THE TIME VALUE OF MONEY

Save this PDF as:
Size: px
Start display at page:

Download "THE TIME VALUE OF MONEY"

Transcription

1 QRMC04 9/17/01 4:43 PM Page 51 CHAPTER FOUR THE TIME VALUE OF MONEY 4.1 INTRODUCTION AND FUTURE VALUE The perspective ad the orgaizatio of this chapter differs from that of chapters 2 ad 3 i that topics are arraged by fiace applicatio rather tha mathematics area. The mathematics tools preseted i chapters 2 ad 3 are applied i this chapter to closely examie the aalytical aspects uderlyig what might be the sigle most importat topic i fiace the time value of moey. I this chapter, we study how ivestors ad borrowers iteract to value ivestmets ad determie iterest rates o loas ad fixed icome securities. Iterest is paid by borrowers to leders for the use of leders moey. The level of iterest charged is typically stated as a percetage of the pricipal (the amout of the loa). Whe a loa matures, the pricipal must be repaid alog with ay upaid accumulated iterest. I a free market ecoomy, iterest rates are determied joitly by the supply of ad demad for moey. Thus, leders will usually attempt to impose as high a iterest rate as possible o the moey they led; borrowers will attempt to obtai the use of moey at the lowest iterest rates available to them. Competitio amog borrowers ad competitio amog leders will ted to lead iterest rates toward some competitive level. Factors affectig the levels of iterest rates will do so by affectig supply ad demad coditios for moey. Amog these factors are iflatio rates, loa risks, ivestor itertemporal moetary prefereces (how much idividuals ad istitutios prefer to have moey ow rather tha have to wait for it), govermet policies, ad the admiistrative costs of extedig credit. 4.2 SIMPLE INTEREST (Backgroud readig: sectios 2.4, 2.7, ad 4.1) Iterest is computed o a simple basis if it is paid oly o the pricipal of the loa. Compoud iterest is paid o accumulated loa iterest as well as o the pricipal. Thus, if a sum of moey (X 0 ) were borrowed at a aual iterest rate i ad repaid at the

2 QRMC04 9/17/01 4:43 PM Page The time value of moey ed of years with accumulated iterest accruig o a simple basis, the total sum repaid (FV or Future Value at the ed of year ) is determied as follows: FV = X 0 (1 + i) (4.1) The subscripts ad 0 merely desigate time; they do ot imply ay arithmetic fuctio. The product i whe multiplied by X 0 reflects the value of iterest paymets to be made o the loa; the value 1 accouts for the fact that the pricipal of the loa must be repaid. If the loa duratio icludes some fractio of a year, the value of will be fractioal; for example, if the loa duratio were oe year ad three moths, would be The total amout paid (or, the future value of the loa) will be a icreasig fuctio of the legth of time the loa is outstadig () ad the iterest rate (i) charged o the loa. For example, if a cosumer borrowed $1,000 at a iterest rate of 10% for oe year, his total repaymet would be $1,100, determied from equatio (4.1) as follows: FV 1 = $1,000( ) = $1, = $1,100. If the loa were to be repaid i two years, its future value would be determied as follows: FV 2 = $1,000( ) = $1, = $1,200. Cotiuig our example, if the loa were to be repaid i five years, its Future Value would be FV 5 = $1,000( ) = $1, = $1,500. The loger the duratio of a loa, the higher will be its future value. Thus, the loger leders must wait to have their moey repaid, the greater will be the total iterest paymets made by borrowers. 4.3 COMPOUND INTEREST (Backgroud readig: sectios 2.7, 3.1, ad 4.2) Iterest is computed o a compoud basis whe the borrower pays iterest o accumulated iterest as well as o the loa pricipal. If iterest o a give loa must accumulate for a full year before it is compouded, the future value of this loa is determied as follows: FV = X 0 (1 + i). (4.2) For example, if a idividual were to deposit $1,000 ito a savigs accout payig aually compouded iterest at a rate of 10% (here, the bak is borrowig moey), the future value of the accout after five years would be $1,610.51, determied by equatio (4.2) as follows:

3 QRMC04 9/17/01 4:43 PM Page 53 Fractioal period compoudig of iterest 53 FV 5 = $1,000( ) 5 = $1, = $1, = $1, Notice that this sum is greater tha the future value of the loa ($1,500) whe iterest is ot compouded. The compoud iterest formula ca be derived ituitively from the simple iterest formula. If iterest must accumulate for a full year before it is compouded, the the future value of the loa after oe year is $1,100, exactly the same sum as if iterest had bee computed o a simple basis: FV = X 0 (1 + i) = X 0 (1 + 1 i) = X 0 (1 + i) 1 = $1,000( ) = $1,100. (4.3) The future values of loas where iterest is compouded aually ad whe iterest is computed o a aual basis will be idetical oly whe equals oe. Sice the value of this loa is $1,100 after oe year ad iterest is to be compouded, iterest ad future value for the secod year will be computed o the ew balace of $1,100: FV 2 = X 0 (1 + 1 i)(1 + 1 i) = X 0 (1 + i)(1 + i) = X 0 (1 + i) 2, FV 2 = $1,000( )( ) = $1,000( ) 2 = $1,210. (4.4) This process ca be cotiued for five years: FV 5 = $1,000( )( )( )( )( ) = $1,000( ) 5 = $1, More geerally, the process ca be applied for a loa of ay maturity. Therefore: FV = X 0 (1 + i)(1 + i) (1 + i) = X 0 (1 + i), FV = $1,000( )( ) ( ) = $1,000( ). (4.5) 4.4 FRACTIONAL PERIOD COMPOUNDING OF INTEREST I the previous examples, iterest is compouded aually; that is, iterest must accumulate at the stated rate i for a etire year before it ca be compouded or recompouded. I may savigs accouts ad other ivestmets, iterest ca be compouded semiaually, quarterly, or eve daily. If iterest is to be compouded more tha oce per year (or oce every fractioal part of a year), the future value of such a ivestmet will be determied as follows: FV = X 0 (1 + i/m) m, (4.6) where iterest is compouded m times per year. The iterpretatio of this formula is fairly straightforward. For example, if m is 2, the iterest is compouded o a semiaual basis. The semiaual iterest rate is simply i/m or i/2. If the ivestmet is held

4 QRMC04 9/17/01 4:43 PM Page The time value of moey for periods, the it is held for 2 semiaual periods. Thus, we compute a semiaual iterest rate i/2 ad the umber of semiaual periods the ivestmet is held 2. If $1,000 were deposited ito a savigs accout payig iterest at a aual rate of 10% compouded semiaually, its future value after five years would be $1,628.90, determied as follows: FV 5 = $1,000( /2) 2 5 = $1,000(1.05) 10 = $1,000( ) = $1, Notice that the semiaual iterest rate is 5% ad that the accout is outstadig for te six-moth periods. This sum of $1, exceeds the future value of the accout if iterest is compouded oly oce aually ($1,610.51). I fact, the more times per year iterest is compouded, the higher will be the future value of the accout. For example, if iterest o the same accout were compouded mothly (12 times per year), the accout s future value would be $1,645.31: FV 5 = $1,000( /12) 12 5 = $1,000( ) 60 = $1, The mothly iterest rate is ad the accout is ope for m or 60 moths. With daily compoudig, the accout s value would be $1,648.61: FV 5 = $1,000( /365) = $1, Therefore, as m icreases, future value icreases, as i table 4.1. However, this rate of icrease i future value becomes smaller with larger values for m; that is, the icreases i FV iduced by icreases i m evetually become quite small. Thus, the differece i the future values of two accouts where iterest is compouded hourly i oe ad every miute i the other may actually be rather trivial. Table 4.1 Future values ad aual percetage yields of accouts with iitial $10,000 deposits at 10% Years to Future value Future value Future value Future value Future value maturity, simple compouded compouded compouded compouded iterest ($) aually ($) mothly ($) daily ($) cotiuously ($) 1 11,000 11,000 11,047 11,052 11, ,000 12,100 12,204 12,214 12, ,000 13,310 13,481 13,498 13, ,000 14,641 14,894 14,917 14, ,000 16,105 16,453 16,486 16, ,000 25,937 27,070 27,179 27, ,000 67,275 73,281 73,870 73, , , , , , ,000 1,173,909 1,453,699 1,483,116 1,484,140 Aual Varies percetage with yield

5 QRMC04 9/17/01 4:43 PM Page 55 Fractioal period compoudig of iterest 55 APPLICATION 4.1: APY AND BANK ACCOUNT COMPARISONS Fiacial istitutios ofte have may ways of defiig the terms or rules associated with their loas, accouts, ad other ivestmets. Such large umbers of terms ad rules frequetly lead to cofusio amog ivestors ad cosumers, particularly whe tryig to compare their various alteratives. For this reaso, there exist several covetios which are iteded to stadardize the disclosure of these terms. For example, we have see i the previous two sectios the impact that chagig the compoudig itervals has o future value. Compariso betwee ivestmets is more complicated whe their umbers of compoudig itervals differ. To simplify the compariso betwee loas with varyig compoudig itervals, it is ofte useful to compute aual percetage yields, also kow as equivalet aual rates. The aual percetage yield (APY) represets the yield that, if compouded oce per year, will produce the same future value as the stated rate i compouded m times per year: 1 Thus, we ca compute APY as follows: m i FV = X 0 1+ X ( APY). m = 0 1+ m i APY = 1+. m 1 (4.7) Because the aual percetage yield simplifies compariso betwee accouts with differet compoudig itervals, U.S. baks are ormally required by law to disclose APYs alog with their stated iterest rates i their advertisemets solicitig bak accouts. Cosider a example where a savigs accout at bak X pays 6% iterest compouded daily ad a similar accout at bak Y pays 6 % iterest, compouded semi- 1 4 aually. Which accout will pay more to a ivestor who leaves a $100 deposit for oe year? Based o equatio (4.6), we ca obtai the followig future values: FV X = $ = $ , 365. FV Y = $ = Thus, a accout payig a stated rate of 6% compouded daily yields a future value equivalet to a accout payig slightly more tha 6.18% compouded aually. A accout payig a stated rate of 6.25% compouded semiaually yields a future value equivalet to a accout payig slightly more tha 6.437% compouded aually If iterest is ot compouded, APY = (1 + i) 1.

