TIME VALUE OF MONEY (TVM) IEG2H2-w2 1

Transcription

1 TIME VALUE OF MONEY (TVM) IEG2H2-w2 1

2 After studying TVM, you should be able to: 1. Understand what is meant by "the time value of money." 2. Understand the relationship between present and future value. 3. Describe how the interest rate can be used to adjust the value of cash flows both forward and backward to a single point in time. 4. Calculate both the future and present value of: (a) an amount invested today; (b) a stream of equal cash flows (an annuity); and (c) a stream of mixed cash flows. 5. Use interest factor tables and understand how they provide a shortcut to calculating present and future values. 6. Use interest factor tables to find an unknown interest rate or growth rate when the number of time periods and future and present values are known. IEG2H2-w2 2

3 The Time Value of Money The Interest Rate Simple Interest Compound Interest Compounding More Than Once peryear IEG2H2-w2 3

4 The Interest Rate Which would you prefer -- $10,000 today or $10,000 in 5 years? Obviously, $10,000 today. You already recognize that there is TIME VALUE TO MONEY!! IEG2H2-w2 4

5 Why TIME? Why is TIME such an important element in your decision? TIME allows you the opportunity to postpone consumption and earn INTEREST. IEG2H2-w2 5

6 Types of Interest Simple Interest Interest paid (earned) on only the original amount, or principal, borrowed (lent). Compound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). IEG2H2-w2 6

7 Simple Interest Formula Formula SI = P 0 (i)(n) SI: Simple Interest P 0 : Deposit today (t=0) i: Interest Rate per Period n: Number of Time Periods IEG2H2-w2 7

8 Simple Interest Example Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year? SI = P 0 (i)(n) = $1,000(.07)(2) = $140 IEG2H2-w2 8

9 Simple Interest (FV) What is the Future Value (FV) of the deposit? F = P 0 + SI = $1,000 + $140 = $1,140 Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate. IEG2H2-w2 9

10 Simple Interest (PV) What is the Present Value (PV) of the previous problem? The Present Value is simply the $1,000 you originally deposited. That is the value today! Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. IEG2H2-w2 10

11 Compound interest reflects both the remaining principal and any accumulated interest. For $1,000 at 10% Period (1) Amount owed at beginning of period (2)=(1)x10% Interest amount for period (3)=(1)+(2) Amount owed at end of period 1 $1,000 $100 $1,100 2 $1,100 $110 $1,210 3 $1,210 $121 $1,331 Compound interest is commonly used in personal and professional financial transactions. IEG2H2-TVM 11

12 Why Compound Interest? Future Value of a Single $1,000 Deposit Future Value (U.S. Dollars) st Year 10th Year 20th Year 30th Year 10% Simple Interest 7% Compound Interest 10% Compound Interest

13 Future Value Single Deposit (Graphic) Assume that you deposit $1,000 at a compound interest rate of 7% for 2 years % 7% $1,000 FV 2 IEG2H2-w2 13

14 F 1 = P 0 (1+i) 1 = $1,000 (1.07) = $1,070 Compound Interest You earned $70 interest on your $1,000 deposit over the first year. This is the same amount of interest you would earn under simple interest. IEG2H2-w2 14

15 F 1 = P 0 (1+i) 1 = $1,000 (1.07) = $1,070 F 2 = F 1 (1+i) 1 = P 0 (1+i)(1+i) = $1,000(1.07)(1.07) = P 0 (1+i) 2 = $1,000(1.07) 2 = $1, You earned an EXTRA $4.90 inyear 2 with compound over simple interest. IEG2H2-w2 15

16 General Future Value Formula F 1 = P 0 (1+i) 1 F 2 = P 0 (1+i) 2 General Future Value Formula: or F n F n = P 0 (1+i) n = P 0 (F/P, i, n) -- SeeTable IEG2H2-w2 16

17 Valuation Using Table (F/P,i,n ) isfound on Table at the end of the book. Period 6% 7% 8%

18 Using Future Value Tables F 2 = $1,000 * (F/P, 7%, 2)) = $1,000 (1.145) = $1,145 [Due to Rounding] Period 6% 7% 8%

19 Story Problem Example John wants to know how large his deposit of $10,000 today will become at a compound annual interest rate of 10% for 5 years. IEG2H2-w2 19

20 Story Problem Solution Calculation based on general formula: F n = P 0 (1+i) n F 5 = $10,000 ( ) 5 = $16, Calculation based on Table: F 5 = $10,000 (F/P, 7%, 5) = $10,000 (1.611) = $16,110 IEG2H2-w2 20

21 Double Your Money!!! Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)? We will use the Rule-of-72. IEG2H2-w2 21

22 The Rule-of-72 Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)? Approx. Years to Double = 72 / i% 72 / 12 = 6 Years [Actual Time is 6.12Years] IEG2H2-w2 22

23 Present Value Single Deposit (Graphic) Assume that you need $1,000 in 2 years. Let s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually. IEG2H2-w2 23

