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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "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"

Transcription

1 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 meas by which a project s aual et cash flows ca be calculated were discussed ad illustrated. The cash flow complexities with regard to taxes, depreciatio, capital gais, ivestmet credit, ad salvage values were also icorporated ito the cash flow computatio. Our earlier discussio did ot, however, suggest a meas for evaluatig alterative ivestmet projects, or did it provide the agribusiess maager with a fiite criterio by which a sigle project could be judged acceptable or uacceptable. This issue shall attempt to address each of the latter two maagemet eeds. It presumes first, that the aual et cash flows attributed to each alterative capital ivestmet project have already bee computed. It presumes secod, that the agribusiess maager is capable of idetifyig his firm s opportuity cost of capital, i.e., the true rate of retur associated with ivestig capital i alterative projects. Iteral Rate of Retur Oe of the most commo criteria by which alterative ivestmet projects are compared ad/or judged acceptable is kow as the iteral rate of retur. I techical terms, the IRR is that iterest rate (aual average) that reders the sum of a ivestmet s aual et cash flows whe discouted to time period zero, equal to zero. Expressed mathematically, the IRR is that iterest rate which satisfies the followig equatio: where: = T = X ( + i) X = et cash flow i year T = umber of years of cash flow i ivestmet's life = the year i which the cash flow X occurs i = IRR = the iteral rate of retur To simplify what appears to may to costitute a rather complex formula, let s cosider a illustrious capital ivestmet project which costs $, ad geerates a aual et cash flow of $ for each of the subsequet six years (see Table 6). TABLE 6 Net Aual Cash Flows ($) Ed of Year Net Cash Flows -, By applyig this stream of cash flows to our formula ad solvig for i we obtai:

2 =, ( + i) ( +i) ( +i) 2 6 =,+ PAi,6 where PA = preset worth factor, PAi,6 = = i = 2% as determied from a discrete compoudig table where PA = ad =6 I this illustratio, of course, our aalysis has bee simplified by the fact that our cash flows are discrete ad uiform. More commoly, however, agribusiess ivestmets will geerate aual et cash flows of uequal amouts such as those show i Table 7. I such cases the applicatio of our formula becomes more complex ad we are forced to approximate the IRR by a trail-ad-error procedure. Typically, we would select two iterest rates such that the discouted cash flows sum to a positive ad egative value (see Table 8) ad the approximate the IRR through liear iterpolatio as show below. Quite obviously this trial-ad-error process ca be both difficult ad time cosumig. Oe eed ot be frighteed, however, as may electroic had calculators ow have built-i programs for makig such IRR calculatios directly. TABLE 7 Net Aual Cash Flows ($) Ed of Year Net Cash Flows -,, 2 2, 3 4, 4 4, 5 4, Accept or Reject Criteria Now that the IRR for the capital ivestmet project has bee determied, how is it used ad what does it mea? As we oted i the March issue, most agribusiess maagers have some geeral coceptio of their true cost of capital ad/or their opportuity cost of capital. Give either or both of these cocepts, that maager also has some policy with regard to a miimum acceptable rate of retur (MARR), i.e., that rate of retur o ivestmet below which the maager or the firm is uwillig to cosider as a attractive project. Give these miimum criteria, if IRR exceeds the MARR, the project is judged attractive or acceptable; if this is ot the case, the project is rejected. If alterative capital ivestmet projects are beig assessed, quite obviously they may ow be preferetially raked by maagemet i accordace with the level of IRR geerated by each. Returig to our earlier illustratio (Table 6) we ca further expad o the true meaig of the IRR. I Table 9 we have demostrated a alterative meaig of IRR = 2%, where IRR is show to be the retur o urecovered capital (allowig for the full recovery of ivested capital over the ivestmet s life). I this cocept, the agribusiess firm is loaig $, to the ivestmet project ad askig for a 2% rate of iterest o those moies remaiig urecovered at the ed of each of the six years of active life. I this case the iteral rate of retur is that retur geerated iterally by the project as a result of its et cash flows. 2

