# COMPOUND INTEREST AND ANNUITY TABLES

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 COMPOUND INTEREST AND ANNUITY TABLES COMPOUND INTEREST AND ANNUITY TABLES 8 Percent VALUE OF AN NO. OF PRESENT PRESENT VALUE OF AN COM- AMORTIZ ANNUITY - ONE PER YEARS VALUE OF ANNUITY POUND ATION YEAR HENCE 1.00 Present Future Increasing Decreasing , , , Interest and annuity tables provide a reference to enable the user to properly account for the effects of interest and time in making an economic analysis. The basic principles of the time value of money, and the use of interest factors in making comparisons between values that occur at different points in time are presented. Interest and annuity problems have four elements in common: (a) an amount, (b) an interest rate, (c) a term, and (d) a payment. If any three of these elements are known, then the fourth can be derived from the tables. Procedures for discounting future benefits and costs or otherwise converting benefits and costs to a common time basis are also presented. BASIC DEFINITIONS Value In economics, value represents any quantity expressing the worth of something. In resource development projects, value is used to express benefits arising from effects of project measures, or it could be the cost for providing such measures.

2 Number of Years Hence Annuity Interest Simple Interest Compound Interest This is the number of periods (years, months, or days) in which calculations are considered. There may be many conditions which influence this determination: (1) a benefit may last a year or indefinitely (perpetuity), (2) the measures may have a short or long useful life, (3) the period of evaluation may be set by policy, (4) an individual may want to recover his costs in a certain time period, or (5) costs or returns may occur over varying time periods or at varying rates for the same period. Annuity is a series of equal payments made at equal intervals of time. The most common type of annuity is our paychecks, at least those that meet the equal payment requirement. Annuity may be a benefit (to those receiving equal sums of money) or a cost (to those making the payment). Interest is economic rent of money. When money is borrowed, the amount borrowed must be repaid along with a use charge called Or, said another way, interest is money paid for the use of money. The appropriate rate of interest will depend upon the situation or the reason for the analysis. Demand, time, and risk (includes inflation) determine the rate of interest charged or paid in commercial lending establishments. If personal money is used in lieu of borrowed or lent funds, an opportunity cost 1 should be taken into consideration. Interest rate is expressed as a percent of the principal amount and is understood to be an interest rate per year. There are two kinds of interest-- simple and compound Simple interest is rent on the principle amount only. If 100 is loaned or borrowed and a year later 108 is repaid, the 8 is interest and, since it was for one year, the interest rate is 8 percent. The amount of interest is computed by the following formula: i= (p)(r)(n), where i = interest, p = principle (100), r = periodic interest rate (8%), and n = number of periods (1 year). i = (100)(.08)(1) = 8.00 Compound interest is interest that is earned for one period and immediately added to the principle, yielding a larger principle on which interest is computed for the following period. This means the accrued unpaid interest is actually converted to additional principle. Problem: What will 500 grow to in 5 years at 8 percent interest? Solution: (500)( ) = The following represents the compound interest factor Formula: (1 + i) n, where n is the number of periods, i is the periodic rate of interest, and 1 represents one dollar since the formula results in a factor that is multiplied by the principle dollar amount. NOTE: Compound interest factors are not shown by column heading in the tables, but the same answer can be obtained by dividing the appropriate "present value of 1" factor since the present value of 1 factor is the reciprocal of the compound interest factor. Using the preceding problem: 1 0pportunity cost is the return forgone from the most likely alternative use of money. 2 Compound interest, 5 years hence, and 8 percent interest (not shown in tables).

3 What will 500 grow to in 5 years at 8 percent interest? The solution is as follows: Solution: 500 / = Present Value of 1 Sometimes called present worth of 1 is what 1.00 due in the future is worth today or at present. The present value of a specified single sum of money due at some named future date is that sum of money which, if put at compound interest for the same time period would have a compound amount equal to the specified amount. Hence, this is the reason for the factor being occasionally called the "discount factor." Delayed cost or benefits can be reduced to present worth at year 0 with this factor. Problem: At 8 percent interest find the present value of 1,000 to be received in 5 years. Solution: (1,000)( ) = In other words, 1,000 to be received in 5 years is worth today. The present value of 1 factor is represented by the following formula. 1 (1 + 1) n NOTE: The present value of 1 factor is the reciprocal of the compound interest factor. Amortization Amortization is sometimes called partial payment or capital recovery factor. This factor will convert capital or initial cost to annual cost. It will determine what annual payment including interest must be made to payoff the initial cost over a given number of years. Problem: At 8 percent interest, find the annual equivalent investment cost over a period of 10 years of a facility with an initial cost of 1,000. Solution: Annual cost = initial cost multiplied by the amortization factor for 10 years at 8 percent Annual cost = (1,000)( ) Annual cost = The formula for the amortization factor is expressed as: Amortization = i (1 + i) n or. i (1 + i) n (1 + i) n NOTE: The amortization factor is the reciprocal of the "present value of an annuity of 1 per year" factor which means that the same answer can be obtained by dividing by the present value of an annuity of 1 per year factor. For example, using the above problem the solution is as follows: 1,000 / = Present value of 1, 5 years hence, 8 percent 2 Present Value of 1, 5 years hence, 8 percent 3 Amortization, 10 years hence, 8 percent 4 Present value of an annuity of 1 per year, 10 years hence, 8 percent