6 QRMC04 9/17/01 4:43 PM Page The time value of moey Therefore, the accout i bak Y is preferred to that at bak X. We ca arrive at the same preferece rakig by examiig aual percetage yields: APY X = +., APYY = = = Because the accout at bak Y has the higher APY, it is preferred. The accout with the higher APY will produce a higher future value. However, it is ot ecessarily true that the accout with the highest stated rate also has the highest APY. A 1997 advertisemet i a New York ewspaper offered a five-year certificate of deposit accout payig iterest at a aual rate of 5.83%, compouded daily. The aual percetage yield (APY) o this accout was advertised at 6.00%. Give these details, the future value of $100 deposited ito this accout ca be computed to be $133.84: FV = $100( /365) = $ The APY of this accout is determied as follows: APY = ( /365) = The 6% APY advertised by the bak was approximately correct; such advertisemets are ofte rouded slightly. I ay case, the future value of this accout ca be determied with the 6.003% accout APY as follows: FV = $100( ) 5 = $ A $100 iitial deposit ito a five-year CD accout payig iterest at a aual rate of 5.85%, compouded quarterly, would have a future value of $133.69: FV = $100( /4) 4 5 = $ The APY of this accout is , determied as follows: APY = ( /4) 4 1 = Note that the future value ad the APY of the secod accout are lower tha those of the first accout eve though the stated iterest rate o the secod accout is higher. Compoudig ca have a sigificat effect o both future value ad APY. 4.5 CONTINUOUS COMPOUNDING OF INTEREST (Backgroud readig: sectios 2.5 ad 4.4) If iterest were to be compouded a ifiite umber of times per period, we would say that iterest is compouded cotiuously. However, we caot obtai a umerical

7 QRMC04 9/17/01 4:43 PM Page 57 Auity future values 57 solutio for future value by merely substitutig i for m i equatio (4.6) calculators have o key. I the previous sectio, we saw that icreases i m cause the future value of a ivestmet to icrease. As m approaches ifiity, FV cotiues to icrease, however at decreasig rates. More precisely, as m approaches ifiity (m ), the future value of a ivestmet ca be defied as follows: FV = X 0 e i, (4.8) where e is the atural log whose value ca be approximated at 2.718, or derived as i sectio 2.5. If a ivestor were to deposit $1,000 ito a accout payig iterest at a rate of 10%, cotiuously compouded (or compouded a ifiite umber of times per year), the accout s future value would be approximately $1,648.64: FV 5 = $1,000 e 1 5 $1, = $1, The Future Value of this accout exceeds oly slightly the value of the accout if iterest were compouded daily. Also, ote that cotiuous compoudig simply meas that iterest is compouded a ifiite umber of times per time period. 4.6 ANNUITY FUTURE VALUES (Backgroud readig: sectios 2.8, 3.4, ad 4.3) A auity is a series of equal paymets made at equal itervals. Suppose that paymets are to be made ito a iterest-bearig accout. The future value of that accout will be a fuctio of iterest accruig o prior deposits as well as the deposits themselves. A future value auity factor (fvaf ) is used to determie the future value of a auity. This auity is a series of equal paymets made at idetical itervals. The future value auity factor may be derived through the use of the geometric expasio procedure discussed i sectio 3.4. This techique is very useful for future value computatios whe a large umber of time periods are ivolved. The geometric expasio eables us to reduce a repetitive expressio requirig may calculatios to a expressio that ca be computed much more quickly. Suppose that we wish to determie the future value of a accout based o a paymet of X made at the ed of each year t for years, where that accout pays a aual iterest rate equal to i: FVA = X[(1 + i) 1 + (1 + i) (1 + i) 2 + (1 + i) 1 + 1]. (4.9) The paymet made at the ed of the first year will accumulate iterest for a total of 1 years, the paymet at the ed of the secod year will accumulate iterest for 2 years, ad so o. Clearly, determiig the future value of this accout with equatio (4.9) will be very time-cosumig if is large. The first step i the geometric expasio to simplify equatio (4.9) is to multiply both of its sides by 1 + i:

8 QRMC04 9/17/01 4:43 PM Page The time value of moey FVA(1 + i) = X[(1 + i) + (1 + i) (1 + i) 3 + (1 + i) 2 + (1 + i)]. (4.10) The secod step i this geometric expasio is to subtract equatio (4.9) from equatio (4.10), to obtai: FVA(1 + i) FVA = X[(1 + i) 1]. (4.11) Notice that the subtractio led to the cacellatio of may terms, reducig the equatio that we wish to compute with to a much more maageable size. Fially, we rearrage terms i equatio (4.12) to obtai equatios (4.12) ad (4.13): FVA 1 + FVA i FVA = X[(1 + i) 1] = FVA i = X[(1 + i) 1], (4.12) FVA X i [( 1+ ) 1] =. i (4.13) Practicig derivatios such as this is a excellet way to uderstad the ituitio behid fiacial formulas. Uderstadig the derivatios is ecessary i order to be able to modify the formulas for a variety of more complex (ad realistic) scearios. Cosider a example applicable to may idividuals who ope Idividual Retiremet Accouts (I.R.A. s), from which they may withdraw whe they reach the age of 1 59 years. Cosider a idividual who makes a $2,000 cotributio to his I.R.A. at 2 the ed of each year for 20 years. All of his cotributios receive a 10% aual rate of iterest, compouded aually. What will be the total value of this accout, icludig accumulated iterest, at the ed of the 20-year period? Equatio (4.13) ca be used to evaluate the future value of this auity, where X is the aual cotributio made at the ed of each year by the ivestor to his accout, i is the iterest rate o the accout, ad FVA is the future value of the auity. The future value of this idividual s I.R.A. is $114,550: 20 ( ) 1 FVA = $ 2000, = $ 114, This future value auity equatio ca be used wheever idetical periodic cotributios are made toward a accout. Sectio 4.8 will preset a discussio o determiig the preset value of such a series of cash flows. (The term preset value is also defied later i sectio 4.8.) Note that each of the above calculatios assumes that cash flows are paid at the ed of each period. If, istead, cash flows were realized at the begiig of each period, the auity would be referred to as a auity due. The auity due would geerate a extra year of iterest o each cash flow. Hece, the future value of a auity due is determied by simply multiplyig the future value auity formula by (1 + i): FVA, due = ( + i) ( + i) ( 1+ i) X ( 1 + i) = X. i i (4.14)

9 QRMC04 9/17/01 4:43 PM Page 59 Auity future values 59 From the above example, we fid that the future value of the idividual s I.R.A. is $126,005 if paymets to the I.R.A. are made at the begiig of each year: 21 ( ) ( ) FVA, due = $ 2, 000 = $ 126, APPLICATION 4.2: PLANNING FOR RETIREMENT (Backgroud readig: sectios 2.5, 3.1, ad 4.5) Suppose that a 23-year-old accoutat wishes to retire as a millioaire based o her retiremet savigs accout. She iteds to ope ad cotribute to a tax-deferred 401k retiremet accout sposored by her employer each year util she retires with $1,000,000 i that accout. Would she meet her retiremet goal if she deposited $10,000 ito that accout at the ed of each year util she is 65 years of age? Assume that her accout will geerate a aual rate of iterest equal to 5% for each of the ext 42 years. Equatio (4.13) will be used to solve this problem: FV. = $ 10, = $ 1, 352, Now, suppose that she would like to retire as soo as possible with $1,000,000 i her accout. Assumig that othig else associated with her situatio chages, what is the earliest age at which she ca retire? Now, we will use equatio (4.13) to algebraically solve for, the umber of years that the accoutat must wait to retire: FV ( 1+ i) 1 ( +. ) = X = $ 10, i 005. ( 1+ i) 1 FV i = X, = ( 1+ i) 1, i X FV i FV i log + log( i), log log( i), X 1 = 1+ + X 1 1+ = log $ 1000,, log( ) = + log(. ).. $, + = log( 105. ) = Sice paymets are made at the ed of each year, the accoutat must wait 37 years whe she is 60 before she ca retire as a millioaire. Note that we were able to fid a closed-form solutio (put o oe side aloe) usig simple algebra. I may time value problems, the exact placemet of the expoet will prevet us from obtaiig a solutio so easily.

