TIME VALUE OF MONEY PROBLEMS CHAPTERS THREE TO TEN


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1 TIME VLUE OF MONEY PROBLEMS CHPTERS THREE TO TEN Probles In how any years $ will becoe $265 if = %? 265 ln n years ln( 2 In how any years will an aount double if = 76%? ln 2 n 9 46 years ln76 3 In how any years will an aount triple if = 9%? ln3 n 2 75years ln9 4 t what rate of interest $ will becoe $265 in 2 years? % 5 t what rate of interest an aount will double in 5 years? % 6 t what an aount will triple in 6 years? % 7 We are going to receive $2 at the beginning of the 3 th year If = 3%, what is the present value? 3 PV ( 3 29 $2754 run J Praash Probles 
2 8 You will receive $3 in the 25 th year What is the present value at the beginning of year 2 if =5%? PV 3 ( 5 4 $ We receive $3 in the 5 th year We will put that oney in an account earning = 5% What will the account be worth at the 4 th year? PV 3 $44353 $443 5 (5 FV 3(5 25 $,59648 $,59 FV4 443,59 $,622 What is the future value of $5 in year 4 if = Year FV 5 ((5(2( $2465 Obtain the future value of $ if % for the first 5 years, next five years =% and next 3 years 3% in 3 years FV ( 5 ( 5 (3 3 $396 2 In how any years an account will double at a given rate of interest? = 5% (Use Rule n (( t what rate of interest an account will double in 2 years? ((2 36% run J Praash Probles  2
3 4 ssue that during the first 3 years is %, next 3 years is 2%, and next 6 years is 2% If you deposit $ today, how uch will you have in years? FV ( 3 (2 3 (2 5 $ We receive $ at the beginning of 5 th year If = 6% for first 2 years, 2% next years and reaining period is 6% What is the PV of the aount? PV $5984 $6 2 9 (6 (2 (6 6 We receive $ at the end of year If = % what is the PV today? PV ( $ 7 In how any years $24 will becoe $24 if = 3%? 24 ln 24 n 3 2years ln( 3 8 The current price of the island of Manhattan is $5trillion 275 years ago the island was sold by native Indians for $24 If =5% per year, was the island sold cheaply? FV (5 $864 x 8 Yes, the island was sold too cheaply!! 9 Obtain the PV of deposits of $, $2, $ 3 in years, 2, 3 if rate of interest is 5 % PV ( $ (5 (5 2 Obtain the PV of deposits of $, $2, $ 3 in beginning of years, 2, 3 if rate of interest is 5 % 2 3 PV $ (5 (5 run J Praash Probles  3
4 2 Obtain the PV of deposits of $, $2, $ 3 a deposits at end of the year b in beginning of years, 2, 3 if siple rate of interest is 5 % 2 3 a PV $53 79 (5 (5(2 (5(3 2 3 b PV $56 32 (5 (5(2 22 You deposit $ in an account every other year for 6 years If = %, what is the PV of these deposits assuing a siple rate of interest b copound rate of interest a PV $2 72 ((2 ((4 ((6 b PV $ ( ( ( 23 You deposit $ in an account every other year for 6 years If = %, what is the FV of these deposits assuing a siple rate of interest b copound rate of interest a FV (4( (2( 4 2 $ b FV ( ( $ Consider the following series S S because coon ratio (4 is > run J Praash Probles  4
5 Practice Probles {Solutions ppear Below} You deposit a lup su of $5, in an account today ssuing the following interest rate structure: K=6% for the first years, 8% for the second years and % for the last years What is the future value of the deposit after 3 years if the account offers (a siple interest rates? (b annually copounded interest rates? (c onthly copounded interest rates? (d continuously copounded interest rates? 2 You expect to receive $,, ten years fro today Suppose the interest rates will be 8% for the first 5 years and % for the next five years, what is the present value if we have a siple interest rates? b annually copounded interest rates? c onthly copounded interest rates? d continuously copounded interest rates? 3 You have the choice between two accounts pays 8% copounded continuously and account B pays 8% copounded seiannually Which one offers a better deal and why? 4 What is the actual yield (or PR of (a 5% copounded quarterly? (b 8% copounded onthly (c 2% copounded seiannually (d % copounded continuously? 5 What is the equivalent noinal quarterly copounded rate of (a 8% copounded annually? (b % copounded onthly? (c 2% copounded seiannually? (d % copounded continuously? 6 What is the equivalent noinal continuously copounded rate of interest of (a 8% copounded annually? (b % copounded onthly? (c 2% copounded seiannually? (d % copounded quarterly? 7 Suppose you deposit $, at the end of each year for 2 years Find the FV at the end of the 3 th year with (a % siple interest rate? (b % annually copounded interest rate? run J Praash Probles  5