3 Ed of Year Net Cash Flow TABLE 8 Selectig IRR Bouds Discouted Cash Flows PF,2% PF,5% $-,. $-,.. $-,., ,.7972, , ,.78 2, , , , , , ,988.8 Sum Iterpolated IRR = 2+ 3 = 2.5% TABLE 9 IRR as Urecovered Capital ($) (Step 3) (Step ) (Step2) Retur o Ed of Year Net Cash Flow Urecovered Capital Urecovered Capital (2%) Capital Recovered $-, -,,+,7 = 8,993.2(,) = 2, 2, =,7 2 8,993+,28= 7,785.2( 8,993) =,799,799=,28 3 7,785+,45 = 6,335.2( 7,785) =,557,557 =,45 4 6,335+,74= 4,595.2( 6,335) =,267,267 =,74 5 4,595+ 2,88= 2,57.2( 4,595) = 99 99= 2,88 6 2,57+ 2,56= *.2( 2,57) = 5 5= 2,56 *=roudig error Observig year i Table 9 we see that as of the ed of that period the firm expects a retur of $2, (2%) for the project s oe-year use of the $, loaed it by the firm. Sice the project actually returs $, this reduces the balace of urecovered capital by $,7 such that the firm s ivestmet i the project durig the secod year is $8,993, upo which a retur of $,799 (2%) is expected durig the secod year. Agai, $ is actually retured, reducig the balace of the firm s urecovered capital from $8,993 to $7,785 for employmet by the project durig year 3. 3

4 Uderstadig of Table 9 may be aided by reviewig each row of the table via steps -3, i order. For those readers who ejoy the mathematical rigor associated with IRR computatios, it should be oted that for certai types of cash flows, IRR either caot be determied or the computatio results i multiple solutios. For example, whe all cash flows associated with a capital ivestmet have the same sig, IRR caot be determied. Coversely, whe the sigs associated with a sequece of aual et cash flows chage more ofte tha oce, a algebraic pheomeo kow as Descarte s Rule of Sigs dictates that there may exist a multiple solutio to the IRR computatio. Whe this situatio arises, there does exist a alterate method for determiig IRR. However, such matters fall beyod the iteded scope of this paper. Net Preset Value A secod commo criterio by which alterative capital ivestmet projects may be compared or judged acceptable is kow as the et preset value. Agai, i techical terms the NPV is the sum of the aual et cash flows associated with a ivestmet project discouted to time zero at the MARR. Mathematically it is expressed as: NPV = T = X ( + k ) where k = MARR Returig oce agai to our earlier example i Table 6 we fid that for a MARR of %: NPV =, ( +.) ( +.) ( +.) ( PA ) =,+,.,6 =, = $3, uacceptable otherwise. Similarly, alterative ivestmets may be preferetially raked i accordace with the magitude of the NPV geerated. A better uderstadig of the true meaig of NPV is facilitated by Table. You will ote the similarity betwee Table ad Table 9. They are similar i computatioal base except that Table 9 uses a IRR of 2% while Table uses a MARR of %. As show below, all capital is fully recovered i the fifth year of the ivestmet period, with a $2,252 surplus remaiig i the fifth year, plus the full cash flow of $ i the sixth year. If these surplus cash flows are discouted to time zero the result is: 2,252 ( PF,.,5) + ( PF,.,6) 2,252 ((.629) + (.5645) = $3,95 This $3,95 is idetical (except for roudig error) to that solutio obtaied from the formula computatio above. Table shows that a NPV > implies that all the capital is recovered over the life of the project (or a shorter period), or a retur (MARR) is received each year o the urecovered capital, ad a surplus or bous (NPV) is also received. If NPV =, this implies that MARR = IRR. IRR ad NPV Relatioship By ow the reader must woder whether or ot the use of IRR or NPV would result i the same maagemet decisio regardig the acceptability of the capital ivestmet project. I fact, for those sigle project assessmets comprised of cash flows with o more tha oe sig chage, either method will produce the same accept/reject decisio. The IRR ad NPV relatioship is diagramatically show i Figure. Give a series of et cash flows, the ivestmet project is judged acceptable if NPV, ad 4

5 TABLE NPVas Urecovered Capital ($) (Step 3) (Step ) (Step2) Ed of Year Net Cash Flow Urecovered Capital Retur o Urecovered Capital Capital Recovered $-, -, -- --,+ 2,7 = 7,993.(,) =,,= 2,7 2 7,993+ 2,28= 5,785.( 7,993) = = 2,28 3 5,785+ 2,428= 3,357.( 5,785) = = 2, ,357+ 2,67= 686.( 3,357) = = 2, ,938= 2,252.( 686 ) =.69 69= 2, The Pay-Back Period Criteria Referrig to Figure, it ca be see that if i is set NPV i Figure IRR as the MARR, the NPV is greater tha zero. I this situatio, IRR is greater tha MARR. Cosequetly, the cash flows are deemed to be acceptable uder both NPV ad IRR criteria. If i 2 is set as the MARR, the NPV is egative ad IRR is less tha MARR. Hece, the cash flows are judged uacceptable by both criteria. Where NPV is set as, the IRR = MARR ad cash flows are acceptable. i i 2 i Oe fial criterio is sometimes applied i maagemet s evaluatio of a capital ivestmet project. This criterio is referred to as the payback period ad refers to the umber of years required for the cash flows to completely recover the origial ivestmet. Expressed mathematically, the pay-back period () is: = X = Usig those data provided i Table, we would calculate that the pay-back period lies somewhere betwee the third ad fourth years, e.g.: Third year =, 5,+ 7,+ 7, = $, Fourth year =, 5,+ 7,+ 7,+ 7, = $6, 5

