(s) made n the past Dscounted Value s a present value pertanng to payment(s) to be made n the future dscountng determnes how much must be nvested ntally (Z) sothatx wll be accumulated after t years Z (1 + ) t = X Z = X = X(1 + ) t (1 + ) t Z represents the present value of X to be pad n t years letv = 1, v s called a dscount factor or present value factor 1+ Z = X v t Dscount Functon (Present Value Factor): PVF t smple nterest: PVF t = 1 1+t compound nterest: PVF t = 1 (1 + ) t = vt compound nterest produces smaller Dscount Values than smple nterest when t>1 4

7 1.6 Rate of Dscount: d Defnton an effectve rate of nterest s taken as a percentage of the balance at the begnnng of the year, whle an effectve rate of dscount s at the end of the year. eg. f 1 s nvested and 6% nterest s pad at the end of the year, then the Accumulated Value s 1.06 eg. f 0.94 s nvested after a 6% dscount s pad at the begnnng of the year, then the Accumulated Value at the end of the year s 1.00 d s also defned as the rato of the amount of nterest (amount of dscount) earned durng the perod to the amount nvested at the end of the perod d = AV t+1 AV t AV t+1 f nterest s constant, then dscount s constant the amount of dscount earned from tme t to t + s s AV t+s AV t Relatonshps Between and d f 1 s borrowed and nterest s pad at the begnnng of the year, then 1 d remans the accumulated value of 1 d at the end of the year s 1: (1 d)(1 + ) =1 nterest rate s the rato of the dscount pad to the amount at the begnnng of the perod: = d 1 d dscount rate s the rato of the nterest pad to the amount at the end of the perod: d = 1+ the present value of 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

8 Dscount Factors: PVF t and AV F t under the smple dscount model the Dscount Present Value Factor s: PVF t =1 dt for 0 t<1/d under the smple dscount model the Dscount Accumulated Value Factor s: AV F t =(1 dt) 1 for 0 t<1/d under the compound dscount model, the Dscount Present Value Factor s: PVF t =(1 d) t = v t for t 0 under the compound dscount model, the Dscount Accumulated Value Factor s: AV F t =(1 d) t for t 0 a constant rate of smple dscount mples an ncreasng effectve rate of dscount a constant rate of compound dscount mples a constant effectve rate of dscount 6

9 1.7 Constant Force of Interest: δ Defntons annual effectve rate of nterest s appled over a 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

10 1.8 Varyng Force of Interest let the constant force of nterest δ now vary at each nftesmal pont n tme and be denoted as δ t a change from tme t 1 to t 2, due to nterest rate δ t, where the accumulated value at tme t 1 s X, can be defned as follows: t2 δ t = d dt AV t AV t = d dt ln(av t) t2 d δ t dt = t 1 t 1 dt ln(av t) dt = ln(av t2 ) ln(av t1 ) ( ) AVt2 δ t dt =ln t2 t 1 t2 AV t1 e t 1 δ t dt = AV t 2 AV t1 Varyng Force of Interest Accumulaton Factor - AV F t1,t 2 let AV F t1,t 2 = e t2 t 1 δ t dt represent an accumulaton factor over the perod t 1 to t 2, where the force of nterest s varyng ft 1 = 0, then the notaton smplfes from AV F 0,t2 to AV F t.e. AV F t = e fδ t s readly ntegrable, then AV F t1,t 2 can be derved easly t 0 δ t dt fδ t s not readly ntegrable, then approxmate methods of ntegraton are requred Varyng Force of Interest Present Value Factor - PVF t1,t 2 let 1 1 PVF t1,t 2 = = AV F t2 = AV δ t dt t 1 = e t 1 t1,t 2 AV t2 δ t dt e t 1 represent a present value factor over the perod t 1 to t 2, where the force of nterest s varyng ft 1 = 0, then the notaton smplfes from PVF 0,t2 t2 to PVF t.e. PVF t = e t 0 δ t dt 8

11 1.9 Dscrete Changes n Interest Rates the most common applcaton of the accumulaton and present value factors over a perod of t years s t AV F t = (1 + k ) and PFV t = k=1 t k=1 1 (1 + k ) where k s the constant rate of nterest between tme k 1andtmek 9

12 Exercses and Solutons 1.2 Smple Interest Exercse (a) At what rate of smple nterest wll 500 accumulate to 615 n 2.5 years? Soluton (a) Exercse (b) 500[1 + (2.5)] = 615 = =9.2% 2.5 In how many years wll 500 accumulate to 630 at 7.8% smple nterest? Soluton (b) Exercse (c) 500[ (n)] = 630 = =3.33 years.078 At a certan rate of smple nterest 1,000 wll accumulate to 1,100 after a certan perod of tme. Fnd the accumulated value of 500 at a rate of smple nterest three fourths as great over twce as long a perod of tme. Soluton (c) 1, 000[1 + n] =1, 100 n =.11 Exercse (d) 500[ n] = 500[1 + (1.5)(.11)] = Smple nterest of = 4% s beng credted to a fund. In whch perod s ths equvalent to an effectve rate of 2.5%? Soluton (d) 0.25 = n = 1+(n 1) (n 1) n =16 10

13 1.3 Compound Interest Exercse (a) Fund A s nvested at an effectve annual nterest rate of 3%. Fund B s nvested at an effectve annual nterest rate of 2.5%. At the end of 20 years, the total n the two funds s 10,000. At the end of 31 years, the amount n Fund A s twce the amount n Fund B. Calculate the total n the two funds at the end of 10 years. Soluton (a) Let the ntal funds be A and B. Therefore, we have two equatons and two unknowns: A(1.03) 20 + B(1.025) 20 =10, 000 A(1.03) 31 =2B(1.025) 31 Solvng for B n the second equaton and pluggng t nto the frst equaton gves us A = 3, and B =2, We seek A(1.03) 10 + B(1.025) 10 whch equals Exercse (b) 3, (1.03) 10 +2, (1.025) 10 =7, Carl puts 10,000 nto a bank account that pays an annual effectve nterest rate of 4% for ten years. If a wthdrawal s made durng the frst fve and 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) K(1.04) K(1.04) 5 K(1.04) 4 K(1.04) 3 =10, , 802 K[(1.05)(1.04) 6 +(1.05)(1.04) 5 +(1.04) 4 +(1.04) 3 ]=10, 000 4, 802 = K 4.9 K =

14 1.4 Accumulated Value Exercse (a) 100 s deposted nto an account at the begnnng of every 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+) (1+) (1+) 40 = X = 5[100(1+) (1+) (1+) 20 ] 100(1+) 4 [1+100(1+) (1+) 36 ]=5 100(1+) 4 [1+100(1+) (1+) 16 ] 100(1 + ) 4 [ 1 [(1 + ) 4 ] 10 1 (1 + ) 4 ] [ 1 [(1 + ) =5 100(1 + ) 4 4 ] 5 ] 1 (1 + ) 40 =5[1 (1 + ) 20 ] 1 (1 + ) 4 (1 +) 40 5(1+) = 0 [(1+) 20 1][(1+) 20 4] = 0 (1 +) 20 =1or(1+) 20 =4 (1 + ) 20 =1 =0% AV 40 = 1000 and AV 20 = 500, mpossble snce AV 40 =5AV 20 therefore, (1 + ) 20 =4 [ ] [ ] 1 (1 + ) X = 100(1 + ) 4 40 = 100( ) = 100( ) = (1 + )

