ANNUITIES SOFTWARE ASSIGNMENT TABLE OF CONTENTS... 1 ANNUITIES SOFTWARE ASSIGNMENT... 2 WHAT IS AN ANNUITY?... 2 EXAMPLE QUESTIONS...

Transcription

1 ANNUITIES SOFTWARE ASSIGNMENT TABLE OF CONTENTS ANNUITIES SOFTWARE ASSIGNMENT TABLE OF CONTENTS... 1 ANNUITIES SOFTWARE ASSIGNMENT... WHAT IS AN ANNUITY?... EXAMPLE 1... QUESTIONS... EXAMPLE BRANDON S PROBLEM... 3 Solutio... 3 GEOMETRIC SERIES... 5 SUM OF A GEOMETRIC SERIES... 6 BACK TO BRANDON S PROBLEM... 6 What if Bao makes Deposits at the Begiig of eah Moth?... 6 EXAMPLE 3 INTEREST RATE CALCULATOR SOFTWARE... 7 WHAT HAPPENS IF THE COMPOUNDING FREQUENCY DOES NOT EQUAL THE PAYMENT FREQUENCY?... 8 Calulatig the Peioi Rate whe Paymet Fequey equals Compouig Fequey... 9 Calulatig the Peioi Rate whe Paymet Fequey oes ot equal Compouig Fequey... 9 Example... 9 How to Covet a Geeal Auity ito a Simple Auity SOLVING ANY ANNUITY IN WHICH EITHER P = 0 OR F = DERIVATION OF ANNUITY FORMULAS Auity Algoithm PRACTICE EXERCISES... 14

2 ANNUITIES SOFTWARE ASSIGNMENT What is a Auity? A auity is a fiaial istumet that ivolves makig paymets o eposits at egula itevals. Auities a be use fo both savig a boowig. If a auity is use fo savig, eposits ae mae at egula itevals. If a auity is use fo boowig, paymets ae mae at egula itevals. Sie this esiptio may be somewhat iffiult to uesta at fist, it s best to ivestigate a few simple examples. Example 1 I have esige a Exel speasheet (Auity Exploatio.xls) to help you eview/lea the followig oepts: Simple Iteest Compou Iteest Aual Iteest Rate Peioi Iteest Rate Oiay Simple Auities Oiay Geeal Auities (iluig Motgages) The speasheet is available fom I:\Out\Is3u0\Auities a a also be owloae fom Ope the speasheet Stuy it aefully Expeimet with iffeet values (e.g. hage aual ate of iteest, umbe of paymets pe yea, et) Aswe the followig questios Questios 1. What is the iffeee betwee a aual ate of iteest a a peioi ate of iteest?. I what ways oes simple iteest iffe fom ompou iteest? Whih oe allows a ivestmet to gow moe quikly? 3. What is the iffeee betwee a oiay simple auity a a oiay geeal auity? 4. What is the effet of ieasig the umbe of ompouig peios pe yea? What is the effet of eeasig the umbe of ompouig peios pe yea?

3 Example Bao s Poblem Bao, who is ow sixtee, woul like to be a poke hampio some ay. At the age of twety-oe, he woul like to make a ame fo himself i the wol of poke by eteig a wiig a big-time poke touamet i Las Vegas. The oly poblem is that he ees a lot of ash to ealize this eam. Besies payig fo the $5000 touamet ety fee, Bao also ees to save a aitioal $10000 to ove tavellig osts a livig expeses. He osies vaious ivestmet possibilities as outlie below. (a) Fist, Bao osies buyig a $10000 Caaa Savigs Bo that pays iteest at a ate of 4% pe aum ompoue aually. Woul this allow him to e up with eough moey by the age of twety-oe? (b) Bao the osies makig mothly eposits to a auity. To ahieve his goal by the age of twetyoe, how muh woul he ee to eposit at the e of eah moth if the auity pays iteest at the ate of 6% pe aum ompoue mothly? Woul the amout iffe if the eposit wee mae at the begiig of eah moth istea of the e of eah moth? Solutio Beause the give ifomatio is iomplete, we ee to make a easoable assumptio befoe we attempt to solve this poblem. Although we ae tol that Bao is ow sixtee, we o ot kow his exat age. He oul have just tue sixtee, he may have tue sixtee a few moths ago o he may be about to tu sevetee. To avoi this ofusio, let s assume that he has just tue sixtee a that he woul like to have the moey by his twety-fist bithay. This gives him exatly five yeas to save the moey. (a) By usig the Auity Exploatio speasheet, we see that Bao will fall fa shot of his goal! How woul we solve this poblem without the help of the Auity Exploatio speasheet? Befoe we a ivestigate this issue, we ee to lea some impotat temiology. Temiology Value i this Example Explaatio P = Peset Value $10, Amout of moey with whih Bao state F = Futue Value $1, The value of the Caaa Savigs Bo afte 5 yeas = Total Numbe of Compouig Peios 5 This is the total umbe of ompouig peios ove the ouse of the ivestmet (o loa). = Aual Iteest Rate 4% = 0.04 Rate of iteest payable pe yea i = Peioi Iteest Rate 4% = 0.04 Rate of iteest payable pe ompouig peio. I this example, i = beause the iteest ompous aually. p = Paymets pe Yea (Paymet Fequey) 1 The umbe of paymets/eposits mae pe yea. = Compouig Peios pe Yea (Compouig Fequey) 1 The umbe of times pe yea that the iteest ompous. Ofte, p = but this oes ot ee to be the ase. The table o the ext page shoul help you uesta how to alulate F without the help of a speasheet.

