# Chapter 2 Factors: How Time and Interest Affect Money

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 2 Factors: How Time and Interest Affect Money Session Dr Abdelaziz Berrado 1

2 Topics to Be Covered in Today s Lecture Section 2: How Time and Interest Affect Money Single-Payment Factors (F/P and P/F) Interest Tables Present Value/Future worth of an uneven payment series Uniform-Series Present Worth Factor and Capital Recovery Factor P/A and A/P Factors Sinking Fund Factor and Uniform-Series compound amount factor 2

3 Cash Flow Diagrams - Exercise An engineer wants to deposit an amount P now such that she can withdraw an equal annual amount \$2500 per year for the first 3 years starting one year after the deposit and a different annual withdrawal of \$3000 per year for the following 5 years. How would the cash flow diagram appear if the interest rate were 7% per year? 3

4 Single-payment Factors (F/P and P/F) F/P: To find F given P To find a certain future amount of money F (compound amount), given the present amount of money P, a number of interest periods n, and an interest rate i% F n?. n P 0 4

5 Single-payment Factors (F/P and P/F) P/F: To find P given F To find the present worth P, given a future amount F, a number of interest periods n, and an interest rate i% P/F factor brings a single future sum back to a specific point in time. F n. n P? 5

6 Single-payment Factors (F/P and P/F) F = P(1+i) n and P = F (1/(1+i) n ) (1+i) n : F/P factor, Single Payment Compound Amount Factor (SPCAF), (F/P, i,n) 1/(1+i) n : P/F factor, Single Payment Present Worth Factor (SPPWF), (P/F, i,n) 6

7 Example- F/P Analysis Example: P= \$1,000;n=3;i=10% What is the future value, F? F =?? P=\$1, i=10%/year F 3 = \$1,000[F/P,10%,3] = \$1,000[1.10] 3 = \$1,000[1.3310] = \$1,

8 Example P/F Analysis Assume F = \$100,000, 9 years from now. What is the present worth of this amount now if i =15%? i = 15%/yr F 9 = \$100, P=?? P 0 = \$100,000(P/F, 15%,9) = \$100,000(1/(1.15) 9 ) = \$100,000(0.2843) = \$28,430 at time t = 0 8

9 Interest Tables Formulas can be used for calculating P or F, but interest tables can make some calculations easier Interest tables given at the back of the book Look up P/F (Present Worth), or F/P (Compound Amount) factors for a given interest rate and given number of interest periods When a given interest rate is not given at the back of the book, can use 1) Formulas (exact) 2) Interpolation (approximation) 9

10 Interest Tables Exercise If you had \$2000 now and invested it at 10% how much would it be worth in 8 years? Suppose that \$1000 is to be received in 5 years. At an annual interest rate of 12%, what is the present worth of this amount? 10

11 Present Value/Future worth of an uneven payment series What if payments are not uniform some amount needs to be withdrawn 2 years from now, a different amount 5 years from now? How will you calculate future worth or present value in this case? 11

12 Present Value/Future worth of an uneven payment series exercise Calculate the worth 5 years from now of an investment of \$200 made 2 years from now and an investment of \$100 made 4 years from now 12

13 Present Value/Future worth of an uneven payment series exercise Wilson Technology a growing machine shop, wishes to set aside money now to invest over the next 4 years in automating its customer service department. The company can earn 10% on a lump sum deposited now and wishes to withdraw the money in the following increments 13

14 Present Value of an uneven payment series exercise 1) Year 1: \$25000 to purchase a computer and database software designed for customer service use 2) Year 2: \$3000 to purchase additional hardware to accommodate the anticipated growth in use of the system 3) Year 3: No expenses 4) Year 4: \$5000 to purchase software upgrades How much money should the company set aside now? 14

15 Present Worth of an even payment series exercise What is the present worth of a series of annual receipts of \$100 that you will be receiving for three years starting next year? 15

16 Uniform Series Present Worth Annuity Cash Flow P =?? n-1 n \$A per period 16

17 Uniform Series Present Worth Desire an expression for the present worth P of a stream of equal, end of period cash flows - A P =?? n-1 n A = given 17

18 Uniform Series Present Worth Write a Present worth expression P = A (1 ) (1 ) (1 ) n n + i + i + i (1 + i) [1] Term inside the brackets is a geometric progression. Mult. This equation by 1/(1+i) to yield a second equation 18

19 Uniform Series Present Worth The second equation P = A i (1 + i) (1 + i) (1 + i) (1 + i) 2 3 n n+ 1 [2] To isolate an expression for P in terms of A, subtract Eq [1] from Eq. [2]. Note that numerous terms will drop out. 19

