CHAPTER 4 MONEY-TIME RELATIONSHIPS AND EQUIVALENCE.

Slides:



Advertisements
Similar presentations
Chapter 3 Mathematics of Finance
Advertisements

Example 1: In the following cash flow diagram, A8=A9=A10=A11=5000, and
William F. Bentz1 Session 11 - Interest Cost. William F. Bentz2 Interest A.Interest is the compensation that must be paid by a borrower for the use of.
APPLICATIONS OF MONEY-TIME RELATIONSHIPS
Copyright © 2008 Pearson Education Canada 7-1 Chapter 7 Interest.
Chapter 2 Interest and Future Value The objectives of this chapter are to enable you to:  Understand the relationship between interest and future value.
Engineering Economics I
Econ. Lecture 3 Economic Equivalence and Interest Formula’s Read 45-70
1 Warm-Up Review Homepage Rule of 72 Single Sum Compounding Annuities.
Chapter 4,5 Time Value of Money.
Chapter 2 Applying Time Value Concepts Copyright © 2012 Pearson Canada Inc. Edited by Laura Lamb, Department of Economics, TRU 1.
TIME VALUE OF MONEY Chapter 5. The Role of Time Value in Finance Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-2 Most financial decisions.
Chapter 5 Mathematics of Finance
(c) 2002 Contemporary Engineering Economics 1 Chapter 4 Time Is Money Interest: The Cost of Money Economic Equivalence Development of Interest Formulas.
Copyright © 2011 Pearson Education, Inc. Managing Your Money.
State University of New York WARNING All rights reserved. No part of the course materials used in the instruction of this course may be reproduced in any.
3-1 Copyright  2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by Peirson Slides prepared by Farida Akhtar and Barry Oliver, Australian.
(c) 2002 Contemporary Engineering Economics
(c) 2002 Contemporary Engineering Economics
Flash Back from before break The Five Types of Cash Flows (a) Single cash flow (b) Equal (uniform) payment series (c) Linear gradient series (d) Geometric.
Reporting and Interpreting Liabilities
Copyright  2006 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 9e by Peirson, Brown, Easton, Howard and Pinder Prepared by Dr Buly Cardak 3–1.
Engineering Economic Analysis Canadian Edition
Lecture 2 Engineering Economics ENGR 3300 Department of Mechanical Engineering Inter American University of Puerto Rico Bayamon Campus Dr. Omar E. Meza.
PRINCIPLES OF MONEY-TIME RELATIONSHIPS
Topic 9 Time Value of Money.
Copyright © 2012 Pearson Prentice Hall. All rights reserved. Chapter 5 Time Value of Money.
Naval Postgraduate School Time Value of Money Discounted Cash Flow Techniques Source: Raymond P. Lutz, “Discounted Cash Flow Techniques,” Handbook of Industrial.
Intro to Engineering Economy
Summer Time Value of Money Session 2 07/02/2015.
PRINCIPLES OF MONEY-TIME RELATIONSHIPS. MONEY Medium of Exchange -- Means of payment for goods or services; What sellers accept and buyers pay ; Store.
Copyright ©2012 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved. Engineering Economy, Fifteenth Edition By William.
Economic System Analysis January 15, 2002 Prof. Yannis A. Korilis.
Professor John Zietlow MBA 621
Chapter 3 Mathematics of Finance
MATHEMATICS OF FINANCE Adopted from “Introductory Mathematical Analysis for Student of Business and Economics,” (Ernest F. Haeussler, Jr. & Richard S.
© 2009 Cengage Learning/South-Western The Time Value Of Money Chapter 3.
Chapter 4: The Time Value of Money
Equivalence and Compound interest
Summary of Interest Formula. Relationships of Discrete Compounding.
Engineering Economic Analysis Canadian Edition
Copyright © 2009 Pearson Prentice Hall. All rights reserved. Chapter 4 Time Value of Money.
© 2004 by Nelson, a division of Thomson Canada Limited Contemporary Financial Management Chapter 4: Time Value of Money.
© 2009 Cengage Learning/South-Western The Time Value Of Money Chapter 3.
Matakuliah: D0762 – Ekonomi Teknik Tahun: 2009 Factors - Extra Problems Course Outline 3.
ENGINEERING ECONOMICS ISE460 SESSION 3 CHAPTER 3, May 29, 2015 Geza P. Bottlik Page 1 OUTLINE Questions? News? Recommendations – no obligation Chapter.
Copyright © 2009 Pearson Prentice Hall. All rights reserved. Chapter 4 Time Value of Money.
Engineering Economic Analysis Canadian Edition Chapter 3: Interest and Equivalence.
8/25/04 Valerie Tardiff and Paul Jensen Operations Research Models and Methods Copyright All rights reserved Time Value of Money Don’t put your.
Engineering Economic Analysis Canadian Edition Chapter 4: More Interest Formulas.
Barnett/Ziegler/Byleen Finite Mathematics 11e1 Chapter 3 Review Important Terms, Symbols, Concepts 3.1. Simple Interest Interest is the fee paid for the.
Chapter 12 Long-Term Liabilities
MER Design of Thermal Fluid Systems INTRODUCTION TO ENGINEERING ECONOMICS Professor Bruno Winter Term 2005.
ECONOMIC EQUIVALENCE Established when we are indifferent between a future payment, or a series of future payments, and a present sum of money. Considers.
1 Engineering Economics.  Money has a time value because it can earn more money over time (earning power).  Money has a time value because its purchasing.
(c) 2002 Contemporary Engineering Economics 1. Engineers must work within the realm of economics and justification of engineering projectsEngineers must.
Faculty of Applied Engineering and Urban Planning Civil Engineering Department Engineering Economy Lecture 1 Week 1 2 nd Semester 20015/2016 Chapter 3.
1 Increasing Speed of Exponential Functions The story of “A Grain of Rice” F(t) = 2 t-1.
1 Engineering Economics Engineering Economy It deals with the concepts and techniques of analysis useful in evaluating the worth of systems,
CHAPTER 3 MONEY-TIME RELATIONSHIPS AND EQUIVALENCE.
Chapter 4: The Time Value of Money
Chapter 11 Introduction to Finance and Review of Financial Mathematics
CHAPTER 4 THE TIME VALUE OF MONEY.
MONEY-TIME RELATIONSHIPS AND EQUIVALENCE
Chapter 2 Time Value of Money
Chapter 3 Mathematics of Finance
By Muhammad Shahid Iqbal
Chapter 4: The Time Value of Money
Chapter 4: The Time Value of Money
Presentation transcript:

CHAPTER 4 MONEY-TIME RELATIONSHIPS AND EQUIVALENCE

MONEY Medium of Exchange -- Means of payment for goods or services; What sellers accept and buyers pay ; Store of Value -- A way to transport buying power from one time period to another; Unit of Account -- A precise measurement of value or worth; Allows for tabulating debits and credits; Medium of Exchange -- Means of payment for goods or services; What sellers accept and buyers pay ; Store of Value -- A way to transport buying power from one time period to another; Unit of Account -- A precise measurement of value or worth; Allows for tabulating debits and credits;

CAPITAL Wealth in the form of money or property that can be used to produce more wealth.