4 Iteest Paymet # Iteest Paymet Aout Balae $10, Total Iteest Pai 1 $ $10, $ $10, How Aout Balae is Calulate $ $10, $ $10, = ( $10, ) 1.04 = $10, $43.64 $11,48.64 $1,48.64 $10, = ($10, ) 1.04 = $10, $ $11, $1, $11, = ($10, ) 1.04 = $10, $ $1, $, $11, = ($10, ) 1.04 = $10, We a see that the aout balae is alulate by multiplyig the pevious balae by The fial balae of $1, (i.e. the futue value) is alulate by multiplyig the oigial amout of $10, (i.e. the peset value) by 1.04, five times. I othe wos, Substitutig ito this fomula we obtai $1, = $10, (1.04)(1.04)(1.04)(1.04)(1.04) = $10, (1.04) 5 Usig the above example as a guie, we a make the followig obsevatio. If P epesets the peset value, F epesets the futue value, i epesets the peioi ate a epesets the umbe of ompouig peios, the F = P(1 + i) This fomula is kow as the ompou iteest fomula a is ofte witte A= P(1 + i). F = P(1 + i) = 10000(1 +.04) 5 = 10000(1.04) = (b) To uesta the this pat of the example, it s impotat to ompehe the temiology i the followig table: 5 Temiology Value i this Example Explaatio P = Peset Value $0.00 Amout of moey with whih Bao state. F = Futue Value $15, = Total Numbe of Compouig Peios 1 5 = 60 The amout of moey Bao wats to have at the e of the five yea peio. This is the total umbe of ompouig peios ove the ouse of the ivestmet (o loa). = Aual Iteest Rate 6% = 0.06 Rate of iteest payable pe yea i = Peioi Iteest Rate 0.06 = = Rate of iteest payable pe ompouig peio. 1 = Deposit/Paymet mae at Regula Itevals Ukow The amout eposite/pai at egula itevals. p = Paymets pe Yea (Paymet Fequey) 1 The umbe of paymets/eposits mae pe yea. = Compouig Peios pe Yea (Compouig Fequey) 1 The umbe of times pe yea that the iteest ompous.