6 (c % continuously copounded interest rate? 8 Suppose you deposit $, at the end of each year for years Find the present value of the annuity with (a % siple interest rate? (b % annually copounded interest rate? (c % continuously copounded interest rate? 9 You deposit $5, in an account that pays 6% copounded annually for 6 years and 8% thereafter What will be your balance after 5 years? What is the present value (cash value of an endowent of $, that you expect to receive at your retireent at age 65? You are 4 years old today The interest rate is expected to be 8% What will be the cash value if interest rates are expected to be 5% You expect to pay $5, for your child s education when she is 8 How uch ust you deposit when the child is 2 years old to insure availability? K=8% 2 You deposit $7, in an account today What is the future value after 25 years with (a 85% siple interest (b 85% interest copounded annually (c 85% interest copounded quarterly (d 85% interest copounded continuously What can you iply fro this exercise? 3 What is the future value of $5, per year after 5 years with (a 95% siple interest (b 95 interest copounded annually (c 95% interest copounded continuously? 4 What is the present value of $2, per year paid for 8 years with (a 9% siple interest (b 9% interest copounded annually (c 9% interest copounded continuously What can you iply fro this exercise? 5 What is the effective annual rate (ER of a 5%, 8% copounded quarterly b 2%, 8% copounded onthly run J Praash Probles  6
7 c 9% and % copounded continuously? For questions 67 find the present value of the cash flows K=8% 6 $6,/year for 2 years and then perpetual growth of 2%/year 7 $9,/year for 5 years and then negative growth of 45%/year for another 6 years 8 Miai utos offers a loan at % on a new $, listed car You pay $, down and then $3 a onth for the next 3 onths Kendall utos does not offer a loan but will give you $, off the list price Which copany is offering the better deal? 9 You own an oil pipeline that generates $2 illion cash flow over the next year The pipeline s operating costs are negligible and it's is expected to last for a very long tie Unfortunately, the volue of oil shipped is declining and cash flows are expected to decline by 4% per year K=% (a What is the PV of the cash flow if it is assued to last forever? (b What is the PV of the cash flow if the pipeline is scrapped after 2 years? 2 Your fir expects to earn $2, this year and projects a growth rate in earnings of 5% /year thereafter K=% What is the PV of the earnings if: (a It expects growth forever (b It expects growth for another year and then it shuts down (c It expects growth for another years and then level earnings forever 2 You have ade the following proposition: Pay us $2/year for years and we will pay you $2/year thereafter in perpetuity Is this a fair deal, what is the interest rate? 22 BC corp borrows $5 illion for 2 years at 2% It is required to set up an escrow account to pay off the principal at aturity What will de the initial annual contribution to the fund with the following stipulations (a level payents each year (b payents grow at 6% per year ( find payents and 2 23 You need to set up a college fund at birth for your child that will pay $6, at age 8 K=2% (a How uch ust you deposit at the end of each year with annual copounding (b How uch ust you deposit at the end of each onth with onthly copounding (c What will be the payents if they are at the beginning of the year (8 payents? 24 You contributed $2,4 per year for 4 years and then retired You expect to live another 2 years K=% What will be your annual pension chec after retireent? Suppose you ade run J Praash Probles  7