4 Present Value of an Annuity of 1 Per Year Present value of an annuity of 1 per year also referred to as constant annuity, present worth of an annuity, or capitalization factors. This factor represents the present value or worth of a series of equal deposits over a period of time. It tells us what an annual deposit of 1.00 is worth today. If a fixed sum is to be deposited or earned annually for "n" years, this factor will determine the present worth of those deposits or earnings. Problem: If 600 will be placed in your savings account each year for 10 years, what is the present worth of the total amount or if you want to give 600 per year to someone for 10 years what sum would be required at present? The interest rate is 8 percent. Solution: 600 x = 4,026 This is the present value of receiving 600 per year for 10 years or it is the amount that would need to be deposited today to make annual withdrawals of 600 for 10 years. The present value of an annuity of 1 per year factor is expressed as follows: (1 + i) n -1 1 (1 + i) n NOTES: a. The factor is the reciprocal of the "amortization" factor. Therefore, the same answer can be obtained by dividing by the amortization factor: Solution: 600 / = 4,026 b. "Amount of an annuity of 1 per year" multiplied by "present value of 1" = "present value of an annuity of 1 per year." Amount of an Annuity of 1 Per Year Amount of an Annuity of 1 per year is also called "accumulation of an annuity. As stated earlier an annuity is a sequence of equal payments made at uniform intervals with each payment earning compound interest during it s respective earning term. The amount of an annuity of 1 per year factor shows how much an annuity, invested each year, will grow over a period of years. It can also show how much it is worth to provide protection against losing the opportunity of investing an annuity each year. For example, preventing erosion that causes a loss of net income of 25 per year for 50 years has an accumulated value of 14,344 at the end of 50 years at 8 percent interest (25 x ). The factor for an amount of an annuity of 1 per year is expressed as follows: (1 + 1) n -1 1 Another example is a person establishing a retirement reserve. If 500 is placed in a savings account, each year earning 8 percent, what is the amount of the reserve at the end of 20 years? Solution: (500)( ) = 22,881 1 Amount of an annuity of 1 per year, SO years hence, and 8 percent 2 Amount of an annuity of 1 per year, 20 years, 8 percent interest

5 Sinking Fund This factor is used to determine what size annual deposit will be required to accumulate to a certain amount (given) in a certain number (given) of years at compound Problem: If 25,000 is needed to meet a term note due in 10 years, what amount will need to be deposited each year at 8 percent compound interest to reach the goal? Solution: (25,000)( ) = 1,726 The sinking fund factor is expressed as: Sinking Fund = 1 (1 + i) n -l NOTE: The sinking fund factor is not shown by column heading in the tables but the same answer can be obtained by dividing by the appropriate "amount of an annuity of 1 per year" factor since the amount of an annuity of 1 per year factor is the reciprocal of the sinking fund factor. Solution: 25,000 / = 1,726 Present Value of an Increasing Annuity This factor shows how much something is presently worth that will provide increasing sums of money over a period of years. It should be noted that to meet the definition of an annuity (equal payments at equal intervals) the increases must be uniform. For example, seeding a field to grass may eventually be worth 18 per acre in increased net income, but it will take 6 years before this value is realized. The annuity is increasing 3 per year (18 / 6) so the definition of an annuity is satisfied. The total annuity we will receive is not level or the same each year since We will receive 3 the first year, 6 the second year, and 9 the third year until at the end of the sixth year the 18 per year will be realized. At 8 percent interest the present value or present worth of something that will provide this amount of flow or build up of funds is (3 x ). The factor for "present value of an increasing annuity" is expressed in the following formula: (1 + i) n (1+ I) n(i) (1 + i) n (i) 2 Present Value of a Decreasing Annuity This factor tells us how much something is presently worth that will provide an annuity that gets less each year. The decrease must be uniform to meet the definition of an annuity. For example, a sediment pool will return 1,000 recreation benefits the first year but at the end of 40 years, because of sediment accumulation, it will have no value. The flow of funds is decreasing uniformly at 25 per year (1,000 / 40 years). Therefore, the basic definition of annuity is met. Benefits would be 1,000 the first year, 975 the second year, 950 the third year, etc., until at the end of 40 years there would be no returns. 1 Sinking fund factor 10 years hence, 8 percent interest (not shown in tables). 2 Amount of an annuity of 1 per year, 8 percent interest, 10 years hence. 3 Present value of an increasing annuity, 6 years hence, 8 percent

6 The present value of this sediment pool at 8 percent interest is 8,774 (25 x ). The 8,774 invested at 8 percent interest would return the decreasing annuities described. The factor for "present value of a decreasing annuity" is displayed in the following formula: n (i) (1 + I) n (1 + i) n (i) 2 Clues to Problem Solving by the use of Diagrams & Sketches Clue 1 The solution to complicated problems is made easier by the use of a sketch or diagram. There are six basic diagrams that can be combined or used to illustrate all problems involving decreases, increases, delays, or discount by the use of the compound interest and annuity tables. By using the graphic illustrations and lettering the basic parts, all parts of the more complicated problems can be broken down into a series of simple familiar steps, each one easy to understand and solve. You will also save time and make fewer mistakes. Situation: A return of 600 will be received each year for 20 years. What is the average annual value for a 50-year evaluation period at 8 percent interest? 0 20 Years 50 Step 1 - Determine present worth of benefits occurring evenly over each of the 20 years. Multiply 600 by the factor for the "present value of annuity of 1 per year" corresponding to the length of return (20 years). (600)( ) 2 = 5,891 Step 2 - Determine the annual equivalent value of the income stream. Multiply the 5,891 by the amortization" factor corresponding to the length of the evaluation period (50 years). (5,891)(.08174) 3 = 482 \ Conclusion: The average annual value is 482. Clue 2 Situation: A return of 600 will be received for 20 years. However, it will be 30 years before the return will begin. What is the average annual value for a 50-year evaluation period at 8 percent interest? 0 30 Years 50 Step 1 - Determine present worth of benefits occurring evenly over each of the 20 years (50-30). Multiply 600 by the factor for "present value of an annuity of 1 per year" corresponding to the length of return (20 years). 1 Present value of decreasing annuity, 40 years, and 8 percent 2 Present value of an annuity of 1 per year, 20 years, and 8 percent 3 Amortization, 50 years, 8 percent