15 1.5 Present Value Exercse (a) Annual payments are made at the end of each year, forever. The payments at tme n s defned as n 3 for the frst n years. After year n, the payments reman constant at n 2. The present value of these payments at tme 0 s 20n 2 when the annual effectve rate of nterest s 0% for the frst n years and 25% thereafter. Calculate n. Soluton (a) [(1 3 )v 0% +(2 3 )v0% 2 +(33 )v0% (n3 )v0% n ]+vn 0% [(n2 )v25% 1 +(n2 )v25% 2 +(n2 )v25% ] =20n2 [(1 3 )+(2 3 )+(3 3 )+... +(n 3 )] + (n 2 )v 25% [1 + v 1 25% + v2 25% +...]=20n2 [ n 2 (n +1) 2 ] [ ] +(n 2 1 )v 25% =20n v 25% [ (n +1) 2 4 ] [ ] 1 +(.8) 1.8 =20 [ ] (n +1) 2 +4=20 4 (n +1)2 4 =16 (n +1) 2 =4 n =7 Exercse (b) At an effectve annual nterest rate of, >0, each of the followng two sets of payments has present value K: () A payment of 121 mmedately and another payment of 121 at the end of one year. () A payment of 144 at the end of two years and another payment of 144 at the end of three years. Calculate K. Soluton (b) v = 144v v 3 = K 121(1 + v) = 144v 2 (1 + v) v = K = 121( ) K =

16 Exercse (c) The present value of a seres of payments of 2 at the end of every eght years, forever, s equal to 5. Calculate the effectve rate of nterest. Soluton (c) 2v 8 +2v 16 +2v =5 2v 8 [1 + v 8 + v =5 [ ] 2v v 8 =5 2v 8 =5 5v 8 7v 8 =5 v 8 = 5 7 (1 + ) 8 = Rate of Dscount Exercse (a) 1+ = ( ) = A busness permts ts customers to pay wth a credt card or to receve a percentage dscount of r for payng cash. For credt card purchases, the busness receves 97% of the purchase prce 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 =$0.97v 1 24 (1 r) =.97(1 d) 1 24 =.97(1.22) 1 24 =.96 r =.04 = 4% 14

17 Exercse (b) You depost 1,000 today and another 2,000 n fve years nto a fund that pays smple dscountng at 5% per year. Your frend makes the same deposts nto another fund, but at tme n and 2n, respectvely. Ths fund credts nterest at an annual effectve rate of 10%. At the end of 10 years, the accumulated value of your deposts s exactly the same as the accumulated value of your frend s deposts. Calculate n. Soluton (b) 1, 000[1 5%(10)] 1 +2, 000[1 5%(5)] 1 =1, 000(1.10) 10 n +2, 000(1.10) 10 2n 1, , =1, 000(1.10)10 v n +2, 000(1.10) 10 v 2n 4, = 2, v n +5, v 2n 5, v 2n +2, v n 4, =0 } {{ } } {{ } } {{ } a b c v n = 2, , (5, )( 4, ) 2(5, ) (1.10) n = 1 ln( ) = n = = ln(1.10) =.7306 Exercse (c) A depost of X s made nto a fund whch pays an annual effectve nterest rate of 6% for 10 years. At the same tme, X/2 s deposted nto another fund whch pays an annual effectve rate of dscount of d for 10 years. The amounts of nterest earned over the 10 years are equal for both funds. Calculate d. Soluton (c) X(1.06) 10 X = X 2 (1 d) 10 X 2 d =

18 1.7 Constant Force of Interest Exercse (a) You are gven that AV t = Kt 2 + Lt + M, for0 t 2, and that AV 0 = 100, AV 1 = 110, and AV 2 = 136. Determne the force of nterest at tme t = 1 2. Soluton (a) AV 0 = M = 100 AV 1 = K + L + M = 110 K + L =10 AV 2 =4K +2L + M = 136 4K +2L =36 These equatons solve for K =8,L =2,M = 100. We know that δ t = d dt AV t AV t = 2Kt + L Kt 2 + Lt + M δ 1 = (2)(8)( 1 2 ) (8)( 1 2 )2 +(2)( 1 = ) = Exercse (b) Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces of nterest on the two funds are equal. Soluton (b) AV F A t =1+.10t and AV F B t =(1+.05t) 1 δ A t = d dt AV F A t AV F A t = t δ B t = Equatng and solve for t d dt AV F B t AV F B t =.05(1.05t) t) 1 =.05(1.05t) t = t = t t =5 1.05t 16

19 1.8 Varyng Force of Interest Exercse (a) On 15 March 2003, a student deposts X nto a bank account. The account s credted wth smple nterest where =7.5% On the same date, the student s professor deposts X nto a dfferent bank account where nterest s credted at a force of nterest δ t = 2t,t 0. t 2 + k From the end of the fourth year untl the end of the eghth year, both accounts earn the same dollar amount of nterest. Calculate k. Soluton (a) Smple Interest Earned = AV 8 AV 4 = X[1+.075(8)] X[1+.075(4)] = X(0.075)(4) = X(.3) Compound Interest Earned = AV 8 AV 4 = Xe 8 0 δ t dt Xe 4 0 δ t dt Xe 8 0 2t 4 t 2 + k dt Xe 0 2t 8 t 2 + k dt f (t) 4 = Xe 0 f(t) dt f (t) Xe 0 X (8)2 + k (0) 2 + k X (4)2 + k (0) 2 + k = X 48, compound nterest k X(.3) = X 48 k.3 =48 k.30k =48 k = 160 f(t) dt = X f(8) f(4) X f(0) f(0) 17

20 Exercse (b) Payments are made to an account at a contnuous rate of, k 2 +8tk 2 +(tk) 2,where0 t 10 and k>0. Interest s credted at a force of nterest where After 10 years, the account s worth 88, 690. Calculate k. Soluton (b) δ t = 8+2t 1+8t + t 2 (1 + ) 10 t = e t [k 2 +8tk 2 +(tk) 2 ](1 + ) 10 t dt =88, 690 δ s ds = e 10 t 8+2s s + s ds f (s) 2 = e t f(s) ds = f(10) f(t) = 1 + 8(10) + (10) t + t 2 = 1+8t + t [k 2 +8tk 2 +(tk) ] dt =88, t + t2 10 k 2 [1 + 8t + t ] 1+8t + t dt =88, k2 (181)(10) = 88, 690 k 2 =49 k =7 0 18

21 Exercse (c) Fund A accumulates at a constant force of nterest of δt A =.05 at tme t, fort 0, and 1+.05t Fund B accumulates at a constant force of nterest of δ B =5%. Youaregven: () The amount n Fund A at tme zero s 1,000. () The amount n Fund B at tme zero s 500. () The amount n Fund C at any tme t, t 0, s equal to the sum of the amounts n Fund A and Fund B. Fund C accumulates at a force of nterest of δt C,fort0. Calculate δ C 2. Soluton (c) AV C t = AV A t + AV B t =1, 000e t 0 δ s dt + 500e.05t AV C t =1, 000e t s ds [ ] f(t) + 500e.05t =1, e.05t f(0) [ ] 1+.05(t) AVt C =1, (.05)e.05t = 1000[1 +.05t] + 500e.05t 1+.05(0) d dt AV C t =1, 000(.05) + 500(.05)e.05t = t δ t = δ 2 = d dt AV t C AVt C = e.05t 1000[1 +.05t] + 500e.05t e.05(2) =4.697% 1, 000[1 +.05(2)] + 500e.05(2) 19