5 The followig iagam, alle a time-lie illustates this situatio, whih seems somewhat moe ompliate tha the seaio i pat (a). I eality, howeve, this situatio is almost exatly the same as that i (a). The oly iffeee is that i this ase, the metho of pat (a) ees to be applie multiple times. The amout eposite eah moth is ollas. Eah eposit of ollas eas iteest, whih auses its value to gow ove time. Afte m ompouig peios (whih happes to be m moths), the value of a eposit of ollas gows to (1.005) m ollas (by applyig the ompou iteest fomula). Sum of futue values of all the eposits Moth (1.005) + (1.005) + (1.005) + (1.005) + + (1.005) 59 + (1.005) 59 Futue Value of 60 th Deposit Futue Value of 59 th Deposit Futue Value of 58 th Deposit Futue Value of 1 st Deposit So it appeas that the ux of this poblem is to figue out how to a up the futue values of eah eposit of ollas. Sie we kow that Bao ees to have $15, by the time he is 1, the sum of the futue values of eah eposit of ollas will have to be $15, Kowig this will allow us to solve fo a fially etemie exatly how muh moey Bao ees to pay to the auity eah moth. At this poit, howeve, we ee to igess fo a shot time a lea how to fi the sum of the futue values. This equies leaig about geometi seies, whih is the subjet of the ext setio. Geometi Seies A seies is simply a sum of oe o moe umbes. Examples of seies ae show below: = = 6 + 6(3) + 6(9) + 6(7) + 6(81) = (3 ) 6(3 ) 6(3 ) 6(3 ) 6(3 ) These ae examples of aithmeti seies. I a aithmeti seies the tems ae evely spae out. That is, the iffeee of ay two oseutive tems is ostat. Thee is o iseible patte i this seies. This is a example of a geometi seies. I a geometi seies, the tems ae ot evely spae out. Howeve, the atio of ay two oseutive tems is ostat. To obtai the ext tem i this example, simply multiply the pevious tem by thee. I geeal, the lette a is usually use to epeset the fist tem of a seies. Theefoe, i the above geometi seies, a = 6. I aitio, the lette is usually use to epeset the ommo atio of a geometi seies. I the example above, = 3 beause ay tem i the seies a be geeate by multiplyig the pevious tem by 3. A geometi seies with tems, fist tem a a ommo atio is of the fom 1 a + a + a + + a

6 Sum of a Geometi Seies Let 1 S = a + a + a + + a (all this equatio (1)). By multiplyig both sies of equatio (1) by we obtai 1 S = ( a + a + a + + a ). 3 1 S = a + a + a + + a + a (all this equatio ()) By subtatig equatio (1) fom equatio () we obtai 1 1 S S a a a = a ( a + a + a + + a ) = a + a + + a + a a a a a 1 1 = a + a a + a a + + a a a = a a S S = a a S ( 1) = a ( 1) 3 1 The sum of a geometi seies of the fom a + a + a + a + a is give by a ( 1) a ( 1) a(1 ) S = S = = a(1 ) S = (by multiplyig both umeato a eomiato of the ight-ha sie by 1). 1 Bak to Bao s Poblem 59 Reall that we left off with the sum of the futue values + (1.005) + (1.005) + + (1.005). Now we have the theoetial suppot to simplify this expessio. I this geometi seies, a =, = a = 60. Theefoe, ( ) ( ) + (1.005) + (1.005) + + (1.005) = = Also eall that Bao ees a futue value of $15,000 to ealize his eam of beomig a poke hampio. Theefoe, 60 ( ) = (0.005) = Theefoe, Bao must eposit $14.99 at the e of eah moth. What if Bao makes Deposits at the Begiig of eah Moth? Whe eposits ae mae at the e of eah moth, the vey last eposit eas o iteest at all beause it is mae o the last ay of the life of the auity. If eposits ae mae at the begiig of eah moth, howeve, the the fial eposit is mae o the fist ay of the fial moth. This meas that evey eposit eas iteest. The timelie at the ight shows how makig paymets at the begiig of eah moth affets the sum of the futue values. We a see lealy that oly oe value hages, that of a. The values of a emai the same. Summaizig, we see that a = (1.005), = a = a + a + + a + a 1 a + a + a + + a a a Whe the expessio i the seo lie is subtate fom the expessio i the fist lie, all tems ael exept fo a a a. + (1.005) + (1.005) (1.005) + (1.005) + + (1.005) + (1.005) (1.005) 60 + (1.005) 59 60

7 By substitutig these values ito the fomula fo the sum of a geometi seies we obtai (1.005)( ) ( ) (1.005) + (1.005) + + (1.005) = = Agai, sie Bao ees a futue value of $15,000 to ealize his eam of beomig a poke hampio, = 60 (1.005)( ) (0.005) ( ) = Theefoe, if Bao makes eposits at the begiig of eah moth, he oly ees to eposit $13.9. Example 3 Iteest Rate Calulato Softwae The softwae that you evelop will allow the use to alulate values elate to auities as show i the iagam give below. P = Peset Value, F = Futue Value, = paymet/eposit, = Numbe of Paymets, = Aual Iteest Rate, i = Peioi Iteest Rate Note To simplify the softwae that you ee to evelop, we shall assume that P a F aot both be o-zeo. That is, we shall assume that if P > 0, the F = 0 a if F > 0, the P = 0. P Auities Softwae F Auities Softwae P Auities Softwae F Auities Softwae Auities Softwae P Auities Softwae F P Auities Softwae F Auities Softwae