10 QRMC04 9/17/01 4:43 PM Page The time value of moey 4.7 DISCOUNTING AND PRESENT VALUE (Backgroud readig: sectio 4.3) Cash flows realized at the preset time have a greater value to ivestors tha cash flows realized later, for the followig reasos: 1 Iflatio. The purchasig power of moey teds to declie over time. 2 Risk. We ever kow with certaity whether we will actually realize the cash flow that we are expectig. 3 The optio to either sped moey ow or defer spedig it is likely to be worth more tha beig forced to defer spedig the moey. The purpose of the Preset-Value model is to express the value of a future cash flow i terms of cash flows at preset. Thus, the Preset-Value model is used to compute how much a ivestor would pay ow for the expectatio of some cash flow to be received i years. The preset value of this cash flow would be a fuctio of iflatio, the legth of wait before the cash flow is received (), the riskiess associated with the cash flow, ad the time value a ivestor associates with moey (how much he eeds moey ow as opposed to later). Perhaps the easiest way to accout for these factors whe evaluatig a future cash flow is to discout it i the followig maer: CF PV = ( 1 + k), (4.15) where CF is the cash flow to be received i year, k is a appropriate discout rate accoutig for risk, iflatio, ad the ivestor s time value associated with moey, ad PV is the preset value of that cash flow. The discout rate eables us to evaluate a future cash flow i terms of cash flows realized today. Thus, the maximum a ratioal ivestor would be willig to pay for a ivestmet yieldig a $9,000 cash flow i six years assumig a discout rate of 15% would be $3,891, determied as follows: $ 9, 000 PV = ( ) 6 $ 9, 000 = = $ 3, I the above example, we simply assumed a 15% discout rate. Realistically, perhaps the easiest value to substitute for k is the curret iterest or retur rate o loas or other ivestmets of similar duratio ad riskiess. However, this market-determied iterest rate may ot cosider the idividual ivestor s time prefereces for moey. Furthermore, the ivestor may fid difficulty i locatig a loa (or other ivestmet) of similar duratio ad riskiess. For these reasos, more scietific methods for determiig appropriate discout rates will be discussed later. I ay case, the discout rate should accout for iflatio, the riskiess of the ivestmet, ad the ivestor s time value for moey.

11 QRMC04 9/17/01 4:43 PM Page 61 The preset value of a series of cash flows 61 Derivig the preset-value formula The preset-value formula ca be derived easily from the compoud iterest formula. Assume that a ivestor wishes to deposit a sum of moey ito a savigs accout payig iterest at a rate of 15%, compouded aually. If the ivestor wishes to withdraw from his accout $9,000 i six years, how much must he deposit ow? This aswer ca be determied by solvig the compoud iterest formula for X 0 : FV FV = X ( 1 + i), X = ( 1 + i) $ 9, 000 = ( ) $ 9, 000 = = $ 3, Therefore, the ivestor must deposit $3, ow i order to withdraw $9,000 i six years at 15%. Notice that the preset-value equatio (4.15) is almost idetical to the compoud iterest formula where we solve for the pricipal (X 0 ): CF PV = FV X ( + k), 0 = 1 ( 1+ i). Mathematically, these formulas are the same; however, there are some differeces i their ecoomic iterpretatios. I the iterest formulas, iterest rates are determied by market supply ad demad coditios, whereas discout rates are idividually determied by ivestors themselves (although their calculatios may be iflueced by market iterest rates). I the preset-value formula, we wish to determie how much some future cash flow is worth ow; i the iterest formula above, we wish to determie how much moey must be deposited ow to attai some give future value. 4.8 THE PRESENT VALUE OF A SERIES OF CASH FLOWS (Backgroud readig: sectios 2.8 ad 4.7) Suppose that a ivestor eeds to evaluate a series of cash flows. She eeds oly to discout each separately ad the sum the preset values of each of the idividual cash flows. Thus, the preset value of a series of cash flows CF t received i time period t ca be determied by the followig expressio: CFt PV = t ( + k). 1 t= 1 (4.16) For example, if a ivestmet were expected to yield aual cash flows of $200 for each of the ext five years, assumig a discout rate of 5%, its preset value would be $865.90: PV = ( ) ( ) ( ) ( ) ( ) = $

12 QRMC04 9/17/01 4:43 PM Page The time value of moey Therefore, the maximum price a idividual should pay for this ivestmet is $865.90, eve though the cash flows yielded by the ivestmet total $1,000. Because the idividual must wait up to five years before receivig the $1,000, the ivestmet is worth oly $ Use of the preset-value series formula does ot require that cash flows CF t i each year be idetical, as does the auity model preseted i the ext sectio. 4.9 ANNUITY PRESENT VALUES (Backgroud readig: sectios 3.4, 4.6, ad 4.8) The expressio for determiig the preset value of a series of cash flows ca be quite cumbersome, particularly whe the paymets exted over a log period of time. This formula requires that cash flows be discouted separately ad the summed. Whe is large, this task may be rather time-cosumig. If the aual cash flows are idetical ad are to be discouted at the same rate, a auity formula ca be a useful time-savig device. The same problem as discussed i the previous sectio ca be solved usig the followig auity formula: PV A CF 1 = 1 k ( 1 + ) k, (4.17) where CF is the level of the aual cash flow geerated by the auity (or series). Use of this formula does require that all of the aual cash flows be idetical. Thus, the preset value of the cash flows i the problem discussed i the previous sectio is $865.90, determied as follows: $ 200 PV A = ( ) 5 = $ 4, 000( ) = $ As becomes larger, this formula becomes more useful relative to the preset-value series formula discussed i the previous sectio. However, the auity formula requires that all cash flows be idetical ad be paid at the ed of each year. The preset-value auity formula ca be derived easily from the perpetuity formula discussed i sectio 4.11, or from the geometric expasio procedure described later i this sectio. Note that each of the above calculatios assumes that cash flows are paid at the ed of each period. If, istead, cash flows were realized at the begiig of each period, the auity would be referred to as a auity due. Each cash flow geerated by the auity due would, i effect, be received oe year earlier tha if cash flows were realized at the ed of each year. Hece, the preset value of a auity due is determied by simply multiplyig the preset-value auity formula by (1 + k): PVA due CF 1 = 1 k ( 1 + k) ( 1 + k). (4.18)

13 QRMC04 9/17/01 4:43 PM Page 63 Auity preset values 63 The preset value of the five-year auity due discouted at 5% is determied as follows: PVA due 200 = ( +. ) = $4,000[ ](1.05) = ( ) Derivig the preset-value auity formula The preset value auity factor (pvaf ) may be derived through use of the geometric expasio (see sectio 3.4). Cosider the case where we wish to determie the preset value of a ivestmet based o a cash flow of CF made at the ed of each year t for years, where the appropriate discout rate is k: PV A = CF ( + k) ( + k) ( + k) (A) Thus, the paymet made at the ed of the first year is discouted for oe year, the paymet at the ed of the secod year is discouted for two years, ad so o. Clearly, determiig the preset value of this accout will be very time-cosumig if is large. The first step of the geometric expasio is to multiply both sides of (A) by (1 + k): 1 PVA( 1+ k) = CF 1+ ( 1 + k) ( 1 k) + 1. (B) The secod step i the geometric expasio is to subtract equatio (A) from equatio (B), to obtai: 1 PVA( 1+ k) PVA= CF 1 ( 1 + ) k, (C) which simplifies to 1 PVA( 1+ k 1) = PVA() k = CF 1 ( 1 + ) k. (D) Notice that the subtractio led to the cacellatio of may terms, reducig the equatio that we wish to compute to a much more maageable size. Fially, we cacel the oes o the left side ad divide both sides of equatio (D) by k, to obtai: PV A CF 1 = 1 k ( 1 + ) k. (4.17)

14 QRMC04 9/17/01 4:43 PM Page The time value of moey APPLICATION 4.3: PLANNING FOR RETIREMENT, PART II (Backgroud readig: applicatio 4.2 ad sectio 4.9) Suppose that the 23-year-old accoutat from applicatio 4.2 wishes to retire as a millioaire based o her retiremet savigs accout, but eeds to kow what the preset value of that millio-dollar accout is. If the accout is ope for the full 37 years, its future value will be $1,016,282, based o equatio (4.13). Based o a discout rate of 5% ad assumig that the accout is ope for 37 years, its preset value is easily determied from equatio (4.15) as follows: CF $ 1, 016, 282 PV = = = $ ,. 07. ( 1 + k) 37 ( ) I preset-value terms, this millio-dollar accout is obviously worth much less tha $1,000,000. However, what is the preset value of the aual series $10,000 deposits that she will make to that accout? Agai, based o a 5% discout rate, we determie this preset value with equatio (4.17) as follows: $ 10, PV A = ( ) = $ 167, Notice that the preset value of cotributios that she makes to the accout is idetical to the preset value of what she will be able to retire with. 37 APPLICATION 4.4: VALUING A BOND Because the preset value of a series of cash flows is simply the sum of the preset values of the cash flows, the auity formula ca be combied with other preset-value formulas to evaluate ivestmets. Cosider, for example, a 7% coupo bod makig aual iterest paymets for ie years. If this bod has a $1,000 face (or par) value, ad its cash flows are discouted at 6%, its cash flows will be $70 i each of the ie years plus $1,000 i the teth year. The preset value of the bod s cash flows ca be determied as follows: $ 70 PV = ( ) $ 1, ( ) 9 9 $ 1, 000 = $ 1166,. 67( ) = $ = $1, Thus, the value of a bod is simply the sum of the preset values of the cash flow streams resultig from iterest paymets ad from pricipal repaymet.