8 the preiu payents and collected the pension checs at the beginning of each year What will be your annual pension chec? 25 You contributed $3,5 per year for 4 years and the retired You expect to live another 25 years What will be your annual pension chec with (a 9% interest copounded annually (b 9% copounded continuously 26 You ust choose between two projects Project expects to earn nothing for 2 years, $5 illion in the 3 rd year and then grow at 4%/year for another 8 years and then level earnings forever Project B expects to earn nothing for three years, $6 illion in the 4 th year, continue to earn $6/year for another five years, and then have perpetual growth of 2%/year K=% 27 BC corp expects to earn $6, this quarter and expects growth of 6% per year It has a dividend payout ratio of /3 and plans to use the reainder to payoff loans in quarterly installents K=2% It needs to finance a project a year loan How uch can it afford to borrow now? 28 You set up a college fund for your child that will pay $, at age 8 You deposit $5, up front and another $5, after years K=% How uch in addition do you need to deposit each year to accuulate the required aount with (a annual (b onthly (ccontinuous copounding 29 You deposit $, at 9% How long will it tae to double, triple with (a annual (b quarterly (c continuous copounding 3 You deposit $/onth at % How long will it tae for the balance to reach $5, with onthly copounding? 3 You inherited a lupsu of $5, and put it into an account at % How long will the aount last if you with draw (a $25,/year (b 75,/year 32 You deposit $7, lupsu in an account You wish to close the account and with draw the oney when the balance reaches $5, How long will it tae if the interest rate is (a 2% copounded annually (b 2% copounded onthly 33 You invested $6, In six years you get bac $5, What is the interest rate with (a annual (b onthly (c continuous copounding 34 Investent requires $, and proises $8, after 8 years Investent B proises 75% continuously copounded return Which one should you choose? 35 You tae out a $, ortgage at 72% for 3 years Your onthly payent is $6788 However, you decide to pay an extra $ per onth How long will tae to pay off the ortgage? 36 You tae a $2, ortgage at fixed rate for 3 years What are the onthly payents if (a interest rates are 78%, (b interest rates are 9% run J Praash Probles  8
9 37 You tae a $, ortgage at fixed rate of 84% What are the onthly payents if the ortgage is for (a 3 years (b 5 years? Solution To The Probles To 37 (a FV = 5, + (6 ( + (8 ( + ( (= 7, (b FV = 5, (6 (8 ( = 5,4 (c FV = 5, + ( 6/2 2 +(8/2 2 +(/2 2 = 54,6666 (d FV = 5, e (6 ( e (8 ( e ( ( = 5, e 24 = 55,588 2 (a PV =,, / +(8(5 + ((5= 526,3579 (b PV=,, / (8 5 ( 5 = 422,58862 (c PV=,, / (+(8/2 6 (+(/2 6 = 47,9545 (d PV=,, / e (8 (5 e ( (5 =,, / e 9 = 46, (a ccount has PR = e 8  = 833% ccount B has PR = +(8/2 2  = 8,264% Therefor the answer is account 4 (a +(5/4 4  = 5865%, (b +(,8/2 2 = 9,562% (c +(2/2 2 = 2,36%, (d e  = 57% Hint: use K a = +(K /  5 Use both K = (+K a / and K a =+(K /  (a K a = 8%, K 4 =4(+8 /4 = 777% (b K 2 =%, K a =+(/2 2  =,473%, K 4 =4(+473 /4 =,836% (c K 2 =2%, K a =(+(2/2 2 =236%, K 4 =4(236 /4 =8252% (d K c =%, K a =e  =57%, K 4 =4(57 /4 = 26% 6 Use both K a =e Kc  and K c =Ln(+K a (a K a =8%, K c =Ln(8 = 7696% (b K 2 =%, K a =+(/2 2  = 473%, K c =Ln(473= 99586% (c K 2 =2%, K a = +(2/2 2 =236%, K c =Ln(236= 6538% (d K 4 =%, K a =+(/4 4 =383%, K c =Ln(383= 9877% 7 =, n=2 (a K=% Siple interest rate PV= n+((n/2 FV=,(2 +( (2(/2+((=59, (b K a =% FV=,(( 2 /( = 57,274(2,593742=48,5566 (c K c =%, FV= e nkc /e Kc  FV=, e (2 ( /e ( e ( ( = 6,74927(2783=65, =, n= (a K=% siple interest rate PV e = /(+ + /(+2K + + /(+K run J Praash Probles  9