20 Uniform Series Present Worth and Capital Recovery Factors Setting up the subtraction P = A (1 + i) (1 + i) (1 + i) (1 + i) (1 + i) (1 + i) n n+ 1 [2] - P = A (1 ) (1 ) (1 ) n n + i + i + i (1 + i) [1] i P = A 1 1 = [3] n i (1 + i) (1 + i) 20

21 Uniform Series Present Worth and Capital Recovery Factors Simplifying Eq. [3] further i P 1 1 = A n i (1 + i) (1 + i) P A 1 = 1 1 i (1 i) n + + n (1 + i) 1 P = A for i n i(1 + i) 0 21

22 Capital Recovery Factor (A/P) A/P: To find A given P Given an amount P now, what is the annual A worth over n years when the interest rate is i P is known n-1 n \$A??? 22

23 P/A versus A/P P/A factor (P/A,i,n) Uniform Series Present Worth Factor P n (1 + i) 1 = A, i n i(1 + i) 0 n (1 + i) 1 n i(1 + i) A/P factor (A/P,i,n) Capital Recovery Factor A n i(1 + i) = P, i n (1 + i) 1 0 n i(1 + i) n (1 + i) 1 23

24 P/A example How much money should you be willing to pay now for a guaranteed \$600 per year for 9 years starting next year, at a rate of return of 16% per year? (Answer using calculation and interest tables) 24

25 A/P Example Suppose you are entitled a payment of 30,000 a year for the next 9 years starting next year. What is the present worth of this annual payment to you? (Answer using calculation and interest tables) 25

26 More exercises BioGen, a small biotechnology firm, has borrowed \$250,000 to purchase laboratory equipment for gene splicing. The loan carries an interest rate of 8% per year and is to be repaid in equal annual installments over the next 6 years. Compute the amount of this annual installment. 26

27 More exercises Suppose you are entitled a payment of 30,000 a year for the next 9 years starting next year from an investment firm. What is the present worth of this annual payment to you? 27

28 Sinking Fund Factor and Uniform-Series Compound Amount Factor Annuity Cash Flow \$F 0.. N \$A per period Find \$A given the Future amt. - \$F 28

29 Sinking Fund and Series Compound amount factors (A/F and F/A) Take advantage of what we already have Recall: Also: P 1 = F (1 + i) n Substitute P and simplify! A n i(1 + i) = P n (1 + i) 1 29

30 F/A and A/F Derivations \$F Annuity Cash Flow 0.. N \$A per period Find \$F given the \$A amounts 30

31 Example 2.6 How much money must Carol deposit every year starting, l year from now at 5.5% per year in order to accumulate \$6000 seven years from now? 31

32 Solution Example 2.6 The cash How diagram from Carol's perspective fits the A/F factor. A= \$6000 (A/F,5.5%,7) = 6000( ) = \$ per year The A/F factor Value 0f was computed using the A/F factor formula 32

33 Assignments Assignments due at the beginning of next class: Read all the textbooks examples from sections 2.1, 2.2, 2.3 Read sections 2.4, 2.5, 2.6 Hw1 due at the beginning of class one week from today. 33

34 Topics to Be Covered in Today s Lecture Section 2: How Time and Interest Affect Money Interpolation in interest tables Arithmetic Gradient Factors Geometric Gradient Factors 34

35 Arithmetic Gradient Factors In applications, the annuity cash flow pattern is not the only type of pattern encountered Two other types of end of period patterns are common The Linear or arithmetic gradient The geometric (% per period) gradient This section presents the Arithmetic Gradient 35

36 Arithmetic Gradient Factors An arithmetic (linear) Gradient is a cash flow series that either increases or decreases by a constant amount over n time periods. A linear gradient is always comprised of TWO components: The Gradient component The base annuity component We need an expression for the Present Worth of an arithmetic gradient 36

37 Linear Gradient ExampleA 1 +(n-1)g Assume the following: A 1 +(n-2)g A 1 +2G A 1 +G n-1 n This represents a positive, increasing arithmetic gradient 37

38 Example: Linear Gradient Typical Negative, Increasing Gradient: G=\$50 The Base Annuity = \$

39 Example: Linear Gradient Desire to find the Present Worth of this cash flow The Base Annuity = \$

40 Arithmetic Gradient Factors The G amount is the constant arithmetic change from one time period to the next. The G amount may be positive or negative! The present worth point is always one time period to the left of the first cash flow in the series or, Two periods to the left of the first gradient cash flow! 40