15 QRMC04 9/17/01 4:43 PM Page 65 Amortizatio 65 Now, let us revise the above example to value aother 7% coupo bod. This bod will make semiaual (twice yearly) iterest paymets for ie years. If this bod has a $1,000 face (or par) value, ad its cash flows are discouted at the stated aual rate of 6%, its value ca be determied as follows: $ 35 PV = ( ) $ 1, ( ) = $ = $1, $ 1, 000 = $ 1166,. 67( ) Agai, the value of the bod is the sum of the preset values of the cash flow streams resultig from iterest paymets ad from the pricipal repaymet. However, the semiaual discout rate equals 3% ad paymets are made to bodholders i each of 18 semiaual periods AMORTIZATION (Backgroud readig: sectio 4.9) At the begiig of this chapter, we derived the cocept of preset value from that of future value. Amortizatio is essetially a topic relatig to iterest, but the presetvalue auity model preseted i this chapter is crucial to its developmet. Amortizatio is the paymet structure associated with a loa. That is, the amortizatio schedule of a loa is its paymet schedule. Cosider the auity model from equatio (4.17): PV A CF 1 = 1 k ( 1 + ) k. (4.17) Typically, whe a loa is amortized, the loa repaymets will be made i equal amouts; that is, each aual or mothly paymet will be idetical. At the ed of the repaymet period, the balace (amout of pricipal remaiig) o the loa will be zero. Thus, each paymet made by the borrower is applied to the pricipal repaymet as well as to iterest. A bak ledig moey will require that the sum of the preset values of its repaymets be at least as large as the sum of moey it loas. Therefore, if the bak loas a sum of moey equal to PV for years at a iterest rate of i, the amout of the aual loa repaymet will be CF: 1 CF = [ PVA k] 1 ( 1 + ) k. (4.18) For example, if a bak were to exted a $865,895 five year mortgage to a corporatio at a iterest rate of 5%, the corporatio s aual paymet o the mortgage would be $200,000, determied by equatio (4.18):

16 QRMC04 9/17/01 4:43 PM Page The time value of moey Table 4.2 The amortizatio schedule of a $865,895 loa with equal aual paymets for five years at 5% Year Pricipal ($) Paymet ($) Iterest ($) Paymet to pricipal ($) 1 865, ,000 43, , , ,000 35, , , ,000 27, , , ,000 18, , , ,000 9, ,476 The loa is fully repaid by the ed of the fifth year. The pricipal represets the balace at the begiig of the give year. The paymet is made at the ed of the give year, ad icludes oe year of iterest accruig o the pricipal from the begiig of that year. The remaiig part of the paymet is paymet to the pricipal. This paymet to the pricipal is deducted from the pricipal or balace as of the begiig of the followig year. 1 CF = [$ 865, ] 1 ( ) = $ 200, 000. Thus, each year, the corporatio will pay $200,000 toward both the loa pricipal ad iterest obligatios. The amouts attributed to each are give i table 4.2. Notice that as paymets are applied toward the pricipal, the pricipal declies; correspodigly, the iterest paymets declie. Noetheless, total aual paymets are idetical util the pricipal dimiishes to zero i the fifth year. 5 APPLICATION 4.5: DETERMINING THE MORTGAGE PAYMENT A family has purchased a home with $30,000 dow ad a $300,000 mortgage. The mortgage will be amortized over 30 years with equal mothly paymets. The iterest rate o the mortgage will be 9% per year. Based o this data, we would like to determie the mothly mortgage paymet ad compile a amortizatio table decomposig each of the mothly paymets ito iterest ad paymet toward priciple. First, we will express aual data as mothly data. Three hudred ad sixty (12 30) moths will elapse before the mortgage is fully paid, ad the mothly iterest rate will be , or 9% divided by 12. Give this mothly data, mothly mortgage paymets are determied as follows: Paymet = $ 300, ( ) 360 $,.. = Table 4.3 depicts the amortizatio schedule for this mortgage.

17 QRMC04 9/17/01 4:43 PM Page 67 Perpetuity models 67 Table 4.3 The amortizatio schedule of a $300,000 loa with equal mothly paymets for 30 years at 9% iterest per aum (0.0075% per moth) Moth Begiig-of- Total Paymet o Paymet o moth pricipal ($) paymet ($) iterest ($) pricipal ($) 1 300, , , , , , , , , , , , , , , , , , , , , , , , Studets should be able to work through the figures o this table startig from the upper lefthad corer, the workig to the left, the dow. I this particular example, because is large (360), use of a computerized spreadsheet will make computatios substatially more efficiet PERPETUITY MODELS (Backgroud readig: sectio 4.9) As the value of approaches ifiity i the auity formula, the value of the righthad side term i the brackets, 1 ( 1 + k), approaches zero. That is, the cash flows associated with the auity are paid each year for a period approachig forever. Therefore, as approaches ifiity, the value of the ifiite time horizo auity approaches CF PVA = [ 1 0]; k PV CF k P =. (4.19) The auity formula discussed i sectio 4.9 ca be derived ituitively by use of figure 4.1. First, cosider a perpetuity as a series of cash flows begiig at time period oe (oe year from ow) ad extedig idefiitely ito perpetuity. Cosider a secod perpetuity with cash flows begiig i time period ad extedig idefiitely ito perpetuity. If a ivestor is to receive a -year auity, the secod perpetuity represets those cash flows from the first perpetuity that he will ot receive. Thus, the

18 QRMC04 9/17/01 4:43 PM Page The time value of moey Preset value of perpetuity begiig i oe year = CF k Time 1 Preset value of -year auity: PVA Preset value of perpetuity begiig i year ( + 1): (CF/k) (1 + k) Figure 4.1 Derivig auity preset value from perpetuity preset values. The preset value of a perpetuity begiig i oe year mius the preset value of a secod perpetuity begiig i year ( + 1) equals the preset value of a -year auity. Thus, PVA = CF/k (CF/k) (1 + k) = CF/k [1 1/(1 + k) ]. differece betwee the preset values of the first ad secod perpetuities represets the value of the auity that he will receive. Note that the secod perpetuity is discouted a secod time, sice its cash flows do ot begi util year : PV A CF CF/ k CF 1 = = 1 k ( 1 + k) k ( 1 + k) The perpetuity model is useful i the evaluatio of a umber of ivestmets. Ay ivestmet with a idefiite or perpetual life expectacy ca be evaluated with the perpetuity model. For example, the preset value of a stock, if its divided paymets are projected to be stable, will be equal to the amout of the aual divided (cash flow) geerated by the stock divided by a appropriate discout rate. I Europea fiacial markets, a umber of perpetual bods have bee traded for several ceturies. I may regios i the Uited States, groud rets (perpetual leases o lad) are traded. The proper evaluatio of these ad may other ivestmets requires the use of perpetuity models. The maximum price a ivestor would be willig to pay for a perpetual bod geeratig a aual cash flow of $200, each discouted at a rate of 5%, ca be determied from equatio (4.19): PV = $ 200 P. =$ 4 005, SINGLE-STAGE GROWTH MODELS (Backgroud readig: sectios 4.9 ad 4.11) If the cash flow associated with a ivestmet were expected to grow at a costat aual rate of g, the amout of the cash flow geerated by that ivestmet i year t would be

