5.3 Annuities. Question 1: What is an ordinary annuity? Question 2: What is an annuity due? Question 3: What is a sinking fund?

Transcription

1 5.3 Auities Questio 1: What is a ordiary auity? Questio : What is a auity due? Questio 3: What is a sikig fud? A sequece of paymets or withdrawals made to or from a accout at regular time itervals is called a auity. The term of the auity is legth of time over which the paymets or withdrawals are made. There are several differet types of auities. A auity whose term is fixed is called a auity certai. A auity that begis at a defiite date but exteds idefiitely is called a perpetuity. If a auities term is ot fixed, it is called a cotiget auity. Auities that are created to fud a purchase at a later date like some equipmet or a college educatio are called sikig fuds. The paymets for a auity may be made at the begiig or ed of the paymet period. I a ordiary auity, the paymets are made at the ed of the paymet period. If the paymet is made at the begiig of the paymet period, it is called a auity due. I this text we ll oly examie auities i which the paymet period coicides with the iterest coversio period. This type of auity is called a simple auity. 1

2 Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally, the paymets to the auity are made at the ed of the paymet period. Suppose a paymet of $1000 is made semiaually to the auity over a term of three years. If the auity ears 4% per year compouded semiaually, the paymet made at the ed of the first six-moth period will accumulate This meas $1000 is multiplied by 1.0 five times, oce for each of the remaiig sixmoth periods. The ext paymet also ears iterest, but over 4 six-moth periods. This paymet has a future value of This process cotiues util we have the future value for each paymet. First paymet period Secod paymet period Third paymet period Fourth paymet period Fifth paymet period Sixth paymet period $ $ $1000 grows to 5 $1000 grows to 4 $ $ $1000 grows to 3 $ $ $1000 grows to $ $ $ $ $1000 grows to 1 $1000

3 The last paymet occurs at the ed of the last period ad ears o iterest. Examiig each expressio for the future value, it appears that there is a patter to the idividual future values. Each future value forms a umber correspodig to a geometric sequece. A geometric sequece is a ifiite list of umbers with the form aarar ar ar 3,,,,,, The amouts above correspod to the first six umbers i the geometric sequece , , , , , , I this case, a 1000 ad r 1.0. The umbers i the geometric sequece are called terms. For this geometric sequece, we ca umber the terms to make them easy to refer to First term Secod term Third term Fourth term Fifth term Sixth term I this geometric sequece, each term is 1.0 times larger tha the term before it. I fact, the ratio of ay two adjacet terms is 1.0. For istace, the ratio of the fourth ad fifth terms is Similarly, the ratio of the first ad secod terms is

4 I a geometric series, dividig a term by the precedig oe should results i the same umber. This umber is called the commo ratio for the geometric sequece ad correspods to r. Notice that the power o the 1.0 factor is oe less tha the term umber. This patter holds i geeral ad leads us to the followig patter. For ay 1, the th term of a geometric sequece is 1 ar. It might seem as though this geeral expressio might ot apply to the first term. I this case, 1, so the first term is or Other terms may be idetified as log as the values for a, r, ad are kow. Example 1 Terms of a Geometric Sequece Fid the fourth ad teth terms of the geometric sequece with a first term 500 ad r Solutio If the first term is 500, the a 500. The fourth term ( 4 ) i the geometric sequece is 3 ar or This is equal to The teth term is Let s cotiue to look at the future value of a paymet of $1000 made semiaually to a auity over a term of three years. The future value of the auity is the sum of the future values of each paymet ad correspodig iterest is

5 This sum represets the future value of the auity. For a sum with few terms, it is easy to add these amouts to give a future value of about $ This icludes six paymets of $1000 ad iterest of $ We ca fid the same amout usig a alterate strategy. Start by multiplyig both sides of the expressio for the sum by 1.0 to yield Subtract the sides of this equatio ad the previous equatio: Each equatio cotais idetical terms that are highlighted i red. Whe those terms are subtracted, may terms drop out leavig us with Each side of this equatio may be divided by 0.0 ad simplified to yield This expressio yields the same amout as addig the terms directly, $ If the paymets to the auity icrease i frequecy or take place over a loger period of time, it is ot coveiet to add all of the terms directly. This strategy allows us to fid the sum of ay umber of terms of a geometric sequece. 5

6 I the terms of the geometric sequece we have bee examiig, we ca recogize the paymet PMT ad iterest rate per period i. If the paymets are paid at the ed of periods, the sum of the accumulated amout of each paymet is 3 PMTPMT 1i PMT 1i PMT 1i PMT 1i Usig the strategy, this sum may be writte i a simplified form. Future Value of a Ordiary Auity If equal paymets of PMT are made ito a ordiary auity for periods at a iterest rate of i per period, the future value of the auity is 1 i 1 PMT i We use this expressio to calculate the sum whe there are ay umber of terms. Example Fid the Future Value of the Auity A ivestor deposits $500 i a simple auity at the ed of each sixmoth paymet period. This auity ears 10% per year, compouded semiaually. a. Fid the future value if paymets are made for three years. Solutio Fid the future value of this ordiary auity usig 1i 1 PMT. I this case, PMT 500, i 0.05, ad 6. i This gives 6