10 PV=/ + /2 +/3+ +/9+/2 =6,68773 (b K a =% PV=((  /=6,445 (c K c =%, PV = (e nkc /e Kc PV= (e ( ( /(e = 6,4 9 F 5 =5,(6 6 (8 9 =4,783 K=8% PV=,/(8 25 = 4,679 K=5% PV =,/(5 25 = 29,5328 ount to be deposit = PV of future requireent = 5,/(8 6 = 4, (a FV= (+nk = 7,+25(85=2,875 (b FV= (+K=7,(85 25 =53,8733 (c FV=+(/ n = 7,(225 =57,3284 (d FV = e Kn = 7,e 225 =58,62 FV is higher with copound interest and as FV It is higher with continuous copounding 3 (a FV with siple interest = 5, (5 +4(95/2=24,875 (b FV with annual copounding = 5((95 5 /95=52,75 (c FV with continuous copounding = 5(e 425 /(e 95 =58, (a PV with siple interest = 2/9 +2/8+2/27+ +2/72=,643 (b PV with annual copounding = 2 ((/9 8 /9=2 (PVF=,69 (c PV with continuous copounding = 2((/e 72 /e 9 =,9 s PVlso PV is greater with siple interest 5 (I K a = ER = (+(K /  = ( = 5865=5865% and equally 8% copounded quarterly is equivalent to ER of 925% (ii For 2% copounded onthly K a = ER = (+ 2 =268= 2,68% For 8% copounded onthly K a = ER =(5 2 = 9,56% (iii For 9% copounded continuously K a = ER = e (9 = 9,42% For 8% copounded continuously K a = ER = e (8  = 9,72% 6 PV = 6, ((/8 2 /8+ (6,(2/(82/(8 2 = 45, ,556 = 85,7227 PV = PV of C 2 + PV of C 37 +g=45 +g=955 PV = 9, ((/8 5 /8+ 9,(955((955/8 6 /(845/(8 5 = 6, K/=8333 PV of payents to Miai uto = + 3(PVF PV of Miai uto Payents =, + 3((/ /8333= 8,93452 PV of Kendall uto Payents = 9, cash So the better deal will be with Miai uto 9 (a PV = C /(g= 2/((4 = 2/4 =429 illion (b PV = 2(PVFC = 2 ((96/ 2 /4= illion 2 (a PV = C /(g = 2,/(5 = 4,, run J Praash Probles 
11 (b PV = 2, ((5/ /(5=,62,4596 (c Earnings for the th year = 2, (5 = 325,77893 This are also the earnings for years 2 to PV =,62, (325,77893//( =,62,4596 +,4,83527 = 2,743, Find K PV of 2/year for years = 2((/(+K /K PV of 2/year starting in the th year and going forever is = 2/((+ Putting this together and solving for K : 2 ((/(+K /K = 2/(K(+K = 2/((+K 2= (+K K = 2 /  K = 78 =78% 22 (a FV = 5,, = P*FVF = P ((2 2 /2 then PMT= 5,,/725 PMT = 693,96253 (b FVC = 5,, = P * FVCF = P /((+K n (+g n /(g= P /(7393 =5,,/7393 =465, P 2 * FVCF = P 2 /((+K n (+g n /(g= P 2 /(9393 = 5,,/9393 = 536, (a FV = 6, = P* FVF = P ((2 8 /2= P( P= FV/FVF then P= 6,/ = 7624/year (b With onthly copounding /=, n=26 onths FV = 6, = P (( 26 /= P ( P = 6,/ = 7972 (c FV due at age 8 = 6, = P((2 8 /2(2 = P = 6,/ = 9692/year FV onthly payents due = 6, = P(( 26 /( = P = 6,/ = 78386/onth 24 (a FV of preiu payents = PV of annuity of pension checs at t=4 (at retireent FV 4 = 2,4 (( 4 /=,62, = PV = P ((/ 2 / P =,62, / = 24,768232/ year (bfv due = PV due 2,4(( 4 /( = P ((/ 2 /( P = 24, still the sae 25 (a FV preiu =3,5 ((9 4 /9= PV of checs = P((/9 25 /9 nnual pension chec = P =,82,588558/ = 2,39496 /year (b FV with continuous copounding = 3,5 (e (9 (4 /(e (9 =,323,32 PV = P ((/e (9 (25 /(e (9 = run J Praash Probles 
12 P=,323,32 / = 2,27396 /year 26 PV of project = 5(((4/ 9 /(4/( 2 + ((5 (4 8 //( = illion PV of project B = 6(((/ 6 //( 3 + (6(2/( 2/( 9 = illion The best choice is project B because it has a higher PV 27 ount available for repayent in the st quarter = 2/3 * 6, = 4, n=4 g=6/4=5/quarter /=,2/4= PV = 4, ((5/3 4 /(35=,837, (afv of 5, deposit up front = 5, ( 8 = 27,79959 at age 8 FV of 5, deposit at age = 5,( 8 =,7794 at age 8 ount required =, 27, ,7794 = 6,48247 FV = 6,48247 = PMT*FVF = P (( 8 / P = 6,48247 / = 348,32 / year (b FV of 5, up front = 5,( = 3,23467 t age 8 FV of 5, deposit at age = 5,( =,9878 ount required = 58, P = FV/FVF = 58,885655/(( /8333= 9855 (c With continuous copounding the reaining requireent =,  5,e ( (85,e ( (8 = 58,6246 P = 58,6246/ (e ( (8 /(e ( =22,98 /year 29 (a n = (Ln (F//Ln(+ = Ln(2/Ln (9 = 843 years to double with annual copounding n = Ln (3/Ln(9=27482 years to triple with annual copounding (b n = Ln (F// Ln(+ = Ln(2/4Ln(225 = years to double with quarterly copounding n = Ln(3/4Ln(225=23436 years to triple with quarterly copounding (c n = (ln(f// = Ln(3/(9 = 2268 years to double with quarterly copounding 3 (a n = (Ln+((/(F/P/(Ln(+(/ = (Ln +833(5// (2Ln(8333 = (Ln467/2Ln(8333 = years 3 n = (Ln ( (K/P/(Ln(+K (a It will last forever because K P where K is interest inflow and P is the withdraw 5,25, (b With P=75, the inherited lupsu will last 5267 years 32 =7, F=5, (a n = (Ln (5,/7,/(Ln(2 = 6725 years with annual copounding (b n = (Ln (5,/7,/2(Ln = years with onthly copounding 33 =6, F=5, n=6 years (a =(5,/6, /6  = 6499 = 6,5% (b = (5,/6, /72 = 2(28 = 5369% run J Praash Probles  2
