CHAPTER 3 THE TIME VALUE OF MONEY


 Harriet Weaver
 2 years ago
 Views:
Transcription
1 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 the techiques used i fiace, oe is more importat tha the cocept of the time value of moey, or discouted cash flow (DCF) aalysis. The priciples of time value aalysis that are developed i this chapter have may applicatios, ragig from settig up schedules for payig off loas to decisios about whether to acquire ew equipmet. Future value ad preset value techiques ca be applied to a sigle cash flow (lump sum), ordiary auities, auities due, ad ueve cash flow streams. Future ad preset values ca be calculated usig a regular calculator or a calculator with fiacial fuctios. Whe compoudig occurs more frequetly tha oce a year, the effective rate of iterest is greater tha the quoted rate. OUTLINE The cash flow time lie is oe of the most importat tools i time value of moey aalysis. Cash flow time lies help to visualize what is happeig i a particular problem. Cash flows are placed directly below the tick marks, ad iterest rates are show directly above the time lie; ukow cash flows are idicated by questio marks. Thus, to fid the future value of $100 after 5 years at 5 percet iterest, the followig cash flow time lie ca be set up: Time: % Cash flows: 100 FV 5 =? A cash outflow is a paymet, or disbursemet, of cash for expeses, ivestmets, ad so o. A cash iflow is a receipt of cash from a ivestmet, a employer, or other sources. Compoudig is the process of determiig the value of a cash flow or series of cash flows some time i the future whe compoud iterest is applied. The future value is the amout to which a cash flow or series of cash flows will grow over a give period of time whe compouded at a give iterest rate. The future value ca be calculated as
2 40 FV = PV(1 + k), where PV = preset value, or begiig amout; k = iterest rate per period; ad = umber of periods ivolved i the aalysis. This equatio ca be solved i oe of two ways: umerically or with a fiacial calculator. For calculatios, assume the followig data that were preseted i the time lie above: preset value (PV) = $100, iterest rate (k) = 5%, ad umber of years () = 5. Compouded iterest is iterest eared o iterest. To solve umerically, use a regular calculator to fid 1 + k = 1.05 raised to the fifth power, which equals Multiply this figure by PV = $100 to get the fial aswer of FV 5 = $ With a fiacial calculator, the future value ca be foud by usig the time value of moey iput keys, where N = umber of periods, I = iterest rate per period, PV = preset value, PMT = auity paymet, ad FV = future value. By eterig N = 5, I = 5, PV = 100, ad PMT = 0, ad the pressig the FV key, the aswer is displayed. Some fiacial calculators require that all cash flows be desigated as either iflows or outflows, thus a outflow must be etered as a egative umber (for example, PV = 100 istead of PV = 100). Some calculators require you to press a Compute key before pressig the FV key. A graph of the compoudig process shows how ay sum grows over time at various iterest rates. The greater the rate of iterest, the faster is the rate of growth. The iterest rate is, i fact, a growth rate. The time value cocepts ca be applied to aythig that is growig. Fidig the preset value of a cash flow or series of cash flows is called discoutig, ad it is simply the reverse of compoudig. I geeral, the preset value is the value today of a future cash flow or series of cash flows. By solvig for PV i the future value equatio, the preset value, or discoutig, equatio ca be developed ad writte i several forms: PV = FV (1 k) FV 1 (1 k). To solve for the preset value of $ discouted back 5 years at a 5% opportuity cost rate, oe ca utilize either of the two solutio methods: Numerical solutio: Divide $ by 1.05 five times to get PV = $100. Fiacial calculator solutio: Eter N = 5, I = 5, PMT = 0, ad FV = , ad the