7 (600)( ) = 5,891 Step 2 - Discount the benefits that are delayed 30 years. Multiply the 5,891 by the discount factor "present value of 1" corresponding to the length of delay (30 years). (5.891)( ) = 585 Step 3 - Determine the annual equivalent value of the capital sum. Multiply the 585 by the "amortization" factor corresponding to the length of the evaluation period (50 years). (585)( ) = 48 Conclusion: The average annual value is 48. Clue 3 Situation: A return will begin to build from 0 in year 0 to 600 in 20 years then cease. What is the average annual benefit at 8 percent interest for a 50-year evaluation period? 0 20 Years 50 Step 1 - To meet the definition of an annuity where values are uniformly increasing, divide the ending value reached by the number of years to reach the value = 30 per year Step 2 - Determine present worth of the income stream. Multiply the 30 per year by the "present value of an increasing annuity" corresponding to the number of years the return is received (20 years). (30) ( ) = 2,367 Step 3 -Determine annual equivalent value of the income stream. Multiply the 2,367 by the amortization" factor corresponding to the length of the evaluation period (5O years). (2.367)( ) = 193 Conclusion: The average annual benefit is 193. Clue 4 Situation: A return will begin to build from 0 in year 30 to 600 in year 50. What is the average annual benefit at 8 percent interest for a 50-year evaluation period? 1 Present. value of an annuity of 1 per year, 20 years, and 8 percent Years 50 Present value of 1, 30 years, 8 percent 3 Present value of an increasing annuity of 1 per year, 20 years, -8 percent interest 4 Amortization. 50 years, 8 percent

8 Step 1 - To meet the definition of an annuity where values are uniformly increasing, divide the ending value reached by the number of years to reach the value = 30 per year Step 2 - Determine present worth of the income stream beginning in year 30. Multiply the 30 per year by the "present value of an increasing annuity" corresponding to the number of years the return is received (20 years). (30)( ) = 2,367 Step 3 -Discount the income stream that is delayed 30 years. Multiply the 2,367 by the discount factor (present value of 1) corresponding to the length of delay (30 years). (2,367)( ) = 235 Step 4 - Determine the annual equivalent value of the income stream. Multiply the 235 by the "amortization factor corresponding to the length of the evaluation period (50 years). (235)( ) = 19 Conclusion: The average annual benefit is 19 Clue 5 Situation: A present return of 600 will uniformly decrease to 0 at year 20. What is the average annual value of 8 percent interest rate for a 50-year evaluation period? 0 20 Years 50 Step 1 -To meet the definition of an annuity where values are uniformly decreasing, divide the beginning value by the number of years to reach = 30 per year Step 2 - Determine the present worth of the income stream. Multiply the 30 per year by the "present value of a decreasing annuity" corresponding to the number of years the return is to be received (20 years). (30)( ) = 3,818 Step 3 - Determine the annual equivalent value of the income stream. Multiply the 3,818 by the "amortization" factor corresponding to the length of the evaluation period (50 years). (3.818)( ) = 312 Conclusion: The average annual benefit is Present value of an increasing annuity, 20 years, 8 percent 2 Present value of 1, 30 years, 8 percent interest 3 Present value of a decreasing annuity, 20 years, 8 percent 4 Amortization, 50 years, 8 percent

9 Clue 6 Situation: After year 30, return of 600 uniformly decreases to 0 at year 50. What is the average annual value at 8 percent interest rate? 0 30 Years 50 Step 1 - To meet the definition of an annuity where values are uniformly decreasing, divide the beginning value by the number of years to reach = 30 per year Step 2 - Determine present worth of the income stream beginning in year 30. Multiply the 30 per year by the "present value of a decreasing annuity" corresponding to the number of years the return is received (20 years). (30)( ) = 3,818 Step 3 - Discount the income stream that's delayed 30 years. Multiply the 3,818 by the discount factor (present value of 1) corresponding to the length of delay (30 years). (3,818)( ) = 379 Step 4 -Determine the annual equivalent value of the income stream. Multiply the 379 by the "amortization" factor corresponding to the length of the evaluation period (50 years). (379)( ) = 31 Conclusion: The average annual value is 31 1 Present value of 1, 30 years, 8 percent interest

10 Example Problems Problem 1 The following problems illustrate the use of the compound interest and annuity tables. The purpose of the problems is to illustrate the principles used in applying each of the annuity factors described in the previous section A net return will increase in uniform amounts to 3,000 for a 15-year period. Thereafter, through the remaining 35 years of the evaluation period, the benefits will remain constant at 8 percent Determine the annual equivalent value over the 50-year evaluation period. First diagram the problem. (a) (b) 0 15 Years 50 Solution: a. 3,000 / 15 = 200 per year increase (200)( ) = 11,289 b. (3,000)( )( ) = 11,022 Annual Equivalent Value: 11, ,022(/ 22,311 (22,311)( ) = 1,824 Shortcut -Table A (page 14) provides straight-line lag discount factors that can be used directly for selected percents of The illustration of the shortcut method in solving the above problem is as follows: (3,000)( ) = 1,824 1 Present value of an increasing annuity, 15 years hence, 8 percent interest 2 Present value of 1I, 15 years hence, 8 percent interest 3 Present value of an annuity of 1 per year, 35 years hence, 8 percent 4 Amortization, 50 years, 8 percent 5 Table I, page 14,15 year lag, 50-year evaluation period, and 8 percent

11 Problem 2 Land use will become more intensive in a flood prevention-drainage project as the operators accept and utilize the protection made possible by the project. Presently net benefits per acre are 50. They are expected to uniformly increase to 95 per acre in 20 years. Thence, to 149 by the end of 50 years and then remain uniform for the remainder of the project life. Determine the average value per acre if the interest rate is 8 percent and the evaluation period is 100 years. Diagram the problem: (d) (e) (b) (c) (a) 0 50 Years 100 Solution: a. (50)( ) = 625 b = / 20 2 = 2.25 per year (2. 25)( ) = 178 c. (45)( )( ) = 120 d = / 03 6 = 1.80 per year (1.80)( )( ) = 44 e. (54)( )( ) = 14 The present value of receiving the above benefits is 981 ( ). The average annual value per acre is (981 x ) 1 Present value of an annuity of 1 per year, 100 years, and 8 percent 2 Necessary to meet the definition of an annuity. 3 Present value of an increasing annuity, 20 years, 8 percent 4 Present value of an annuity of 1 per year, 80 years, and 8 percent 5 Present value of 1, 20 years, 8 percent 6 Necessary to meet the definition of an annuity, (50-20) = 30 years. 7 Present value of an increasing annuity, 30 years, 8 percent 8 Present value of an annuity of 1 per year, 50 years, and 8 percent 9 Present value of 1, 50 years, 8 percent 10 Amortization, 100 years, 8 percent