41 Derivation: Gradient Component Only Focus Only on the gradient Component (n-1)g 0 G (n-2)g +2G G Removed Base annuity n-1 N 41

42 Present Worth Point \$100 \$200 \$300 \$400 \$500 \$600 \$700 X The Present Worth Point of the Gradient 42

43 Gradient Component The Gradient Component \$400 \$300 \$200 \$100 \$0 \$500 \$600 X The Present Worth Point of the Gradient 43

44 Present Worth Point PW of the Base Annuity is at t = 0 Base Annuity A = \$100 X The Present Worth Point of the Gradient PWBASE Annuity=\$100(P/A,i%,7) 44

45 Present Worth: Linear Gradient The present worth of a linear gradient is the present worth of the two components: 1. The Present Worth of the Gradient Component and, 2. The Present Worth of the Base Annuity flow Requires 2 separate calculations! 45

46 Present Worth: Gradient Component The PW of the Base Annuity is simply the Base Annuity A(P/A, i%, n) factor What is needed is a present worth expression for the gradient component cash flow. We need to derive a closed form expression for the gradient component! 46

47 Present Worth: Gradient Component General CF Diagram Gradient Part Only 1G 2G 3G (n-2)g (n-1)g 0G We want the PW at time t = 0 (2 periods to the left of 1G) n-1 n 47

48 To Begin- Derivation of (P/G,i%,n) P = G( P / F, i%, 2) + 2 G( P / F, i%,3) +... [(n-2)g](p/f,i%,n-1) + [(n-1)g](p/f,i%,n) Next Step: Factor out G and re-write as.. 48

49 Factoring G out. P/G factor P=G ( P/F,i%,2)+2( P/F,i%, 3) +...( n-1)( P/F,i%,n) { } What is inside of the { } s? 49

50 Replace (P/F s) with closed-form 1 2 n-2 n-1 P=G (1+i) (1+i) (1+i) (1+i) 2 3 n-1 n [1] Multiply both sides by (1+i) 50

51 Mult. Both Sides By (n+1) n-2 n-1 (1+i) (1+i) (1+i) (1+i) [2] 1 P(1+i) =G n-2 n-1 We have 2 equations [1] and [2]. Next, subtract [1] from [2] and work with the resultant equation. 51

52 Subtracting [1] from [2] n-2 n-1 (1+i) (1+i) (1+i) (1+i) 1 P(1+i) =G n-2 n n-2 n-1 P=G (1+i) (1+i) (1+i) (1+i) 2 3 n-1 n P n G (1 + i) 1 n = i i(1 i) n (1 i) n

53 The P/G factor for i and N P n G (1 + i) 1 n = i i(1 i) n (1 i) n + + ( P / G, i%, N) factor 53

54 Further Simplification on P/G ( P / G, i%, N) = (1 + i) in 1 i 2 N (1 + i) N Remember, the present worth point of any linear gradient is 2 periods to the left of the 1-G cash flow or, 1 period to the left of the 0-G cash flow. P=G(P/G,i,n) 54

55 Gradient Example Consider the following cash flow \$500 \$400 \$300 \$200 \$ Present Worth Point is here! And the G amt. = \$100/period Find the present worth if i = 10%/yr; n = 5 yrs 55

56 Gradient Example- Base Annuity First, The Base Annuity of \$100/period A = +\$ PW(10%) of the base annuity = \$100(P/A,10%,5) PW Base = \$100(3.7908)= \$ Not Finished: We need the PW of the gradient component and then add that value to the \$ amount 56

57 Focus on the Gradient Component \$0 \$100 \$200 \$300 \$ We desire the PW of the Gradient Component at t = 0 P = G( P/G,10%,5 ) = \$100( P/G,10%,5 ) 57

58 The Set Up \$0 \$100 \$200 \$300 \$ P = G(P/G,10%,5) = \$100(P/G,10%,5) N G (1 + i) 1 N P= i (1 ) N i + i (1 + i ) N Could substitute n=5, i=10% and G = \$100 into the P/G closed form to get the value of the factor. 58

59 PW of the Gradient Component P = G(P/G,10%,5) = \$100(P/G,10%,5) P/G,10%,5) Sub. G=\$100;i=0.10;n=5 N G (1 + i) 1 N P= i (1 ) N i + i (1 + i ) N Calculating or looking up the (P/G,10%,5) factor yields the following: P t=0 = \$100(6.8618) = \$ for the gradient PW 59

60 Gradient Example: Final Result PW(10%) Base Annuity = \$ PW(10%) Gradient Component = \$ Total PW(10%) = \$ \$ Equals \$ Note: The two sums occur at t =0 and can be added together concept of equivalence 60