19 QRMC04 9/17/01 4:43 PM Page 69 Sigle-stage growth models 69 CF t = CF 1 (1 + g) t 1, (4.20) where CF 1 is the cash flow geerated by the ivestmet i year oe. Thus, if a stock payig a divided of $100 i year oe were expected to icrease its divided paymet by 10% each year thereafter, the divided paymet i the fourth year would be $133.10: CF 4 = CF 1 ( ) 4 1 Similarly, the cash flow geerated by the ivestmet i the followig year (t + 1) will be CF t+1 = CF 1 (1 + g) t. The stock s divided i the fifth year will be $146.41: CF 4+1 = CF 1 ( ) 4 = $ If the stock had a ifiite life expectacy (as most stocks might be expected to), ad its divided paymets were discouted at a rate of 13%, the value of the stock would be determied by PV gp $ 100 $ 100 = = = $ 3, This expressio is ofte called the Gordo Stock Pricig Model. It assumes that the cash flows (divideds) associated with the stock are kow i the first period ad will grow at a costat compoud rate i subsequet periods. More geerally, this growig perpetuity expressio ca be writte as follows: PV CF1 = k g gp. (4.21) The growig perpetuity expressio simply subtracts the growth rate from the discout rate; the growth i cash flows helps to cover the time value of moey. This formula for evaluatig growig perpetuities ca be used oly whe k > g. If g > K, either the growth rate or discout rate has probably bee calculated improperly. Otherwise, the ivestmet would have a ifiite value (eve though the formula would geerate a egative value). The formula (4.22) for evaluatig growig auities ca be derived ituitively from the growig perpetuity model. I figure 4.2, the differece betwee the preset value of a growig perpetuity with cash flows begiig i time period is deducted from the preset value of a perpetuity with cash flows begiig i year oe, resultig i the preset value of a -year growig auity. Notice that the amout of the cash flow geerated by the growig auity i year ( + 1) is CF(1 + g). This is the first of the

20 QRMC04 9/17/01 4:43 PM Page The time value of moey Preset value of growig perpetuity begiig i oe year: CF 1 (k g) Time 1 Preset value of -year growig auity: PV GA Preset value of growig perpetuity begiig i year ( + 1): [CF 1(1 + g) /(k g)] (1 + k) Figure 4.2 Derivig growig auity preset value from growig perpetuity pereset value. The preset value of a growig perpetuity begiig i oe year mius the preset value of a secod growig perpetuity begiig i year ( + 1) equals the preset value of a -year growig auity: CF 1 /(k g) [CF 1 /(k g)] (1 + k) ; PV GA = [CF 1 /(k g)][1 (1 + g) ] (1 + k). cash flows ot geerated by the growig auity; it is geerated after the auity is sold or termiated. Because the cash flow is growig at the rate g, the iitial amout of the cash flow geerated by the secod perpetuity is exceeded by the iitial cash flow of the perpetuity begiig i year oe: PV GA = CF1 g k g ( 1 + ) 1. ( 1 + k) (4.22) Cash flows geerated by may ivestmets will grow at the rate of iflatio. For example, cosider a project udertake by a corporatio whose cash flow i year oe is expected to be $10,000. If cash flows were expected to grow at the iflatio rate of 6% each year util year six, the termiate, the project s preset value would be $48,320.35, assumig a discout rate of 11%: 6 $ 10, 000 ( +. ) PV GA = = $, (. ) = $, ( ) Cash flows are geerated by this ivestmet through the ed of the sixth year. No cash flow was geerated i the seveth year. Verify that the amout of cash flow that would have bee geerated by the ivestmet i the seveth year if it had cotiued to grow would have bee $10,000(1.06) 6 = $14,185. APPLICATION 4.6: STOCK VALUATION MODELS Cosider a stock whose aual divided ext year is projected to be $50. This paymet is expected to grow at a aual rate of 5% i subsequet years. A ivestor has

21 QRMC04 9/17/01 4:43 PM Page 71 Sigle-stage growth models 71 determied that the appropriate discout rate for this stock is 10%. The curret value of this stock is $1,000, determied by the growig perpetuity model: PV gp $ 50 = = $ 1, This model is ofte referred to as the Gordo Stock Pricig Model. It may seem that this model assumes that the stock will be held by the ivestor forever. But what if the ivestor iteds to sell the stock i five years? Its value would be determied by the sum of the preset values of cash flows the ivestor does expect to receive: PV GA DIV1 ( 1 + g) = 1 k g ( 1 + k), where P is the price the ivestor expects to receive whe he sells the stock i year ; ad DIV 1 is the divided paymet the ivestor expects to receive i year oe. The preset value of the divideds the ivestor expects to receive is $207.53: PV GA $ 50 ( +. ) = ( ) 5 5 = $ The sellig price of the stock i year five will be a fuctio of the divided paymets that the prospective purchaser expects to receive begiig i year six. Thus, i year five, the prospective purchaser will pay $1, for the stock, based o his iitial divided paymet of $63.81, determied by the followig equatios: DIV 6 = DIV 1 ( ) 6 1 = $63.81, stock value i year five = 63.81/( ) = $1, The preset value of the $1, that the ivestor will receive whe he sells the stock at the ed of the fifth year is $792.47: $ 1, PV = = $ ( 1+ 0.) 1 The total stock value will be the sum of the preset values of the divideds received by the ivestor ad his cash flows received from the sale of the stock. Thus, the curret value of the stock is $ plus $792.47, or $1,000. This is exactly the same sum determied by the growig perpetuity model earlier; therefore, the growig perpetuity model ca be used to evaluate a stock eve whe the ivestor expects to sell it.

22 QRMC04 9/17/01 4:43 PM Page The time value of moey 4.13 MULTIPLE-STAGE GROWTH MODELS (Backgroud readig: sectio 4.12) The Gordo Stock Pricig Model may be urealistic i may scearios, i that it assumes that oe growth rate applies to the firm s cash flows ad that this growth rate exteds forever. Multiple-stage growth models eable the user to allow for differet growth rates i differet periods. For example, a growth compay might geerate cash flows that are expected to grow at a high rate i the short term ad the declie as the firm matures. The multistage growth model ca accommodate this patter. Suppose, for example, that a ivestor has the opportuity to ivest i a stock curretly sellig for $100 per share. The stock is expected to pay a $3 divided ext year (at the ed of year 1). I each subsequet year util the seveth year, the aual divided is expected to grow at a rate of 20%. Startig i the eighth year, the aual divided will grow at a aual rate of 3% forever. All cash flows are to be discouted at a aual rate of 10%. Should the stock be purchased at its curret price? The followig Two-Stage Growth Model ca be used to evaluate this stock: 1 ( 1+ g1 ) P0 = DIV1 k g 1 ( k g1 )( 1 + k) 1 DIV1( 1+ g1) ( 1+ g2) +. ( k g2)( 1 + k) (4.23) Note that this model begis with a -year auity at growth rate g 1 ad accommodates the ew growth rate g 2 i the growig perpetuity that follows. The perpetuity is discouted a secod time because it is deferred; it does ot commece paymets util year. Substitutig values from the problem statemet yields the followig: ( ) ( )( ) 7 P 0 = $ $ 31 ( ) ( ) + = ( )( ) Sice the $100 purchase price of the stock exceeds its value, the stock should ot be purchased. The followig represets a Three-Stage Growth Model which is based o a growig auity, a deferred growig auity, ad a deferred growig perpetuity: ( 1) 1 ( 1+ g1 ) P0 = DIV1 k g 1 ( k g1 )( 1 + k) + DIV 1 ( 1) ( 1+ g1 ) ( 1+ g2) ( 1+ g1 ) ( 1+ g2) ( 1) ( 1 + k) ( k g2) ( k g2)( 1 + k) ( ) ( 2) ( 1) + 1 ( 1) 1 ( 2) ( 1) DIV1( 1+ g1) ( 1+ g2) ( 1+ g3) +. ( 2) ( k g )( 1 + k) 3 ( 1) ( 2)

23 QRMC04 9/17/01 4:43 PM Page 73 Exercises 73 There are three stages here, the first edig at time (1), the secod edig at time (2), ad the third extedig ito perpetuity. It may be a useful exercise to closely examie this expressio to determie why it is structured i this maer. Try to determie why the growth rates ad discout rates are structured as they are. Be certai to first be comfortable with the Preset-Value Growig Auity ad Perpetuity Models ad the Two-Stage Growth Model. EXERCISES 4.1. The Ruth Compay borrowed $21,000 at a aual iterest rate of 9%. What is the future value of this loa assumig iterest is accumulated o a simple basis? 4.2. The Cobb Compay has issued te millio dollars i 10% coupo bods maturig i five years. Iterest paymets o these bods will be made semiaually. (a) (b) (c) How much are Cobb s semiaual iterest paymets? What will be the total paymet made by Cobb o the bods i each of the first four years? What will be the total paymet made by Cobb o the bods i the fifth year? 4.3. I have the opportuity to deposit $10,000 ito my savigs accout today, which pays iterest at a aual rate of 5.5%, compouded daily. What will be the edig balace of my accout i five years if I make o additioal deposits or withdrawals? 4.4. What would be the future value of the loa i problem 4.1 if iterest were compouded: (a) (b) (c) (d) (e) aually? semiaually? mothly? daily? cotiuously? 4.5. A cosumer has the opportuity to deposit $10,000 ito his savigs accout today, which pays iterest at a aual rate of 5.5%, compouded daily. What will be the edig balace of his accout i five years if he makes o additioal deposits or withdrawals?