13 (c = (/n(ln(f/p = (/6(Ln(5,/6,= 89 = 89% 34 Investent is paying: K=(Ln(8,/,/8 = 7347 = 73473% copounding continuously Investent B proises 75% continuously copounding return Then the best deal is Investent B 35 =, /= 72/2= 6 P= = 7788 n = (Ln(  (((//P/(Ln(+(/ = Ln  (,*6/(7788/(2Ln(6 = years 36 (a =2, n=3*2=36 onths /=78/2 = 65 MP= principal plus interest MP= /PVF= 2,/((/65 36 /65= 2,/ =,43974 /onth (b MP = 2,/((/75 36 /75= 2,/24288 =,69246/onth where /=9/2 =75 37 (a =, /=84/2 =7 n=3*2 =36 onths MP = /PVF =,/((/7 36 /7=,/32656 = 7684/onth (b =, /= 7 n= 5*2 =8 onths MP =,/((/7 8 /7=,/25687 = 97889/onth dditional Probles Mr I M Thrifty expects to retire at age 65 Today is his fortieth birthday, and he has planned to ae the following deposits in his savings account, which earns 6 percent copounded yearly (ll deposits will be on his birthday ge $ 454, 586, , dditionally, he expects his ban to increase the rate of interest to 8 percent after his fiftysixth birthday How uch oney will be in his ban account at age 65? 2 With reference to the proble above, how uch can he withfraw fro his ban account per year on his sixthsixth through his eightyeighth birthdays if he wants his account to be depleted by his eightyeighth birthday? 3 Continuing with the proble above, how uch could he withdraw fro his ban account on his sixthsixth through his eightyeighth birthdays if he wants to leave $2,5 to be withdrawn on his ninetyfifth birthday? run J Praash Probles  3
14 4 an arranges to repay a 6 percent, $, loan in ten equal annual installents fter his second payent, he borrows an additional $, with the following understanding: he is to pay nothing for the next two years and then to repay the balance on both loans in six equal annual payents What will his annual payent be? 5 XYZ Loan Copany has agreed to lend you $4, for eight years, if you agree to ae one lup su payent of $7,725 in eight years What annual rate of interest is involved on his loan? 6 Mr and Mrs Debt recently borrowed $64, The interest on the loan is 2 percent How uch are the annual payents necessary to aortize this ortgage if the payent schedule is as follows: payent 2 is triple payent; payent 3 is twice payent 2; payent 4 is onethird of payent3; and payent 5 and 6 are twice payent 4? 7 bright finance student would lie to buy a prograable calculator She presently has $55 in her ban account fter carefully shopping for this calculator, she discovers that the calculator she wants costs $8 ssuing that calculators of this type are expected to increase in price at a 3 percent annual rate, at what rate of interest ust she invest her $55 in order to be able to purchase this calculator eight years fro now? 8 Ms I Travel would lie to have $2, to go on the Love Boat If she presently has $,4 in a savings account earning 525 percent, how long ust she wait before she can ae her trip? Solutions To dditional Probles Solutions to Coputational Probles = 6 percent = 8 percent Interest Rate ge,,5 2, Deposits Since 6 percent will be earned until his fiftysixth birthday: $2,57 (6 2 = $23,6253 Since the interest rate earned becoes 8 percent fro his fiftysixth to his sixtyfifth birthday: $23,6253 (8 9 = $47,256 ( b run J Praash Probles (8 $,5 $4,8696 8
15 Since this oney will continue earning interest at 8 percent until his sixtyfifth birthday: $4,8696 (8 5 = $7,554 ( c 3 (8 $2, $6, gain, this oney will earn 8 percent for one additional period, therefore, $6,4928 (8 = $7,222 nswer: ount available at age 65 will be $47,256 + $7,554 + $7,222 = $6, This is basically a present value of an annuity proble Mr Thrifty has $6,36882 available fro which he wants to ae 23 equal withdrawals By the present value or annuity forula, we have ( 8 $6,36882 P 8 23 Solving for P yields: $5,973 3 This proble becoes soewhat ore coplicated because at age 65, he ust now leave enough oney in the ban account (present value of an aount for hi to have $2,5 available at age 95 Therefore, the aount available for the annuity now becoes $6,36882 inus the present value of $2,5 in 3 years $2,5 $6, (8 $6,36882 $24844 $6,238 $6, 238 (8 P 8 23 Solving for P, it now equals $5, Initially, this is a present value ($, of an annuity (payents proble, P 6 6 run J Praash Probles  5 ; P $3587