61 Example Summarized This Cash Flow \$100 \$200 \$300 \$400 \$ Is equivalent to \$ at time 0 if the interest rate is 10% per year! 61

62 Shifted Gradient Example: i = 10% Consider the following Cash Flow \$600 \$550 \$500 \$ This is a shifted negative, decreasing gradient. 2. The PW point in time is at t = 3 (not t = o) 62

63 Shifted Gradient Example Consider the following Cash Flow \$600 \$550 \$500 \$450 The t = 0 requires getting the t =3; Then using the P/F factor move PW 3 back to t=0 63

64 Shifted Gradient Example Consider the following Cash Flow \$600 \$550 \$500 \$450 The base annuity is a \$600 cash flow for 3 time periods 64

65 Shifted Gradient Example: Base Annuity PW of the Base Annuity: 2 Steps P 3 =-600(P/A,10%,4) P 0 =P 3 (P/F,10%,3) P 3 P 0 A = -\$ P 0 = [-600(P/A,10%,4)](P/F,10%,3) P 0-base annuity = -\$

66 Shifted Gradient Example: Gradient PW of Gradient Component: G = -\$ P 3-Grad = +50(P/G,10%,4) P 0 =P 3 (P/F,10%,3) P 3 P 0 0G 1G 2G 3G P 0-grad = {+50(P/G,10%,4)}(P/F,10%,3) =-\$

67 Extension The A/G factor A/G converts a linear gradient to an equivalent annuity cash flow. Remember, at this point one is only working with gradient component There still remains the annuity component that you must also handle separately! 67

68 The A/G Factor Convert G to an equivalent A A = G( P / G, i, n)( A/ P, i, n) How to do it 68

69 A/G factor using A/P with P/G P n G (1 + i) 1 n = n n i i(1 + i) (1 + i) n i(1 + i) n (1 + i) 1 (A/P,i,n) The results follow.. 69

70 Resultant A/G factor P n G (1 + i) 1 n = n n i i(1 + i) (1 + i) n i(1 + i) n (1 + i) 1 (A/P,i,n) (A/G,i,n) = 1 n A = G i (1 i ) n

71 Terminology Let P A be the present worth of the base annual amount A A (which is a series of equal end-of-period cash flows). Let P G be the present worth of the gradient cash flow Let P T be the total present worth, P T = P A + P G if the gradient is positive P T = P A - P G if the gradient is negative Let A T be the total annual worth for n years of the series of gradient cash flows including the base annual amount A A 71

72 P G 1 (1 + i) 1 n = G i i(1 + i) n (1 + i) n n n 1 (1 + i) 1 n n i i(1 + i) (1 + i) n (P/G,i,n) factor P/G factor Arithmetic Gradient Present Worth Factor 72

73 Arithmetic Gradients P T = P A + P G for positive gradients and P T = P A - P G for negative gradients So, how would you find P A? Given that you have the P T how would you find A T? 73

74 Assignments Assignments due at the beginning of next class: Read all the textbooks examples from sections 2.4, 2.5 Read sections 2.6, 2.7,

75 Topics to Be Covered in Today s Lecture Section 2: How Time and Interest Affect Money The A/G factor Geometric Gradient Factors Determination of an unknown interest rate Determination of an unknown number of periods 75

76 Extension The A/G factor A/G converts a linear gradient to an equivalent annuity cash flow. Remember, at this point one is only working with gradient component There still remains the annuity component that you must also handle separately! 76

77 The A/G Factor Convert G to an equivalent A A = G( P / G, i, n)( A/ P, i, n) How to do it 77

78 A/G factor using A/P with P/G P A= n G (1 + i) 1 n = n n i i(1 + i) (1 + i) n i(1 + i) n (1 + i) 1 (A/P,i,n) The results follow.. 78

79 Resultant A/G factor P A= n G (1 + i) 1 n = n n i i(1 + i) (1 + i) n i(1 + i) n (1 + i) 1 (A/P,i,n) (A/G,i,n) = 1 n A = G i (1 i ) n

80 Terminology Let P A be the present worth of the base annual amount A A (which is a series of equal end-of-period cash flows). Let P G be the present worth of the arithmetic gradient cash flow Let P T be the total present worth, P T = P A + P G if the gradient is positive P T = P A - P G if the gradient is negative Let A T be the total annual worth for n years of the series of gradient cash flows including the base annual amount A A 80