3 41 press the PV key to get PV = The opportuity cost rate is the rate of retur o the best available alterative ivestmet of equal risk. A graph of the discoutig process shows how the preset value of ay sum to be received i the future dimiishes ad approaches zero as the paymet date is exteded farther ito the future. At relatively high iterest rates, fuds due i the future are worth very little today, ad eve at a relatively low discout rate, the preset value of a sum due i the very distat future is quite small. The compoudig ad discoutig processes are reciprocals, or iverses, of oe aother. I additio, there are four variables i the time value of moey equatios: PV, FV, k, ad. If three of the four variables are kow, you ca fid the value of the fourth. If we are give PV, FV, ad, we ca determie k by substitutig the kow values ito either the preset value or future value equatios, ad the solvig for k. Thus, if you ca buy a security at a price of $78.35 which will pay you $100 after 5 years, what is the iterest rate eared o the ivestmet? Numerical solutio: Use a trial ad error process to reach the 5% value for k. This is a tedious ad iefficiet process. Alteratively, you could use algebra to solve the time value equatio. Fiacial calculator solutio: Eter N = 5, PV = , PMT = 0, ad FV = 100, the press the I key, ad I = 5 is displayed. Likewise, if we are give PV, FV, ad k, we ca determie by substitutig the kow values ito either the preset value or future value equatios, ad the solvig for. Thus, if you ca buy a security with a 5 percet iterest rate at a price of $78.35 today, how log will it take for your ivestmet to retur $100? Numerical solutio: Use a trial ad error process to reach the value of 5 for. This is a tedious ad iefficiet process. The equatio ca also be solved algebraically. Fiacial calculator solutio: Eter I = 5, PV = , PMT = 0, ad FV = 100, the press the N key, ad N = 5 is displayed. A auity is a series of equal paymets made at fixed itervals for a specified umber of periods. If the paymets occur at the ed of each period, as they typically do, the auity is a ordiary, or deferred, auity. If the paymets occur at the begiig of each period, it is called a auity due. The future value of a ordiary auity, FVA, is the total amout oe would have at the ed of the auity period if each paymet were ivested at a give iterest rate ad held to the ed of the auity period.
4 42 Defiig FVA as the future value of a ordiary auity of years, ad PMT as the periodic paymet, we ca write FVA = PMT t 1 t ( 1 k) = PMT 1 t (1 k) 1 ( 1 k) = PMT. t 0 k Usig a fiacial calculator, eter N = 3, I = 5, PV = 0, ad PMT = The press the FV key, ad is displayed. For a auity due, each paymet is compouded for oe additioal period, so the future value of the etire auity is equal to the future value of a ordiary auity compouded for oe additioal period. Thus: (1 k) FVA (DUE) = PMT k 1 (1 k). Most fiacial calculators have a switch, or key, marked DUE or BEG that permits you to switch from edofperiod paymets (a ordiary auity) to begiigofperiod paymets (a auity due). Switch your calculator to BEG mode, ad calculate as you would for a ordiary auity. Do ot forget to switch your calculator back to END mode whe you are fiished. The preset value of a ordiary auity, PVA, is the sigle (lump sum) paymet today that would be equivalet to the auity paymets spread over the auity period. It is the amout today that would permit withdrawals of a equal amout (PMT) at the ed (or begiig for a auity due) of each period for periods. Defiig PVA as the preset value of a ordiary auity of years ad PMT as the periodic paymet, we ca write PVA = PMT t1 1 (1 t k) = PMT 1 (1 k) k 1 1 (1 k) = PMT. k Usig a fiacial calculator, eter N = 3, I = 5, PMT = 100, ad FV = 0, ad the press the PV key, for a aswer of $ Oe especially importat applicatio of the auity cocept relates to loas with costat paymets, such as mortgages ad auto loas. With these amortized loas the amout borrowed is the preset value of a ordiary auity, ad the paymets costitute the auity stream. The preset value for a auity due is