6 Usig liear iterpolatio, the pay-back period ca be approximated as: 3, 4 6,, = 3+ = 3.4years 7, Pay-Back Limitatios Used aloe as a criterio by which to judge the acceptace or rejectio of a series of cash flows, the pay-back period method has true limitatios. It is ot difficult to geerate illustrative cash flows where the IRR ad NPV criterio dictate a maagemet decisio opposite that geerated by the pay-back period. This coflict results because the pay-back period igores the magitude of cash flows followig the poit of full recovery. Neither does the pay-back criterio ackowledge i ay way the time value of moey. Obviously for a small agribusiess firm, the time required to recover its origial ivestmet is a importat cosideratio. However, ay attempt to base the acceptability of a series of cash flows o the payback period criterio aloe could lead to faulty decisios. TABLE Cash Flows Pay-Back ($) Ed of Year Net Cash Flows -, -5, 2 7, 3 7, 4 7, 5 7, 6 7, 7 7, 6

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

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

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

.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

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

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

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

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

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

9.8: THE POWER OF A TEST

9.8: THE POWER OF A TEST 9.8: The Power of a Test CD9-1 9.8: THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based

More information

MESSAGE TO TEACHERS: NOTE TO EDUCATORS:

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

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

Confidence Intervals for One Mean with Tolerance Probability

Confidence 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 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

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 and Investment Analysis

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

More information

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.

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

More information

ARITHMETIC AND GEOMETRIC PROGRESSIONS

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

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

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

Geometric Sequences and Series. Geometric Sequences. Definition of Geometric Sequence. such that. a2 4

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

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

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

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

INTRODUCTION TO ENGINEERING ECONOMICS. Types of Interest

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

More information

Module 4: Mathematical Induction

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

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

15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011

15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011 15.075 Exam 3 Istructor: Cythia Rudi TA: Dimitrios Bisias November 22, 2011 Gradig is based o demostratio of coceptual uderstadig, so you eed to show all of your work. Problem 1 A compay makes high-defiitio

More information

Literal Equations and Formulas

Literal 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 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

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

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

Sequences II. Chapter 3. 3.1 Convergent Sequences

Sequences II. Chapter 3. 3.1 Convergent Sequences Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,

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

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

Numerical Solution of Equations

Numerical 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 information

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

More information

Power Factor in Electrical Power Systems with Non-Linear Loads

Power Factor in Electrical Power Systems with Non-Linear Loads Power Factor i Electrical Power Systems with No-Liear Loads By: Gozalo Sadoval, ARTECHE / INELAP S.A. de C.V. Abstract. Traditioal methods of Power Factor Correctio typically focus o displacemet power

More information

Checklist. Assignment

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

More information

Confidence Intervals and Sample Size

Confidence Intervals and Sample Size 8/7/015 C H A P T E R S E V E N Cofidece Itervals ad Copyright 015 The McGraw-Hill Compaies, Ic. Permissio required for reproductio or display. 1 Cofidece Itervals ad Outlie 7-1 Cofidece Itervals for the

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

Lecture Notes CMSC 251

Lecture Notes CMSC 251 We have this messy summatio to solve though First observe that the value remais costat throughout the sum, ad so we ca pull it out frot Also ote that we ca write 3 i / i ad (3/) i T () = log 3 (log ) 1

More information

Chapter Gaussian Elimination

Chapter 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 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

Bond Pricing Theorems. Floyd Vest

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

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

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required.