KINDS OF CAPITAL Equity capital is that owned by individuals who have invested their money or property in a business project or venture in the hope of receiving a profit. Debt capital, often called borrowed capital, is obtained from lenders (e.g., through the sale of bonds) for investment. Equity capital is that owned by individuals who have invested their money or property in a business project or venture in the hope of receiving a profit. Debt capital, often called borrowed capital, is obtained from lenders (e.g., through the sale of bonds) for investment.

Exchange money for shares of stock as proof of partial ownership

INTEREST The fee that a borrower pays to a lender for the use of his or her money. INTEREST RATE The percentage of money being borrowed that is paid to the lender on some time basis. The fee that a borrower pays to a lender for the use of his or her money. INTEREST RATE The percentage of money being borrowed that is paid to the lender on some time basis.

HOW INTEREST RATE IS DETERMINED Interest Rate Quantity of Money

HOW INTEREST RATE IS DETERMINED Interest Rate Quantity of Money Money Demand

HOW INTEREST RATE IS DETERMINED Interest Rate Quantity of Money Money Demand Money Supply MS 1

HOW INTEREST RATE IS DETERMINED Interest Rate Quantity of Money ieie Money Demand Money Supply MS 1

HOW INTEREST RATE IS DETERMINED Interest Rate Quantity of Money ieie Money Demand Money Supply MS 1 MS 2 i2i2

HOW INTEREST RATE IS DETERMINED Interest Rate Quantity of Money ieie Money Demand Money Supply MS 1 MS 2 i2i2 MS 3 i3i3

SIMPLE INTEREST The total interest earned or charged is linearly proportional to the initial amount of the loan (principal), the interest rate and the number of interest periods for which the principal is committed. When applied, total interest “I” may be found by I = ( P ) ( N ) ( i ), where –P = principal amount lent or borrowed –N = number of interest periods ( e.g., years ) –i = interest rate per interest period The total interest earned or charged is linearly proportional to the initial amount of the loan (principal), the interest rate and the number of interest periods for which the principal is committed. When applied, total interest “I” may be found by I = ( P ) ( N ) ( i ), where –P = principal amount lent or borrowed –N = number of interest periods ( e.g., years ) –i = interest rate per interest period

COMPOUND INTEREST Whenever the interest charge for any interest period is based on the remaining principal amount plus any accumulated interest charges up to the beginning of that period. Period Amount Owed Interest Amount Amount Owed Beginning of for Period at end of period 10% ) period 1$1,000$100$1,100 2$1,100$110$1,210 3$1,210$121$1,331 Whenever the interest charge for any interest period is based on the remaining principal amount plus any accumulated interest charges up to the beginning of that period. Period Amount Owed Interest Amount Amount Owed Beginning of for Period at end of period 10% ) period 1$1,000$100$1,100 2$1,100$110$1,210 3$1,210$121$1,331

ECONOMIC EQUIVALENCE Established when we are indifferent between a future payment, or a series of future payments, and a present sum of money. Considers the comparison of alternative options, or proposals, by reducing them to an equivalent basis, depending on: –interest rate; –amounts of money involved; –timing of the affected monetary receipts and/or expenditures; –manner in which the interest, or profit on invested capital is paid and the initial capital is recovered. Established when we are indifferent between a future payment, or a series of future payments, and a present sum of money. Considers the comparison of alternative options, or proposals, by reducing them to an equivalent basis, depending on: –interest rate; –amounts of money involved; –timing of the affected monetary receipts and/or expenditures; –manner in which the interest, or profit on invested capital is paid and the initial capital is recovered.

CASH FLOW DIAGRAMS / TABLE NOTATION i = effective interest rate per interest period N = number of compounding periods (e.g., years) P = present sum of money; the equivalent value of one or more cash flows at the present time reference point F = future sum of money; the equivalent value of one or more cash flows at a future time reference point A = end-of-period cash flows (or equivalent end-of- period values ) in a uniform series continuing for a specified number of periods, starting at the end of the first period and continuing through the last period G = uniform gradient amounts -- used if cash flows increase by a constant amount in each period i = effective interest rate per interest period N = number of compounding periods (e.g., years) P = present sum of money; the equivalent value of one or more cash flows at the present time reference point F = future sum of money; the equivalent value of one or more cash flows at a future time reference point A = end-of-period cash flows (or equivalent end-of- period values ) in a uniform series continuing for a specified number of periods, starting at the end of the first period and continuing through the last period G = uniform gradient amounts -- used if cash flows increase by a constant amount in each period

CASH FLOW DIAGRAM NOTATION = N 1 1 Time scale with progression of time moving from left to right; the numbers represent time periods (e.g., years, months, quarters, etc...) and may be presented within a time interval or at the end of a time interval.

CASH FLOW DIAGRAM NOTATION = N 1 1 Time scale with progression of time moving from left to right; the numbers represent time periods (e.g., years, months, quarters, etc...) and may be presented within a time interval or at the end of a time interval. P =$8, Present expense (cash outflow) of $8,000 for lender.

CASH FLOW DIAGRAM NOTATION = N 1 1 Time scale with progression of time moving from left to right; the numbers represent time periods (e.g., years, months, quarters, etc...) and may be presented within a time interval or at the end of a time interval. P =$8, Present expense (cash outflow) of $8,000 for lender. A = $2, Annual income (cash inflow) of $2,524 for lender.

CASH FLOW DIAGRAM NOTATION = N 1 1 Time scale with progression of time moving from left to right; the numbers represent time periods (e.g., years, months, quarters, etc...) and may be presented within a time interval or at the end of a time interval. P =$8, Present expense (cash outflow) of $8,000 for lender. A = $2, Annual income (cash inflow) of $2,524 for lender. i = 10% per year 4 4 Interest rate of loan.

CASH FLOW DIAGRAM NOTATION = N 1 1 Time scale with progression of time moving from left to right; the numbers represent time periods (e.g., years, months, quarters, etc...) and may be presented within a time interval or at the end of a time interval. P =$8, Present expense (cash outflow) of $8,000 for lender. A = $2, Annual income (cash inflow) of $2,524 for lender. i = 10% per year 4 4 Interest rate of loan. 5 5 Dashed-arrow line indicates amount to be determined.