16 fter the first payent, the aount owed is $, inus $7587, or $9243 ($6 of the payent was interest fter the second payent, the outstanding balance is $ $842 = $8437 ($5545 of the second payent was interest Since $, is borrowed, the balance owed is $,8437 Furtherore, since no payents will be ade for two years, the aount owed in two years will be $,8437 (6 2 = $2,759 Therefore, payents will be Solving for P, it equals $4228 (6 $2,759 P 6 5 To solve this proble we use the forula FV = PV( + n and solve for the only unnown ( 7,9725 4,( ( ( This is a present value proble with the addition of unequal payents Using a little bit of algebra, this proble is easily solved: F 3F $64, ( 4F ( F 3F $64, 2 (2 6 4F ( F ( 6F ( F ( 2F ( F ( 4F (2 5 5 run J Praash Probles  6
17 , (2 (2 (2 (2 (2 $64, F [ $ 6 $64, F [322387] Therefore, ] Payent : F = $4,8776 Payent 2: 3F = $4,6348 Payent 3: 6F = $29,26296 Payent 4: 2F = $9,75432 Payent 5 & 6: 4F = $9, The cost of his calculator in eight years will be $8(l3 8 = $34 Therefore, he ust invest his $55 at a rate such that it will grow to $34 $55( + 8 = $34 Using the sae procedure as in proble 5, is equal to This proble involves the unnown period F ln( T / P n ln( 2, ln,4 n ln( n 5683 n 697 or approxiately seven years THEORETICL PROBLEMS: THE TIME VLUE OF MONEY Show that the noinal rate of interest per year for an aount P to becoe F in n years in equal to F n / P 2 Show that the nuber of years it will tae an aount P to becoe F at a noinal rate of interest when there are copoundings per year is equal to run J Praash Probles  7
18 Ln( F / P n Ln P P 3 Show that the nuber of years in which an annuity of $ per period will grow to $, assuing the noinal interest rate with copoundings per year, is equal to Ln n Ln 4 Show that if equal withdrawals of $P are ade at the end of each period of length / year with an initial deposit of P, the nuber of years required to exhaust the account will be equal to P Ln P n Ln for = for 5 Show that the relationship between the annual effective and noinal rates of interest is independent for the aount of oney deposited and nuber of years left in an account That is, show that the relationship between i and does not depend on $ (the aount deposited or n 6 Show that in the case of continuous copounding the effective rate of interest i and the noinal rate of interest are related as follows: i = e  and = Ln( + i P 7 ssue that a deposit of $ was ade today If P the noinal rates of interest are, 2,, during periods,2,,n show that the future value of $ at the end of year n, assuing copoundings per year, is run J Praash Probles  8
19 2 n and assuing continuous copounding in each year is e +2++n 8 Using interest rates and periods as in the pervious question, show that the present value of an aount to be received at the end of n years is 2 n for copoundings per year and e (+2++n for the case of continuous copounding 9 Show that for the continuous copounding case and FV c = (PV c e n (PV c = (FV c e n Show that the noinal rate of interest that is required for an aount P to be worth a value of F in n years, assuing continuous copounding, is equal to Ln( F / P n Show that the nuber of years required for an aount P to be worth a value of F copounded continuously is equal to n Ln( F / P 2 Show that the nuber of years in which an annuity of $ will grow to $, assuing a noinal rate of interest, copounding continuously is n Ln ( e 3 Show that the nuber of years required for equal withdrawals of at the end of each year to exhaust an initial deposit of is run J Praash Probles  9