12 Problem 3 It will take 5 years for seeding on rangeland to become established after which it will yield a value of 500 on a continuing basis. What is the average annual benefit for the 25-year evaluation period at 8 percent interest? Diagram the problem: 0 5 Years 25 Solution: (500)( )( )( ) = 313 Problem 4 The recreational use of a reservoir is expected to be 3,000 visits the first year of use and will remain constant for 35 years. They are then expected to decline to 0 at the 60th year of the 100-year evaluation period. Using an 8 percent interest rate, determine the average annual recreational benefits if visits are valued at 1.25 each. (a) (b) 0 35 Years Solution: a. (3,000}(l.25) = 3,750 (3,750)( ) = 43,705 b. 3,750 / 25 5 = 150 per year (150)( )( ) = 1,817 43, ,817 = 45,522 (45,522}( ) = 3,644 1 Present value of an annuity of 1, 20 years, 8 percent 2 Present value of 1, 5 years, 8 percent 3 Amortization, 25 years, 8 percent 4 Present value of an annuity of 1 per year, 35 years, and 8 percent 5 Necessary to meet the definition of an annuity. (60 years - 35 years = 25 years.) 6 Present value of a decreasing annuity, 25 years hence, 8 percent 7 Present value of 1, 35 years hence, 8 percent 8 Amortization, 100 years, 8 percent

13 Problem 5 This problem combines all of the previous principles into one example. Assume this problem involved benefits that would accrue in the following manner: begin with 300; grow to 800 in 5 years; grow to 1500 by year 15; remain constant until year 25, then decrease to 800 by year 35; decrease to 300 by year 50 and remain constant to year 100. To solve this complicated problem, first break the problem into segments that can be easily computed. (b) (e) (f) (g (c) (d) (a) Years a. Is a constant annuity 1 for the duration of the evaluation period. b. Is a buildup period or increasing annuity for 5 years with no lag or delay. c. Is a constant annuity 1 for the duration of the evaluation period. d. Is a decreasing annuity for 15 years delayed 35 years. e. Is an increasing annuity for 10 years delayed 5 years. f. Is a constant annuity 1 for 10 years delayed 15 years. g. Is a decreasing annuity for 10 years delayed 25 years. Dollar values listed in the problem and the factors from the tables are multiplied together. The values for (a) through (g) are then added together and amortized over the evaluation period to determine the average annual values. Remember where increasing and decreasing annuities are involved, you must divide the dollar value by the number of years, to get an annual rate of increase or decrease. 1 Constant annuity is referred to in the tables as "present value of an annuity of 1 per year."

14 Solution: The average annual value at 8 percent interest is as follows: a. (300)( ) = 3,748 b = / 5 2 = 100 (100)( ) = 1,137 c. (500)( )( ) = 3,831 d. 500 / 15 2 = (33.33)( )( ) = 181 e = / 10 2 = 70 (70)( )( ) = 1,557 f. (700)( )( ) = 1,481 g. 700 / 10 = 70 (70)( )( ) = 420 (3, , , , , = 12,355 (12,355)( ) = 989 Average Annual Value 1 Present value of an annuity of 1 per year, 100 years hence, 8 percent 2 Necessary to meet the definition of an annuity 3 Present value of an increasing annuity, 5 years hence, 8 percent 4 Present value of an annuity of 1 per year, 30 years hence, 8 percent 5 Present value of 1, 5 years, 8 percent 6 Present value of a decreasing annuity, 15 years hence, 8 percent 7 Present value of 1, 35 years hence, 8 percent 8 Present value of an increasing annuity, 10 years hence, 8 percent 9 Present value of an annuity of 1 per year, 10 years hence, 8 percent 10 Present value of 1, 15 years, 8 percent 11 Present value of a decreasing annuity, 10 years hence, 8 percent 12 Present value of 1, 25 years, 8 percent 13 Amortization, 100 years, 8 percent

### Six Functions of a Dollar. Made Easy! Business Statistics AJ Nelson 8/27/2011 1

Six Functions of a Dollar Made Easy! Business Statistics AJ Nelson 8/27/2011 1 Six Functions of a Dollar Here's a list. Simple Interest Future Value using Compound Interest Present Value Future Value of

### Chapter The Time Value of Money

Chapter The Time Value of Money PPT 9-2 Chapter 9 - Outline Time Value of Money Future Value and Present Value Annuities Time-Value-of-Money Formulas Adjusting for Non-Annual Compounding Compound Interest

### The Institute of Chartered Accountants of India

CHAPTER 4 SIMPLE AND COMPOUND INTEREST INCLUDING ANNUITY APPLICATIONS SIMPLE AND COMPOUND INTEREST INCLUDING ANNUITY- APPLICATIONS LEARNING OBJECTIVES After studying this chapter students will be able

### CHAPTER 6. Accounting and the Time Value of Money. 2. Use of tables. 13, 14 8 1. a. Unknown future amount. 7, 19 1, 5, 13 2, 3, 4, 6

CHAPTER 6 Accounting and the Time Value of Money ASSIGNMENT CLASSIFICATION TABLE (BY TOPIC) Topics Questions Brief Exercises Exercises Problems 1. Present value concepts. 1, 2, 3, 4, 5, 9, 17, 19 2. Use