81 P G 1 (1 + i) 1 n = G i i(1 + i) n (1 + i) n n n 1 (1 + i) 1 n n i i(1 + i) (1 + i) n (P/G,i,n) factor P/G factor Arithmetic Gradient Present Worth Factor 81

82 Arithmetic Gradients P T = P A + P G for positive gradients and P T = P A - P G for negative gradients So, how would you find P A? Given that you have the P T how would you find A T? 82

83 Geometric Gradients An arithmetic (linear) gradient changes by a fixed dollar amount each time period. A GEOMETRIC gradient changes by a fixed percentage each time period. We define a UNIFORM RATE OF CHANGE (%) for each time period Define g as the constant rate of change in decimal form by which amounts increase or decrease from one period to the next 83

84 Geometric Gradients: Increasing Typical Geometric Gradient Profile Let A 1 = the first cash flow in the series n-1 n A 1 A1 (1+g) A 1 (1+g) 2 A 1 (1+g) 3 A 1 (1+g) n-1 84

85 Geometric Gradients: Decreasing Typical Geometric Gradient Profile Let A 1 = the first cash flow in the series n-1 n A 1 (1-g) 2 A 1 (1-g) 3 A 1 (1-g) n-1 A 1 (1-g) A 1 85

86 Geometric Gradients: Derivation First Major Point to Remember: A 1 does NOT define a Base Annuity; There is no BASE ANNUITY for a Geometric Gradient! The objective is to determine the Present Worth one period to the left of the A 1 cash flow point in time Remember: The PW point in time is one period to the left of the first cash flow A 1! 86

87 Geometric Gradients: Derivation For a Geometric Gradient the following parameters are required: The interest rate per period i The constant rate of change g No. of time periods n The starting cash flow A 1 87

88 Geometric Gradients: Starting P g = The A j s time the respective (P/F,i,j) factor Write a general present worth relationship to find P g. P g A A (1 + g) A (1 + g) A (1 + g) = (1 + i) (1 + i) (1 + i) (1 + i) 2 n n Now, factor out the A1 value and rewrite as.. 88

89 Geometric Gradients P g 1 (1 + g) (1 + g) (1 + g) = A (1 + i) (1 + i) (1 + i) (1 + i) 1 2 n 1 n (1) (1+g) Multuply both sides by to create another equation (1+i) P g (1+g) (1+g) 1 (1 + g) (1 + g) (1 + g) = A (1+i) (1+i) (1 + i) (1 + i) (1 + i) (1 + i) 1 2 n 1 n (2) Subtract (1) from (2) and the result is.. 89

90 90 Geometric Gradients g (1 ) i (1 ) 1 n g n g P A i i + + = + + Solve for P g and simplify to yield g i n g g i P A i g + + =

91 91 Geometric Gradient P/A factor This is the (P/A,g,i,n) factor and is valid if g is not equal to i g i n g g i P A i g + + =

92 Geometric Gradient P/A factor Note: If g = i we have a division by 0 undefined. For g = i we can derive the closed form PW factor for this special case. We substitute i for g into the P g relationship to yield: 92

93 Geometric Gradient: i = g Case P g =A (1+i) (1+i) (1+i) (1+i) P g = na 1 (1 + i) For the case i = g 93

94 94 Geometric Gradients: Summary P g = A 1 (P/A,g,i,n) 1 (1 ) g na P i = g i n g g i P A i g + + = g not = to i Case: g = i

95 Geometric Gradient: Notes The geometric gradient requires knowledge of: A 1, i, n, and g There exist an infinite number of combinations for i, n, and g: Hence one will not find tabulated tables for the (P/A, g,i,n) factor. 95

96 Geometric Gradient: Notes You have to calculated either from the closed form for each problem or apply a pre-programmed spreadsheet model to find the needed factor value No spreadsheet built-in function for this factor! 96

97 Geometric Gradient: Example Assume maintenance costs for a particular activity will be \$1700 one year from now. Assume an annual increase of 11% per year over a 6-year time period. 97

98 Geometric Gradient: Example If the interest rate is 8% per year, determine the present worth of the future expenses at time t = 0. First, draw a cash flow diagram to represent the model. 98

99 Geometric Gradient Example (+g) g = +11% per period; A1 = \$1700; i = 8%/yr \$1700 \$1700(1.11) 1 \$1700(1.11) 2 \$1700(1.11) 3 PW(8%) =?? \$1700(1.11) 6 99

100 Solution P = \$1700(P/A,11%,8%,7) Need to calculate the P/A factor from the closed-form expression for a geometric gradient. \$ P g n 1+ g 1 1+ i = A 1 g i g i 100