5 43 PVA (DUE) = PMT 1 (1 k) k 1 (1 k). Usig a fiacial calculator, switch to the BEG mode, ad the eter N = 3, I = 5, PMT = 100, ad FV = 0, ad the press PV to get the aswer, $ Agai, do ot forget to switch your calculator back to END mode whe you are fiished. You ca solve for the iterest rate (rate of retur) eared o a auity. To solve umerically, you must use the trialaderror process ad plug i differet values for k i the auity equatio to solve for the iterest rate. You ca use the fiacial calculator by eterig the appropriate values for N, PMT, ad either FV or PV, ad the pressig I to solve for the iterest rate. You ca solve for the umber of periods (N) i a auity. To solve umerically, you must use the trialaderror process ad plug i differet values for N i the auity equatio to solve for the umber of periods. You ca use the fiacial calculator by eterig the appropriate values for I, PMT, ad either FV or PV, ad the pressig N to solve for the umber of periods. A perpetuity is a stream of equal paymets expected to cotiue forever. The preset value of a perpetuity is: PVP = Paymet Iterest rate PMT. k For example, if the iterest rate were 12 percet, a perpetuity of $1,000 a year would have a preset value of $1,000/0.12 = $8, A cosol is a perpetual bod issued by the British govermet to cosolidate past debts; i geeral, ay perpetual bod. The value of a perpetuity chages dramatically whe iterest rates chage. May fiacial decisios require the aalysis of ueve, or ocostat, cash flows rather tha a stream of fixed paymets such as a auity. A ueve cash flow stream is a series of cash flows i which the amout varies from oe period to the ext. The term paymet, PMT, desigates costat cash flows, while the term CF desigates cash flows i geeral, icludig ueve cash flows.
6 44 The preset value of a ueve cash flow stream is the sum of the PVs of the idividual cash flows of the stream. The PV is foud by applyig the followig geeral preset value equatio: 1 PV = CF t. t t1 (1 k) With a fiacial calculator, eter each cash flow (begiig with the t = 0 cash flow) ito the cash flow register, CF j, eter the appropriate iterest rate, ad the press the NPV key to obtai the PV of the cash flow stream. Be sure to clear the cash flow register before startig a ew problem. Similarly, the future value of a ueve cash flow stream, or termial value, is the sum of the FVs of the idividual cash flows of the stream. The FV ca be foud by applyig the followig geeral future value equatio: t FV = CF t (1 k). t1 Some calculators have a et future value (NFV) key which allows you to obtai the FV of a ueve cash flow stream. We geerally are more iterested i the preset value of a asset s cash flow stream tha i the future value because the preset value represets today s value, which we ca compare with the price of the asset. Oce we kow its preset value, we ca fid the future value of a ueve cash flow stream by treatig the preset value as a lump sum amout ad compoudig it to the future period. If oe kows the relevat cash flows, the effective iterest rate ca be calculated efficietly with a fiacial calculator. Eter each cash flow (begiig with the t = 0 cash flow) ito the cash flow register, CF j, ad the press the IRR key to obtai the iterest rate of a ueve cash flow stream. IRR stads for iteral rate of retur, which is the retur o a ivestmet. Aual compoudig is the arithmetic process of determiig the fial value of a cash flow or series of cash flows whe iterest is added oce a year. Semiaual, quarterly, ad other compoudig periods more frequet tha o a aual basis are ofte used i fiacial trasactios. Compoudig o a oaual basis requires a adjustmet to both the compoudig ad discoutig procedures discussed previously. Moreover, whe comparig securities with differet compoudig periods, they eed to be put o a commo basis. This requires distiguishig betwee the simple, or quoted, iterest rate ad the effective aual rate.