22 Exercse (d) Fund F accumulates at the rate δ t = 1 1+t. Fund G accumulates at the rate δ t = 4t 1+2t. 2 F (t) s the amount n Fund F at tme t, and G(t) s the amount n fund G at tme t, wth F (0) = G(0). Let H(t) =F (t) G(t). Calculate T, the value of tme t when H(t) sa maxmum. Soluton (d) Snce F (0) = G(0), we can assume an ntal depost of 1. Then we have, F (t) =e t r dr = e ln(1+t) ln(1) =1+t G(t) =e t 0 4r 1+2r 2 dr = e ln(1+2t2 ) ln(1) =1+2t 2 H(t) =t 2t 2 and d dt H(t) =1 4t =0 t =

23 2 Level Annutes Overvew Defnton of An Annuty a seres of payments made at equal ntervals of tme (annually or otherwse) payments made for certan over a fxed perod of tme are called an 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 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 v n 2 + v n 1 ) ( )( ) 1 1 v n = 1+ 1 v ( )( ) 1 1 v n = 1+ d ( ) ( ) 1 1 v n = 1+ = 1 vn 1+ 21

24 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 n - 1 n The present value of ths multple payment ncome stream at t =0s a n +(1)v n. Therefore, the present value of both nvestment opportuntes are equal. Also note that a n = 1 vn 1= a n + v n. 22

25 Basc Relatonshp 2:PV(1 + ) n = FV and PV = FV v n f the future value at tme n, s n, s dscounted back to tme 0, then you wll have ts present value, a n [ (1 + ) s n v n n ] 1 = v n = (1 + )n v n v n = 1 vn = a n f the present value at tme 0, a n, s accumulated forward to tme n, then you wll have ts future value, s n [ ] 1 v a n (1 + ) n n = (1 + ) n = (1 + )n v n (1 + ) n = (1 + )n 1 = s n 23

26 1 Basc Relatonshp 3: = 1 + a n s n Consder a loan of 1, to be pad back over n years wth equal annual payments of P made at the end of each year. An annual effectve rate of nterest,, s used. The present value of ths sngle payment loan must be equal to the present value of the multple payment ncome stream. P a n =1 P = 1 a n Alternatvely, consder a loan of 1, where the annual nterest due on the loan, (1), spadat the end of each year for n years and the loan amount s pad back at tme n. In order to produce the loan amount at tme n, annual payments at the end of each year, for n years, wll be made nto an account that credts nterest at an annual effectve rate of nterest. The future value of the multple depost ncome stream must equal the future value of the sngle payment, whch s the loan of 1. D s n =1 D = 1 s n The total annual payment wll be the nterest payment and account payment: + 1 s n Therefore, a level annual annuty payment on a loan s the same as makng an annual nterest payment each year plus makng annual deposts n order to save for the loan repayment. Also note that 1 (1 + )n (1 + )n = = a n 1 vn (1 + ) n (1 + ) n 1 = (1 + )n + (1 + ) n 1 = [(1 + )n 1] + (1 + ) n 1 = + (1 + ) n 1 = + 1 s n 24

27 2.2 Annuty Due Defnton payments of 1 are made at the begnnng of every year for n years n - 1 n 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 + ) (1)(1 + ) n 1 + (1)(1 + ) n =(1+)[1 + (1 + )+ +(1+) n 2 +(1+) n 1 ] [ ] 1 (1 + ) n =(1+) 1 (1 + ) [ ] 1 (1 + ) n =(1+) [ (1 + ) n ] 1 =(1+) = (1 + )n 1 d 25

28 Basc Relatonshp 1:1=d ä n + v n Consder an n year nvestment where 1 s nvested at tme 0. The present value of ths sngle payment ncome stream at t =0s1. Alternatvely, consder a n year nvestment where 1 s nvested at tme 0 and produces annual nterest payments of (1) d at the begnnng of each year and then have the 1 refunded at t = n. d d d... d n - 1 n The present value of ths multple payment ncome stream at t =0sd ä n +(1)v n. Therefore, the present value of both nvestment opportuntes are equal. Also note that ä n = 1 vn d 1=d ä n + v n. Basc Relatonshp 2:PV(1 + ) n = FV and PV = FV v n f the future value at tme n, s n, s dscounted back to tme 0, then you wll have ts present value, ä n [ (1 + ) s n v n n ] 1 = v n = (1 + )n v n v n = 1 vn =ä n d d d f the present value at tme 0, ä n, s accumulated forward to tme n, then you wll have ts future value, s n [ ] 1 v ä n (1 + ) n n = (1 + ) n = (1 + )n v n (1 + ) n d d = (1 + )n 1 d = s n 26

29 1 Basc Relatonshp 3: = 1 + d ä n s n Consder a loan of 1, to be pad back over n years wth equal annual payments of P made at the begnnng of each year. An annual effectve rate of nterest,, s used. The present value of the sngle payment loan must be equal to the present value of the multple payment stream. P ä n =1 P = 1 ä n Alternatvely, consder a loan of 1, where the annual nterest due on the loan, (1) d, spad at the begnnng of each year for n years and the loan amount s pad back at tme n. In order to produce the loan amount at tme n, annual payments at the begnnng of each year, for n years, wll be made nto an account that credts nterest at an annual effectve rate of nterest. The future value of the multple depost ncome stream must equal the future value of the sngle payment, whch s the loan of 1. D s n =1 D = 1 s n The total annual payment wll be the nterest payment and account payment: d + 1 s n Therefore, a level annual annuty payment s the same as makng an annual nterest payment each year and makng annual deposts n order to save for the loan repayment. Also note that 1 = d (1 + )n d(1 + )n = ä n 1 vn (1 + ) n (1 + ) n 1 = d(1 + )n + d d (1 + ) n 1 = d[(1 + )n 1] + d (1 + ) n 1 = d + d (1 + ) n 1 = d + 1 s n Basc Relatonshp 4: AnnutyDue= Annuty Immedate (1 + ) ä n = 1 vn d s n = (1 + )n 1 d = 1 vn (1 + ) =a n (1 + ) [ (1 + ) n 1 = ] (1 + ) =s n (1 + ) An annuty due starts one perod earler than an annuty-mmedate and as a result, earns one more perod of nterest, hence t wll be larger. 27

30 Basc Relatonshp 5:ä n =1+a n 1 ä n =1+[v + v v n 2 + v n 1 ] =1+v[1 + v + + v n 3 + v n 2 ] ( ) 1 v n 1 =1+v 1 v ( )( ) 1 1 v n 1 =1+ 1+ d ( )( ) 1 1 v n 1 =1+ 1+ /1+ =1+ 1 vn 1 =1+a n 1 Ths relatonshp can be vsualzed wth a tme lne dagram. a n n - 1 n becomng n payments that now com- An addtonal payment of 1 at tme 0 results n a n 1 mence at the begnnng of each year whch s ä n. 28