101 Geometric Gradient ( -g ) Consider the following problem with a negative growth rate g. A 1 = \$1000 \$900 \$810 \$ P 0 =?? g = -10%/yr; i = 8%; n = 4 We simply apply a g value =

102 Geometric Gradient (-g value) Evaluate: For a negative g value = =\$ P g n 1 + g i = A 1 g i i g 102

103 Determination of an unknown interest rate and number of periods Interest Rate Unknown Given: P, F,n Determine i Number of periods unknown Given: P, F, i Determine n Use formulas to calculate 103

104 Determination of an unknown interest rate exercise If Laurel can make an investment in a friend s business of \$3000 now in order to receive \$5000 five years from now, determine the rate of return. If Laurel can receive 7% per year interest on a certificate of deposit, which investment should be made? 104

105 Determination of an unknown number of interest periods exercise How long will it take for \$1000 to double if the interest rate is 5% per year? Calculate the exact value. How different are the estimates if the rate is 25%? 105

106 Assignments and Announcements Assignments due at the beginning of next class: Read all the textbook examples from sections 2.6, 2.7, 2.8 Read chapter 3 We have a quiz next time 106

### CE 314 Engineering Economy. Interest Formulas

METHODS OF COMPUTING INTEREST CE 314 Engineering Economy Interest Formulas 1) SIMPLE INTEREST - Interest is computed using the principal only. Only applicable to bonds and savings accounts. 2) COMPOUND

### 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 17 ENGINEERING COST ANALYSIS

CHAPTER 17 ENGINEERING COST ANALYSIS Charles V. Higbee Geo-Heat Center Klamath Falls, OR 97601 17.1 INTRODUCTION In the early 1970s, life cycle costing (LCC) was adopted by the federal government. LCC

### PRESENT VALUE ANALYSIS. Time value of money equal dollar amounts have different values at different points in time.

PRESENT VALUE ANALYSIS Time value of money equal dollar amounts have different values at different points in time. Present value analysis tool to convert CFs at different points in time to comparable values

### COURSE SUMMARY CASH FLOW \$4500

COURSE SUMMARY This chapter is a brief review of engineering economic analysis/engineering economy. The goal is to give you a better grasp of the major topics in a typical first course. Hopefully, this

### 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

### Chapter 3 Equivalence A Factor Approach

Chapter 3 Equivalence A Factor Approach 3-1 If you had \$1,000 now and invested it at 6%, how much would it be worth 12 years from now? F = 1,000(F/P, 6%, 12) = \$2,012.00 3-2 Mr. Ray deposited \$200,000

### 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

### Chapter 4: Nominal and Effective Interest Rates

Chapter 4: Nominal and Effective Interest Rates Session 9-10-11 Dr Abdelaziz Berrado 1 Topics to Be Covered in Today s Lecture Section 4.1: Nominal and Effective Interest Rates statements Section 4.2:

### Cash Flow and Equivalence

Cash Flow and Equivalence 1. Cash Flow... 51-1 2. Time Value of Money... 51-2 3. Discount Factors and Equivalence... 51-2 4. Functional Notation... 51-5 5. Nonannual Compounding... 51-5 6. Continuous Compounding...

### ISE 2014 Chapter 3 Section 3.11 Deferred Annuities

ISE 2014 Chapter 3 Section 3.11 Deferred Annuities If we are looking for a present (future) equivalent sum at time other than one period prior to the first cash flow in the series (coincident with the

### 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

### 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

### 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

### 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

### CONSTRUCTION EQUIPMENT

Equipment is a critical resource in the execution of most construction projects. The equipment fleet may represent the largest long-term capital investment in many construction companies. Consequently,

### 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

### The Time Value of Money (contd.)

The Time Value of Money (contd.) February 11, 2004 Time Value Equivalence Factors (Discrete compounding, discrete payments) Factor Name Factor Notation Formula Cash Flow Diagram Future worth factor (compound

### CHAPTER 4 MORE INTEREST FORMULAS

CHAPTER 4 MORE INTEREST FORMULAS After Completing This Chapter the student should be able. Solve problems modeled by Uniform Series (very common) Use arithmetic and geometric gradients Apply nominal and

### 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

### 9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

### 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

### COMPOUND INTEREST AND ANNUITY TABLES

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

### 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

### 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

### 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

### 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

### Example 1 - Solution. Since the problém is of the form "find F when given P" the formula to use is F = P(F/P, 8%, 5) = \$10,000(1.4693) = \$14,693.