### CHAPTER 6 Accounting and the Time Value of Money

CHAPTER 6 Accounting and the Time Value of Money 6-1 LECTURE OUTLINE This chapter can be covered in two to three class sessions. Most students have had previous exposure to single sum problems and ordinary

### Present Value (PV) Tutorial

EYK 15-1 Present Value (PV) Tutorial The concepts of present value are described and applied in Chapter 15. This supplement provides added explanations, illustrations, calculations, present value tables,

### Introduction to Real Estate Investment Appraisal

Introduction to Real Estate Investment Appraisal Maths of Finance Present and Future Values Pat McAllister INVESTMENT APPRAISAL: INTEREST Interest is a reward or rent paid to a lender or investor who has

### Calculations for Time Value of Money

KEATMX01_p001-008.qxd 11/4/05 4:47 PM Page 1 Calculations for Time Value of Money In this appendix, a brief explanation of the computation of the time value of money is given for readers not familiar with

### REVIEW MATERIALS FOR REAL ESTATE ANALYSIS

REVIEW MATERIALS FOR REAL ESTATE ANALYSIS 1997, Roy T. Black REAE 5311, Fall 2005 University of Texas at Arlington J. Andrew Hansz, Ph.D., CFA CONTENTS ITEM ANNUAL COMPOUND INTEREST TABLES AT 10% MATERIALS

### Reference: Gregory Mankiw s Principles of Macroeconomics, 2 nd edition, Chapter 15. The Banking System and the Money Supply

Macroeconomics Topic 6: Explain how the Federal Reserve and the banking system create money (i.e., the supply of money) Explain the factors that affect the demand for money. Reference: Gregory Mankiw s

### Chapter 6. Learning Objectives Principles Used in This Chapter 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams

Chapter 6 Learning Objectives Principles Used in This Chapter 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams 1. Distinguish between an ordinary annuity and an annuity due, and calculate present

### CHAPTER 4. The Time Value of Money. Chapter Synopsis

CHAPTER 4 The Time Value of Money Chapter Synopsis Many financial problems require the valuation of cash flows occurring at different times. However, money received in the future is worth less than money

### Time Value of Money. Work book Section I True, False type questions. State whether the following statements are true (T) or False (F)

Time Value of Money Work book Section I True, False type questions State whether the following statements are true (T) or False (F) 1.1 Money has time value because you forgo something certain today for

### Chapter 6 Contents. Principles Used in Chapter 6 Principle 1: Money Has a Time Value.

Chapter 6 The Time Value of Money: Annuities and Other Topics Chapter 6 Contents Learning Objectives 1. Distinguish between an ordinary annuity and an annuity due, and calculate present and future values

### The Time Value of Money C H A P T E R N I N E

The Time Value of Money C H A P T E R N I N E Figure 9-1 Relationship of present value and future value PPT 9-1 \$1,000 present value \$ 10% interest \$1,464.10 future value 0 1 2 3 4 Number of periods Figure

### Chapter 22: Borrowings Models

October 21, 2013 Last Time The Consumer Price Index Real Growth The Consumer Price index The official measure of inflation is the Consumer Price Index (CPI) which is the determined by the Bureau of Labor

### CHAPTER 1. Compound Interest

CHAPTER 1 Compound Interest 1. Compound Interest The simplest example of interest is a loan agreement two children might make: I will lend you a dollar, but every day you keep it, you owe me one more penny.

### Compound Interest Formula

Mathematics of Finance Interest is the rental fee charged by a lender to a business or individual for the use of money. charged is determined by Principle, rate and time Interest Formula I = Prt \$100 At

### MAT116 Project 2 Chapters 8 & 9

MAT116 Project 2 Chapters 8 & 9 1 8-1: The Project In Project 1 we made a loan workout decision based only on data from three banks that had merged into one. We did not consider issues like: What was the

### SOCIETY OF ACTUARIES FINANCIAL MATHEMATICS. EXAM FM SAMPLE QUESTIONS Interest Theory

SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS Interest Theory This page indicates changes made to Study Note FM-09-05. January 14, 2014: Questions and solutions 58 60 were

### Learning Objectives. Learning Objectives. Learning Objectives. Principles Used in this Chapter. Simple Interest. Principle 2:

Learning Objectives Chapter 5 The Time Value of Money Explain the mechanics of compounding, which is how money grows over a time when it is invested. Be able to move money through time using time value

### TIME VALUE OF MONEY (TVM)

TIME VALUE OF MONEY (TVM) INTEREST Rate of Return When we know the Present Value (amount today), Future Value (amount to which the investment will grow), and Number of Periods, we can calculate the rate

### 1. If you wish to accumulate \$140,000 in 13 years, how much must you deposit today in an account that pays an annual interest rate of 14%?

Chapter 2 - Sample Problems 1. If you wish to accumulate \$140,000 in 13 years, how much must you deposit today in an account that pays an annual interest rate of 14%? 2. What will \$247,000 grow to be in

### ICASL - Business School Programme

ICASL - Business School Programme Quantitative Techniques for Business (Module 3) Financial Mathematics TUTORIAL 2A This chapter deals with problems related to investing money or capital in a business

### Introduction to the Hewlett-Packard (HP) 10BII Calculator and Review of Mortgage Finance Calculations

Introduction to the Hewlett-Packard (HP) 10BII Calculator and Review of Mortgage Finance Calculations Real Estate Division Sauder School of Business University of British Columbia Introduction to the Hewlett-Packard

### Numbers 101: Cost and Value Over Time

The Anderson School at UCLA POL 2000-09 Numbers 101: Cost and Value Over Time Copyright 2000 by Richard P. Rumelt. We use the tool called discounting to compare money amounts received or paid at different

### SOCIETY OF ACTUARIES FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS

SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS This page indicates changes made to Study Note FM-09-05. April 28, 2014: Question and solutions 61 were added. January 14, 2014:

### Lecture Notes on the Mathematics of Finance

Lecture Notes on the Mathematics of Finance Jerry Alan Veeh February 20, 2006 Copyright 2006 Jerry Alan Veeh. All rights reserved. 0. Introduction The objective of these notes is to present the basic aspects

### 2. How would (a) a decrease in the interest rate or (b) an increase in the holding period of a deposit affect its future value? Why?

