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

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 the tricks I. Why is there a time value to moey (TVM)? Which would you prefer: a cheque from me to you for \$,, dated today or a cheque from me to you for \$,, dated oe year from ow Why? What does the TVM mea?

2 Time Value alculatios Basics so far: Sigle cash flow over oe period PV ( + r) FV ( + r) Objective: Exted to multiple, or, periods II. The Basic Time-Value-of-Moey Relatioships for a Sigle ash Flow where FV t+ t ( + r) PV (+ r) t+ t r is the iterest rate per period is the duratio of the ivestmet, stated i the time uits of the period for r t is the cash flow at period t t+ is the cash flow at period t+ FV t+ is the future value at time period t+ PV t is the preset value at time period t 2

3 Ituitio for the Future Value formula Future Value of \$ ivested for 2 years at 8% per year compouded yearly Year 2 FV 2 (+r) 2 \$ (+.8) 2 \$.664 Note: simple iterest is just 6 over the two years but compoud iterest is 6.64 which icludes the simple iterest plus iterest o iterest. The Power of ompoudig Give a iterest rate of 8% per year ad a iitial \$, ivestmet, compare the compoud iterest i the 2 d year ad the 2 th year. What is the total compoud iterest over the 2 year ivestmet? Explai the power of compoudig 3

4 Ituitio for the Preset Value Formula Preset value of \$ received i 2 years give a discout rate of 8% per year. Year 2 PV 2 (+r) 2 \$ (+.8) 2 \$.8573 Practice exercises Fid the value i 3 years of \$, received i 8 years. Fid the value i years of \$25, ivested i 2 years. How may years was the moey ivested if \$, grew to \$,? If you ivested \$5 for 7 years ad ow have \$, how much did your ivestmet retur? 4

5 Housekeepig fuctios:. Set to 8 decimal places: TVM i your HP B alculator Yellow DISP 8 2. lear previous TVM data: Yellow ALL 3. Set paymet at Begiig/Ed of Period: Yellow MAR BEG/END Always keep set o Ed the display should ever say Begi 3. Set # of times iterest is calculated (compouded) per year to : Yellow PMT P/YR Usig the HP B TVM fuctios N umber of periods or umber of paymets I/YR the effective iterest rate or discout rate per period PV preset value or a preset cash flow PMT repeatig cash flow paymets (ot yet discussed) FV future value or a future cash flow 5

6 III. Groups of ash Flows osider the followig series of cash flows: Year 2 3 \$, \$,5 \$,2.5 \$,55.33 What is PV ; what is FV? We could apply our PV ad FV formulae to each idividual cash flow but that would be too paiful! Mathematics of Perpetuities ad Auities Fortuately, math provides a simplified way osider a growig perpetuity: Goes o forever 2 3 t \$, \$,5 \$,2.5 2 (+g) 3 (+g) 2 t (+g) t- 6

7 Sum of a ifiite series PV of the growig perpetuity is mathematically equivalet to the sum of a ifiite geometric series. The sum is defied ad is fiite as log as the PV of each subsequet cash flow is a fractio (less tha ) of the PV of the previous cash flow. I.e., as log as r > g PV of a growig perpetuity This sums the PV s of each idividual cash flow i the growig perpetuity. is the first cash flow PV r g PV is the PV oe period before the first cash flow This formula is oly correct whe r > g. If r g or if r < g, the the PV is ifiite. 7

8 Growig Perpetuity Example To service the curret atioal debt, the govermet plas to make the followig series of paymets begiig i oe year ad cotiuig i perpetuity: \$8 billio iitially ad the growig by 4% each year. The iterest rate o log-term debt is 6%. What is the PV of these paymets? Note: the PV of all future debt paymets is equal to the pricipal amout curretly outstadig. PV of a Growig Auity A auity is a fiite series of cash flows i.e., a series that has a ed assume the ed as at time. We ca determie the PV of the growig auity by subtractig off the latter part from a growig perpetuity (+g) (+g) - + (+g) 8

9 PV of a Growig Auity so the PV of the growig auity is just the PV of the whole growig perpetuity mius the PV of the latter part of the growig perpetuity. The latter part of the growig perpetuity is just aother growig perpetuity that starts at a later time with a differet iitial cash flow. PV of a Growig Auity PV r g r g + ( + r) Subtract off the PV of the latter part of the growig perpetuity PV r g ( + g) r g ( + r) PV of the whole growig perpetuity PV ( + g) r g ( + r) PV is the PV oe period before the first cash flow 9