Example 1 Ms. Smith loans Mr. Brown \$10,000 with interest compounded at a rate of 8% per year. How much will Mr. Brown owe Ms. Smith if he repays the loan at the end of 5 years? Example 1 - Solution Since

### Chapter 5 Present Worth

Chapter 5 Present Worth 5-1 Emma and her husband decide they will buy \$1,000 worth of utility stocks beginning one year from now. Since they expect their salaries to increase, they will increase their

### 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

### 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

### ENGINEERING ECONOMICS AND FINANCE

CHAPTER Risk Analysis in Engineering and Economics ENGINEERING ECONOMICS AND FINANCE A. J. Clark School of Engineering Department of Civil and Environmental Engineering 6a CHAPMAN HALL/CRC Risk Analysis

### Chapter 2 Time value of money

Chapter 2 Time value of money Interest: the cost of money Economic equivalence Interest formulas single cash flows Equal-payment series Dealing with gradient series Composite cash flows. Power-Ball Lottery

### 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

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

CHAPTER 2 Time Value of Money 2-1 Time Value of Money (TVM) Time Lines Future value & Present value Rates of return Annuities & Perpetuities Uneven cash Flow Streams Amortization 2-2 Time lines 0 1 2 3

### 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

### 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

### Ordinary Annuities Chapter 10

Ordinary Annuities Chapter 10 Learning Objectives After completing this chapter, you will be able to: > Define and distinguish between ordinary simple annuities and ordinary general annuities. > Calculate

### 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.

### 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

### 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

### 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

### Geometric Series and Annuities

Geometric Series and Annuities Our goal here is to calculate annuities. For example, how much money do you need to have saved for retirement so that you can withdraw a fixed amount of money each year for

### 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

### Time Value of Money (TVM) A dollar today is more valuable than a dollar sometime in the future...

Lecture: II 1 Time Value of Money (TVM) A dollar today is more valuable than a dollar sometime in the future...! The intuitive basis for present value what determines the effect of timing on the value

### Texas Instruments BAII Plus Tutorial for Use with Fundamentals 11/e and Concise 5/e

Texas Instruments BAII Plus Tutorial for Use with Fundamentals 11/e and Concise 5/e This tutorial was developed for use with Brigham and Houston s Fundamentals of Financial Management, 11/e and Concise,

### 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

### 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

### 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

### EXAM 2 OVERVIEW. Binay Adhikari

EXAM 2 OVERVIEW Binay Adhikari FEDERAL RESERVE & MARKET ACTIVITY (BS38) Definition 4.1 Discount Rate The discount rate is the periodic percentage return subtracted from the future cash flow for computing

### The Time Value of Money

C H A P T E R6 The Time Value of Money When plumbers or carpenters tackle a job, they begin by opening their toolboxes, which hold a variety of specialized tools to help them perform their jobs. The financial

### Chapter F: Finance. Section F.1-F.4

Chapter F: Finance Section F.1-F.4 F.1 Simple Interest Suppose a sum of money P, called the principal or present value, is invested for t years at an annual simple interest rate of r, where r is given

### 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

### 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

### GROWING ANNUITIES by Albert L. Auxier and John M. Wachowicz, Jr. Associate Professor and Professor, The University of Tennessee

GROWING ANNUITIES by Albert L. Auxier and John M. Wachowicz, Jr. Associate Professor and Professor, The University of Tennessee An article in the Journal of Financial Education by Richard Taylor [1] provided

### Key Concepts and Skills. Multiple Cash Flows Future Value Example 6.1. Chapter Outline. Multiple Cash Flows Example 2 Continued

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

### 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

### Chapter 6 Equivalent Annual Worth

Chapter 6 Equivalent Annual Worth 6-1 Deere Construction just purchased a new track hoe attachment costing \$12,500. The CFO, John, expects the implement will be used for five years when it is estimated

### Finance Unit 8. Success Criteria. 1 U n i t 8 11U Date: Name: Tentative TEST date

1 U n i t 8 11U Date: Name: Finance Unit 8 Tentative TEST date Big idea/learning Goals In this unit you will study the applications of linear and exponential relations within financing. You will understand

### Discounted Cash Flow Valuation

Discounted Cash Flow Valuation Chapter 5 Key Concepts and Skills Be able to compute the future value of multiple cash flows Be able to compute the present value of multiple cash flows Be able to compute

### Time Value of Money. Nature of Interest. appendix. study objectives

2918T_appC_C01-C20.qxd 8/28/08 9:57 PM Page C-1 appendix C Time Value of Money study objectives After studying this appendix, you should be able to: 1 Distinguish between simple and compound interest.