7 45 The simple, or quoted, iterest rate is the cotracted, or quoted, iterest rate that is used to calculate the iterest paid per period. The periodic rate is the iterest rate charged per period. Periodic rate = Stated aual iterest rate/number of periods per year. The aual percetage rate, APR, is the periodic rate times the umber of periods per year. The effective aual rate, EAR, is the rate that would have produced the fial compouded value uder aual compoudig. The effective aual rate is give by the followig formula: k SIMPLE Effective aual rate (EAR) = 1 1.0, m where k SIMPLE is the simple, or quoted, iterest rate (that is, the APR), ad m is the umber of compoudig periods (iterest paymets) per year. The EAR is useful i comparig securities with differet compoudig periods. m For example, to fid the effective aual rate if the simple rate is 6 percet ad semiaual compoudig is used, we have: EAR = ( /2) = 6.09%. For aual compoudig use the formula to fid the future value of a sigle paymet (lump sum): FV = PV(1 + k). Whe compoudig occurs more frequetly tha oce a year, use this formula: m k SIMPLE FV = PV1. m Here m is the umber of times per year compoudig occurs, ad is the umber of years. The amout to which $1,000 will grow after 5 years if quarterly compoudig is applied to a omial 8 percet iterest rate is foud as follows: FV = $1,000( /4) (4)(5) = $1,000(1.02) 20 = $1, Fiacial calculator solutio: Eter N = 20, I = 2, PV = 1000, ad PMT = 0, ad the press the FV key to fid FV = $1,
8 46 The preset value of a 5year future ivestmet equal to $1,485.95, with a 8 percet omial iterest rate, compouded quarterly, is foud as follows: $1, PV PV(1 0.08/4) $1, (1.02) (4)(5) $1,000. Fiacial calculator solutio: Eter N = 20, I = 2, PMT = 0, ad FV = , ad the press the PV key to fid PV = $1, I geeral, oaual compoudig ca be hadled oe of two ways. State everythig o a periodic rather tha o a aual basis. Thus, = 6 periods rather tha = 3 years ad k = 3% istead of k = 6% with semiaual compoudig. Fid the effective aual rate (EAR) with the equatio below ad the use the EAR as the rate over the give umber of years. k SIMPLE EAR = m A importat applicatio of compoud iterest ivolves amortized loas, which are paid off i equal istallmets over the life of the loa. m The amout of each paymet, PMT, is foud usig a fiacial calculator by eterig N (umber of years), I (iterest rate), PV (amout borrowed), ad FV = 0, ad the pressig the PMT key to fid the periodic paymet. Each paymet cosists partly of iterest ad partly of repaymet of the amout borrowed (pricipal). This breakdow is ofte developed i a loa amortizatio schedule. The iterest compoet is largest i the first period, ad it declies over the life of the loa as the outstadig balace of the loa decreases. The repaymet of pricipal is smallest i the first period, ad it icreases thereafter. The text discussio has ivolved three differet iterest rates. It is importat to uderstad their differeces. The simple, or quoted, rate, k SIMPLE, is the iterest rate quoted by borrowers ad leders. This quotatio must iclude the umber of compoudig periods per year. This rate is ever show o a time lie, ad it is ever used as a iput i a fiacial calculator uless compoudig occurs oly oce a year. k SIMPLE = Periodic rate m = Aual percetage rate = APR. The periodic rate, k PER, is the rate charged by a leder or paid by a borrower each iterest period. Periodic rate = k PER = k SIMPLE /m.
9 47 The periodic rate is used for calculatios i problems where two coditios hold: (1) paymets occur o a regular basis more frequetly tha oce a year, ad (2) a paymet is made o each compoudig (or discoutig) date. The APR, or aual percetage rate, represets the periodic rate stated o a aual basis without cosiderig iterest compoudig. The APR ever is used i actual calculatios; it is simply reported to borrowers. The effective aual rate, EAR, is the rate with which, uder aual compoudig, we would obtai the same result as if we had used a give periodic rate with m compoudig periods per year. EAR is foud as follows: k SIMPLE EAR = m m
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 timevalue
More information5.4 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 informationTime 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 informationChapter 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 informationI. 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 informationFI 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 informationPresent 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 informationTerminology 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 Fixedpaymet loa: series of (ofte equal) repaymets Bod is issued at some
More informationTime Value of Money, NPV and IRR equation solving with the TI86
Time Value of Moey NPV ad IRR Equatio Solvig with the TI86 (may work with TI85) (similar process works with TI83, TI83 Plus ad may work with TI82) Time Value of Moey, NPV ad IRR equatio solvig with
More informationSimple Annuities Present Value.
Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX9850GB PLUS to efficietly compute values associated with preset value auities.
More information2 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 informationFM4 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 informationBENEFITCOST ANALYSIS Financial and Economic Appraisal using Spreadsheets
BENEITCST 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.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 informationQuestion 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 informationCDs 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 informationCHAPTER 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 informationInstitute 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 informationVALUATION 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 informationTHE TIME VALUE OF MONEY
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
More informationMMQ 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 informationBond 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 fixedicome security that typically pays periodic coupo paymets, ad a pricipal
More informationCHAPTER 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 informationINTRODUCTION TO ENGINEERING ECONOMICS. Types of Interest
INTRODUCTION TO ENGINEERING ECONOMICS A. J. Clark School of Egieerig Departmet of Civil ad Evirometal Egieerig by Dr. Ibrahim A. Assakkaf Sprig 2000 Departmet of Civil ad Evirometal Egieerig Uiversity
More informationSECTION 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 informationwhere: 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 informationBond Pricing Theorems. Floyd Vest
Bod Pricig Theorems Floyd Vest The followig Bod Pricig Theorems develop mathematically such facts as, whe market iterest rates rise, the price of existig bods falls. If a perso wats to sell a bod i this
More informationTO: 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 informationA Resource for Freestanding Mathematics Qualifications Working with %
Ca you aswer these questios? A savigs accout gives % iterest per aum.. If 000 is ivested i this accout, how much will be i the accout at the ed of years? A ew car costs 16 000 ad its value falls by 1%
More informationINVESTMENT 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 informationUnderstanding 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 isketu tadeoff ad time value of
More informationTime Value of Money and Investment Analysis
Time Value of Moey ad Ivestmet Aalysis Explaatios ad Spreadsheet Applicatios for Agricultural ad Agribusiess Firms Part I. by Bruce J. Sherrick Paul N. Elliger David A. Lis V 1.2, September 2000 The Ceter
More informationFramingham State College Department of Economics and Business Managerial Finance Practice Final Exam Spring 2006
Name Framigham State College Departmet of Ecoomics ad Busiess Maagerial Fiace Practice Fial Exam Sprig 2006 This exam provides questios that are represetative of those cotaied o your exam. This test should
More informationDiscounting. 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 informationMESSAGE TO TEACHERS: NOTE TO EDUCATORS:
MESSAGE TO TEACHERS: NOTE TO EDUCATORS: Attached herewith, please fid suggested lesso plas for term 1 of MATHEMATICS Grade 12. Please ote that these lesso plas are to be used oly as a guide ad teachers
More informationSoving 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 informationGeometric Sequences and Series. Geometric Sequences. Definition of Geometric Sequence. such that. a2 4
3330_0903qxd /5/05 :3 AM Page 663 Sectio 93 93 Geometric Sequeces ad Series 663 Geometric Sequeces ad Series What you should lear Recogize, write, ad fid the th terms of geometric sequeces Fid th partial
More informationSolving 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 informationChecklist. Assignment
Checklist Part I Fid the simple iterest o a pricipal. Fid a compouded iterest o a pricipal. Part II Use the compoud iterest formula. Compare iterest growth rates. Cotiuous compoudig. (Math 1030) M 1030
More information*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 informationHow 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 shortterm assets ito chequig ad savigs accouts.