8 What happes if the Compouig Fequey oes ot equal the Paymet Fequey? The followig table summaizes what we have leae thus fa. I all of the followig, it is assume that the umbe of ompouig peios pe yea (ompouig fequey) equals the umbe of paymets pe yea (paymet fequey). Fiaial Coept Impotat Temiology Relatioships Simple Iteest * Compou Iteest P, F a ae the same as fo ompou iteest. t = Time i Yeas Slow, liea gowth of moey. P = Peset Value = Cuet Value of Moey F = Futue Value = Value of Moey oe all Iteest is Pai = Aual Iteest Rate = Total Numbe of Compouig Peios i = Peioi Iteest Rate = Rate of Iteest pe Compouig Peio F = P(1 + t) F = P(1 + i) = Fast, expoetial gowth of moey. Sum of a Geometi Seies A geometi seies has the fom a = Fist Tem of the Seies = Commo Ratio = Total Numbe of Tems 1 a + a + a + + a. a ( 1) a(1 ) S = = 1 1 Auities P, F, a i ae the same as fo ompou iteest. = Deposit/Paymet mae at Regula Itevals Fast, expoetial gowth of moey. See pages * The otes o simple iteest ae ilue fo efeee puposes oly. Simple iteest fomulas ae ot equie fo auity alulatios.

9 Calulatig the Peioi Rate whe Paymet Fequey equals Compouig Fequey As show i the followig examples, alulatig the peioi ate is quite easy whe the umbe of ompouig peios pe yea equals the umbe of paymets pe yea. Compouig Peios Aual Rate () Paymets pe Yea (p) Peioi Rate (i) pe Yea () 1% = 1% = 0.1 p = 1 (mothly) = 1 i = = 1% = % = 6% = 0.06 p = (semi-aually) = i = = 3% = % = 3% = 0.03 p = 4 (quately) = 4 i = = 0.75% = % =.6% = 0.06 p = 5 (weekly) = 5 i = = 0.05% = Calulatig the Peioi Rate whe Paymet Fequey oes ot equal Compouig Fequey Whe the paymet fequey is ot equal to the ompouig fequey, a moe etaile aalysis is eee to alulate the peioi ate. As show i the table below, it s ot immeiately obvious how to alulate the peioi ate! Aual Rate () Paymets pe Yea (p) Compouig Peios pe Yea () Peioi Rate (i) = 1% = 0.1 p = 1 (mothly) = i =? = 6% = 0.06 p = (semi-aually) = 1 i =? = 3% = 0.03 p = 4 (quately) = 3 i =? =.6% = 0.06 p = 5 (weekly) = 4 i =? Sie it s vey iffiult to uesta what is meat by havig a ompouig fequey that is ot equal to the paymet fequey (i.e. p ), pehaps we shoul fi a way to ovet this poblem ito oe i whih the two values ae equal (i.e. p= ). The followig example shows how this a be oe. Example Suppose that a auity has iteest ompouig thee times pe yea but paymets ae mae twie pe yea (i.e. p =, = 3). If we ajust the iteest ate, we a ovet this auity ito oe that has the same futue value but whih ompous at the same time as the paymets ae mae (i.e. p = = ). Futue Value of Dollas afte 1 Moths with Iteest Compouig Twie pe Yea (Peioi Rate i ) Futue Value of Dollas afte 1 Moths with Iteest Compouig Thee Times pe Yea (Peioi Rate i 1 ) As log as (1 + i) = (1 + i1), the the oigial ivestmet of ollas will gow to the same futue value afte 1 moths egaless of whethe the iteest ompous twie pe yea o thee times pe yea. 3 (1 + i) = (1 + i1) 3 (1 + i ) = (1 + i ) i1 (1 + i ) = (1 + ) 3 i i1 3 (1 i1) = (1 + ) i = + Theefoe, a auity i whih paymets ae mae twie pe yea with iteest ompouig thee times pe yea at peioi ate i 1 is equivalet to a auity i whih paymets ae mae twie pe yea with iteest ompouig twie pe yea at peioi ate i (1 + i ) 3 i1 = (1 + ) 1. (1 + i ) (1 + i ) 1 (1 ) 3 + i1 (1 + i1 )