RELATING PRESENT AND FUTURE EQUIVALENT VALUES OF SINGLE CASH FLOWS Finding F when given P: Finding future value when given present value F = P ( 1+i ) N –(1+i) N single payment compound amount factor –functionally expressed as F = ( F / P, i%,N ) –predetermined values of this are presented in column 2 of Appendix C of text. Finding F when given P: Finding future value when given present value F = P ( 1+i ) N –(1+i) N single payment compound amount factor –functionally expressed as F = ( F / P, i%,N ) –predetermined values of this are presented in column 2 of Appendix C of text. P 0 N = F = ?

Finding P when given F: Finding present value when given future value P = F [1 / (1 + i ) ] N – (1+i) -N single payment present worth factor – functionally expressed as P = F ( P / F, i%, N ) –predetermined values of this are presented in column 3 of Appendix C of text; Finding P when given F: Finding present value when given future value P = F [1 / (1 + i ) ] N – (1+i) -N single payment present worth factor – functionally expressed as P = F ( P / F, i%, N ) –predetermined values of this are presented in column 3 of Appendix C of text; RELATING PRESENT AND FUTURE EQUIVALENT VALUES OF SINGLE CASH FLOWS P = ? 0N = F

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding F given A:

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding F given A: Finding future equivalent income (inflow) value given a series of uniform equal Payments Finding F given A: Finding future equivalent income (inflow) value given a series of uniform equal Payments

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding F given A: Finding future equivalent income (inflow) value given a series of uniform equal Payments ( 1 + i ) N - 1 F = A i Finding F given A: Finding future equivalent income (inflow) value given a series of uniform equal Payments ( 1 + i ) N - 1 F = A i

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding F given A: Finding future equivalent income (inflow) value given a series of uniform equal Payments ( 1 + i ) N - 1 F = A i –uniform series compound amount factor in [ ] Finding F given A: Finding future equivalent income (inflow) value given a series of uniform equal Payments ( 1 + i ) N - 1 F = A i –uniform series compound amount factor in [ ]

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding F given A: Finding future equivalent income (inflow) value given a series of uniform equal Payments ( 1 + i ) N - 1 F = A i –uniform series compound amount factor in [ ] –functionally expressed as F = A ( F / A,i%,N ) Finding F given A: Finding future equivalent income (inflow) value given a series of uniform equal Payments ( 1 + i ) N - 1 F = A i –uniform series compound amount factor in [ ] –functionally expressed as F = A ( F / A,i%,N )

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding F given A: Finding future equivalent income (inflow) value given a series of uniform equal Payments ( 1 + i ) N - 1 F = A i –uniform series compound amount factor in [ ] –functionally expressed as F = A ( F / A,i%,N ) –predetermined values are in column 4 of Appendix C of text Finding F given A: Finding future equivalent income (inflow) value given a series of uniform equal Payments ( 1 + i ) N - 1 F = A i –uniform series compound amount factor in [ ] –functionally expressed as F = A ( F / A,i%,N ) –predetermined values are in column 4 of Appendix C of text

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding F given A: Finding future equivalent income (inflow) value given a series of uniform equal Payments ( 1 + i ) N - 1 F = A i –uniform series compound amount factor in [ ] –functionally expressed as F = A ( F / A,i%,N ) –predetermined values are in column 4 of Appendix C of text Finding F given A: Finding future equivalent income (inflow) value given a series of uniform equal Payments ( 1 + i ) N - 1 F = A i –uniform series compound amount factor in [ ] –functionally expressed as F = A ( F / A,i%,N ) –predetermined values are in column 4 of Appendix C of text F = ? A =

( F / A,i%,N ) = (P / A,i,N ) ( F / P,i,N ) ( F / A,i%,N ) =    F / P,i,N-k ) N k = 1

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding P given A:

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding P given A: Finding present equivalent value given a series of uniform equal receipts Finding P given A: Finding present equivalent value given a series of uniform equal receipts

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding P given A: Finding present equivalent value given a series of uniform equal receipts ( 1 + i ) N - 1 P = A i ( 1 + i ) N Finding P given A: Finding present equivalent value given a series of uniform equal receipts ( 1 + i ) N - 1 P = A i ( 1 + i ) N

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding P given A: Finding present equivalent value given a series of uniform equal receipts ( 1 + i ) N - 1 P = A i ( 1 + i ) N –uniform series present worth factor in [ ] Finding P given A: Finding present equivalent value given a series of uniform equal receipts ( 1 + i ) N - 1 P = A i ( 1 + i ) N –uniform series present worth factor in [ ]

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding P given A: Finding present equivalent value given a series of uniform equal receipts ( 1 + i ) N - 1 P = A i ( 1 + i ) N –uniform series present worth factor in [ ] –functionally expressed as P = A ( P / A,i%,N ) Finding P given A: Finding present equivalent value given a series of uniform equal receipts ( 1 + i ) N - 1 P = A i ( 1 + i ) N –uniform series present worth factor in [ ] –functionally expressed as P = A ( P / A,i%,N )

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding P given A: Finding present equivalent value given a series of uniform equal receipts ( 1 + i ) N - 1 P = A i ( 1 + i ) N –uniform series present worth factor in [ ] –functionally expressed as P = A ( P / A,i%,N ) –predetermined values are in column 5 of Appendix C of text Finding P given A: Finding present equivalent value given a series of uniform equal receipts ( 1 + i ) N - 1 P = A i ( 1 + i ) N –uniform series present worth factor in [ ] –functionally expressed as P = A ( P / A,i%,N ) –predetermined values are in column 5 of Appendix C of text

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding P given A: Finding present equivalent value given a series of uniform equal receipts ( 1 + i ) N - 1 P = A i ( 1 + i ) N –uniform series present worth factor in [ ] –functionally expressed as P = A ( P / A,i%,N ) –predetermined values are in column 5 of Appendix C of text Finding P given A: Finding present equivalent value given a series of uniform equal receipts ( 1 + i ) N - 1 P = A i ( 1 + i ) N –uniform series present worth factor in [ ] –functionally expressed as P = A ( P / A,i%,N ) –predetermined values are in column 5 of Appendix C of text P = ? A =

( P / A,i%,N ) =    P / F,i,k ) N k = 1

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding A given F:

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding A given F: Finding amount A of a uniform series when given the equivalent future value Finding A given F: Finding amount A of a uniform series when given the equivalent future value