More informationCHAPTER 2. Time Value of Money 61
CHAPTER 2 Tme Value of Moey 6 Tme Value of Moey (TVM) Tme Les Future value & Preset value Rates of retur Autes & Perpetutes Ueve cash Flow Streams Amortzato 62 Tme les 0 2 3 % CF 0 CF CF 2 CF 3 Show
More informationExample 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 informationExponential function: For a > 0, the exponential function with base a is defined by. f(x) = a x
MATH 11011 EXPONENTIAL FUNCTIONS KSU AND THEIR APPLICATIONS Defiitios: Expoetial fuctio: For a > 0, the expoetial fuctio with base a is defied by fx) = a x Horizotal asymptote: The lie y = c is a horizotal
More informationDefinition. 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 informationA 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 informationA NOTE ON THE CALCULATION OF THE AFTERTAX COST OF DEBT
INTERNATIONAL JOURNAL OF BUSINESS, 1(1), 1996 ISSN:10834346 A NOTE ON THE CALCULATION OF THE AFTERTAX COST OF DEBT Wm R McDaiel, Daiel E. McCarty, ad Keeth A. Jessell Whe oe examies stadard fiacial maagemet
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More information8.1 Arithmetic Sequences
MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first
More informationIntroduction to Financial Derivatives
550.444 Itroductio to Fiacial Derivatives Iterest Rates Week of September 3, 013 Where we are Previously: Fudametals of Forward ad Futures Cotracts (Chapters 3, OFOD) This week: Itroductio to Iterest
More informationSubject 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 informationNATIONAL 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 informationCHAPTER Factors: How Time and Interest Affect Money
2 CHAPTER Factors: How Time ad Iterest Affect Moey I the previous chapter we leared the basic cocepts of egieerig ecoomy ad their role i decisio makig. The cash flow is fudametal to every ecoomic study.
More informationNEW HIGH PERFORMANCE COMPUTATIONAL METHODS FOR MORTGAGES AND ANNUITIES. Yuri Shestopaloff,
NEW HIGH PERFORMNCE COMPUTTIONL METHODS FOR MORTGGES ND NNUITIES Yuri Shestopaloff, Geerally, mortgage ad auity equatios do ot have aalytical solutios for ukow iterest rate, which has to be foud usig umerical
More informationThe Mathematics of Amortization Schedules on the TI83. Floyd Vest, Nov (Preliminary Version)
The Mathematics of Amortizatio Schedules o the TI83 Note: TI 83/84 letters are ot i italics. Floyd Vest, Nov. 2011 (Prelimiary Versio) Moey wisely ivested i a home ca provide a secure form of ivestmet.
More informationChapter Gaussian Elimination
Chapter 04.06 Gaussia Elimiatio After readig this chapter, you should be able to:. solve a set of simultaeous liear equatios usig Naïve Gauss elimiatio,. lear the pitfalls of the Naïve Gauss elimiatio
More informationINVESTMENT 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 informationPreSuit Collection Strategies
PreSuit 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 informationIntroducing 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 easytoread statemet. It provides a overview of your accout ad a complete
More informationComparing 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 informationSavings 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 informationLiteral Equations and Formulas
. Literal Equatios ad Formulas. OBJECTIVE 1. Solve a literal equatio for a specified variable May problems i algebra require the use of formulas for their solutio. Formulas are simply equatios that express
More informationCHAPTER 3: FINANCIAL ANALYSIS WITH INFLATION
Up to ow, we have mostly igored iflatio. However, iflatio ad iterest are closely related. It was oted i the last chapter that iterest rates should geerally cover more tha iflatio. I fact, the amout of
More informationAnnuities 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 informationStatement 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 informationPVA(DUE) n PMT { o 2.1 PVP PMT 2.2. PVCF n CF Equity multiplier ) Total assets Common equity. r Treasury r RF MRP [r* IP] MRP 5.
Chapter 2 Aalysis of Fiacial Statemets et cash et Depreciatio ad flow icome amortizatio ROA et profit margi turover et icome 2.1 2.2 PVADUE PMT { o t1 PVP PMT 1 r ] Equatio Card 1 1 r t ]1 r} 1 1 _ 1 r
More informationDivide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016
CS 6, Lecture 3 Divide ad Coquer, Solvig Recurreces, Iteger Multiplicatio Scribe: Juliaa Cook (05, V Williams Date: April 6, 06 Itroductio Today we will cotiue to talk about divide ad coquer, ad go ito
More informationArithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition of Arithmetic Sequence. Example 1. 7, 11, 15, 19,..., 4n 3,...