10 Growig Auity Example I 2 years you pla o retirig ad you would like icome each year that grows at the expected iflatio rate of 3%. You desire your year 2 icome to be \$5, ad you expect to eed 3 years of retiremet icome. If you are cofidet your savigs will ear 8% per year, how much do you eed saved by year 2? FV of a Growig Auity If PV discouts all the cash flows to time zero ad sums up the discouted amouts the FV, the future value of all the cash flows take to time, is just PV (+r) ( + g) FV ( + r) r g ( + r) [( + r) ( g ] FV ) r g + I effect, this takes all cash flows of the growig auity, icludig the last cash flow, forward to the time period of the last cash flow

11 FV of Growig Auity Example Give your retiremet plas of the previous example, how much do you eed to save each year begiig i oe year ad edig with year 9? Assume your savigs will ear 8% ad you icrease your cotributios by 6% each year. Simple (o-growig) series of cash flows For costat auities ad costat perpetuities, the time value formulas are simplified by settig g. We ca use the PMT butto o the fiacial calculator for the auity cash flows, PV r PV FV r (+ r) r regular perpetuity [(+ r) ] regular auity

12 Regular Auities & Perpetuities Examples Your father loaed you \$2, as your dow paymet o your ew house. If you repay him i equal amouts of \$2,6 each of the ext years, what rate of iterest are you, i effect, payig him? Regular Auities & Perpetuities Examples You have wo the lottery ad are offered cash paymets of \$ millio per year for the ext 2 years (first paymet is oe year from today). If you could ivest at a rate of %, how much as a sigle lump sum would you be willig to receive today i exchage for the 2 yearly cash flows? 2

13 Regular Auities & Perpetuities Examples You have just doated to the Uiversity of Maitoba ad your doatio stipulates that the Uiversity must sped the icome eared from your doatio each year. If your doatio is \$ millio ad it ears a 6% rate of retur, how much ca be spet each year ad for how log ca this cotiue? IV. Some fial warigs Eve though the time value calculatios look easy there are may potetial pitfalls you may experiece Be careful of the followig: PV of auities or perpetuities that do ot begi i period ; remember the PV formulas give always discout to exactly oe period before the first cash flow. If the cash flows begi at period t, the you must divide the PV from our formula by (+r) t- to get PV. Note: this works eve if t is a fractio. 3

14 Be careful of auity paymets out the umber of paymets i a auity. If the first paymet is i period ad the last is i period 2, there are obviously 2 paymets. How may paymets are there if the st paymet is i period 2 ad the last paymet is i period 2 (aswer is use your figers). How about if the st paymet is ow (period ) ad the last paymet is i period 5 (aswer is 6 paymets). If the first cash flow is at period t ad the last cash flow is at period T, the there are T-t+ cash flows i the auity. Be careful of wordig A cash flow occurs at the ed of the third period. A cash flow occurs at time period three. A cash flow occurs at the begiig of the fourth period Each of the above statemets refers to the same poit i time! If i doubt, draw a time lie. 4

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### .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,

### 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

### Geometric 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

### 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

### 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

### 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.

### 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

### THE 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

### 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

### 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

### 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

### 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

### 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.

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%

### 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

### 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

### 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,

### INTRODUCTION 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

### The 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.

### 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

### 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

### 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

### 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

### 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,

### 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

### 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

### Checklist. 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

### Introduction 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

### 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.

### 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

### Bond 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

### 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

### when 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

### 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

### Framingham 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

### 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

### 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

### Since valuation models are quantitative, valuation is objective. A well-researched and well-done valuation is timeless

Sessio #6 Valuatio Damodara - Chapter 23: 4,8,1,15,2 Refereces: Damodara: Damodara o Valuatio (1994: Joh Wiley) Copelad, Koller & Murri: Valuatio (199: Joh Wiley) Geeralities (myths) about valuatio: Sice

### 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

### 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

### 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

### 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

### Time 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

### 2.3. GEOMETRIC SERIES

6 CHAPTER INFINITE SERIES GEOMETRIC SERIES Oe of the most importat types of ifiite series are geometric series A geometric series is simply the sum of a geometric sequece, Fortuately, geometric series