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding A given F: Finding amount A of a uniform series when given the equivalent future value i A = F ( 1 + i ) N -1 Finding A given F: Finding amount A of a uniform series when given the equivalent future value i A = F ( 1 + i ) N -1

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding A given F: Finding amount A of a uniform series when given the equivalent future value i A = F ( 1 + i ) N -1 –sinking fund factor in [ ] Finding A given F: Finding amount A of a uniform series when given the equivalent future value i A = F ( 1 + i ) N -1 –sinking fund factor in [ ]

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding A given F: Finding amount A of a uniform series when given the equivalent future value i A = F ( 1 + i ) N -1 –sinking fund factor in [ ] –functionally expressed as A = F ( A / F,i%,N ) Finding A given F: Finding amount A of a uniform series when given the equivalent future value i A = F ( 1 + i ) N -1 –sinking fund factor in [ ] –functionally expressed as A = F ( A / F,i%,N )

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding A given F: Finding amount A of a uniform series when given the equivalent future value i A = F ( 1 + i ) N -1 –sinking fund factor in [ ] –functionally expressed as A = F ( A / F,i%,N ) –predetermined values are in column 6 of Appendix C of text Finding A given F: Finding amount A of a uniform series when given the equivalent future value i A = F ( 1 + i ) N -1 –sinking fund factor in [ ] –functionally expressed as A = F ( A / F,i%,N ) –predetermined values are in column 6 of Appendix C of text

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding A given F: Finding amount A of a uniform series when given the equivalent future value i A = F ( 1 + i ) N -1 –sinking fund factor in [ ] –functionally expressed as A = F ( A / F,i%,N ) –predetermined values are in column 6 of Appendix C of text Finding A given F: Finding amount A of a uniform series when given the equivalent future value i A = F ( 1 + i ) N -1 –sinking fund factor in [ ] –functionally expressed as A = F ( A / F,i%,N ) –predetermined values are in column 6 of Appendix C of text F = A =?

( A / F,i%,N ) = 1 / ( F / A,i%,N ) ( A / F,i%,N ) = ( A / P,i%,N ) - i

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding A given P:

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding A given P: Finding amount A of a uniform series when given the equivalent present value Finding A given P: Finding amount A of a uniform series when given the equivalent present value

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding A given P: Finding amount A of a uniform series when given the equivalent present value i ( 1+i ) N A = P ( 1 + i ) N -1 Finding A given P: Finding amount A of a uniform series when given the equivalent present value i ( 1+i ) N A = P ( 1 + i ) N -1

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding A given P: Finding amount A of a uniform series when given the equivalent present value i ( 1+i ) N A = P ( 1 + i ) N -1 –capital recovery factor in [ ] Finding A given P: Finding amount A of a uniform series when given the equivalent present value i ( 1+i ) N A = P ( 1 + i ) N -1 –capital recovery factor in [ ]

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding A given P: Finding amount A of a uniform series when given the equivalent present value i ( 1+i ) N A = P ( 1 + i ) N -1 –capital recovery factor in [ ] –functionally expressed as A = P ( A / P,i%,N ) Finding A given P: Finding amount A of a uniform series when given the equivalent present value i ( 1+i ) N A = P ( 1 + i ) N -1 –capital recovery factor in [ ] –functionally expressed as A = P ( A / P,i%,N )

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding A given P: Finding amount A of a uniform series when given the equivalent present value i ( 1+i ) N A = P ( 1 + i ) N -1 –capital recovery factor in [ ] –functionally expressed as A = P ( A / P,i%,N ) –predetermined values are in column 7 of Appendix C of text Finding A given P: Finding amount A of a uniform series when given the equivalent present value i ( 1+i ) N A = P ( 1 + i ) N -1 –capital recovery factor in [ ] –functionally expressed as A = P ( A / P,i%,N ) –predetermined values are in column 7 of Appendix C of text

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding A given P: Finding amount A of a uniform series when given the equivalent present value i ( 1+i ) N A = P ( 1 + i ) N -1 –capital recovery factor in [ ] –functionally expressed as A = P ( A / P,i%,N ) –predetermined values are in column 7 of Appendix C of text Finding A given P: Finding amount A of a uniform series when given the equivalent present value i ( 1+i ) N A = P ( 1 + i ) N -1 –capital recovery factor in [ ] –functionally expressed as A = P ( A / P,i%,N ) –predetermined values are in column 7 of Appendix C of text P = A =?

( A / P,i%,N ) = 1 / ( P / A,i%,N )

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO FUTURE EQUIVALENTS Find F when given G:

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO FUTURE EQUIVALENTS Find F when given G: Find the future equivalent value when given the uniform gradient amount Find F when given G: Find the future equivalent value when given the uniform gradient amount

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO FUTURE EQUIVALENTS Find F when given G: Find the future equivalent value when given the uniform gradient amount (1+i) N-1 -1 (1+i) N-2 -1 (1+i) 1 -1 F = G Find F when given G: Find the future equivalent value when given the uniform gradient amount (1+i) N-1 -1 (1+i) N-2 -1 (1+i) 1 -1 F = G i i i

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO FUTURE EQUIVALENTS Find F when given G: Find the future equivalent value when given the uniform gradient amount (1+i) N-1 -1 (1+i) N-2 -1 (1+i) 1 -1 F = G Functionally represented as (G/ i) (F/A,i%,N) - (NG/ i) Find F when given G: Find the future equivalent value when given the uniform gradient amount (1+i) N-1 -1 (1+i) N-2 -1 (1+i) 1 -1 F = G Functionally represented as (G/ i) (F/A,i%,N) - (NG/ i) i i i

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO FUTURE EQUIVALENTS Find F when given G: Find the future equivalent value when given the uniform gradient amount (1+i) N-1 -1 (1+i) N-2 -1 (1+i) 1 -1 F = G Functionally represented as (G/ i) (F/A,i%,N) - (NG/ i) Usually more practical to deal with annual and present equivalents, rather than future equivalent values Find F when given G: Find the future equivalent value when given the uniform gradient amount (1+i) N-1 -1 (1+i) N-2 -1 (1+i) 1 -1 F = G Functionally represented as (G/ i) (F/A,i%,N) - (NG/ i) Usually more practical to deal with annual and present equivalents, rather than future equivalent values i i i

Cash Flow Diagram for a Uniform Gradient Increasing by G Dollars per period 1234N-2N-1N G 2G 3G (N-3)G (N-2)G (N-1)G i = effective interest rate per period End of Period

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find A when given G:

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find A when given G: Find the annual equivalent value when given the uniform gradient amount Find A when given G: Find the annual equivalent value when given the uniform gradient amount