3330_090.qxd 1/5/05 11:9 AM Page 653 Sectio 9. Arithmetic Sequeces ad Partial Sums 653 9. Arithmetic Sequeces ad Partial Sums What you should lear Recogize,write, ad fid the th terms of arithmetic sequeces.
More informationClassic Problems at a Glance using the TVM Solver
C H A P T E R 2 Classc Problems at a Glace usg the TVM Solver The table below llustrates the most commo types of classc face problems. The formulas are gve for each calculato. A bref troducto to usg the
More informationARITHMETIC AND GEOMETRIC PROGRESSIONS
Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives
More informationwhen n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.
Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have
More informationMathematical 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 informationOutput 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 informationSum of Exterior Angles of Polygons TEACHER NOTES
Sum of Exterior Agles of Polygos TEACHER NOTES Math Objectives Studets will determie that the iterior agle of a polygo ad a exterior agle of a polygo form a liear pair (i.e., the two agles are supplemetary).
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationHere 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 informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 009() MARKS: 50 TIME: 3 hours This questio paper cosists of 0 pages, a iformatio sheet ad diagram sheet. Please tur over Mathematics/P DoE/November
More informationFor 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 informationHow 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 informationLesson 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 informationReview for College Algebra Final Exam
Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 14. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i
More informationNumerical Solution of Equations
School of Mechaical Aerospace ad Civil Egieerig Numerical Solutio of Equatios T J Craft George Begg Buildig, C4 TPFE MSc CFD Readig: J Ferziger, M Peric, Computatioal Methods for Fluid Dyamics HK Versteeg,
More informationLesson 15 ANOVA (analysis of variance)
Outlie Variability betwee group variability withi group variability total variability Fratio Computatio sums of squares (betwee/withi/total degrees of freedom (betwee/withi/total mea square (betwee/withi
More informationRecursion and Recurrences
Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,
More informationConfidence Intervals for One Mean with Tolerance Probability
Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) with
More informationTHE 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 informationSubject CT1 Financial Mathematics Core Technical Core Reading
Subject CT1 Fiacial Mathematics Core Techical Core Readig for the 2016 exams 1 Jue 2015 Copyright i this Core Readig is the property of the Istitute ad Faculty of Actuaries who are the sole distributors.
More informationFor Educational Purposes Only
PCSYBLF For Educatioal Purposes ly The materials preseted i this course represet the opiios ad views of the developers ad/or reviewers. Although these materials may have bee reviewed, the views ad opiios
More informationPresent 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 informationChapter One BASIC MATHEMATICAL TOOLS
Chapter Oe BAIC MATHEMATICAL TOOL As the reader will see, the study of the time value of moey ivolves substatial use of variables ad umbers that are raised to a power. The power to which a variable is
More informationSole 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 informationECONOMICS. Calculating loan interest no. 3.758
F A M & A N H S E E S EONOMS alculatig loa iterest o. 3.758 y Nora L. Dalsted ad Paul H. Gutierrez Quick Facts... The aual percetage rate provides a coo basis to copare iterest charges associated with
More informationDetermining 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 information8.3 POLAR FORM AND DEMOIVRE S THEOREM
SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM 48 8. POLAR FORM AND DEMOIVRE S THEOREM Figure 8.6 (a, b) b r a 0 θ Complex Number: a + bi Rectagular Form: (a, b) Polar Form: (r, θ) At this poit you ca add,
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationManaging 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 informationSearching Algorithm Efficiencies
Efficiecy of Liear Search Searchig Algorithm Efficiecies Havig implemeted the liear search algorithm, how would you measure its efficiecy? A useful measure (or metric) should be geeral, applicable to ay
More informationValuing Firms in Distress
Valuig Firms i Distress Aswath Damodara http://www.damodara.com Aswath Damodara 1 The Goig Cocer Assumptio Traditioal valuatio techiques are built o the assumptio of a goig cocer, I.e., a firm that has
More informationThe Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 0 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages, diagram sheet ad iformatio sheet. Please tur over Mathematics/P DBE/November 0
More information