31 Basc Relatonshp 6:s n =1+ s n 1 s n =1+[(1+)+(1+) 2 + +(1+) n 2 +(1+) n 1 ] =1+(1+)[1 + (1 + )+ +(1+) n 3 +(1+) n 2 ] [ ] 1 (1 + ) n 1 =1+(1+) 1 (1 + ) [ ] 1 (1 + ) n 1 =1+(1+) [ (1 + ) n 1 ] 1 =1+(1+) =1+ (1 + )n 1 1 d =1+ s n 1 Ths relatonshp can also the vsualzed wth a tme lne dagram s n n - 1 n An addtonal payment of 1 at tme n results n s n 1 commerce at the end of each year whch s s n becomng n payments that now 29

32 2.3 Deferred Annutes There are three alternatve dates to valung annutes rather than at the begnnng of the term (t =0)orattheendoftheterm(t = n) () present values more than one perod before the frst payment date () accumulated values more than one perod after the last payment date () current value between the frst and last payment dates The followng example wll be used to llustrate the above cases. Consder a seres of payments of 1 that are made at tme t =3tot =9,nclusve Present Values More than One Perod Before The Frst Payment Date At t = 2, there exsts 7 future 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

33 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

34 Wth the annuty due verson, one can pretend that there are 10 payments beng made. Ths can be done by addng 3 payments to the exstng 7 payments. In ths case, let the 3 addtonal payments be made at t = 0, 1 and 2 and be denoted as At t = 0, there now exsts 10 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

35 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

36 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

37 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

38 2.4 Contnuously Payable Annutes payments are made contnuously every year for the next n years (.e. m = ) Contnuously Payable Annuty Present Value Factor the present value (at t = 0) of a contnuous annuty, where the annual effectve rate of nterest s, shall be denoted as ā n and s calculated as follows: ā n = = n 0 n 0 v t dt e δt dt = 1 ] n δ e δt 0 = 1 [ e δn e δ0] δ = 1 [ ] 1 e δn δ = 1 vn δ Contnuously Payable Annuty Accumulated Value Factor the accumulated value (at t = n) of a contnuous annuty, where the annual effectve rate of nterest s, shall be denoted as s n and s calculated as follows: s n = = = n 0 n 0 n 0 = 1 δ eδt dt = 1 δ (1 + ) n t dt (1 + ) t dt e δt dt ] n 0 [ e δn e δ0] = (1 + )n 1 δ 36

39 Basc Relatonshp 1:1=δ ā n + v n Basc Relatonshp 2:PV(1 + ) n = FV and PV = FV v n f the future value at tme n, s n, s dscounted back to tme 0, then you wll have ts present value, ā n [ (1 + ) s n v n n ] 1 = v n δ = (1 + )n v n v n δ = 1 vn δ =ā n f the present value at tme 0, ā n, s accumulated forward to tme n, then you wll have ts future value, s n [ ] 1 v ā n (1 + ) n n = (1 + ) n δ = (1 + )n v n (1 + ) n δ = (1 + )n 1 δ = s n 37

40 1 Basc Relatonshp 3: = 1 + δ ā n s n Consder a loan of 1, to be pad back over n years wth annual payments of P that are pad contnuously each year, for the next n years. An annual effectve rate of nterest,, and annual force of nterest, δ, s used. The present value of ths sngle payment loan must be equal to the present value of the multple payment ncome stream. P ā n =1 P = 1 ā n Alternatvely, consder a loan of 1, where the annual nterest due on the loan, (1) δ, s pad contnuously durng the year for n years and the loan amount s pad back at tme n. In order to produce the loan amount at tme n, annual payments of D are pad contnuously each year, for the next n years, nto an account that credts nterest at an annual force of nterest, δ. The future value of the multple depost ncome stream must equal the future value of the sngle payment, whch s the loan of 1. D s n =1 D = 1 s n The total annual payment wll be the nterest payment and account payment: δ + 1 s n Notethat 1 ā n = δ 1 v (1 + )n δ(1 + )n = n (1 + ) n (1 + ) n 1 = δ(1 + )n + δ δ (1 + ) n 1 = δ[(1 + )n 1] + δ (1 + ) n 1 = δ + δ (1 + ) n 1 = δ + 1 s n Therefore, a level contnuous annual annuty payment on a loan s the same as makng an annual contnuous nterest payment each year plus makng level annual contnuous deposts n order to save for the loan repayment. 38

41 Basc Relatonshp 4:ā n = δ a n, s n = δ s n Consder annual payments of 1 made contnuously each year for the next n years. Over a 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

42 2.5 Perpetutes Defnton Of A Perpetuty-Immedate payments of 1 are made at the end of every year forever.e. n = 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+ d ( ) ( ) 1 1 = = 1 one could also derve the above formula by smply substtutng n = nto the orgnal present value formula: a = 1 v = 1 0 = 1 Notethat 1 represents an ntal amount that can be nvested at t = 0. The annual nterest ( ) 1 payments, payable at the end of the year, produced by ths nvestment s =1. s s not defned snce t would equal 40

43 Basc Relatonshp 1:a n = a v n a The present value formula for an 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

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

45 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

46 2.6 Equatons of Value the value at any gven pont n tme, t, wll be ether a present value or a future value (sometmes referred to as the tme value of money) the tme value of money depends on the calculaton date from whch payment(s) are ether accumulated or dscounted to Tme Lne Dagrams t helps to draw out a tme lne and plot the payments and wthdrawals accordngly P 1 P 2... P t... P n-1 P n t... n-1 n W 1 W 2 W t W n-1 W n 44

47 Example a payment of 600 s due n 8 years; the alternatve s to receve 100 now,200 n 5 years and X n 10 years. If = 8%, fnd $X, such that the value of both optons s equal X compare the values at t = X v 8 8% = v5 8% + Xv10 8% X = 600v8 8% v5 8% v 10 8% =

48 compare the values at t = X v 3 = 100(1 + ) Xv 5 X = 600v3 100(1 + ) v 5 = compare the values at t = X (1 + ) 2 = 100(1 + ) (1 + ) 5 + X X = 600(1 + ) 2 100(1 + ) (1 + ) 5 = all 3 equatons gave the same answer because all 3 equatons treated the value of the payments consstently at a gven pont of tme. 46

49 Unknown Rate of Interest Assumng that you do not have a fnancal calculator Lnear Interpolaton need to fnd the value of a n at two dfferent nterest rates where a n 1 = P 1 <P and a n 2 = P 2 >P. } a n 1 = P 1 a n = P a n 2 = P P 1 P P 1 P 2 ( 2 1 ) 47

50 Exercses and Solutons 2.1 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 t + t2 a 15 [ a = Y 36 a 45 a 9 a 15 [ ] a 36 a 15 1 a 45 a 9 s 4 =1+(1+) 4 3 +(1+) 4 2 +(1+) δ s ds δ s ds δ s ds =1+e 3 + e 2 + e s 4 =1+e 3 1+8s + s ds 4+s e 2 1+8s + s ds 2 + e 1 4 f (s) =1+e 3 f(s) ds f (s) e 2 f(s) ds f (s) e 1 f(s) ds [ ] 1 [ ] 1 [ ] 1 f(4) 2 f(4) 2 f(4) 2 = f(3) f(2) f(1) [ ] 1 [ ] 1 [ 1 + 8(4) + (4) (4) + (4) (4) + (4) 2 = (3) + (3) (2) + (2) (1) + (1) 2 = = or ] 4+s 1+8s + s 2 ]