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find A when given G: Find the annual equivalent value when given the uniform gradient amount 1 N A = G - i(1 + i ) N - 1 Find A when given G: Find the annual equivalent value when given the uniform gradient amount 1 N A = G - i(1 + i ) N - 1

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find A when given G: Find the annual equivalent value when given the uniform gradient amount 1 N A = G - i(1 + i ) N - 1 Functionally represented as A = G ( A / G, i%,N ) Find A when given G: Find the annual equivalent value when given the uniform gradient amount 1 N A = G - i(1 + i ) N - 1 Functionally represented as A = G ( A / G, i%,N )

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find A when given G: Find the annual equivalent value when given the uniform gradient amount 1 N A = G - i(1 + i ) N - 1 Functionally represented as A = G ( A / G, i%,N ) The value shown in [ ] is the gradient to uniform series conversion factor and is presented in column 9 of Appendix C (represented in the above parenthetical expression). Find A when given G: Find the annual equivalent value when given the uniform gradient amount 1 N A = G - i(1 + i ) N - 1 Functionally represented as A = G ( A / G, i%,N ) The value shown in [ ] is the gradient to uniform series conversion factor and is presented in column 9 of Appendix C (represented in the above parenthetical expression).

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given G:

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given G: Find the present equivalent value when given the uniform gradient amount Find P when given G: Find the present equivalent value when given the uniform gradient amount

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given G: Find the present equivalent value when given the uniform gradient amount 1(1 + i ) N -1N P = G- ii (1 + i ) N (1 + i ) N Find P when given G: Find the present equivalent value when given the uniform gradient amount 1(1 + i ) N -1N P = G- ii (1 + i ) N (1 + i ) N

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given G: Find the present equivalent value when given the uniform gradient amount 1(1 + i ) N -1N P = G- ii (1 + i ) N (1 + i ) N Functionally represented as P = G ( P / G, i%,N ) Find P when given G: Find the present equivalent value when given the uniform gradient amount 1(1 + i ) N -1N P = G- ii (1 + i ) N (1 + i ) N Functionally represented as P = G ( P / G, i%,N )

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given G: Find the present equivalent value when given the uniform gradient amount 1(1 + i ) N -1N P = G- ii (1 + i ) N (1 + i ) N Functionally represented as P = G ( P / G, i%,N ) The value shown in{ } is the gradient to present equivalent conversion factor and is presented in column 8 of Appendix C (represented in the above parenthetical expression). Find P when given G: Find the present equivalent value when given the uniform gradient amount 1(1 + i ) N -1N P = G- ii (1 + i ) N (1 + i ) N Functionally represented as P = G ( P / G, i%,N ) The value shown in{ } is the gradient to present equivalent conversion factor and is presented in column 8 of Appendix C (represented in the above parenthetical expression).

RELATING GE0METRIC SEQUENCE OF CASH FLOWS TO PRESENT AND ANNUAL EQUIVALENTS Projected cash flow patterns changing at an average rate of f each period;

RELATING GE0METRIC SEQUENCE OF CASH FLOWS TO PRESENT AND ANNUAL EQUIVALENTS Projected cash flow patterns changing at an average rate of f each period; Resultant end-of-period cash-flow pattern is referred to as a geometric gradient series; Projected cash flow patterns changing at an average rate of f each period; Resultant end-of-period cash-flow pattern is referred to as a geometric gradient series;

RELATING GE0METRIC SEQUENCE OF CASH FLOWS TO PRESENT AND ANNUAL EQUIVALENTS Projected cash flow patterns changing at an average rate of f each period; Resultant end-of-period cash-flow pattern is referred to as a geometric gradient series; A 1 is cash flow at end of period 1 Projected cash flow patterns changing at an average rate of f each period; Resultant end-of-period cash-flow pattern is referred to as a geometric gradient series; A 1 is cash flow at end of period 1

RELATING GE0METRIC SEQUENCE OF CASH FLOWS TO PRESENT AND ANNUAL EQUIVALENTS Projected cash flow patterns changing at an average rate of f each period; Resultant end-of-period cash-flow pattern is referred to as a geometric gradient series; A 1 is cash flow at end of period 1 A k = (A k-1 ) ( 1 +f ),2 < k < N Projected cash flow patterns changing at an average rate of f each period; Resultant end-of-period cash-flow pattern is referred to as a geometric gradient series; A 1 is cash flow at end of period 1 A k = (A k-1 ) ( 1 +f ),2 < k < N

RELATING GE0METRIC SEQUENCE OF CASH FLOWS TO PRESENT AND ANNUAL EQUIVALENTS Projected cash flow patterns changing at an average rate of f each period; Resultant end-of-period cash-flow pattern is referred to as a geometric gradient series; A 1 is cash flow at end of period 1 A k = (A k-1 ) ( 1 +f ),2 < k < N A N = A 1 ( 1 + f ) N-1 Projected cash flow patterns changing at an average rate of f each period; Resultant end-of-period cash-flow pattern is referred to as a geometric gradient series; A 1 is cash flow at end of period 1 A k = (A k-1 ) ( 1 +f ),2 < k < N A N = A 1 ( 1 + f ) N-1

RELATING GE0METRIC SEQUENCE OF CASH FLOWS TO PRESENT AND ANNUAL EQUIVALENTS Projected cash flow patterns changing at an average rate of f each period; Resultant end-of-period cash-flow pattern is referred to as a geometric gradient series; A 1 is cash flow at end of period 1 A k = (A k-1 ) ( 1 +f ),2 < k < N A N = A 1 ( 1 + f ) N-1 f = (A k - A k-1 ) / A k-1 Projected cash flow patterns changing at an average rate of f each period; Resultant end-of-period cash-flow pattern is referred to as a geometric gradient series; A 1 is cash flow at end of period 1 A k = (A k-1 ) ( 1 +f ),2 < k < N A N = A 1 ( 1 + f ) N-1 f = (A k - A k-1 ) / A k-1

RELATING GE0METRIC SEQUENCE OF CASH FLOWS TO PRESENT AND ANNUAL EQUIVALENTS Projected cash flow patterns changing at an average rate of f each period; Resultant end-of-period cash-flow pattern is referred to as a geometric gradient series; A 1 is cash flow at end of period 1 A k = (A k-1 ) ( 1 +f ),2 < k < N A N = A 1 ( 1 + f ) N-1 f = (A k - A k-1 ) / A k-1 f may be either positive or negative Projected cash flow patterns changing at an average rate of f each period; Resultant end-of-period cash-flow pattern is referred to as a geometric gradient series; A 1 is cash flow at end of period 1 A k = (A k-1 ) ( 1 +f ),2 < k < N A N = A 1 ( 1 + f ) N-1 f = (A k - A k-1 ) / A k-1 f may be either positive or negative