CHAPTER 3 CONCEPT REVIEW QUESTIONS 1. Will a deposit made into an account paying compound interest (assuming compounding occurs once per year) yield a higher future value after one period than an equal-sized

### Time Value Conepts & Applications. Prof. Raad Jassim

Time Value Conepts & Applications Prof. Raad Jassim Chapter Outline Introduction to Valuation: The Time Value of Money 1 2 3 4 5 6 7 8 Future Value and Compounding Present Value and Discounting More on

### Determinants of Valuation

2 Determinants of Valuation Part Two 4 Time Value of Money 5 Fixed-Income Securities: Characteristics and Valuation 6 Common Shares: Characteristics and Valuation 7 Analysis of Risk and Return The primary

### PREVIEW OF CHAPTER 6-2

6-1 PREVIEW OF CHAPTER 6 6-2 Intermediate Accounting IFRS 2nd Edition Kieso, Weygandt, and Warfield 6 Accounting and the Time Value of Money LEARNING OBJECTIVES After studying this chapter, you should

### Time Value of Money. Background

Time Value of Money (Text reference: Chapter 4) Topics Background One period case - single cash flow Multi-period case - single cash flow Multi-period case - compounding periods Multi-period case - multiple

### Chapter 6. Time Value of Money Concepts. Simple Interest 6-1. Interest amount = P i n. Assume you invest \$1,000 at 6% simple interest for 3 years.

6-1 Chapter 6 Time Value of Money Concepts 6-2 Time Value of Money Interest is the rent paid for the use of money over time. That s right! A dollar today is more valuable than a dollar to be received in

### Present Value Concepts

Present Value Concepts Present value concepts are widely used by accountants in the preparation of financial statements. In fact, under International Financial Reporting Standards (IFRS), these concepts

### FIN 3000. Chapter 6. Annuities. Liuren Wu

FIN 3000 Chapter 6 Annuities Liuren Wu Overview 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams Learning objectives 1. Distinguish between an ordinary annuity and an annuity due, and calculate

### Problem Set: Annuities and Perpetuities (Solutions Below)

Problem Set: Annuities and Perpetuities (Solutions Below) 1. If you plan to save \$300 annually for 10 years and the discount rate is 15%, what is the future value? 2. If you want to buy a boat in 6 years

### Annuities Certain. 1 Introduction. 2 Annuities-immediate. 3 Annuities-due

Annuities Certain 1 Introduction 2 Annuities-immediate 3 Annuities-due Annuities Certain 1 Introduction 2 Annuities-immediate 3 Annuities-due General terminology An annuity is a series of payments made

### Dick Schwanke Finite Math 111 Harford Community College Fall 2013

Annuities and Amortization Finite Mathematics 111 Dick Schwanke Session #3 1 In the Previous Two Sessions Calculating Simple Interest Finding the Amount Owed Computing Discounted Loans Quick Review of

### CHAPTER 6. Accounting and the Time Value of Money. 2. Use of tables. 13, 14 8 1. a. Unknown future amount. 7, 19 1, 5, 13 2, 3, 4, 7

CHAPTER 6 Accounting and the Time Value of Money ASSIGNMENT CLASSIFICATION TABLE (BY TOPIC) Topics Questions Brief Exercises Exercises Problems 1. Present value concepts. 1, 2, 3, 4, 5, 9, 17 2. Use of

### Integrated Case. 5-42 First National Bank Time Value of Money Analysis

Integrated Case 5-42 First National Bank Time Value of Money Analysis You have applied for a job with a local bank. As part of its evaluation process, you must take an examination on time value of money

### Lecture Notes on Actuarial Mathematics

Lecture Notes on Actuarial Mathematics Jerry Alan Veeh May 1, 2006 Copyright 2006 Jerry Alan Veeh. All rights reserved. 0. Introduction The objective of these notes is to present the basic aspects of the

### MODULE 2. Finance An Introduction

MODULE 2 Finance An Introduction The functions of finance in an organization is interlinked with other managerial responsibilities and in many instances, the finance manager could also done the role of

### Formulas and Approaches Used to Calculate True Pricing

Formulas and Approaches Used to Calculate True Pricing The purpose of the Annual Percentage Rate (APR) and Effective Interest Rate (EIR) The true price of a loan includes not only interest but other charges

### The Time Value of Money

The following is a review of the Quantitative Methods: Basic Concepts principles designed to address the learning outcome statements set forth by CFA Institute. This topic is also covered in: The Time

### Chapter 4 Time Value of Money ANSWERS TO END-OF-CHAPTER QUESTIONS

Chapter 4 Time Value of Money ANSWERS TO END-OF-CHAPTER QUESTIONS 4-1 a. PV (present value) is the value today of a future payment, or stream of payments, discounted at the appropriate rate of interest.

### CHAPTER 6 DISCOUNTED CASH FLOW VALUATION

CHAPTER 6 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1. The four pieces are the present value (PV), the periodic cash flow (C), the discount rate (r), and

### the Time Value of Money

8658d_c06.qxd 11/8/02 11:00 AM Page 251 mac62 mac62:1st Shift: 6 CHAPTER Accounting and the Time Value of Money he Magic of Interest T Sidney Homer, author of A History of Interest Rates, wrote, \$1,000

### If I offered to give you \$100, you would probably

File C5-96 June 2013 www.extension.iastate.edu/agdm Understanding the Time Value of Money If I offered to give you \$100, you would probably say yes. Then, if I asked you if you wanted the \$100 today or

### Finding the Payment \$20,000 = C[1 1 / 1.0066667 48 ] /.0066667 C = \$488.26

Quick Quiz: Part 2 You know the payment amount for a loan and you want to know how much was borrowed. Do you compute a present value or a future value? You want to receive \$5,000 per month in retirement.