### 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

### Bond Mathematics & Valuation

Bod Mathematics & Valuatio Below is some legalese o the use of this documet. If you d like to avoid a headache, it basically asks you to use some commo sese. We have put some effort ito this, ad we wat

### PVA(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

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

### 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

### 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

### 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

### Exponential 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

### 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:

### CHAPTER 2. Time Value of Money 6-1

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 6-2 Tme les 0 2 3 % CF 0 CF CF 2 CF 3 Show

### PFF2 2015/16. Assessment of Financial Circumstances For parents and partners of students. /SFEngland. /SF_England SFE/PFF2/1516/B

PFF2 2015/16 Assessmet of Fiacial Circumstaces For parets ad parters of studets SFE/PFF2/1516/B /SF_Eglad /SFEglad Who should complete this form? Complete this form if you are: The studet s atural or adoptive

### Now here is the important step

LINEST i Excel The Excel spreadsheet fuctio "liest" is a complete liear least squares curve fittig routie that produces ucertaity estimates for the fit values. There are two ways to access the "liest"

### NATIONAL 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

### One-step equations. Vocabulary

Review solvig oe-step equatios with itegers, fractios, ad decimals. Oe-step equatios Vocabulary equatio solve solutio iverse operatio isolate the variable Additio Property of Equality Subtractio Property

### *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.

### 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

### SEQUENCES AND SERIES CHAPTER

CHAPTER SEQUENCES AND SERIES Whe the Grat family purchased a computer for \$,200 o a istallmet pla, they agreed to pay \$00 each moth util the cost of the computer plus iterest had bee paid The iterest each

### ARITHMETIC 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

### 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

### 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.

### 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

### 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

### CHAPTER 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

### 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

### MESSAGE 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

Radicals ad Roots Radicals ad Fractioal Expoets I math, may problems will ivolve what is called the radical symbol, X is proouced the th root of X, where is or greater, ad X is a positive umber. What it

### UNIT 3 SUMMARY STATIONS THROUGHOUT THE NEXT 2 DAYS, WE WILL BE SUMMARIZING THE CONCEPT OF EXPONENTIAL FUNCTIONS AND THEIR VARIOUS APPLICATIONS.

Name: Group Members: UNIT 3 SUMMARY STATIONS THROUGHOUT THE NEXT DAYS, WE WILL BE SUMMARIZING THE CONCEPT OF EXPONENTIAL FUNCTIONS AND THEIR VARIOUS APPLICATIONS. EACH ACTIVITY HAS A COLOR THAT CORRESPONDS

### Chapter 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

### 3.2 Introduction to Infinite Series

3.2 Itroductio to Ifiite Series May of our ifiite sequeces, for the remaider of the course, will be defied by sums. For example, the sequece S m := 2. () is defied by a sum. Its terms (partial sums) are

### Sum 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).

### Arithmetic 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.

### 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

### Review for College Algebra Final Exam

Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 1-4. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i

### NATIONAL SENIOR CERTIFICATE GRADE 11

NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 007 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad a -page formula sheet. Please tur over Mathematics/P DoE/November

### 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

### 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.

### 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

### ORDERS OF GROWTH KEITH CONRAD

ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus

### 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

### Module 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

### 3. If x and y are real numbers, what is the simplified radical form

lgebra II Practice Test Objective:.a. Which is equivalet to 98 94 4 49?. Which epressio is aother way to write 5 4? 5 5 4 4 4 5 4 5. If ad y are real umbers, what is the simplified radical form of 5 y

### HSAs the American Fidelity Way:

Health Ser vices Admiistratio, LLC HSAs the America Fidelity Way: Kowledge, Experiece, Commitmet With over 30 years experiece i providig pre-tax services, America Fidelity is a pioeer i Sectio 125 flexible

### Installment Joint Life Insurance Actuarial Models with the Stochastic Interest Rate

Iteratioal Coferece o Maagemet Sciece ad Maagemet Iovatio (MSMI 4) Istallmet Joit Life Isurace ctuarial Models with the Stochastic Iterest Rate Nia-Nia JI a,*, Yue LI, Dog-Hui WNG College of Sciece, Harbi