COMPOUND INTEREST (30 August 2003)

Transcription

1 COMPOUND INTEREST 30 August 2003 We note the mateial used in this section is patially based on the efeence book: EF Haeussle and RS Paul Intoductoy Mathematical Analysis fo Business, Economics and the Life and Social Sciences 9th ed, Pentice-Hall, 1999 Some copies ae available in the eseve section of the libay Suppose an amount of money $1, 000 is deposited in a bank with an annual inteest ate of 10% If inteest is calculated only on the deposited amount, then at the end of the fist yea we obtain $1, % = $100 as inteest, and at the end of the second yea we obtain $1, % + $1, % = $100 + $100 = $200 as inteest When the inteest is calculated as above, it is called simple inteest In eality, fo the second yea, most banks will calculate inteest on the inteest $100 obtained in the fist yea togethe with the inteest on the pincipal $1,000, that is, $1, % = $110 So the total inteest obtained at the end of the second yea is $210 This is called compound inteest The total amount of the pincipal and the inteest to be obtained at a futue time is called the futue value of the pincipal Example Suppose the pincipal $10,000 is deposited in a bank account fo thee yeas with a fixed annual inteest ate of 6% Find the futue value of the pincipal at the end of the thid yea At the end of the fist yea, we have $10, $10, 000 6% = $10, = $10, 600 At the end of the second yea, we have $10, $10, 600 6% = $10, = $11, 236 At the end of the thid yea, we have $11, $11, 236 6% = $11, = $11, We can simplify the above calculation to obtain the compounded amount at the end of thid yea as $10, = $11,

2 Example Repeat the above question, assuming that the inteest ate is compounded quately We need to know the inteest ate fo each quate Since the annual inteest ate is 6%, so the inteest ate fo each quate is 15% The inteest on the pincipal $10,000 will be compounded 3 4 = 12 times We ae asked to compute the futue value To cut the stoy shot, we condense the steps of computation as The answe is accuate to two decimal places $10, $10, = $11, We emak that the amount $11,95618 is highe than the amount $ 11,91016 obtained in the pevious example We now look at the geneal case Let the pincipal to be P and the annual inteest ate to be Suppose futhe that the inteest is compounded m times a yea, then the futue value fo P afte t yeas is given by P 1 + m tm We take m to be 4 if inteest is compounded quately, m = 12 if the inteest is compounded monthly We take m = 1 if inteest is calculated once a yea In geneal, we adjust the values of m accoding to the poblem being consideed Example Let the annual inteest ate be 4% and say it is compounded annually How many yeas ae needed so that the compounded amount will double the pincipal P? We need to solve 2P = t P, whee t stands fo the numbes of yeas Taking logaithm see Chapte 2, section 2 3 on both sides of the equation yields t = log yeas log 104 Execises 1 5%, 2 8% Repeat the above poblems with inteest ates eplaced by A Convention: We shall assume that the inteest ates given in all questions ae meant to be annual ate unless the question stated othewise Execises A pincipal of $100 is invested at an inteest ate 9% fo one yea Detemine the futue value if the inteest is compounded 1 annually, $109 2 semi-annually, $ quately, $ monthly, $

3 5 weekly $ Ponde! What do you think will happen if the inteest is compounded in shote and shote intevals days, minutes, seconds,? Can the inteest obtained fo a yea at the above ate be made abitaily lage by suitably adjusting the peiod of compounding? PRESENT VALUES Suppose we pay a peson an amount of S afte 3 months unde the annual inteest ate What amount of money, $P, should we invest ight now so as to obtain the amount S at the end of thee months? We call P the pesent value of S Clealy, the amount P depends on S, the length of time involved, the inteest ate, and the peiod of compounding Assuming that the annual inteest ate is and that inteest is compounded monthly, then the elationship between S and P is given by S = P Hence P = S Note that P is less than S In geneal, if t yeas fom now a commodity will woth S, then its pesent value P, unde the annual inteest ate compounded m times a yea, is given by P = S 1 + m tm Thus, the pesent value P of S is obtained by discounting S by the inteest ate fo an appopiate numbe of peiods Example Suppose we ae to be paid $15,000 at the end of six yeas time Find the pesent value of the $15,000 assuming that the inteest ate has been fixed at 6%, compounded monthly fo the next six yeas So Let P be the pesent value Then we have $15, 000 = P P = $15, = $15, $15, = $10, The answe is given to one decimal place So the pesent value of $15,000 discounted fo six yeas unde the annual inteest ate 6% compounded monthly is about $10, Execises Find the pesent value of the given futue payment at the specified inteest ate: 1 $6,500 due in 20 yeas at 5% compounded annually; $2, $4,500 due in 12 yeas at 7% compounded semiannually; $1,