20 ln ( e for ( e n = for ( e Solutions to Theoretical Probles To show this, we begin with the definition of the future value of an aount for the general copounding case [equation (26] F P Using the rules for exponents, we solve for as n or or / n F P F P / n F P / n 2 Starting once again with equation (26 but this tie solving for n, we have F P Taing the natural log of both sides, we have n n ln F ln P n ln Solving for n yields ln F ln P ln( F / P n ln ln run J Praash Probles  2
21 run J Praash Probles We use basically the sae procedure as in the previous proble but we start with the definition of the future value of an annuity Taing the log of both sides and solving for n, we have 4 gain we apply the sae procedure, this tie starting with the present value of an annuity forula Let P represent the equal withdrawals and P the initial aount Then Rearranging ters, we have n n n ln ln ln ln n or n / P P n n P P
22 Taing logs and solving for n, we get P ln P nln P ln P P n for P ln otherwise 5 To show this, let be the noinal rate of interest and i the annual effective rate of interest The future value, F, of any aount can be written as ( i n Equating these, we have n F F for the no in al int erest rate; and for the effective annual rate n ( i n Solving this expression for yields or ( i [( i / / ] Since this expression for contains no ters with or n in it, the relationship ust be independent of these variables 6 To show this, let us start with the definition of future value as in the previous proble, but now assuing continuous copounding: e n = F for the continuous copounding case; and ( + i n = F to deterine the annual effective rate run J Praash Probles  22
23 Equating the two expressions and canceling the s, we have e n = ( + i n Taing logs and solving for, we have n ln( i ln( i n Instead of solving for, we could have solved for i as ( e n / n i i e 7 Note that each period here represents one year The future value of an aount at the end of one year, F, can be written as F The future value of the aount at the end of the second year will be the future value of F after one year of earning interest at a rate 2 Thus, F F Carrying this out for n years, we would get 2 F Perfor the sae steps to show this in the case of continuous copounding That is, F e 2 F2 F e e e For n periods, the future value, F, is K 2 e K 2 F e 2 n run J Praash Probles  23
24 8 The present value of an aount to be received one year fro now with an interest rate of is given by PV The present value of this aount if it were received after another year (two years fro tie at interest rate 2 for year 2 would be PV 2 2 PV Substituting for PV, we have PV 2 2 Carrying this out for n years would yield PV 2 n For the case of continuous copounding, PV 2 PV PV e e 2 e ; and e 2 e ( 2 Extending this for n years, we have PV 9 Fro equation (222, FV c e P( e e ( 2 n n Multiplying both sides by e n, we have c FV e n P( e n e n run J Praash Probles  24 n e e
25 ( n P e e e n P( e e hence, PV c c FV e n P( e ; e n PV c, or by dividing through by e n, PV c e n FV c Fro equation (22, the present value of an aount to be received in the future is given by or P Fe e n n F / P Taing the log of both sides and solving for, we have or n ln e ln( F / P ln( F / P n Fro the above proble, we siply solve for n instead of That is, run J Praash Probles  25
26 so nln e ln( F / P, n ln( F / P 2 The future values of an annuity of dollars (call its value is given by equation (222: e e n Rearranging ters, we have e n ( e Taing the log of both sides and solving for n yields n ln e ln ( e or n ln ( e 3 The present value of an annuity of dollars is given in equation (223 as Rearranging ters, we have ( e e n or e e n n ( e ( e run J Praash Probles  26
27 Taing the log of both sides and solving for n, we have or n ln n ln ( e ( e Note that if the ter ( e is less than or equal to zero, the log function is identified; hence, this expression for n is only valid if the above ter is greater than zero If the ter is less than or equal to zero, it iplies that withdrawals of can never exhaust the initial deposit run J Praash Probles  27
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