7 We could also fid this same amout by addig the terms directly, The six paymets of $500 have eared $ $3400 or $ i iterest over the life of the auity. b. Fid the future value if paymets are made for 30 years. Solutio I this ordiary auity, the term is much loger. The geometric series would have 30 terms ad it would ot be practical to add the terms directly. However, if we set R 500, i 0.05, ad 60 i the formula for the future value of a auity, we get , c. How much iterest is eared over the 30 year term i part b? Solutio Over the term of the auity, sixty paymets of $500 are made for a total of $30,000. This yields $176, $30,000 or $146, i iterest. The future value of a ordiary auity formula assumes that the auity starts out with a balace of zero. However, the auity may have a existig balace ad the paymets are added to that amout. I this case, the balace grows accordig to the compoud iterest formula. The paymets grow accordig to the future value of the auity. The sum of these amouts is the future value of both ivestmets combied. 7

8 Example 3 Fid the Future Value of a Retiremet Accout A employee s retiremet accout curretly has a balace of $61,000. Suppose the employee cotributes $33 at the ed of each moth. If the accout ears a retur of 5% compouded mothly, what will the future value of the accout i 15 years? Solutio The origial balace grows accordig to the compoud iterest formula, PV 1 i. The origial amout is PV 61000, the iterest rate per period is years is i, ad the umber of periods over The future value of the paymets ito the auity grow accordig to 1i 1 PMT i ad yields. For this accout, the paymet is PMT The sum of future value for the compouded amout ad the future value of the auity is $,181,

9 We ca combie the two amouts ito a sigle formula that accouts for paymets ito a existig accout that has some balace. I this cotext, the balace is called the preset value. It correspods to the value of the accout at the time the paymets commece. Future Value of a Ordiary Auity Whose Preset Value Is Not Zero If paymets PMT are made to a ordiary auity whose preset value is PV, the future value is 1 i 1 PV1i PMT i If paymets PMT are made from a ordiary auity whose preset value is PV, the future value is 1 i 1 PV1i PMT i We could have used the first formula to calculate the future value i Example 3, ,181, I Examples 4 ad 5, four of the five values i the future value formula for a ordiary auity are kow. This allows us to solve for the remaiig value. 9

10 Example 4 Fid the Amout Needed to Establish a Trust Fud A wealthy idividual wishes to create a trust fud for his gradso so that he may withdraw $5000 at the ed of every quarter for te years. At the ed of te years, the gradso will receive the rest of the trust which cotais $50,000. If the trust ears 8% iterest compouded quarterly, how much should be put ito the trust iitially? Solutio I this problem, the amout i the auity is decreasig sice withdrawals are beig made. However, we wish the future value of the auity to be $50,000 i te years. This meas that a larger amout must be placed i the trust ow so that paymet may be made from it Substitute 50000, PMT 5000, i 4 0.0, ad ito 1 i 1 PV1i PMT i to give PV Now solve this equatio for the preset value PV PV PV The trust must be established with a iitial deposit of $159,

11 Example 5 Fid the Paymet from a Auity A savvy ivestor has accumulated $1,000,000 i a ordiary auity. The auity ears 4% iterest compouded quarterly. She wishes to receive paymets from the auity each quarter for the ext 30 years. a. If the auity will ed up with o moey i 30 years, what paymet should she receive? Solutio Sice the paymets are made from a ordiary auity, we ll start from the future value of a ordiary auity formula, 1 i 1 PV1i PMT i The auity curretly cotais $1,000,000, so PV 1,000,000. The ivestor wats the value of the auity to be $0 i 30 years. This meas 0 0. Substitute these values, the iterest rate per period i, ad the umber of periods 430 to yield ,000, PMT 0 4 To solve for the paymet PMT, simplify ad isolate the paymet: ,000, PMT PMT 1,000, PMT 1,000, PMT 14, Multiply both sides by the reciprocal of the fractio i brackets The quarterly paymet is $14,

12 The paymet has bee rouded dow to isure all of the paymets are equal. This will leave a small amout of moey i the auity at the ed of 30 years. However, you could also roud up if you realize the last paymet might be differet from the earlier paymets. b. The ivestor wishes to leave a balace i the auity to leave to her heirs. If the auity is to ed cotai $100,000 i 30 years, what paymet should she receive? Solutio I this case, she wishes 100, 000. Chage the future value to this amout a solve for the paymet i ,000 1,000, PMT PMT 1,000, ,000 PMT 1,000, , PMT 13, Reducig the paymet by $ isures that the auity will cotai $100,000 i 30 years. 1