24 QRMC04 9/17/01 4:43 PM Page The time value of moey 4.6. The Speaker Compay has the opportuity to purchase a five-year $1,000 certificate of deposit (CD) payig iterest at a aual rate of 12%, compouded aually. The compay will ot withdraw early ay of the moey i its CD accout. Will this accout have a greater future value tha a five-year $1,000 CD payig a aual iterest rate of 10%, compouded daily? 4.7. The Waer Compay eeds to set aside a sum of moey today for the purpose of purchasig, for $10,000, a ew machie i three years. Moey used to fiace this purchase will be placed i a savigs accout payig iterest at a rate of 8%. How much moey must be placed i this accout ow to assure the Waer compay $10,000 i three years if iterest is compouded yearly? 4.8. A give savigs accout pays iterest at a aual rate of 3% compouded quarterly. Fid the aual percetage yield (APY) for this accout. 4.9.* Assumig o withdrawals or additioal deposits, how much time is required for $1,000 to double if placed i a savigs accout payig a aual iterest rate of 10% if iterest were: (a) (b) (c) (d) computed o a simple basis? compouded aually? compouded mothly? compouded cotiuously? What is the preset value of a security promisig to pay $10,000 i five years if its associated discout rate is: (a) 20%? (b) 10%? (c) 1%? (d) 0%? What is the preset value of a security to be discouted at a 10% rate promisig to pay $10,000 i: (a) (b) (c) (d) (e) 20 years? te years? oe year? six moths? 73 days? The Gehrig Compay is cosiderig a ivestmet that will result i a $2,000 cash flow i oe year, a $3,000 cash flow i two years, ad a $7,000 cash

25 QRMC04 9/17/01 4:43 PM Page 75 Exercises 75 flow i three years. What is the preset value of this ivestmet if all cash flows are to be discouted at a 8% rate? Should Gehrig Compay maagemet be willig to pay $10,000 for this ivestmet? The Cramde Compay has the opportuity to pay $30,000 for a security which promises to pay $6,000 i each of the ext ie years. Would this be a wise ivestmet if the appropriate discout rate were: (a) 5%? (b) 10%? (c) 20%? The Larse Compay is sellig preferred stock which is expected to pay a $50 aual divided per share. What is the preset value of divideds associated with each share of stock if the appropriate discout rate were 8% ad its life expectacy were ifiite? The Dikis Compay has purchased a machie whose output will result i a $5,000 cash flow i its first year of operatio. This cash flow is projected to grow at the aual 10% rate of iflatio over each of the ext te years. What will be the cash flow geerated by this machie i: (a) (b) (c) (d) its secod year of operatio? its third year of operatio? its fifth year of operatio? its teth year of operatio? The Wager Compay is cosiderig the purchase of a asset that will result i a $5,000 cash flow i its first year of operatio. Aual cash flows are projected to grow at the 10% aual rate of iflatio i subsequet years. The life expectacy of this asset is seve years, ad the appropriate discout rate for all cash flows is 12%. What is the maximum price that Wager should be willig to pay for this asset? What is the preset value of a stock whose $100 divided paymet ext year is projected to grow at a aual rate of 5%? Assume a ifiite life expectacy ad a 12% discout rate Which of the followig series of cash flows has the highest preset value at a 5% discout rate: (a) (b) (c) (d) $500,000 ow? $100,000 per year for eight years? $60,000 per year for 20 years? $30,000 each year forever?

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

More information

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth 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,

More information

I. Why is there a time value to money (TVM)?

I. Why is there a time value to money (TVM)? Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios

More information

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized? 5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso

More information

Terminology for Bonds and Loans

Terminology for Bonds and Loans ³ ² ± Termiology for Bods ad Loas Pricipal give to borrower whe loa is made Simple loa: pricipal plus iterest repaid at oe date Fixed-paymet loa: series of (ofte equal) repaymets Bod is issued at some

More information

FI A CIAL MATHEMATICS

FI A CIAL MATHEMATICS CHAPTER 7 FI A CIAL MATHEMATICS Page Cotets 7.1 Compoud Value 117 7.2 Compoud Value of a Auity 118 7.3 Sikig Fuds 119 7.4 Preset Value 122 7.5 Preset Value of a Auity 122 7.6 Term Loas ad Amortizatio 123

More information

FM4 CREDIT AND BORROWING

FM4 CREDIT AND BORROWING FM4 CREDIT AND BORROWING Whe you purchase big ticket items such as cars, boats, televisios ad the like, retailers ad fiacial istitutios have various terms ad coditios that are implemeted for the cosumer

More information

Time Value of Money. First some technical stuff. HP10B II users

Time Value of Money. First some technical stuff. HP10B II users Time Value of Moey Basis for the course Power of compoud iterest $3,600 each year ito a 401(k) pla yields $2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle

More information

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig

More information

Learning objectives. Duc K. Nguyen - Corporate Finance 21/10/2014

Learning objectives. Duc K. Nguyen - Corporate Finance 21/10/2014 1 Lecture 3 Time Value of Moey ad Project Valuatio The timelie Three rules of time travels NPV of a stream of cash flows Perpetuities, auities ad other special cases Learig objectives 2 Uderstad the time-value

More information

Simple Annuities Present Value.

Simple Annuities Present Value. Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX-9850GB PLUS to efficietly compute values associated with preset value auities.

More information

2 Time Value of Money

2 Time Value of Money 2 Time Value of Moey BASIC CONCEPTS AND FORMULAE 1. Time Value of Moey It meas moey has time value. A rupee today is more valuable tha a rupee a year hece. We use rate of iterest to express the time value

More information

CHAPTER 11 Financial mathematics

CHAPTER 11 Financial mathematics CHAPTER 11 Fiacial mathematics I this chapter you will: Calculate iterest usig the simple iterest formula ( ) Use the simple iterest formula to calculate the pricipal (P) Use the simple iterest formula

More information

TO: Users of the ACTEX Review Seminar on DVD for SOA Exam FM/CAS Exam 2

TO: Users of the ACTEX Review Seminar on DVD for SOA Exam FM/CAS Exam 2 TO: Users of the ACTEX Review Semiar o DVD for SOA Exam FM/CAS Exam FROM: Richard L. (Dick) Lodo, FSA Dear Studets, Thak you for purchasig the DVD recordig of the ACTEX Review Semiar for SOA Exam FM (CAS

More information

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig

More information

Institute of Actuaries of India Subject CT1 Financial Mathematics

Institute of Actuaries of India Subject CT1 Financial Mathematics Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i

More information

CDs Bought at a Bank verses CD s Bought from a Brokerage. Floyd Vest

CDs Bought at a Bank verses CD s Bought from a Brokerage. Floyd Vest CDs Bought at a Bak verses CD s Bought from a Brokerage Floyd Vest CDs bought at a bak. CD stads for Certificate of Deposit with the CD origiatig i a FDIC isured bak so that the CD is isured by the Uited

More information

BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets

BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets BENEIT-CST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal - Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts

More information

MMQ Problems Solutions with Calculators. Managerial Finance

MMQ Problems Solutions with Calculators. Managerial Finance MMQ Problems Solutios with Calculators Maagerial Fiace 2008 Adrew Hall. MMQ Solutios With Calculators. Page 1 MMQ 1: Suppose Newma s spi lads o the prize of $100 to be collected i exactly 2 years, but

More information

Bond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond

Bond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond What is a bod? Bod Valuatio I Bod is a I.O.U. Bod is a borrowig agreemet Bod issuers borrow moey from bod holders Bod is a fixed-icome security that typically pays periodic coupo paymets, ad a pricipal

More information

A Guide to the Pricing Conventions of SFE Interest Rate Products

A Guide to the Pricing Conventions of SFE Interest Rate Products A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios

More information

How to use what you OWN to reduce what you OWE

How to use what you OWN to reduce what you OWE How to use what you OWN to reduce what you OWE Maulife Oe A Overview Most Caadias maage their fiaces by doig two thigs: 1. Depositig their icome ad other short-term assets ito chequig ad savigs accouts.

More information

CHAPTER 4: NET PRESENT VALUE

CHAPTER 4: NET PRESENT VALUE EMBA 807 Corporate Fiace Dr. Rodey Boehe CHAPTER 4: NET PRESENT VALUE (Assiged probles are, 2, 7, 8,, 6, 23, 25, 28, 29, 3, 33, 36, 4, 42, 46, 50, ad 52) The title of this chapter ay be Net Preset Value,

More information

Question 2: How is a loan amortized?