4 3 $8,500 due in 75 yeas at 6% compounded quately; $5, $760 due in thee yeas at 18% compounded monthly; $ $10,000 due in fou yeas at 95% compounded daily; $6, A bank account pays 12% annual inteest, compounded monthly How much must be deposited now so that the account contains exactly $10,000 at the end of the one yea? $8,87449 Moe Examples Example Suppose John owes Tim two sums of money: one being $1,000 due in two yeas, and anothe one being $600 due in 5 yeas How much should John pay Tim if he wants to pay back all the debts at pesent unde the annual inteest ate 8% compounded quately? We note that all values ae to be bought to the pesent Let $x be the pesent value of the total debt to be paid at pesent Then $1,000 will be discounted ove 2 yeas that is, 8 quates and the $600 will be discounted ove 5 yeas 20 quates Thus x = 1, in $ Example A debt of $3,000 due in 6 yeas is to be paid off in 3 installments: $500 at pesent, $1,500 in 3 yeas, and a final payment in 5 yeas Find the last payment if the inteest ate is 6% compounded annually? Method I All values ae bought back to pesent fo compaison Let $x be the payment at the end of the fifth yea Thus 3, = , x Multiplying on both sides of this equation by and solving fo x yields x = 3, , Method II All values ae bought to the end of the fifth yea Then x , = 3, That is, x = 3, , which is the same as in method I but we have saved one step Example Suppose a pincipal of $20,000 is invested, and we get back etuns $10,000, $8,000 and $6,000 at the ends of the second, thid and fifth yea espectively If the annual inteest ate is 7%, and the pesent values of the etuns ae compounded annually, is this a good investment? Since the sum of the pesent values of the etuns is which is less than $20,000; a bad investment 10, , , ,

5 Execises 1 A debt of $550 due in fou yeas and $550 due in five yeas is to be epaid by a single payment now Find the payment due if an annual inteest ate of 10%, compounded quately, is assumed ans $ A debt of $600 due in thee yeas and $800 due in fou yeas is to be epaid by a single payment two yeas fom now If the inteest ate is 8% compounded semiannually, how much is the payment? ans $ 1, A debt of $5,000 due five yeas fom now and $5,000 due ten yeas fom now is to be epaid by a payment of $2,000 at the end of two yeas, a payment of $4,000 at the end of fou yeas, and a final payment at the end of six yeas If the inteest ate is 7% compounded annually, how much is the final payment? ans $ 1,96328 GEOMETRIC SEQUENCES We efe to the Appendix B2 pp textbook , but excluding the applications in the execises of the ANNUITIES An annuity is a sequence of payments at fixed intevals The pesent value of an annuity is the sum of the pesent values of the payments We usually assume that the payments ae made at the end of the peiods We now deive a fomula fo the pesent value of an annuity Suppose an annuity of amount R will be paid ove n peiods ie, the amount R will be paid at the end of each peiod Suppose is the inteest ate pe peiod Then the pesent value of the annuity is given by R R R1 + n n n n n 5

6 Example Find the pesent value of an annuity of monthly payment $100 fo of 6% compounded monthly yeas at an inteest ate Hee we have R = $100, = 006/12 = 0005, and the numbe of compounding peiods is = 42 Thus, the annuity is equal to A n = $ $3, Example Suppose an annuity spead ove seven yeas consists of $2, 000 being paid at the ends of the fist thee yeas and $5, 000 being paid at the ends of the next fou yeas Given the inteest ate is 5% compounded annually, find the pesent value of the annuity The pesent value in dollas of the fist thee yeas is given by and that of the next fou yeas can be witten as Thus, the pesent value of the annuity is given by 2, , Pesent value fo 7 yeas Pesent value fo 3 yeas 5, 000 = , , , , , 000 = , , , = 20, in $ Execises Find the pesent value of the given annuity: 1 $ 500 pe yea fo five yeas at the ate of 7% compounded annually, 2 $ 2,000 pe quate fo yeas at the ate of 8% compounded quately ans $2,05010, $29,98406 The following example intoduces the concept of the pesent value of an annuity due in compaison with the pesent value of an annuity Example The pemiums on an insuance policy ae $50 pe quate, payable, ove one yea, at the beginning of each quate If the policy holde wishes to pay off all the pemiums in advance, and suppose the inteest ate is 4% compounded quately, how much should be paid? 6