51 Exercse (c) You are gven the followng nformaton: () The present value of a 6n-year annuty-mmedate of 1 at the end of every year s () The present value of a 6n-year annuty-mmedate of 1 at the end of every second year s () 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) = a 6n = a 3n (1+) 2 1 = a 6n s Annuty-Due Exercse (a) K = a 2n (1+) 3 1 = a 6n s = s 2 =2.05 = 1 + (1 + ) =5% K = s 3 5% = =3.095 Smplfy a 15 (1 + ) 45 ä 3 j to one actuaral symbol, gven that j =(1+) Soluton (a) a 15 (1 + ) 45 ä 3 j = s 15 (1 + ) 30 (1 + v 1 j + v 2 j ) = s 15 (1 + ) 30 (1 + v 15 + v 30 )=s 15 [(1 + ) 30 +(1+) 15 +1] There are 3 sets of 15 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

52 Exercse (b) A person deposts 100 at the begnnng of each year for 20 years. Smple nterest at an annual rate of s credted to each depost from the date of depost to the end of the twenty year perod. The total amount thus accumulated s 2,840. If nstead, compound nterest had been credted at an effectve annual rate of, what would the accumulated value of these deposts have been at the end of twenty years? Soluton (b) 100[(1 + )+(1+2)+(1+3)+... +(1+20)] = 2, [20 + ( )] = 2, 840 Exercse (c) 20 + ( )=28.40 =.04 and d = s 20 4% =3, 097 You plan to accumulate 100,000 at the end of 42 years by makng the followng deposts: X at the begnnng of years 1-14 No deposts at the begnnng of years 15-32; and Y at the begnnng of years The annual effectve nterest rate s 7%. X Y = 100. Calculate Y. Soluton (c) }{{} X s 14 7% (1.07) 28 + Y s 10 7% = 100, Y (100 + Y )( ) + Y ( ) = 100, 000 Y =

53 2.3 Deferred Annutes Exercse (a) Usng an annual effectve nterest rate j 0, you are gven: () The present value of 2 at the end of each year for 2n years, plus an addtonal 1 at the end of each of the frst n years, s 36. () The present value of an 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

54 Exercse (b) A loan of 1,000 s to be repad by annual payments of 100 to commence at the end of the ffth year and to contnue thereafter for as long as necessary. The effectve rate of dscount s 5%. Fnd the amount of the fnal payment f t s to be larger than the regular payments. Soluton (b) PV 0 =1, 000 = 100a n 100a 4 a n = 1, a = v n = = v n 3.47 = ( ) n n = ln(3.47) ln( ) = n =24 1, 000 = 100a 24 + Xv a 4 X = 1, (a 24 a 4 ) v 24 X = ( ) 24 Thus, the total fnal payment s X =

55 2.4 Contnuously Payable Annutes Exercse (a) There s 40,000 n a fund whch s accumulatng at 4% per annum convertble contnuously. If money s wthdrawn contnuously at the rate of 2,400 per annum, how long wll the fund last? Soluton (a) If the fund s exhausted at t = n, then the accumulated value of the fund at that tme must equal the accumulated value of the wthdrawals. Thus we have: 40, 000e.04n =2, 400 s n = 2400( e.04n 1 ).04 40, 000 (.04) = 1 e.04n 2, 400 n = 40,000 ln[1 (.04) 2,400 ] =27.47 years.04 Exercse (b) If ā n =4and s n = 12 fnd δ. Soluton (b) ā n = 1 vn δ =4 v n =1 4δ s n = (1 + )n 1 δ (1 + ) n =1+12δ =12 (1 + ) n = v n then δ = 1 1 4δ 1+8δ 48δ 2 =1orδ = 4 48 =

56 2.5 Perpetutes Exercse (a) A 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 =

57 Exercse (b) A perpetuty pays 1 at the begnnng of every year plus an addtonal 1 at the begnnng of every second year. Determne the present value of ths annuty. Soluton (b) K =ä + v ä (1+) 2 1 = 1 d d (1+) 2 1 = 1+ + (1 + ) (1 + ) 2 1 Exercse (c) K = (1 + ) 1+(1+)2 1 (1 + ) 2 = (1 + ) (1 + ) 2 (1 + )(1 + ) 2 1 K = 1+ K =(1+) + [ ] (1 + ) 1 (1 + ) 2 1 =(1+) + 1 (1 + ) 2 1 [ (1 + ) 2 ] [ ] 1+ =(1+) [(1 + ) 2 1] [ [ ] [ ] K =(1+) =(1+) [2 + 2 ] [ ] 3+ =(1+) = 3+ (2 + ) d(2 + ) A 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

58 2.6 Equaton of Value Exercse (a) An nvestment requres an ntal payment of 10,000 and annual payments of 1,000 at the end of each of the frst 10 years. Startng at the end of the eleventh year, the nvestment returns fve equal annual payments of X. Determne X to yeld an annual effectve rate of 10% over the 15-year perod. Soluton (a) PV of cash flow n=pv of cash flow out 10, , 000a 10 10% = v 10 X a 5 10% X = 10, , 000a 10 10% v 10 a 5 10% X = 10, , 000( ) ( )( ) X =11,

59 Exercse(b) At a certan nterest rate the present value of the followng two payment patterns are equal: () 200 at the end of 5 years plus 500 at the end of 10 years. () at the end of 5 years. At the same nterest rate, 100 nvested now plus 120 nvested at the end of 5 years wll accumulate to P at the end of 10 years. Calculate P. Soluton (b) 200v v 10 = v 5 500v 10 = v 5 v 5 = v 5 = (1 + ) 5 = Exercse (c) P = 100(1 + ) (1 + ) 5 = 100( ) ( ) = Whereas the choce of a comparson date has no effect on the answer obtaned wth compound nterest, the same cannot be sad of smple nterest. Fnd the amount to be pad at the end of 10 years whch s equvalent to two payments of 100 each, the frst to be pad mmedately and the second to be pad at the end of 5 years. Assume 5% smple nterest s earned from the date each payment s made and use a comparson date of the end of 10 years. Soluton (c) Equatng at t =10 X = 100(1 + 10) + 100(1 + 5) = [100(1 + 10(.05)) + 100(1 + 5(.05)) =

60 3 Varyng Annutes Overvew n ths secton, payments wll now vary; but the nterest converson perod wll contnue to concde wth the payment frequency annutes can vary n 3 dfferent ways () where the payments ncrease or decrease by a fxed amount (sectons 3.1, 3.2, 3.3, 3.4 and 3.5) () where the payments ncrease or decrease by a fxed rate (secton 3.6) () where the payments ncrease or decrease by a varable amount or rate (secton 3.7) 3.1 Increasng 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 n - 1 n 58