### Index Numbers ja Consumer Price Index

1 Excel and Mathematics of Finance Index Numbers ja Consumer Price Index The consumer Price index measures differences in the price of goods and services and calculates a change for a fixed basket of goods

### CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY

CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 1. The simple interest per year is: \$5,000.08 = \$400 So after 10 years you will have: \$400 10 = \$4,000 in interest. The total balance will be

### Chapter 6. Discounted Cash Flow Valuation. Key Concepts and Skills. Multiple Cash Flows Future Value Example 6.1. Answer 6.1

Chapter 6 Key Concepts and Skills Be able to compute: the future value of multiple cash flows the present value of multiple cash flows the future and present value of annuities Discounted Cash Flow Valuation

### Chapter 7 SOLUTIONS TO END-OF-CHAPTER PROBLEMS

Chapter 7 SOLUTIONS TO END-OF-CHAPTER PROBLEMS 7-1 0 1 2 3 4 5 10% PV 10,000 FV 5? FV 5 \$10,000(1.10) 5 \$10,000(FVIF 10%, 5 ) \$10,000(1.6105) \$16,105. Alternatively, with a financial calculator enter the

### Vilnius University. Faculty of Mathematics and Informatics. Gintautas Bareikis

Vilnius University Faculty of Mathematics and Informatics Gintautas Bareikis CONTENT Chapter 1. SIMPLE AND COMPOUND INTEREST 1.1 Simple interest......................................................................

### Finite Mathematics. CHAPTER 6 Finance. Helene Payne. 6.1. Interest. savings account. bond. mortgage loan. auto loan

Finite Mathematics Helene Payne CHAPTER 6 Finance 6.1. Interest savings account bond mortgage loan auto loan Lender Borrower Interest: Fee charged by the lender to the borrower. Principal or Present Value:

### Chapter 03 - Basic Annuities

3-1 Chapter 03 - Basic Annuities Section 7.0 - Sum of a Geometric Sequence The form for the sum of a geometric sequence is: Sum(n) a + ar + ar 2 + ar 3 + + ar n 1 Here a = (the first term) n = (the number

### 9. Time Value of Money 1: Present and Future Value

9. Time Value of Money 1: Present and Future Value Introduction The language of finance has unique terms and concepts that are based on mathematics. It is critical that you understand this language, because

### Module 5: Interest concepts of future and present value

Page 1 of 23 Module 5: Interest concepts of future and present value Overview In this module, you learn about the fundamental concepts of interest and present and future values, as well as ordinary annuities

### Chapter 02 How to Calculate Present Values

Chapter 02 How to Calculate Present Values Multiple Choice Questions 1. The present value of \$100 expected in two years from today at a discount rate of 6% is: A. \$116.64 B. \$108.00 C. \$100.00 D. \$89.00

### APPENDIX. Interest Concepts of Future and Present Value. Concept of Interest TIME VALUE OF MONEY BASIC INTEREST CONCEPTS

CHAPTER 8 Current Monetary Balances 395 APPENDIX Interest Concepts of Future and Present Value TIME VALUE OF MONEY In general business terms, interest is defined as the cost of using money over time. Economists

### The Basics of Interest Theory

Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest

### The time value of money: Part II

The time value of money: Part II A reading prepared by Pamela Peterson Drake O U T L I E 1. Introduction 2. Annuities 3. Determining the unknown interest rate 4. Determining the number of compounding periods

### Annuity is defined as a sequence of n periodical payments, where n can be either finite or infinite.

1 Annuity Annuity is defined as a sequence of n periodical payments, where n can be either finite or infinite. In order to evaluate an annuity or compare annuities, we need to calculate their present value

### 4 Annuities and Loans

4 Annuities and Loans 4.1 Introduction In previous section, we discussed different methods for crediting interest, and we claimed that compound interest is the correct way to credit interest. This section

### Dick Schwanke Finite Math 111 Harford Community College Fall 2013

Annuities and Amortization Finite Mathematics 111 Dick Schwanke Session #3 1 In the Previous Two Sessions Calculating Simple Interest Finding the Amount Owed Computing Discounted Loans Quick Review of

### Chapter 2 Factors: How Time and Interest Affect Money

Chapter 2 Factors: How Time and Interest Affect Money Session 4-5-6 Dr Abdelaziz Berrado 1 Topics to Be Covered in Today s Lecture Section 2: How Time and Interest Affect Money Single-Payment Factors (F/P

### 5. Time value of money

1 Simple interest 2 5. Time value of money With simple interest, the amount earned each period is always the same: i = rp o We will review some tools for discounting cash flows. where i = interest earned

### Engineering Economy. Time Value of Money-3

Engineering Economy Time Value of Money-3 Prof. Kwang-Kyu Seo 1 Chapter 2 Time Value of Money Interest: The Cost of Money Economic Equivalence Interest Formulas Single Cash Flows Equal-Payment Series Dealing

### Chapter 2. CASH FLOW Objectives: To calculate the values of cash flows using the standard methods.. To evaluate alternatives and make reasonable

Chapter 2 CASH FLOW Objectives: To calculate the values of cash flows using the standard methods To evaluate alternatives and make reasonable suggestions To simulate mathematical and real content situations

Sample Examination Questions CHAPTER 6 ACCOUNTING AND THE TIME VALUE OF MONEY MULTIPLE CHOICE Conceptual Answer No. Description d 1. Definition of present value. c 2. Understanding compound interest tables.

### Solutions to Problems: Chapter 5

Solutions to Problems: Chapter 5 P5-1. Using a time line LG 1; Basic a, b, and c d. Financial managers rely more on present value than future value because they typically make decisions before the start

### BUSI 121 Foundations of Real Estate Mathematics

Real Estate Division BUSI 121 Foundations of Real Estate Mathematics SESSION 2 By Graham McIntosh Sauder School of Business University of British Columbia Outline Introduction Cash Flow Problems Cash Flow

### FinQuiz Notes 2 0 1 4

Reading 5 The Time Value of Money Money has a time value because a unit of money received today is worth more than a unit of money to be received tomorrow. Interest rates can be interpreted in three ways.