Question 2: How is a loan amortized? Questio 2: How is a loa amortized? Decreasig auities may be used i auto or home loas. I these types of loas, some amout of moey is borrowed. Fixed paymets are made to pay off the loa as well as ay accrued

More information

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The

More information

INVESTMENT PERFORMANCE COUNCIL (IPC)

INVESTMENT PERFORMANCE COUNCIL (IPC) INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks

More information

Subject CT5 Contingencies Core Technical Syllabus

Subject CT5 Contingencies Core Technical Syllabus Subject CT5 Cotigecies Core Techical Syllabus for the 2015 exams 1 Jue 2014 Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which ca be used to model ad value

More information

Time Value of Money, NPV and IRR equation solving with the TI-86

Time Value of Money, NPV and IRR equation solving with the TI-86 Time Value of Moey NPV ad IRR Equatio Solvig with the TI-86 (may work with TI-85) (similar process works with TI-83, TI-83 Plus ad may work with TI-82) Time Value of Moey, NPV ad IRR equatio solvig with

More information

VALUATION OF FINANCIAL ASSETS

VALUATION OF FINANCIAL ASSETS P A R T T W O As a parter for Erst & Youg, a atioal accoutig ad cosultig firm, Do Erickso is i charge of the busiess valuatio practice for the firm s Southwest regio. Erickso s sigle job for the firm is

More information

Comparing Credit Card Finance Charges

Comparing Credit Card Finance Charges Comparig Credit Card Fiace Charges Comparig Credit Card Fiace Charges Decidig if a particular credit card is right for you ivolves uderstadig what it costs ad what it offers you i retur. To determie how

More information

Information about Bankruptcy

Information about Bankruptcy Iformatio about Bakruptcy Isolvecy Service of Irelad Seirbhís Dócmhaieachta a héirea Isolvecy Service of Irelad Seirbhís Dócmhaieachta a héirea What is the? The Isolvecy Service of Irelad () is a idepedet

More information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean 1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

More information

Present Value Tax Expenditure Estimate of Tax Assistance for Retirement Saving

Present Value Tax Expenditure Estimate of Tax Assistance for Retirement Saving Preset Value Tax Expediture Estimate of Tax Assistace for Retiremet Savig Tax Policy Brach Departmet of Fiace Jue 30, 1998 2 Preset Value Tax Expediture Estimate of Tax Assistace for Retiremet Savig This

More information

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature. Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.

More information

How to read A Mutual Fund shareholder report

How to read A Mutual Fund shareholder report Ivestor BulletI How to read A Mutual Fud shareholder report The SEC s Office of Ivestor Educatio ad Advocacy is issuig this Ivestor Bulleti to educate idividual ivestors about mutual fud shareholder reports.

More information

Investing in Stocks WHAT ARE THE DIFFERENT CLASSIFICATIONS OF STOCKS? WHY INVEST IN STOCKS? CAN YOU LOSE MONEY?

Investing in Stocks WHAT ARE THE DIFFERENT CLASSIFICATIONS OF STOCKS? WHY INVEST IN STOCKS? CAN YOU LOSE MONEY? Ivestig i Stocks Ivestig i Stocks Busiesses sell shares of stock to ivestors as a way to raise moey to fiace expasio, pay off debt ad provide operatig capital. Ecoomic coditios: Employmet, iflatio, ivetory

More information

Savings and Retirement Benefits

Savings and Retirement Benefits 60 Baltimore Couty Public Schools offers you several ways to begi savig moey through payroll deductios. Defied Beefit Pesio Pla Tax Sheltered Auities ad Custodial Accouts Defied Beefit Pesio Pla Did you

More information

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

More information

Solving Logarithms and Exponential Equations

Solving Logarithms and Exponential Equations Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:

More information

Understanding Financial Management: A Practical Guide Guideline Answers to the Concept Check Questions

Understanding Financial Management: A Practical Guide Guideline Answers to the Concept Check Questions Udestadig Fiacial Maagemet: A Pactical Guide Guidelie Aswes to the Cocept Check Questios Chapte 4 The Time Value of Moey Cocept Check 4.. What is the meaig of the tems isk-etu tadeoff ad time value of

More information

Discounting. Finance 100

Discounting. Finance 100 Discoutig Fiace 100 Prof. Michael R. Roberts 1 Topic Overview The Timelie Compoudig & Future Value Discoutig & Preset Value Multiple Cash Flows Special Streams of Cash Flows» Perpetuities» Auities Iterest

More information

INVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology

INVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology

More information

Swaps: Constant maturity swaps (CMS) and constant maturity. Treasury (CMT) swaps

Swaps: Constant maturity swaps (CMS) and constant maturity. Treasury (CMT) swaps Swaps: Costat maturity swaps (CMS) ad costat maturity reasury (CM) swaps A Costat Maturity Swap (CMS) swap is a swap where oe of the legs pays (respectively receives) a swap rate of a fixed maturity, while

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios

More information

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here). BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

More information

Statement of cash flows

Statement of cash flows 6 Statemet of cash flows this chapter covers... I this chapter we study the statemet of cash flows, which liks profit from the statemet of profit or loss ad other comprehesive icome with chages i assets

More information

Get advice now. Are you worried about your mortgage? New edition

Get advice now. Are you worried about your mortgage? New edition New editio Jauary 2009 Are you worried about your mortgage? Get advice ow If you are strugglig to pay your mortgage, or you thik it will be difficult to pay more whe your fixed-rate deal eds, act ow to

More information

Enhance Your Financial Legacy Variable Annuity Death Benefits from Pacific Life

Enhance Your Financial Legacy Variable Annuity Death Benefits from Pacific Life Ehace Your Fiacial Legacy Variable Auity Death Beefits from Pacific Life 7/15 20172-15B As You Pla for Retiremet, Protect Your Loved Oes A Pacific Life variable auity ca offer three death beefits that

More information

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients 652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you

More information

Confidence Intervals for One Mean

Confidence Intervals for One Mean Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

More information

Sequences and Series

Sequences and Series CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

More information

Basic Elements of Arithmetic Sequences and Series

Basic Elements of Arithmetic Sequences and Series MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

More information

Enhance Your Financial Legacy Variable Annuities with Death Benefits from Pacific Life

Enhance Your Financial Legacy Variable Annuities with Death Benefits from Pacific Life Ehace Your Fiacial Legacy Variable Auities with Death Beefits from Pacific Life 9/15 20188-15C FOR CALIFORNIA As You Pla for Retiremet, Protect Your Loved Oes A Pacific Life variable auity ca offer three

More information

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,

More information

Managing Your Money. UNIT 4D Loan Payments, Credit Cards, and Mortgages: We calculate monthly payments and explore loan issues.

Managing Your Money. UNIT 4D Loan Payments, Credit Cards, and Mortgages: We calculate monthly payments and explore loan issues. A fool ad his moey are soo parted. Eglish proverb Maagig Your Moey Maagig your persoal fiaces is a complex task i the moder world. If you are like most Americas, you already have a bak accout ad at least

More information

Finance Practice Problems

Finance Practice Problems Iteest Fiace Pactice Poblems Iteest is the cost of boowig moey. A iteest ate is the cost stated as a pecet of the amout boowed pe peiod of time, usually oe yea. The pevailig maket ate is composed of: 1.

More information

How To Find FINANCING For Your Business

How To Find FINANCING For Your Business How To Fid FINANCING For Your Busiess Oe of the most difficult tasks faced by the maagemet team of small busiesses today is fidig adequate fiacig for curret operatios i order to support ew ad ogoig cotracts.

More information

Incremental calculation of weighted mean and variance

Incremental calculation of weighted mean and variance Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

More information

DC College Savings Plan Helping Children Reach a Higher Potential

DC College Savings Plan Helping Children Reach a Higher Potential 529 DC College Savigs Pla Helpig Childre Reach a Higher Potetial reach Sposored by Govermet of the District of Columbia Office of the Mayor Office of the Chief Fiacial Officer Office of Fiace ad Treasury

More information

Introducing Your New Wells Fargo Trust and Investment Statement. Your Account Information Simply Stated.

Introducing Your New Wells Fargo Trust and Investment Statement. Your Account Information Simply Stated. Itroducig Your New Wells Fargo Trust ad Ivestmet Statemet. Your Accout Iformatio Simply Stated. We are pleased to itroduce your ew easy-to-read statemet. It provides a overview of your accout ad a complete

More information

Forecasting. Forecasting Application. Practical Forecasting. Chapter 7 OVERVIEW KEY CONCEPTS. Chapter 7. Chapter 7

Forecasting. Forecasting Application. Practical Forecasting. Chapter 7 OVERVIEW KEY CONCEPTS. Chapter 7. Chapter 7 Forecastig Chapter 7 Chapter 7 OVERVIEW Forecastig Applicatios Qualitative Aalysis Tred Aalysis ad Projectio Busiess Cycle Expoetial Smoothig Ecoometric Forecastig Judgig Forecast Reliability Choosig the

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

TIAA-CREF Wealth Management. Personalized, objective financial advice for every stage of life

TIAA-CREF Wealth Management. Personalized, objective financial advice for every stage of life TIAA-CREF Wealth Maagemet Persoalized, objective fiacial advice for every stage of life A persoalized team approach for a trusted lifelog relatioship No matter who you are, you ca t be a expert i all aspects

More information

Grow your business with savings and debt management solutions

Grow your business with savings and debt management solutions Grow your busiess with savigs ad debt maagemet solutios A few great reasos to provide bak ad trust products to your cliets You have the expertise to help your cliets get the best rates ad most competitive

More information

Lesson 17 Pearson s Correlation Coefficient

Lesson 17 Pearson s Correlation Coefficient Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig

More information

Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2

Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2 74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is

More information

Determining the sample size

Determining the sample size Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

More information

I apply to subscribe for a Stocks & Shares ISA for the tax year 20 /20 and each subsequent year until further notice.