61 The present value (at t = 0) of ths annual annuty mmedate, where the annual effectve rate of nterest s, shall be calculated as follows: PV 0 =[P ]v +[P + Q]v 2 + +[P +(n 2)Q]v n 1 +[P +(n 1)Q]v n = P [v + v v n 1 + v n ]+Q[v 2 +2v 3 +(n 2)v n 1 +(n 1)v n ] = P [v + v v n 1 + v n ]+Qv 2 [1 + 2v +(n 2)v n 3 +(n 1)v n 2 ] = P [v + v v n 1 + v n ]+Qv 2 d dv [1 + v + v2 + + v n 2 + v n 1 ] = P a n + Qv 2 d dv [ä n ] = P a n + Qv 2 d [ ] 1 v n dv 1 v [ (1 v) ( nv = P a n + Qv 2 n 1 ) (1 v n ] ) ( 1) (1 v) 2 [ Q nv n (v 1 1) + (1 v n ] ) = P a n + (1 + ) 2 (/1+) 2 [ (1 v n ) nv n 1 nv n ] = P a n + Q 2 [ (1 v n ) nv n (v 1 ] 1) = P a n + Q 2 [ (1 v n ) nv n ] (1 + 1) = P a n + Q = P a n + Q (1 v n ) [ a n = P a n + Q nv n ] 2 nvn () The accumulated value (at t = n) of an annuty mmedate, where the annual effectve rate of nterest s, can be calculated usng the same approach as above or calculated by usng the basc prncple where an accumulated value s equal to ts present value carred forward wth nterest: FV n = PV 0 (1 + ) n ( [ a n = P a n + Q nv n ]) (1 + ) n [ a = P a n (1 + ) n n + Q (1 + ) n nv n (1 + ) n ] [ ] s n = P s n + Q n 59

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression. Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook

More information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuity Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

More information

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt. Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. 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

More information

Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money

Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money Ch. 6 - The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21- Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important

More information

Simple Interest Loans (Section 5.1) :

Simple Interest Loans (Section 5.1) : Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

More information

10. (# 45, May 2001). At time t = 0, 1 is deposited into each of Fund X and Fund Y. Fund X accumulates at a force of interest

10. (# 45, May 2001). At time t = 0, 1 is deposited into each of Fund X and Fund Y. Fund X accumulates at a force of interest 1 Exam FM questons 1. (# 12, May 2001). Bruce and Robbe each open up new bank accounts at tme 0. Bruce deposts 100 nto hs bank account, and Robbe deposts 50 nto hs. Each account earns an annual e ectve

More information

0.02t if 0 t 3 δ t = 0.045 if 3 < t

0.02t if 0 t 3 δ t = 0.045 if 3 < t 1 Exam FM questons 1. (# 12, May 2001). Bruce and Robbe each open up new bank accounts at tme 0. Bruce deposts 100 nto hs bank account, and Robbe deposts 50 nto hs. Each account earns an annual effectve

More information

Using Series to Analyze Financial Situations: Present Value

Using Series to Analyze Financial Situations: Present Value 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

Finite Math Chapter 10: Study Guide and Solution to Problems

Finite Math Chapter 10: Study Guide and Solution to Problems Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount

More information

An Overview of Financial Mathematics

An Overview of Financial Mathematics An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take

More information

A) 3.1 B) 3.3 C) 3.5 D) 3.7 E) 3.9 Solution.

A) 3.1 B) 3.3 C) 3.5 D) 3.7 E) 3.9 Solution. ACTS 408 Instructor: Natala A. Humphreys SOLUTION TO HOMEWOR 4 Secton 7: Annutes whose payments follow a geometrc progresson. Secton 8: Annutes whose payments follow an arthmetc progresson. Problem Suppose

More information

10.2 Future Value and Present Value of an Ordinary Simple Annuity

10.2 Future Value and Present Value of an Ordinary Simple Annuity 348 Chapter 10 Annutes 10.2 Future Value and Present Value of an Ordnary Smple Annuty In compound nterest, 'n' s the number of compoundng perods durng the term. In an ordnary smple annuty, payments are

More information

7.5. Present Value of an Annuity. Investigate

7.5. Present Value of an Annuity. Investigate 7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on

More information

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is Payout annutes: Start wth P dollars, e.g., P = 100, 000. Over a 30 year perod you receve equal payments of A dollars at the end of each month. The amount of money left n the account, the balance, earns

More information

Level Annuities with Payments Less Frequent than Each Interest Period

Level Annuities with Payments Less Frequent than Each Interest Period Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Symoblc approach

More information

Section 2.3 Present Value of an Annuity; Amortization

Section 2.3 Present Value of an Annuity; Amortization Secton 2.3 Present Value of an Annuty; Amortzaton Prncpal Intal Value PV s the present value or present sum of the payments. PMT s the perodc payments. Gven r = 6% semannually, n order to wthdraw $1,000.00

More information

Time Value of Money Module

Time Value of Money Module Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the

More information

1. Math 210 Finite Mathematics

1. Math 210 Finite Mathematics 1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

More information

Section 5.3 Annuities, Future Value, and Sinking Funds

Section 5.3 Annuities, Future Value, and Sinking Funds Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme

More information

FINANCIAL MATHEMATICS

FINANCIAL MATHEMATICS 3 LESSON FINANCIAL MATHEMATICS Annutes What s an annuty? The term annuty s used n fnancal mathematcs to refer to any termnatng sequence of regular fxed payments over a specfed perod of tme. Loans are usually

More information

8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value

8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value 8. Annutes: Future Value YOU WILL NEED graphng calculator spreadsheet software GOAL Determne the future value of an annuty earnng compound nterest. INVESTIGATE the Math Chrstne decdes to nvest $000 at

More information

Thursday, December 10, 2009 Noon - 1:50 pm Faraday 143

Thursday, December 10, 2009 Noon - 1:50 pm Faraday 143 1. ath 210 Fnte athematcs Chapter 5.2 and 4.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

More information

EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR

EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR 8S CHAPTER 8 EXAMPLES EXAMPLE 8.4A THE INVESTMENT NEEDED TO REACH A PARTICULAR FUTURE VALUE What amount must you nvest now at 4% compoune monthly

More information

3. Present value of Annuity Problems

3. Present value of Annuity Problems Mathematcs of Fnance The formulae 1. A = P(1 +.n) smple nterest 2. A = P(1 + ) n compound nterest formula 3. A = P(1-.n) deprecaton straght lne 4. A = P(1 ) n compound decrease dmshng balance 5. P = -

More information

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

A Master Time Value of Money Formula. Floyd Vest

A Master Time Value of Money Formula. Floyd Vest A Master Tme Value of Money Formula Floyd Vest For Fnancal Functons on a calculator or computer, Master Tme Value of Money (TVM) Formulas are usually used for the Compound Interest Formula and for Annutes.