### 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

### 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

### 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 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

### Compounding Quarterly, Monthly, and Daily

126 Compounding Quarterly, Monthly, and Daily So far, you have been compounding interest annually, which means the interest is added once per year. However, you will want to add the interest quarterly,

### 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

### Annuities and Sinking Funds

Annuities and Sinking Funds Sinking Fund A sinking fund is an account earning compound interest into which you make periodic deposits. Suppose that the account has an annual interest rate of compounded

### 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

### Chapter 7 Internal Rate of Return

Chapter 7 Internal Rate of Return 7-1 Andrew T. invested \$15,000 in a high yield account. At the end of 30 years he closed the account and received \$539,250. Compute the effective interest rate he received

### Solutions to Supplementary Questions for HP Chapter 5 and Sections 1 and 2 of the Supplementary Material. i = 0.75 1 for six months.

Solutions to Supplementary Questions for HP Chapter 5 and Sections 1 and 2 of the Supplementary Material 1. a) Let P be the recommended retail price of the toy. Then the retailer may purchase the toy at

### Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

FE/EI/EIT REVIEW ENGINEERING ECONOMICS Prof. Adedeji B. Badiru CASH FLOW ANALYSIS The basic reason for performing economic analysis is to make a choice between mutually exclusive projects that are competing

### Lesson 1. Key Financial Concepts INTRODUCTION

Key Financial Concepts INTRODUCTION Welcome to Financial Management! One of the most important components of every business operation is financial decision making. Business decisions at all levels have

### Reducing balance loans

Reducing balance loans 5 VCEcoverage Area of study Units 3 & 4 Business related mathematics In this chapter 5A Loan schedules 5B The annuities formula 5C Number of repayments 5D Effects of changing the

### 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

### Tools for Project Evaluation. Nathaniel Osgood 1.040/1.401J 2/11/2004

Tools for Project Evaluation Nathaniel Osgood 1.040/1.401J 2/11/2004 Basic Compounding Suppose we invest \$x in a bank offering interest rate i If interest is compounded annually, asset will be worth \$x(1+i)

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

CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY Answers to Concepts Review and Critical Thinking Questions 1. The four parts are the present value (PV), the future value (FV), the discount

### 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.

### 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

### 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

### 2.1 The Present Value of an Annuity

2.1 The Present Value of an Annuity One example of a fixed annuity is an agreement to pay someone a fixed amount x for N periods (commonly months or years), e.g. a fixed pension It is assumed that the

### 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 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.

### 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

### 5.1 Simple and Compound Interest

5.1 Simple and Compound Interest Question 1: What is simple interest? Question 2: What is compound interest? Question 3: What is an effective interest rate? Question 4: What is continuous compound interest?

### 3. Time value of money. We will review some tools for discounting cash flows.

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

### The Time Value of Money

CHAPTER 7 The Time Value of Money After studying this chapter, you should be able to: 1. Explain the concept of the time value of money. 2. Calculate the present value and future value of a stream of cash

### CALCULATOR TUTORIAL. Because most students that use Understanding Healthcare Financial Management will be conducting time

CALCULATOR TUTORIAL INTRODUCTION Because most students that use Understanding Healthcare Financial Management will be conducting time value analyses on spreadsheets, most of the text discussion focuses

### IB Maths SL Sequence and Series Practice Problems Mr. W Name

IB Maths SL Sequence and Series Practice Problems Mr. W Name Remember to show all necessary reasoning! Separate paper is probably best. 3b 3d is optional! 1. In an arithmetic sequence, u 1 = and u 3 =

### 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

### The Time Value of Money

The Time Value of Money Time Value Terminology 0 1 2 3 4 PV FV Future value (FV) is the amount an investment is worth after one or more periods. Present value (PV) is the current value of one or more future

### Math 120 Basic finance percent problems from prior courses (amount = % X base)

Math 120 Basic finance percent problems from prior courses (amount = % X base) 1) Given a sales tax rate of 8%, a) find the tax on an item priced at \$250, b) find the total amount due (which includes both

### Bond Price Arithmetic

1 Bond Price Arithmetic The purpose of this chapter is: To review the basics of the time value of money. This involves reviewing discounting guaranteed future cash flows at annual, semiannual and continuously

### Check off these skills when you feel that you have mastered them.

Chapter Objectives Check off these skills when you feel that you have mastered them. Know the basic loan terms principal and interest. Be able to solve the simple interest formula to find the amount of

### Corporate Finance Fundamentals [FN1]

Page 1 of 32 Foundation review Introduction Throughout FN1, you encounter important techniques and concepts that you learned in previous courses in the CGA program of professional studies. The purpose

### 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