10 How to Covet a Geeal Auity ito a Simple Auity Simple Auity: p must equal Geeal Auity: p oes ot ee to equal Suppose that you ae give a geeal auity with p paymets pe yea a ompouig peios pe yea. We woul like to ovet this to a simple auity with p paymets pe yea a p ompouig peios pe yea. To o this, we shall let i 1 = epeset the peioi ate that woul be use if the iteest wee pai a ompoue times pe yea a let i epeset the (ukow) ate to be pai if the iteest wee ompoue a pai p times pe yea. The osie the followig vey simple equatio: Futue Value of Dollas afte Oe Yea with Iteest Compouig p times pe Yea (Peioi Rate i ) = (1 + i ) = (1 + i ) Futue Value of Dollas afte Oe Yea with Iteest Compouig times pe Yea (Peioi Rate i 1 ) p 1 p i i1 1 1 p p i p 1 (1 + ) = (1 + ) (1 + i ) = (1 + ) 1 + i = (1 + i ) 1 1 p p i = (1 + i ) 1 p i = (1 + ) 1 Summay A geeal auity with p paymets pe yea, ompouig peios pe yea a aual ate is equivalet to a simple auity with p paymets pe yea, p ompouig peios pe yea a peioi ate Solvig ay Auity i whih eithe P = 0 o F = 0 To simplify the poblem of solvig auities, we shall assume that eithe P = 0 o F= 0. If P = 0, we a eue that the auity is use fo savig beause we begi with o moey but e up havig moey at some poit i the futue. If F = 0, we a eue that the auity is use fo boowig beause we begi with some moey but e up with o moey afte we have epai the ebt. = # paymets = # ompouig peios, = eposit/paymet amout, P = peset value (piipal), F = futue value, p i = (1 + ) 1. p i = peioi iteest ate = (1 + ) 1, = aual iteest ate, p = #paymets pe yea, = #ompou peios pe yea. Give Paymet mae at Begiig of Iteval (Auity Due) Paymet mae at E of Iteval (Oiay Auity), i, P Fi. (I this ase, F = 0.) Fi. (I this ase, F = 0.), i, F Fi. (I this ase, P = 0.) Fi. (I this ase, P = 0.),, P Fi i. (I this ase, F = 0.) Fi i. (I this ase, F = 0.),, F Fi i. (I this ase, P = 0.) Fi i. (I this ase, P = 0.), i, Fi P o fi F. Fi P o fi F. P, i, Fi. (I this ase, F = 0.) Fi. (I this ase, F = 0.) F, i, Fi. (I this ase, P = 0.) Fi. (I this ase, P = 0.)

11 Deivatio of Auity Fomulas Peset Value Paymet mae at Begiig of Iteval (Auity Due) The peset value P is give by the sum of the followig geometi seies: P = i (1 + i) (1 + i) 1 1 ( 1) (1 + i) + (1 + i) (1 + i) = = 1 + i. Theefoe, i + 1 I this seies, a = a = ( ) 1 a P = (1 + i) (1 + i) (1 + i) ( 1 ) = ( i) ( i) (1 + i) = (1 + i) Solvig fo we obtai, i 1 = P 1 (1 i) i. Solvig fo we obtai, Pi l 1 (1 + i) = l(1 + i) Paymet mae at E of Iteval (Oiay Auity) (1 + i) (1 + i) The peset value P is give by the sum of the followig geometi seies: P = i + i + i 1 (1 ) (1 ) I this seies, a = = ( 1+ i) 1 a i + 1 = 1 = ( 1+ i) 1. Usig a poess simila to that i i + 1 the left olum, we obtai 1 (1 + i) P=. Solvig fo we obtai, i = P 1 (1 i). + Solvig fo we obtai, Pi l 1 = l(1 + i) (1 + i) Table Cotiue o Next Page

12 Table Cotiue fom Pevious Page Paymet mae at Begiig of Iteval (Auity Due) (1 + i) (1 + i) Paymet mae at E of Iteval (Oiay Auity) (1 + i) (1 + i) Futue Value The futue value F is give by the sum of the followig geometi seies: F = + i + + i i (1 ) (1 ) (1 ) I this seies, a = (1 + i) a = 1 + i. Theefoe, (1 + i) 1 F = (1 + i ). Solvig fo we obtai, i 1 = F. (1 + i) 1 1+ i Solvig fo we obtai, Fi l + 1 (1 + i) = l(1 + i) (1 + i) The futue value F is give by the sum of the followig geometi seies: F = + (1 + i) + (1 + i) + + (1 + i) 1 I this seies, a = a = 1 + i. Theefoe, (1 + i) 1 F =. Solvig fo we obtai, i = F (1 + i) 1. Solvig fo we obtai, Fi l + 1 = l(1 + i) (1 + i) 1 Note: Ufotuately, it is ot possible to solve fo i algebaially. We ee to use methos of appoximatio to alulate i.