01234N A1A1 A 2 =A 1 (1+f ) A 3 =A 1 (1+f ) 2 A N =A 1 (1+f ) N - 1 End of Period Cash-flow diagram for a Geometric Sequence of Cash Flows

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given A:

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f )

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) A i P = ( P / A, -1, N ) ( 1 + f ) 1 + f Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) A i P = ( P / A, -1, N ) ( 1 + f ) 1 + f

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) A 1 [1 – (1+i) –N (1+f) N ] P = i - f which may also be written as A 1 [1 - (P/F,i%,N) (F/P,f%,N)] P = i - f Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) A 1 [1 – (1+i) –N (1+f) N ] P = i - f which may also be written as A 1 [1 - (P/F,i%,N) (F/P,f%,N)] P = i - f

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Note that the foregoing is mathematically equivalent to the following (i = f ): A i P = ( P / A -1, N )` 1 + f 1 + f Note that the foregoing is mathematically equivalent to the following (i = f ): A i P = ( P / A -1, N )` 1 + f 1 + f

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS The foregoing may be functionally represented as A = P (A / P, i%,N ) The year zero “base” of annuity, increasing at constant rate f % is A 0 = P ( A / P, f %, N ) The future equivalent of this geometric gradient is F = P ( F / P, i%, N ) The foregoing may be functionally represented as A = P (A / P, i%,N ) The year zero “base” of annuity, increasing at constant rate f % is A 0 = P ( A / P, f %, N ) The future equivalent of this geometric gradient is F = P ( F / P, i%, N )

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given A:

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) P = A 1 N (1+i) -1 which may be written as Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) P = A 1 N (1+i) -1 which may be written as

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) P = A 1 N (P/F,i%,1) Functionally represented as A = P (A / P, i%,N ) Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) P = A 1 N (P/F,i%,1) Functionally represented as A = P (A / P, i%,N )

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) P = A 1 N (i+i)-1 which may be written as P = A 1 N (P/F,i%,1) Functionally represented as A = P (A / P, i%,N ) The year zero “base” of annuity, increasing at constant rate f % is A 0 = P ( A / P, f %, N ) The future equivalent of this geometric gradient is F = P ( F / P, i%, N ) Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) P = A 1 N (i+i)-1 which may be written as P = A 1 N (P/F,i%,1) Functionally represented as A = P (A / P, i%,N ) The year zero “base” of annuity, increasing at constant rate f % is A 0 = P ( A / P, f %, N ) The future equivalent of this geometric gradient is F = P ( F / P, i%, N )

INTEREST RATES THAT VARY WITH TIME Find P given F and interest rates that vary over N

INTEREST RATES THAT VARY WITH TIME Find P given F and interest rates that vary over N Find the present equivalent value given a future value and a varying interest rate over the period of the loan Find P given F and interest rates that vary over N Find the present equivalent value given a future value and a varying interest rate over the period of the loan

INTEREST RATES THAT VARY WITH TIME Find P given F and interest rates that vary over N Find the present equivalent value given a future value and a varying interest rate over the period of the loan F N P =  N (1 + i k ) Find P given F and interest rates that vary over N Find the present equivalent value given a future value and a varying interest rate over the period of the loan F N P =  N (1 + i k ) k + 1

NOMINAL AND EFFECTIVE INTEREST RATES Nominal Interest Rate - r - For rates compounded more frequently than one year, the stated annual interest rate. Effective Interest Rate - i - For rates compounded more frequently than one year, the actual amount of interest paid. i = ( 1 + r / M ) M - 1 = ( F / P, r / M, M ) -1 – M - the number of compounding periods per year Annual Percentage Rate - APR - percentage rate per period times number of periods. –APR = r x M Nominal Interest Rate - r - For rates compounded more frequently than one year, the stated annual interest rate. Effective Interest Rate - i - For rates compounded more frequently than one year, the actual amount of interest paid. i = ( 1 + r / M ) M - 1 = ( F / P, r / M, M ) -1 – M - the number of compounding periods per year Annual Percentage Rate - APR - percentage rate per period times number of periods. –APR = r x M

COMPOUNDING MORE OFTEN THAN ONCE A YEAR Single Amounts Given nominal interest rate and total number of compounding periods, P, F or A can be determined by F = P ( F / P, i%, N ) i% = ( 1 + r / M ) M - 1 Uniform and / or Gradient Series Given nominal interest rate, total number of compounding periods, and existence of a cash flow at the end of each period, P, F or A may be determined by the formulas and tables for uniform annual series and uniform gradient series. Single Amounts Given nominal interest rate and total number of compounding periods, P, F or A can be determined by F = P ( F / P, i%, N ) i% = ( 1 + r / M ) M - 1 Uniform and / or Gradient Series Given nominal interest rate, total number of compounding periods, and existence of a cash flow at the end of each period, P, F or A may be determined by the formulas and tables for uniform annual series and uniform gradient series.

CASH FLOWS LESS OFTEN THAN COMPOUNDING PERIODS Find A, given i, k and X, where: –i is the effective interest rate per interest period –k is the period at the end of which cash flow occurs –X is the uniform cash flow amount Use: A = X (A / F,i%, k ) Find A, given i, k and X, where: –i is the effective interest rate per interest period –k is the period at the beginning of which cash flow occurs –X is the uniform cash flow amount Use: A = X ( A / P, i%, k ) Find A, given i, k and X, where: –i is the effective interest rate per interest period –k is the period at the end of which cash flow occurs –X is the uniform cash flow amount Use: A = X (A / F,i%, k ) Find A, given i, k and X, where: –i is the effective interest rate per interest period –k is the period at the beginning of which cash flow occurs –X is the uniform cash flow amount Use: A = X ( A / P, i%, k )

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS

Continuous compounding assumes cash flows occur at discrete intervals, but compounding is continuous throughout the interval.