13 Questio : What is a auity due? I a ordiary auity, the paymet ito the auity is made at the ed of the period. A auity due differs from this sice the paymets are made at the begiig of the period. However, we ca apply the formula for determiig the future value of a ordiary auity with a little modificatio. Let s look at the paymets ad the iterest eared i a auity due where paymets of $1000 are made. As i our earlier example, these paymets ear 4% iterest compouded semiaually. First paymet period Secod paymet period Third paymet period Fourth paymet period Fifth paymet period Sixth paymet period $1000 grows to 6 $ $ $1000 grows to 5 $ $ $1000 grows to 4 $ $ $1000 grows to 3 $ $ $1000 grows to $ $ $1000 grows to 1 $ $100 The sum of these paymets ad iterest are S This sum cotais the term correspodig to the amout the first paymet grows to over six periods. Ulike the sum from a similar ordiary auity, this sum does ot cotai a term equal to This is because the last paymet is made at the begiig of the sixth paymet period. It ears iterest to grow to $ or $

14 I effect, the future value of a auity due is like a ordiary auity with a extra period at the begiig of the auity ad lackig the fial paymet. This allows us to write the future value for the auity due. Future Value of a Auity Due If equal paymets of PMT are made ito a auity due for periods at a iterest rate of i per period, the future value of the auity is PMT i PMT i This formula is almost idetical to the future value of the ordiary auity. The power of 1 i the umerator icorporates the extra iterest eared by the first paymet. A paymet is subtracted from the ordiary auity formula to accout for the last paymet earig iterest. Example 6 Fid the Future Value of a Auity Due A ivestor deposits $1000 i a simple auity at the begiig of each six-moth paymet period. This auity ears 4% per year, compouded semiaually. Fid the future value if paymets are made for three years. Solutio Sice the paymets are made at the begiig of each period, this is a auity due. The paymet is R 500 ad the iterest rate per period is i Sice iterest is eared semiaually over the term of three years, there are 6 periods. Usig these values, the future value is 14

15 S This results i the same future value as addig the terms of the sum directly, S

16 Questio 3: What is a sikig fud? If the future value of a auity is fixed, the auity is called a sikig fud. For istace, you aticipate that you will eed $80,000 to fud you child s educatio. If you set aside $500 each moth i a iterest bearig accout to reach this goal, you have set up a sikig fud. I problems ivolvig sikig fuds, the future value of the auity is specified ad some other quatity is determied. The formulas used deped o whether the paymets are made at the begiig or ed of each period. Example 7 How Log Will It Take To Accumulate Some Amout? The parets of a ewbor child aticipate that they will eed $80,000 for the child s college educatio. They pla to deposit $500 i a accout at the begiig of each moth i a accout that ears 3% iterest compouded mothly. How log will it take them to reach their goal? Solutio This problem specifies a future value of $80,000. Sice the paymets are made at the begiig of each period, we ll use the auity due formula to calculate the legth of time it will take to reach the future value. The paymet is R 500 ad the iterest rate per period is 0.03 i Whe these values are put ito the auity due 1 formula, we get ,

17 We eed to solve this equatio for the umber of periods. I doig this, we ll avoid roudig ay umber. Roudig too early may lead to large chages i the value of , , log log l l Add 500 to both sides Divide both sides by 500 Multiply both sides by Add 1 to both sides Covert to logarithm form Subtract 1 from both sides Use the chage of base formula to covert to atural logarithms 135 A period of approximately 135 moths or a little over 11 years is eeded to reach the goal. The umber if periods is rouded up to isure the goal is reached. Roudig dow would cause the accumulated amout to be slightly less tha $80,000. I some situatios, we are iterested i determiig the paymet that will yield a fixed future value. Example 8 Fid The Paymet Needed To Accumulate a Fixed Amout A ew car buyer wishes to buy a vehicle with cash i five years. To do this, she wishes to deposit some amout at the ed of each moth i a 17

18 accout earig.5% iterest compouded mothly. If she wishes to accumulate $30,000, what should the paymets be? Solutio Sice the paymets are made at the ed of each period, we ll use the future value for a ordiary auity, 1 i 1 PMT i to fid the paymet. Sice the accout has a balace of zero to begi with, the preset value is zero. She wishes to accumulate $30,000, so 30, 000. Over five years, there are 1 5 or 60 periods with a iterest rate per period of i The value of this rate per period is ot a termiatig decimal so we will use the fractio throughout the calculatio. Substitute the values ito the formula above to give ,000 PMT , PMT PMT Multiply both sides by the reciprocal to the fractio i the brackets The paymet is rouded up to isure the future value is reached. For this rouded paymet, the future value will actually exceed 30,000 by about 58 cets. If the paymets were rouded dow, the future value would be slightly less tha $30,