13 Auity Algoithm 1. The use etes thee of the five values P, F,,,. (If P is etee, F =0. If F is etee, P = 0.). The use must also ete the values of p a. 3. If the use has etee the value of (i.e. the aual ate), it must be ovete to a equivalet peioi ate p usig the fomula i = (1 + ) Base o the ifomatio etee by the use, oe of the ukow values is alulate, as outlie i the table below. Give, i, P (F = 0) Paymet mae at Begiig of Iteval (Auity Due) Paymet mae at E of Iteval (Oiay Auity) i 1 i Calulate usig = P. Calulate usig = P 1 (1 + i) 1+ i 1 (1 + i)., i, F (P = 0),, P (F = 0),, F (P = 0), i, i 1 i Calulate usig = F. Calulate usig = F (1 + i) 1 1+ i (1 + i) 1. Calulate a appoximatio fo i usig a suffiiet umbe of iteatios of the eusive fomula 1 (1 m) ( ) m m+ 1 = im (1 )(1 m) i + i + P i. + i + P Calulate a appoximatio fo i usig a suffiiet umbe of iteatios of the eusive fomula + 1 (1 m) ( ) m m+ 1 = im ( 1)(1 m) i + i F+ i. + + i F 1 (1 + i) Calulate P usig P = (1 + i) a/o alulate F usig (1 + i) 1 F = (1 + i ). Calulate a appoximatio fo i usig a suffiiet umbe of iteatios of the eusive fomula (1 + i ) + Pi. m m m+ 1 = im 1 P (1 + im ) i Calulate a appoximatio fo i usig a suffiiet umbe of iteatios of the eusive fomula (1 + i ) Fi. i F m m m+ 1 = im 1 (1 + m) i 1 (1 + i) Calulate P usig P = a/o (1 + i) 1 alulate F usig F =. P, i, Calulate usig Pi Pi l 1 l 1 (1 + i) =. Calulate usig = l(1 + i) l(1 + i). F, i, Calulate usig Fi Fi l + 1 l + 1 (1 + i) =. Calulate usig = l(1 + i) l(1 + i). 5. The alulate value is isplaye. (If the use asks fo the aual ate to be alulate, a aitioal step is eessay. The algoithm show i the table will alulate the peioi ate i, ot the aual ate. Theefoe, befoe isplayig the fial esult, the peioi ate must be ovete to a aual ate usig the fomula = [(1 + i) 1].) p

14 Patie Exeises 1. Caol iveste $1000 at 6% pe aum, ompoue aually. Without usig the ompou iteest fomula, alulate the futue value of the ivestmet afte 7 yeas (use the give table). Yea Iteest ($) Value of Ivestmet ($) 0 $0.00 $ Fo eah, state the values of P,, a i. (a) a $600 loa at 8%, ompoue quately fo 5 yeas (b) a $000 ivestmet at.4%, ompoue mothly fo yeas 3. Taa boowe $000 at 5.9%, ompoue semi-aually fo a -yea tem. She pomises to epay piipal a iteest at the e of the two yea peio. How muh will Taa have to pay bak altogethe? 4. Detemie whethe the followig ae geometi seies. If they ae, alulate the sum. (a) (b) () () ½ + ¼ (e) (f) (1.0) + 000(1.0) + 000(1.0) 3

15 5. Alliso eposits $100 at the e of eah moth ito a savigs aout that pays iteest at 3%, ompoue mothly. What will he savigs be woth i 1 yea? 6. Rya makes eposits of $000 semi-aually ito a auity ue that pays 4% iteest, ompoue semiaually. How muh will the auity be woth afte the 5-yea tem? 7. Ae bought a ew sou system. He was offee a paymet pla that osists of $00 paymets mae at the e of evey 3 moths, fo yeas. The pla is eally a loa that ivolves iteest that is alulate at 8%, ompoue quately. What is the atual ost of the sou system if he pays fo it ight away? 8. Joh ha $7000 ash fo a ow paymet o a a. He took out a loa at 8%, ompoue mothly, to pay fo the est. His mothly loa paymets will be $ fo the ext 3 yeas. What is the pie of the a?