### Chapter 3 Mathematics of Finance

Chapter 3 Mathematics of Finance Section 3 Future Value of an Annuity; Sinking Funds Learning Objectives for Section 3.3 Future Value of an Annuity; Sinking Funds The student will be able to compute the

### DISCOUNTED CASH FLOW VALUATION and MULTIPLE CASH FLOWS

Chapter 5 DISCOUNTED CASH FLOW VALUATION and MULTIPLE CASH FLOWS The basic PV and FV techniques can be extended to handle any number of cash flows. PV with multiple cash flows: Suppose you need \$500 one

### Statistical Models for Forecasting and Planning

Part 5 Statistical Models for Forecasting and Planning Chapter 16 Financial Calculations: Interest, Annuities and NPV chapter 16 Financial Calculations: Interest, Annuities and NPV Outcomes Financial information

### Chapter 4. The Time Value of Money

Chapter 4 The Time Value of Money 1 Learning Outcomes Chapter 4 Identify various types of cash flow patterns Compute the future value and the present value of different cash flow streams Compute the return

### 1. Annuity a sequence of payments, each made at equally spaced time intervals.

Ordinary Annuities (Young: 6.2) In this Lecture: 1. More Terminology 2. Future Value of an Ordinary Annuity 3. The Ordinary Annuity Formula (Optional) 4. Present Value of an Ordinary Annuity More Terminology

### Chapter 21: Savings Models

October 16, 2013 Last time Arithmetic Growth Simple Interest Geometric Growth Compound Interest A limit to Compounding Problems Question: I put \$1,000 dollars in a savings account with 2% nominal interest

### 14 ARITHMETIC OF FINANCE

4 ARITHMETI OF FINANE Introduction Definitions Present Value of a Future Amount Perpetuity - Growing Perpetuity Annuities ompounding Agreement ontinuous ompounding - Lump Sum - Annuity ompounding Magic?

### Engineering Economics Cash Flow

Cash Flow Cash flow is the sum of money recorded as receipts or disbursements in a project s financial records. A cash flow diagram presents the flow of cash as arrows on a time line scaled to the magnitude

### Lesson 4 Annuities: The Mathematics of Regular Payments

Lesson 4 Annuities: The Mathematics of Regular Payments Introduction An annuity is a sequence of equal, periodic payments where each payment receives compound interest. One example of an annuity is a Christmas

### Discounted Cash Flow Valuation

6 Formulas Discounted Cash Flow Valuation McGraw-Hill/Irwin Copyright 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter Outline Future and Present Values of Multiple Cash Flows Valuing

### Chris Leung, Ph.D., CFA, FRM

FNE 215 Financial Planning Chris Leung, Ph.D., CFA, FRM Email: chleung@chuhai.edu.hk Chapter 2 Planning with Personal Financial Statements Chapter Objectives Explain how to create your personal cash flow

### hp calculators HP 20b Time value of money basics The time value of money The time value of money application Special settings

The time value of money The time value of money application Special settings Clearing the time value of money registers Begin / End mode Periods per year Cash flow diagrams and sign conventions Practice

### Accounting Building Business Skills. Interest. Interest. Paul D. Kimmel. Appendix B: Time Value of Money

Accounting Building Business Skills Paul D. Kimmel Appendix B: Time Value of Money PowerPoint presentation by Kate Wynn-Williams University of Otago, Dunedin 2003 John Wiley & Sons Australia, Ltd 1 Interest

### Real estate investment & Appraisal Dr. Ahmed Y. Dashti. Sample Exam Questions

Real estate investment & Appraisal Dr. Ahmed Y. Dashti Sample Exam Questions Problem 3-1 a) Future Value = \$12,000 (FVIF, 9%, 7 years) = \$12,000 (1.82804) = \$21,936 (annual compounding) b) Future Value

### Finance. Simple Interest Formula: I = P rt where I is the interest, P is the principal, r is the rate, and t is the time in years.

MAT 142 College Mathematics Finance Module #FM Terri L. Miller & Elizabeth E. K. Jones revised December 16, 2010 1. Simple Interest Interest is the money earned profit) on a savings account or investment.

### CHAPTER 4 DISCOUNTED CASH FLOW VALUATION

CHAPTER 4 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1. Assuming positive cash flows and interest rates, the future value increases and the present value

### An Appraisal Tool for Valuing Forest Lands

An Appraisal Tool for Valuing Forest Lands By Thomas J. Straka and Steven H. Bullard Abstract Forest and natural resources valuation can present challenging analysis and appraisal problems. Calculations

### Mathematics. Rosella Castellano. Rome, University of Tor Vergata

and Loans Mathematics Rome, University of Tor Vergata and Loans Future Value for Simple Interest Present Value for Simple Interest You deposit E. 1,000, called the principal or present value, into a savings

### Time Value of Money Revisited: Part 1 Terminology. Learning Outcomes. Time Value of Money

Time Value of Money Revisited: Part 1 Terminology Intermediate Accounting II Dr. Chula King 1 Learning Outcomes Definition of Time Value of Money Components of Time Value of Money How to Answer the Question

### Purpose EL-773A HP-10B BA-II PLUS Clear memory 0 n registers

D-How to Use a Financial Calculator* Most personal finance decisions involve calculations of the time value of money. Three methods are used to compute this value: time value of money tables (such as those

### FinQuiz Notes 2 0 1 5

Reading 5 The Time Value of Money Money has a time value because a unit of money received today is worth more than a unit of money to be received tomorrow. Interest rates can be interpreted in three ways.

### TIME VALUE OF MONEY. In following we will introduce one of the most important and powerful concepts you will learn in your study of finance;

In following we will introduce one of the most important and powerful concepts you will learn in your study of finance; the time value of money. It is generally acknowledged that money has a time value.