I apply to subscribe for a Stocks & Shares ISA for the tax year 20 /20 and each subsequent year until further notice. IFSL Brooks Macdoald Fud Stocks & Shares ISA Trasfer Applicatio Form IFSL Brooks Macdoald Fud Stocks & Shares ISA Trasfer Applicatio Form Please complete usig BLOCK CAPITALS ad retur the completed form

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say

More information

France caters to innovative companies and offers the best research tax credit in Europe

France caters to innovative companies and offers the best research tax credit in Europe 1/5 The Frech Govermet has three objectives : > improve Frace s fiscal competitiveess > cosolidate R&D activities > make Frace a attractive coutry for iovatio Tax icetives have become a key elemet of public

More information

Present Values, Investment Returns and Discount Rates

Present Values, Investment Returns and Discount Rates Preset Values, Ivestmet Returs ad Discout Rates Dimitry Midli, ASA, MAAA, PhD Presidet CDI Advisors LLC dmidli@cdiadvisors.com May 2, 203 Copyright 20, CDI Advisors LLC The cocept of preset value lies

More information

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

More information

Pre-Suit Collection Strategies

Pre-Suit Collection Strategies Pre-Suit Collectio Strategies Writte by Charles PT Phoeix How to Decide Whether to Pursue Collectio Calculatig the Value of Collectio As with ay busiess litigatio, all factors associated with the process

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

Output Analysis (2, Chapters 10 &11 Law)

Output Analysis (2, Chapters 10 &11 Law) B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

More information

You are given that mortality follows the Illustrative Life Table with i 0.06 and that deaths are uniformly distributed between integral ages.

You are given that mortality follows the Illustrative Life Table with i 0.06 and that deaths are uniformly distributed between integral ages. 1. A 2 year edowmet isurace of 1 o (6) has level aual beefit premiums payable at the begiig of each year for 1 years. The death beefit is payable at the momet of death. You are give that mortality follows

More information

A NOTE ON THE CALCULATION OF THE AFTER-TAX COST OF DEBT

A NOTE ON THE CALCULATION OF THE AFTER-TAX COST OF DEBT INTERNATIONAL JOURNAL OF BUSINESS, 1(1), 1996 ISSN:1083-4346 A NOTE ON THE CALCULATION OF THE AFTER-TAX COST OF DEBT Wm R McDaiel, Daiel E. McCarty, ad Keeth A. Jessell Whe oe examies stadard fiacial maagemet

More information

Learning Objectives. Chapter 2 Pricing of Bonds. Future Value (FV)

Learning Objectives. Chapter 2 Pricing of Bonds. Future Value (FV) Leaig Objectives Chapte 2 Picig of Bods time value of moey Calculate the pice of a bod estimate the expected cash flows detemie the yield to discout Bod pice chages evesely with the yield 2-1 2-2 Leaig

More information

Create Income for Your Retirement. What You Can Expect. What to Consider. Page 1 of 7

Create Income for Your Retirement. What You Can Expect. What to Consider. Page 1 of 7 Page 1 of 7 RBC Retiremet Icome Plaig Process Create Icome for Your Retiremet At RBC Wealth Maagemet, we believe maagig your wealth to produce a icome durig retiremet is fudametally differet from maagig

More information

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL. Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory

More information

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern. 5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

More information

For customers Key features of the Guaranteed Pension Annuity

For customers Key features of the Guaranteed Pension Annuity For customers Key features of the Guarateed Pesio Auity The Fiacial Coduct Authority is a fiacial services regulator. It requires us, Aego, to give you this importat iformatio to help you to decide whether

More information

PENSION ANNUITY. Policy Conditions Document reference: PPAS1(7) This is an important document. Please keep it in a safe place.

PENSION ANNUITY. Policy Conditions Document reference: PPAS1(7) This is an important document. Please keep it in a safe place. PENSION ANNUITY Policy Coditios Documet referece: PPAS1(7) This is a importat documet. Please keep it i a safe place. Pesio Auity Policy Coditios Welcome to LV=, ad thak you for choosig our Pesio Auity.

More information

Hypothesis testing. Null and alternative hypotheses

Hypothesis testing. Null and alternative hypotheses Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

More information

Choosing a Mortgage FIXED-RATE MORTGAGES. ADJUSTABLE-RATE MORTGAGES (ARMs)

Choosing a Mortgage FIXED-RATE MORTGAGES. ADJUSTABLE-RATE MORTGAGES (ARMs) Choosig A Mortgage Like homes, home mortgages come i all shapes ad sizes: short-term, log-term, fixed, adjustable, jumbo, balloo these are all terms that will soo be familiar to you, if they re ot already.

More information

A GUIDE TO BUILDING SMART BUSINESS CREDIT

A GUIDE TO BUILDING SMART BUSINESS CREDIT A GUIDE TO BUILDING SMART BUSINESS CREDIT Establishig busiess credit ca be the key to growig your compay DID YOU KNOW? Busiess Credit ca help grow your busiess Soud paymet practices are key to a solid

More information

MainStay Funds IRA/SEP/Roth IRA Distribution Form

MainStay Funds IRA/SEP/Roth IRA Distribution Form MaiStay Fuds IRA/SEP/Roth IRA Distributio Form Do ot use for IRA Trasfers or SIMPLE IRA INSTRUCTIONS Before completig this form, please refer to the applicable Custodial Agreemet ad Disclosure Statemet

More information

MARTINGALES AND A BASIC APPLICATION

MARTINGALES AND A BASIC APPLICATION MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this

More information

Erik Ottosson & Fredrik Weissenrieder, 1996-03-01 CVA. Cash Value Added - a new method for measuring financial performance.

Erik Ottosson & Fredrik Weissenrieder, 1996-03-01 CVA. Cash Value Added - a new method for measuring financial performance. CVA Cash Value Added - a ew method for measurig fiacial performace Erik Ottosso Strategic Cotroller Sveska Cellulosa Aktiebolaget SCA Box 7827 S-103 97 Stockholm Swede Fredrik Weisserieder Departmet of

More information

Sole trader financial statements

Sole trader financial statements 3 Sole trader fiacial statemets this chapter covers... I this chapter we look at preparig the year ed fiacial statemets of sole traders (that is, oe perso ruig their ow busiess). We preset the fiacial

More information

Corporation tax trading profits

Corporation tax trading profits 2 Corporatio tax tradig profits this chapter covers... I this chapter we provide a brief review of the Corporatio Tax computatio, ad examie the step by step procedures for compilig the computatio, before

More information

Hypergeometric Distributions

Hypergeometric Distributions 7.4 Hypergeometric Distributios Whe choosig the startig lie-up for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you

More information

Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:

Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows: Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network

More information

Performance Attribution in Private Equity

Performance Attribution in Private Equity Performace Attributio i Private Equity Austi M. Log, III MPA, CPA, JD Parter Aligmet Capital Group 4500 Steier Rach Blvd., Ste. 806 Austi, TX 78732 Phoe 512.506.8299 Fax 512.996.0970 E-mail alog@aligmetcapital.com

More information

Current Year Income Assessment Form

Current Year Income Assessment Form Curret Year Icome Assessmet Form Academic Year 2015/16 Persoal details Perso 1 Your Customer Referece Number Your Customer Referece Number Name Name Date of birth Address / / Date of birth / / Address

More information

Money Math for Teens. Introduction to Earning Interest: 11th and 12th Grades Version

Money Math for Teens. Introduction to Earning Interest: 11th and 12th Grades Version Moey Math fo Tees Itoductio to Eaig Iteest: 11th ad 12th Gades Vesio This Moey Math fo Tees lesso is pat of a seies ceated by Geeatio Moey, a multimedia fiacial liteacy iitiative of the FINRA Ivesto Educatio

More information

My first gold holdings. My first bank. Simple. Transparent. Individual. Our investment solutions for clients abroad.

My first gold holdings. My first bank. Simple. Transparent. Individual. Our investment solutions for clients abroad. My first gold holdigs. My first bak. Simple. Trasparet. Idividual. Our ivestmet solutios for cliets abroad. The perfect basis for workig together successfully The wheel of time is turig faster tha ever

More information

Abstract. 1. Introduction

Abstract. 1. Introduction Optimal Tax Deferral Choices i the Presece of Chagig Tax Regimes Terrace Jalbert (E-mail: jalbert@hawaii.edu), Uiversity of Hawaii at Hilo Eric Rask (E-mail: rask@hawaii.edu) Uiversity of Hawaii at Hilo

More information