More information

Section 2.2 Future Value of an Annuity

Section 2.2 Future Value of an Annuity Secton 2.2 Future Value of an Annuty Annuty s any sequence of equal perodc payments. Depost s equal payment each nterval There are two basc types of annutes. An annuty due requres that the frst payment

More information

Mathematics of Finance

Mathematics of Finance 5 Mathematcs of Fnance 5.1 Smple and Compound Interest 5.2 Future Value of an Annuty 5.3 Present Value of an Annuty;Amortzaton Chapter 5 Revew Extended Applcaton:Tme, Money, and Polynomals Buyng a car

More information

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount owed is. P (1 + i) A

In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount owed is. P (1 + i) A Amortzed loans: Suppose you borrow P dollars, e.g., P = 100, 000 for a house wth a 30 year mortgage wth an nterest rate of 8.25% (compounded monthly). In ths type of loan you make equal payments of A dollars

More information

IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS

IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,

More information

Financial Mathemetics

Financial Mathemetics Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,

More information

Interest Rate Forwards and Swaps

Interest Rate Forwards and Swaps Interest Rate Forwards and Swaps Forward rate agreement (FRA) mxn FRA = agreement that fxes desgnated nterest rate coverng a perod of (n-m) months, startng n m months: Example: Depostor wants to fx rate

More information

ANALYSIS OF FINANCIAL FLOWS

ANALYSIS OF FINANCIAL FLOWS ANALYSIS OF FINANCIAL FLOWS AND INVESTMENTS II 4 Annutes Only rarely wll one encounter an nvestment or loan where the underlyng fnancal arrangement s as smple as the lump sum, sngle cash flow problems

More information

On some special nonlevel annuities and yield rates for annuities

On some special nonlevel annuities and yield rates for annuities On some specal nonlevel annutes and yeld rates for annutes 1 Annutes wth payments n geometrc progresson 2 Annutes wth payments n Arthmetc Progresson 1 Annutes wth payments n geometrc progresson 2 Annutes

More information

Mathematics of Finance

Mathematics of Finance Mathematcs of Fnance 5 C H A P T E R CHAPTER OUTLINE 5.1 Smple Interest and Dscount 5.2 Compound Interest 5.3 Annutes, Future Value, and Snkng Funds 5.4 Annutes, Present Value, and Amortzaton CASE STUDY

More information

Mathematics of Finance

Mathematics of Finance CHAPTER 5 Mathematcs of Fnance 5.1 Smple and Compound Interest 5.2 Future Value of an Annuty 5.3 Present Value of an Annuty; Amortzaton Revew Exercses Extended Applcaton: Tme, Money, and Polynomals Buyng

More information

Multiple discount and forward curves

Multiple discount and forward curves Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of

More information

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

Intra-year Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error

Intra-year Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error Intra-year Cash Flow Patterns: A Smple Soluton for an Unnecessary Apprasal Error By C. Donald Wggns (Professor of Accountng and Fnance, the Unversty of North Florda), B. Perry Woodsde (Assocate Professor

More information

AS 2553a Mathematics of finance

AS 2553a Mathematics of finance AS 2553a Mathematcs of fnance Formula sheet November 29, 2010 Ths ocument contans some of the most frequently use formulae that are scusse n the course As a general rule, stuents are responsble for all

More information

Joe Pimbley, unpublished, 2005. Yield Curve Calculations

Joe Pimbley, unpublished, 2005. Yield Curve Calculations Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

More information

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

Chapter 15 Debt and Taxes

Chapter 15 Debt and Taxes hapter 15 Debt and Taxes 15-1. Pelamed Pharmaceutcals has EBIT of $325 mllon n 2006. In addton, Pelamed has nterest expenses of $125 mllon and a corporate tax rate of 40%. a. What s Pelamed s 2006 net

More information

DEFINING %COMPLETE IN MICROSOFT PROJECT

DEFINING %COMPLETE IN MICROSOFT PROJECT CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,

More information

Small pots lump sum payment instruction

Small pots lump sum payment instruction For customers Small pots lump sum payment nstructon Please read these notes before completng ths nstructon About ths nstructon Use ths nstructon f you re an ndvdual wth Aegon Retrement Choces Self Invested

More information

Interest Rate Futures

Interest Rate Futures Interest Rate Futures Chapter 6 6.1 Day Count Conventons n the U.S. (Page 129) Treasury Bonds: Corporate Bonds: Money Market Instruments: Actual/Actual (n perod) 30/360 Actual/360 The day count conventon

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

Number of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000

Number of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000 Problem Set 5 Solutons 1 MIT s consderng buldng a new car park near Kendall Square. o unversty funds are avalable (overhead rates are under pressure and the new faclty would have to pay for tself from

More information

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Luby s Alg. for Maximal Independent Sets using Pairwise Independence Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

More information

SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:

SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background: SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and

More information

Texas Instruments 30X IIS Calculator

Texas Instruments 30X IIS Calculator Texas Instruments 30X IIS Calculator Keystrokes for the TI-30X IIS are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the

More information

ADVA FINAN QUAN ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS VAULT GUIDE TO. Customized for: Jason (jason.barquero@cgu.edu) 2002 Vault Inc.

ADVA FINAN QUAN ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS VAULT GUIDE TO. Customized for: Jason (jason.barquero@cgu.edu) 2002 Vault Inc. ADVA FINAN QUAN 00 Vault Inc. VAULT GUIDE TO ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS Copyrght 00 by Vault Inc. All rghts reserved. All nformaton n ths book s subject to change wthout notce. Vault

More information

Texas Instruments 30Xa Calculator

Texas Instruments 30Xa Calculator Teas Instruments 30Xa Calculator Keystrokes for the TI-30Xa are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the tet, check

More information

Hedging Interest-Rate Risk with Duration

Hedging Interest-Rate Risk with Duration FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton

More information

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

More information

Present Values and Accumulations

Present Values and Accumulations Present Values an Accumulatons ANGUS S. MACDONALD Volume 3, pp. 1331 1336 In Encyclopea Of Actuaral Scence (ISBN -47-84676-3) Ete by Jozef L. Teugels an Bjørn Sunt John Wley & Sons, Lt, Chchester, 24 Present

More information

= i δ δ s n and PV = a n = 1 v n = 1 e nδ

= i δ δ s n and PV = a n = 1 v n = 1 e nδ Exam 2 s Th March 19 You are allowe 7 sheets of notes an a calculator 41) An mportant fact about smple nterest s that for smple nterest A(t) = K[1+t], the amount of nterest earne each year s constant =

More information

Problem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative.

Problem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative. Queston roblem Set 3 a) We are asked how people wll react, f the nterest rate on bonds s negatve. When

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

More information

2.4 Bivariate distributions

2.4 Bivariate distributions page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

More information

Stress test for measuring insurance risks in non-life insurance

Stress test for measuring insurance risks in non-life insurance PROMEMORIA Datum June 01 Fnansnspektonen Författare Bengt von Bahr, Younes Elonq and Erk Elvers Stress test for measurng nsurance rsks n non-lfe nsurance Summary Ths memo descrbes stress testng of nsurance

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

6. EIGENVALUES AND EIGENVECTORS 3 = 3 2

6. EIGENVALUES AND EIGENVECTORS 3 = 3 2 EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a non-zero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :

More information

Uncrystallised funds pension lump sum payment instruction

Uncrystallised funds pension lump sum payment instruction For customers Uncrystallsed funds penson lump sum payment nstructon Don t complete ths form f your wrapper s derved from a penson credt receved followng a dvorce where your ex spouse or cvl partner had