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required. S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,

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

NATIONAL SENIOR CERTIFICATE GRADE 11

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

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

Recursion and Recurrences

Recursion 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 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

A probabilistic proof of a binomial identity

A probabilistic proof of a binomial identity A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

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

CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

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

Lesson 12. Sequences and Series

Lesson 12. Sequences and Series Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or

More information

NUMBERS COMMON TO TWO POLYGONAL SEQUENCES

NUMBERS COMMON TO TWO POLYGONAL SEQUENCES NUMBERS COMMON TO TWO POLYGONAL SEQUENCES DIANNE SMITH LUCAS Chia Lake, Califoria a iteger, The polygoal sequece (or sequeces of polygoal umbers) of order r (where r is r > 3) may be defied recursively

More information

Quadratics - Revenue and Distance

Quadratics - Revenue and Distance 9.10 Quadratics - Reveue ad Distace Objective: Solve reveue ad distace applicatios of quadratic equatios. A commo applicatio of quadratics comes from reveue ad distace problems. Both are set up almost

More information

A Resource for Free-standing Mathematics Qualifications Working with %

A Resource for Free-standing 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 information

ME 101 Measurement Demonstration (MD 1) DEFINITIONS Precision - A measure of agreement between repeated measurements (repeatability).

ME 101 Measurement Demonstration (MD 1) DEFINITIONS Precision - A measure of agreement between repeated measurements (repeatability). INTRODUCTION This laboratory ivestigatio ivolves makig both legth ad mass measuremets of a populatio, ad the assessig statistical parameters to describe that populatio. For example, oe may wat to determie

More information

7. Sample Covariance and Correlation

7. Sample Covariance and Correlation 1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

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

Fourier Series and the Wave Equation Part 2

Fourier Series and the Wave Equation Part 2 Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries

More information

Divide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016

Divide 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 information

The Euler Totient, the Möbius and the Divisor Functions

The 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 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

Arithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition of Arithmetic Sequence. Example 1. 7, 11, 15, 19,..., 4n 3,...

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.

More information

One-step equations. Vocabulary

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

More information

The second difference is the sequence of differences of the first difference sequence, 2

The 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 information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

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

The Field of Complex Numbers

The Field of Complex Numbers The Field of Complex Numbers S. F. Ellermeyer The costructio of the system of complex umbers begis by appedig to the system of real umbers a umber which we call i with the property that i = 1. (Note that

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

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

Solving Inequalities

Solving Inequalities Solvig Iequalities Say Thaks to the Authors Click http://www.ck12.org/saythaks (No sig i required) To access a customizable versio of this book, as well as other iteractive cotet, visit www.ck12.org CK-12

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

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015 CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a

More information

1 Correlation and Regression Analysis

1 Correlation and Regression Analysis 1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

More information

Ekkehart Schlicht: Economic Surplus and Derived Demand

Ekkehart Schlicht: Economic Surplus and Derived Demand Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/

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

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

CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homework 4 Solutions CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

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

CHAPTER 3: FINANCIAL ANALYSIS WITH INFLATION

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

More information

Page 1. Real Options for Engineering Systems. What are we up to? Today s agenda. J1: Real Options for Engineering Systems. Richard de Neufville

Page 1. Real Options for Engineering Systems. What are we up to? Today s agenda. J1: Real Options for Engineering Systems. Richard de Neufville Real Optios for Egieerig Systems J: Real Optios for Egieerig Systems By (MIT) Stefa Scholtes (CU) Course website: http://msl.mit.edu/cmi/ardet_2002 Stefa Scholtes Judge Istitute of Maagemet, CU Slide What

More information

NPTEL STRUCTURAL RELIABILITY

NPTEL STRUCTURAL RELIABILITY NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics

More information

Sum of Exterior Angles of Polygons TEACHER NOTES

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).

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

{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers

{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers . Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,

More information

8.3 POLAR FORM AND DEMOIVRE S THEOREM

8.3 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 information

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the. Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).

More information

Hypothesis Tests Applied to Means

Hypothesis Tests Applied to Means The Samplig Distributio of the Mea Hypothesis Tests Applied to Meas Recall that the samplig distributio of the mea is the distributio of sample meas that would be obtaied from a particular populatio (with

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

1 The Binomial Theorem: Another Approach

1 The Binomial Theorem: Another Approach The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets

More information

Standard Errors and Confidence Intervals

Standard Errors and Confidence Intervals Stadard Errors ad Cofidece Itervals Itroductio I the documet Data Descriptio, Populatios ad the Normal Distributio a sample had bee obtaied from the populatio of heights of 5-year-old boys. If we assume

More information

Review for College Algebra Final Exam

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

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

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

More information

8.5 Alternating infinite series

8.5 Alternating infinite series 65 8.5 Alteratig ifiite series I the previous two sectios we cosidered oly series with positive terms. I this sectio we cosider series with both positive ad egative terms which alterate: positive, egative,

More information