7 Notice that the fist payment does not need to be discounted, and the discounts fo the emaining thee payments ae at the beginning of each quate Thus the pesent value of the annuity is given by $50 + $ =$ $19705 The above situation is called the pesent value of an annuity due We now conside the geneal situation of the pesent value of an annuity due An annuity due consists of n peiodic payments, each of amount R, to be paid at the beginning of each peiod Suppose the inteest ate fo a peiod is The pesent value of the annuity due is given by R + R R1 + n n n n It is instuctive to compae this fomula with that of the pesent value of an annuity Execise Suppose we have an annuity due with $800 to be paid at the beginning of each six-month peiod fo six yeas at the inteest ate of 7% compounded semiannually Find the pesent value of the annuity due ans $8,00124 In contast to the pesent value of an annuity, we define the amount of the futue value of an annuity to be the sum of all payments being bought to the end of the whole peiod Suppose thee ae n equal payments, each of amount R, and is the inteest ate fo each peiod Then the amount of the annuity its futue value is R1 + n 1 + R1 + n R R1 + + R =R n n =R n n 1 Notice that the last payment being aleady at the end of the whole peiod is not multiplied by any facto It is clea fom the above desciption that the mathematical content of the amount of an annuity is simila to that of a pesent value of an annuity it involves a geometic seies Simila to the concept of pesent value of an annuity due descibed in a pevious example, thee is also anothe concept known as an amount of an annuity due This is illustated in the following example 7

8 Example At the beginning of each quate, $50 is deposited into a saving account that pays an annual inteest ate 6% compounded quately Find the balance in the account at the end of 3 yeas This final balance is called the amount of the futue value of an annuity due Thus the amount of the annuity due in $ is equal to = = = in $ We now conside the geneal situation of the amount of an annuity due Suppose peiodic payments ae made at the beginning of each peiod fo n peiods and that the inteest ate is pe peiod Then the sum of the futue values of all the payments at the end of the n peiods is called the amount of an annuity due R1 + + R R1 + n n n n 1 It is instuctive to compae this fomula with that of the amount of an annuity obtained ealie Execises Suppose that an annuity due is given such that $ 1,200 pe yea fo 12 yeas at the ate of 8% compounded annually Find the amount of the annuity due ans $24,59436 SINKING FUNDS A sinking fund is a fund into which peiodic payments ae made in ode to satisfy a futue obligation It is like a futue value amount of an annuity due that we have encounteed befoe Fo example, assume that amount S is equied at the vey end of n peiods and that the inteest ate is pe peiod Then S + R1 + + R R1 + n n n 1 So R = S 1 + n 1 8

9 Example Suppose a machine costing $7,000 is to be eplaced at the end of eight yeas, at which time it will have a salvage value of $700 In ode to povide money at that time to puchase a new machine with the same cost, a sinking fund is set up The amount of the fund at that time is expected to be the diffeence between a new machine cost and the salvage value $700 If equal payments ae to be paid into the fund at the end of each quate with inteest ate 8% compounded quately, what should each payment be? At the end the accumulated value in the fund should be equal to $7, 000 $700 = $6, 300 Let R be the peiodic payment The solution is $6, R R Thus R = $6, 300 $ Example A ental fim estimates that, if puchased, a machine will yield an annual net etun of $1,000 fo 6 yeas, afte which the machine will be wothless How much should the fim pay fo the machine if it wants to ean 7% annually on its investment ie, the cost of the machine and also set up a sinking fund to eplace a new machine at the same cost? Fo the fund, assume annual payments and a ate 5% compounded annually Let x to be the cost of the machine Each yea the fim would like to ean 007x The amount left afte we deduct 007x fom the actual etun of $1, 000 is $1, x This amount will be used as the peiodic payment R fo the sinking fund so that the company has enough money to eplace the machine at the end of the six yeas So we let R = $1, x We also note that the inteest pe peiod is = 5% We fist wite down the fomula of the sinking fund We must have But R = $1, x So x + R R It is easy to check that x = $ x So x $ $4,

10 AMORTIZATION OF LOANS optional Suppose one obtains a loan fom bank with the epayment scheduled as n peiodic payments of equal amount unde an inteest ate If A is the amount of the loan and R is the payment pe peiod, we have whee A R1 + n, R = epayment + inteest A loan in the above sense is amotized when pat of each payment is used to pay inteest and the emaining pat is used to educe the outstanding pincipal Example Suppose that a bank lends a boowe $1,500 and chages inteest at the annual ate of 12% The $1,500 plus inteest is to be epaid by thee equal installments R at the end of each month Using the fomula given above, we must have So R $ $51003 $ R R , = 001 The actual schedule can be undestood fom the following table Pincipal Inteest Payment Pincipal Peiod Outstanding at fo at end epaid at beginning of peiod Peiod of peiod end of peiod 1 $1,500 $ 1500 $51003 $ $1,00497 $ 1005 $51003 $ $50499 $ 505 $51004 $ Total $ 0 $ 3010 $ $ 1,50000 At the end of fist peiod, the inteest is 001$1500 = $15, hence only $51003 $15 = $49503 of $51003 was epaid to settle the outstanding loan, and consequently $1500 $49503 = $ is outstanding At the end of second peiod, the inteest is 001$ $1005, and only $51003 $1005 $49998 was left to settle the loan afte the second payment Hence $ $49998 $50499 emains Finally, the inteest fo the thid peiod is 001$50499 = $505 and now $51003 $505 = $50498 is eally paid back to the loan Hence $50499 $50498 = $001 emains, which can always be added to the last payment, ie, the last payment should be $