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Continuous compounding assumes cash flows occur at discrete intervals, but compounding is continuous throughout the interval. Given nominal per year interest rate -- r, compounding per year -- M one unit of principal = [ 1 + (r / M ) ] M Continuous compounding assumes cash flows occur at discrete intervals, but compounding is continuous throughout the interval. Given nominal per year interest rate -- r, compounding per year -- M one unit of principal = [ 1 + (r / M ) ] M

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Continuous compounding assumes cash flows occur at discrete intervals, but compounding is continuous throughout the interval. Given nominal per year interest rate -- r, compounding per year -- M one unit of principal = [ 1 + (r / M ) ] M Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp Continuous compounding assumes cash flows occur at discrete intervals, but compounding is continuous throughout the interval. Given nominal per year interest rate -- r, compounding per year -- M one unit of principal = [ 1 + (r / M ) ] M Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Continuous compounding assumes cash flows occur at discrete intervals, but compounding is continuous throughout the interval. Given nominal per year interest rate -- r, compounding per year -- M one unit of principal = [ 1 + (r / M ) ] M Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp Given lim [ 1 + (1 / p) ] p = e 1 = Continuous compounding assumes cash flows occur at discrete intervals, but compounding is continuous throughout the interval. Given nominal per year interest rate -- r, compounding per year -- M one unit of principal = [ 1 + (r / M ) ] M Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp Given lim [ 1 + (1 / p) ] p = e 1 = p 

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Continuous compounding assumes cash flows occur at discrete intervals, but compounding is continuous throughout the interval. Given nominal per year interest rate -- r, compounding per year -- M one unit of principal = [ 1 + (r / M ) ] M Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp Given lim [ 1 + (1 / p) ] p = e 1 = ( F / P, r%, N ) = e rN Continuous compounding assumes cash flows occur at discrete intervals, but compounding is continuous throughout the interval. Given nominal per year interest rate -- r, compounding per year -- M one unit of principal = [ 1 + (r / M ) ] M Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp Given lim [ 1 + (1 / p) ] p = e 1 = ( F / P, r%, N ) = e rN p 

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Continuous compounding assumes cash flows occur at discrete intervals, but compounding is continuous throughout the interval. Given nominal per year interest rate -- r, compounding per year -- M one unit of principal = [ 1 + (r / M ) ] M Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp Given lim [ 1 + (1 / p) ] p = e 1 = ( F / P, r%, N ) = e rN i = e r - 1 Continuous compounding assumes cash flows occur at discrete intervals, but compounding is continuous throughout the interval. Given nominal per year interest rate -- r, compounding per year -- M one unit of principal = [ 1 + (r / M ) ] M Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp Given lim [ 1 + (1 / p) ] p = e 1 = ( F / P, r%, N ) = e rN i = e r - 1 p 

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Single Cash Flow Finding F given P Finding future equivalent value given present value F = P (e rN ) Functionally expressed as ( F / P, r%, N ) e rN is continuous compounding compound amount Predetermined values are in column 2 of appendix D of text Finding F given P Finding future equivalent value given present value F = P (e rN ) Functionally expressed as ( F / P, r%, N ) e rN is continuous compounding compound amount Predetermined values are in column 2 of appendix D of text

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Single Cash Flow Finding P given F Finding present equivalent value given future value P = F (e -rN ) Functionally expressed as ( P / F, r%, N ) e -rN is continuous compounding present equivalent Predetermined values are in column 3 of appendix D of text Finding P given F Finding present equivalent value given future value P = F (e -rN ) Functionally expressed as ( P / F, r%, N ) e -rN is continuous compounding present equivalent Predetermined values are in column 3 of appendix D of text

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Uniform Series Finding F given A Finding future equivalent value given a series of uniform equal receipts F = A (e rN - 1)/(e r - 1) Functionally expressed as ( F / A, r%, N ) (e rN - 1)/(e r - 1) is continuous compounding compound amount Predetermined values are in column 4 of appendix D of text Finding F given A Finding future equivalent value given a series of uniform equal receipts F = A (e rN - 1)/(e r - 1) Functionally expressed as ( F / A, r%, N ) (e rN - 1)/(e r - 1) is continuous compounding compound amount Predetermined values are in column 4 of appendix D of text

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Uniform Series Finding P given A Finding present equivalent value given a series of uniform equal receipts P = A (e rN - 1) / (e rN ) (e r - 1) Functionally expressed as ( P / A, r%, N ) (e rN - 1) / (e rN ) (e r - 1) is continuous compounding present equivalent Predetermined values are in column 5 of appendix D of text Finding P given A Finding present equivalent value given a series of uniform equal receipts P = A (e rN - 1) / (e rN ) (e r - 1) Functionally expressed as ( P / A, r%, N ) (e rN - 1) / (e rN ) (e r - 1) is continuous compounding present equivalent Predetermined values are in column 5 of appendix D of text

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Uniform Series Finding A given F Finding a uniform series given a future value A = F (e r - 1) / (e rN - 1) Functionally expressed as ( A / F, r%, N ) (e r - 1) / (e rN - 1) is continuous compounding sinking fund Predetermined values are in column 6 of appendix D of text Finding A given F Finding a uniform series given a future value A = F (e r - 1) / (e rN - 1) Functionally expressed as ( A / F, r%, N ) (e r - 1) / (e rN - 1) is continuous compounding sinking fund Predetermined values are in column 6 of appendix D of text

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Uniform Series Finding A given P Finding a series of uniform equal receipts given present equivalent value A = P [e rN (e r - 1) / (e rN - 1) ] Functionally expressed as ( A / P, r%, N ) [e rN (e r - 1) / (e rN - 1) ] is continuous compounding capital recovery Predetermined values are in column 7 of appendix D of text Finding A given P Finding a series of uniform equal receipts given present equivalent value A = P [e rN (e r - 1) / (e rN - 1) ] Functionally expressed as ( A / P, r%, N ) [e rN (e r - 1) / (e rN - 1) ] is continuous compounding capital recovery Predetermined values are in column 7 of appendix D of text

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Continuous flow of funds suggests a series of cash flows occurring at infinitesimally short intervals of time

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Continuous flow of funds suggests a series of cash flows occurring at infinitesimally short intervals of time Given: – a nominal interest rate or r –p is payments per year Continuous flow of funds suggests a series of cash flows occurring at infinitesimally short intervals of time Given: – a nominal interest rate or r –p is payments per year

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Continuous flow of funds suggests a series of cash flows occurring at infinitesimally short intervals of time Given: – a nominal interest rate or r –p is payments per year [ 1 + (r / p ) ] p - 1 P = r [ 1 + ( r / p ) ] p Continuous flow of funds suggests a series of cash flows occurring at infinitesimally short intervals of time Given: – a nominal interest rate or r –p is payments per year [ 1 + (r / p ) ] p - 1 P = r [ 1 + ( r / p ) ] p

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Continuous flow of funds suggests a series of cash flows occurring at infinitesimally short intervals of time Given: – a nominal interest rate or r –p is payments per year [ 1 + (r / p ) ] p - 1 P = r [ 1 + ( r / p ) ] p Given Lim [ 1 + ( r / p ) ] p = e r Continuous flow of funds suggests a series of cash flows occurring at infinitesimally short intervals of time Given: – a nominal interest rate or r –p is payments per year [ 1 + (r / p ) ] p - 1 P = r [ 1 + ( r / p ) ] p Given Lim [ 1 + ( r / p ) ] p = e r p --> oo