More information

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

1 Approximation Algorithms

1 Approximation Algorithms CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons

More information

A Model of Private Equity Fund Compensation

A Model of Private Equity Fund Compensation A Model of Prvate Equty Fund Compensaton Wonho Wlson Cho Andrew Metrck Ayako Yasuda KAIST Yale School of Management Unversty of Calforna at Davs June 26, 2011 Abstract: Ths paper analyzes the economcs

More information

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL

More information

Staff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall

Staff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall SP 2005-02 August 2005 Staff Paper Department of Appled Economcs and Management Cornell Unversty, Ithaca, New York 14853-7801 USA Farm Savngs Accounts: Examnng Income Varablty, Elgblty, and Benefts Brent

More information

Chapter 15: Debt and Taxes

Chapter 15: Debt and Taxes Chapter 15: Debt and Taxes-1 Chapter 15: Debt and Taxes I. Basc Ideas 1. Corporate Taxes => nterest expense s tax deductble => as debt ncreases, corporate taxes fall => ncentve to fund the frm wth debt

More information

IS-LM Model 1 C' dy = di

IS-LM Model 1 C' dy = di - odel Solow Assumptons - demand rrelevant n long run; assumes economy s operatng at potental GDP; concerned wth growth - Assumptons - supply s rrelevant n short run; assumes economy s operatng below potental

More information

ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING

ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 610-519-4390,

More information

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

More information

The Application of Fractional Brownian Motion in Option Pricing

The Application of Fractional Brownian Motion in Option Pricing Vol. 0, No. (05), pp. 73-8 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qng-xn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com

More information

Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide

Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide Reportng Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (ncludng SME Corporate), Soveregn and Bank Instructon Gude Ths nstructon gude s desgned to assst n the completon of the FIRB

More information

Interest Rate Fundamentals

Interest Rate Fundamentals Lecture Part II Interest Rate Fundamentals Topcs n Quanttatve Fnance: Inflaton Dervatves Instructor: Iraj Kan Fundamentals of Interest Rates In part II of ths lecture we wll consder fundamental concepts

More information

LIFETIME INCOME OPTIONS

LIFETIME INCOME OPTIONS LIFETIME INCOME OPTIONS May 2011 by: Marca S. Wagner, Esq. The Wagner Law Group A Professonal Corporaton 99 Summer Street, 13 th Floor Boston, MA 02110 Tel: (617) 357-5200 Fax: (617) 357-5250 www.ersa-lawyers.com

More information

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

More information

SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW.

SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW. SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW. Lucía Isabel García Cebrán Departamento de Economía y Dreccón de Empresas Unversdad de Zaragoza Gran Vía, 2 50.005 Zaragoza (Span) Phone: 976-76-10-00

More information

EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN

EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson - 3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson - 6 Hrs.) Voltage

More information

Ameriprise Financial Services, Inc. or RiverSource Life Insurance Company Account Registration

Ameriprise Financial Services, Inc. or RiverSource Life Insurance Company Account Registration CED0105200808 Amerprse Fnancal Servces, Inc. 70400 Amerprse Fnancal Center Mnneapols, MN 55474 Incomng Account Transfer/Exchange/ Drect Rollover (Qualfed Plans Only) for Amerprse certfcates, Columba mutual

More information

Traffic-light a stress test for life insurance provisions

Traffic-light a stress test for life insurance provisions MEMORANDUM Date 006-09-7 Authors Bengt von Bahr, Göran Ronge Traffc-lght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE-113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax

More information

The Cox-Ross-Rubinstein Option Pricing Model

The Cox-Ross-Rubinstein Option Pricing Model Fnance 400 A. Penat - G. Pennacc Te Cox-Ross-Rubnsten Opton Prcng Model Te prevous notes sowed tat te absence o arbtrage restrcts te prce o an opton n terms o ts underlyng asset. However, te no-arbtrage

More information

Activity Scheduling for Cost-Time Investment Optimization in Project Management

Activity Scheduling for Cost-Time Investment Optimization in Project Management PROJECT MANAGEMENT 4 th Internatonal Conference on Industral Engneerng and Industral Management XIV Congreso de Ingenería de Organzacón Donosta- San Sebastán, September 8 th -10 th 010 Actvty Schedulng

More information

Nordea G10 Alpha Carry Index

Nordea G10 Alpha Carry Index Nordea G10 Alpha Carry Index Index Rules v1.1 Verson as of 10/10/2013 1 (6) Page 1 Index Descrpton The G10 Alpha Carry Index, the Index, follows the development of a rule based strategy whch nvests and

More information

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

More information

Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc.

Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc. Underwrtng Rsk By Glenn Meyers Insurance Servces Offce, Inc. Abstract In a compettve nsurance market, nsurers have lmted nfluence on the premum charged for an nsurance contract. hey must decde whether

More information

greatest common divisor

greatest common divisor 4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no

More information

Hollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA )

Hollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA ) February 17, 2011 Andrew J. Hatnay ahatnay@kmlaw.ca Dear Sr/Madam: Re: Re: Hollnger Canadan Publshng Holdngs Co. ( HCPH ) proceedng under the Companes Credtors Arrangement Act ( CCAA ) Update on CCAA Proceedngs

More information

Effective December 2015

Effective December 2015 Annuty rates for all states EXCEPT: NY Prevous Index Annuty s effectve Wednesday, December 7 Global Multple Index Cap S&P Annual Pt to Pt Cap MLSB Annual Pt to Pt Spread MLSB 2Yr Pt to Pt Spread 3 (Annualzed)

More information

Multiple-Period Attribution: Residuals and Compounding

Multiple-Period Attribution: Residuals and Compounding Multple-Perod Attrbuton: Resduals and Compoundng Our revewer gave these authors full marks for dealng wth an ssue that performance measurers and vendors often regard as propretary nformaton. In 1994, Dens

More information

Abstract # 015-0399 Working Capital Exposure: A Methodology to Control Economic Performance in Production Environment Projects

Abstract # 015-0399 Working Capital Exposure: A Methodology to Control Economic Performance in Production Environment Projects Abstract # 015-0399 Workng Captal Exposure: A Methodology to Control Economc Performance n Producton Envronment Projects Dego F. Manotas. School of Industral Engneerng and Statstcs, Unversdad del Valle.

More information

Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT

Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

More information

Effective September 2015

Effective September 2015 Annuty rates for all states EXCEPT: NY Lock Polces Prevous Prevous Sheet Feld Bulletns Index Annuty s effectve Monday, September 28 Global Multple Index Cap S&P Annual Pt to Pt Cap MLSB Annual Pt to Pt

More information

Kiel Institute for World Economics Duesternbrooker Weg 120 24105 Kiel (Germany) Kiel Working Paper No. 1120

Kiel Institute for World Economics Duesternbrooker Weg 120 24105 Kiel (Germany) Kiel Working Paper No. 1120 Kel Insttute for World Economcs Duesternbrooker Weg 45 Kel (Germany) Kel Workng Paper No. Path Dependences n enture Captal Markets by Andrea Schertler July The responsblty for the contents of the workng

More information

A Critical Note on MCEV Calculations Used in the Life Insurance Industry

A Critical Note on MCEV Calculations Used in the Life Insurance Industry A Crtcal Note on MCEV Calculatons Used n the Lfe Insurance Industry Faban Suarez 1 and Steven Vanduffel 2 Abstract. Snce the begnnng of the development of the socalled embedded value methodology, actuares

More information