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Continuous flow of funds suggests a series of cash flows occurring at infinitesimally short intervals of time Given: – a nominal interest rate or r –p is payments per year [ 1 + (r / p ) ] p - 1 P = r [ 1 + ( r / p ) ] p Given Lim [ 1 + ( r / p ) ] p = e r For one year ( P / A, r%, 1 ) = ( e r - 1 ) / re r Continuous flow of funds suggests a series of cash flows occurring at infinitesimally short intervals of time Given: – a nominal interest rate or r –p is payments per year [ 1 + (r / p ) ] p - 1 P = r [ 1 + ( r / p ) ] p Given Lim [ 1 + ( r / p ) ] p = e r For one year ( P / A, r%, 1 ) = ( e r - 1 ) / re r p --> oo

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding F given A

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding F given A Finding the future equivalent given the continuous funds flow Finding F given A Finding the future equivalent given the continuous funds flow

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding F given A Finding the future equivalent given the continuous funds flow F = A [ ( e rN - 1 ) / r ] Finding F given A Finding the future equivalent given the continuous funds flow F = A [ ( e rN - 1 ) / r ]

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding F given A Finding the future equivalent given the continuous funds flow F = A [ ( e rN - 1 ) / r ] Functionally expressed as ( F / A, r%, N ) Finding F given A Finding the future equivalent given the continuous funds flow F = A [ ( e rN - 1 ) / r ] Functionally expressed as ( F / A, r%, N )

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding F given A Finding the future equivalent given the continuous funds flow F = A [ ( e rN - 1 ) / r ] Functionally expressed as ( F / A, r%, N ) ( e rN - 1 ) / r is continuous compounding compound amount Finding F given A Finding the future equivalent given the continuous funds flow F = A [ ( e rN - 1 ) / r ] Functionally expressed as ( F / A, r%, N ) ( e rN - 1 ) / r is continuous compounding compound amount

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding F given A Finding the future equivalent given the continuous funds flow F = A [ ( e rN - 1 ) / r ] Functionally expressed as ( F / A, r%, N ) ( e rN - 1 ) / r is continuous compounding compound amount Predetermined values are found in column 6 of appendix D of text. Finding F given A Finding the future equivalent given the continuous funds flow F = A [ ( e rN - 1 ) / r ] Functionally expressed as ( F / A, r%, N ) ( e rN - 1 ) / r is continuous compounding compound amount Predetermined values are found in column 6 of appendix D of text.

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding P given A

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding P given A Finding the present equivalent given the continuous funds flow Finding P given A Finding the present equivalent given the continuous funds flow

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding P given A Finding the present equivalent given the continuous funds flow P = A [ ( e rN - 1 ) / re rN ] Finding P given A Finding the present equivalent given the continuous funds flow P = A [ ( e rN - 1 ) / re rN ]

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding P given A Finding the present equivalent given the continuous funds flow P = A [ ( e rN - 1 ) / re rN ] Functionally expressed as ( P / A, r%, N ) Finding P given A Finding the present equivalent given the continuous funds flow P = A [ ( e rN - 1 ) / re rN ] Functionally expressed as ( P / A, r%, N )

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding P given A Finding the present equivalent given the continuous funds flow P = A [ ( e rN - 1 ) / re rN ] Functionally expressed as ( P / A, r%, N ) ( e rN - 1 ) / re rN is continuous compounding present equivalent Finding P given A Finding the present equivalent given the continuous funds flow P = A [ ( e rN - 1 ) / re rN ] Functionally expressed as ( P / A, r%, N ) ( e rN - 1 ) / re rN is continuous compounding present equivalent

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding P given A Finding the present equivalent given the continuous funds flow P = A [ ( e rN - 1 ) / re rN ] Functionally expressed as ( P / A, r%, N ) ( e rN - 1 ) / re rN is continuous compounding present equivalent Predetermined values are found in column 7 of appendix D of text. Finding P given A Finding the present equivalent given the continuous funds flow P = A [ ( e rN - 1 ) / re rN ] Functionally expressed as ( P / A, r%, N ) ( e rN - 1 ) / re rN is continuous compounding present equivalent Predetermined values are found in column 7 of appendix D of text.

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding A given F

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding A given F Finding the continuous funds flow given the future equivalent Finding A given F Finding the continuous funds flow given the future equivalent

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding A given F Finding the continuous funds flow given the future equivalent A = F [ r / ( e rN - 1 )] Finding A given F Finding the continuous funds flow given the future equivalent A = F [ r / ( e rN - 1 )]

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding A given F Finding the continuous funds flow given the future equivalent A = F [ r / ( e rN - 1 )] Functionally expressed as ( A / F, r%, N ) Finding A given F Finding the continuous funds flow given the future equivalent A = F [ r / ( e rN - 1 )] Functionally expressed as ( A / F, r%, N )

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding A given F Finding the continuous funds flow given the future equivalent A = F [ r / ( e rN - 1 )] Functionally expressed as ( A / F, r%, N ) r / ( e rN - 1 ) is continuous compounding sinking fund Finding A given F Finding the continuous funds flow given the future equivalent A = F [ r / ( e rN - 1 )] Functionally expressed as ( A / F, r%, N ) r / ( e rN - 1 ) is continuous compounding sinking fund

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding A given P

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding A given P Finding the continuous funds flow given the present equivalent Finding A given P Finding the continuous funds flow given the present equivalent

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding A given P Finding the continuous funds flow given the present equivalent A = P [ r / ( e rN - 1 )] Finding A given P Finding the continuous funds flow given the present equivalent A = P [ r / ( e rN - 1 )]

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding A given P Finding the continuous funds flow given the present equivalent A = P [ r / ( e rN - 1 )] Functionally expressed as ( A / P, r%, N ) Finding A given P Finding the continuous funds flow given the present equivalent A = P [ r / ( e rN - 1 )] Functionally expressed as ( A / P, r%, N )

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding A given P Finding the continuous funds flow given the present equivalent A = F [ re rN / ( e rN - 1 )] Functionally expressed as ( A / P, r%, N ) re rN / ( e rN - 1 ) is continuous compounding capital recovery Finding A given P Finding the continuous funds flow given the present equivalent A = F [ re rN / ( e rN - 1 )] Functionally expressed as ( A / P, r%, N ) re rN / ( e rN - 1 ) is continuous compounding capital recovery