24 Present Value Single Deposit (Formula) P 0 = F 2 / (1+i) 2 = $1,000 / (1.07) 2 = F 2 / (1+i) 2 = $ IEG2H2-w2 24

25 General Present Value Formula P 0 = F 1 / (1+i) 1 P 0 = F 2 / (1+i) 2 General Present Value Formula: or etc. P 0 = F n / (1+i) n P 0 = FV n (P/F, i, n) -- SeeTable IEG2H2-w2 25

26 Valuation Using Table (P/F, i, n ) is found on Table at the end of the book. Period 6% 7% 8%

27 Using Present Value Tables P 2 = $1,000 *(P/F, 7%, 2)) = $1,000 *.(.873) = $873 $873 [Due to Rounding] Period 6% 7% 8%

28 Story Problem Example Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000 in 5 years at a discount rate of 10%. IEG2H2-w2 28

29 Story Problem Solution Calculation based on general formula: P 0 = F n / (1+i) n P 0 = $10,000 / ( ) 5 = $6, Calculation based on Table I: P 0 = $10,000 (P/F, 10%, 5) = $10,000 (.621) = $6, IEG2H2-w2 29

30 Types of Annuities An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods. Ordinary Annuity: Payments or receipts occur at the end of each period. Annuity Due: Payments or receipts occur at the beginning of each period. IEG2H2-w2 30

31 Examples of Annuities Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings IEG2H2-w2 31

32 Parts of an Annuity IEG2H2-w2 32

33 Parts of an Annuity IEG2H2-w2 33

34 Overview of an Ordinary Annuity - FVA n n+1 R = Periodic Cash Flow Cash flows occur at the end of the period i%... R R R FVA n FVA n = R(1+i) n-1 + R(1+i) n R(1+i) 1 + R(1+i) 0 IEG2H2-w2 34

35 There are interest factors for a series of end-of-period cash flows. How much will you have in 40 years if you save $3,000 each year and your account earns 8% interest each year? IEG2H2-TVM 35

36 Finding the present amount from a series of end-of-period cash flows. How much would is needed today to provide an annual amount of $50,000 each year for 20 years, at 9% interest each year? IEG2H2-TVM 36

37 Finding A when given F. How much would you need to set aside each year for 25 years, at 10% interest, to have accumulated $1,000,000 at the end of the 25 years? IEG2H2-TVM 37

38 Finding A when given P. If you had $500,000 today in an account earning 10% each year, how much could you withdraw each year for 25 years? IEG2H2-TVM 38

39 Example of an Ordinary Annuity -- FVA Cash flows occur at the end of the period % $1,000 $1,000 $1,000 FVA 3 = $1,000(1.07) 2 + $1,000(1.07) 1 + $1,000(1.07) 0 = $1,145 + $1,070 + $1,000 = $3,215 $1,070 $1,145 $3,215 = FVA 3 IEG2H2-w2 39

40 Valuation Using Table FVA n = R (F/A, i, n) FVA 3 = $1,000 (F/A, 7%, 3) = $1,000 (3.215) = $3,215

41 Overview of an Ordinary Annuity -- PVA Cash flows occur at the end of the period n n+1 i%... R R R PVA n R = Periodic Cash Flow PVA n = R/(1+i) 1 + R/(1+i) R/(1+i) n IEG2H2-w2 41

42 Example of an Ordinary Annuity -- PVA $ $ $ % $2, = PVA 3 Cash flows occur at the end of the period $1,000 $1,000 $1,000 PVA 3 = $1,000/(1.07) 1 + $1,000/(1.07) 2 + $1,000/(1.07) 3 = $ $ $ = $2, IEG2H2-w2 42

43 Valuation Using Table PVA n = R (P/A, i, n) PVA 3 = $1,000 (P/A, 7%, 3) = $1,000 (2.624) = $2,624

44 Steps to Solve Time Value of Money Problems 1. Read problem throughly 2. Create a time line 3. Put cash flows and arrows on time line 4. Determine if it is a PV or FV problem 5. Determine if solution involves a single CF, annuity stream(s), or mixed flow 6. Solve the problem 7. Check with financial calculator (optional) IEG2H2-w2 44

45 A cash flow diagram is an indispensable tool for clarifying and visualizing a series of cash flows. IEG2H2-TVM 45

46 Cash flow tables are essential to modeling engineering economy problems in a spreadsheet IEG2H2-TVM 46

47 We can apply compound interest formulas to move cash flows along the cash flow diagram. Using the standard notation, we find that a present amount, P, can grow into a future amount, F, in N time periods at interest rate i according to the formula below. In a similar way we can find P given F by IEG2H2-TVM 47

48 It is common to use standard notation for interest factors. This is also known as the single payment compound amount factor. The term on the right is read F given P at i% interest per period for N interest periods. is called the single payment present worth factor. IEG2H2-TVM 48

49 Mixed Flows Example Julie Miller will receive the set of cash flows below. What is the Present Value at a discount rate of 10% 10%. 0 10% 1 $ $600 $400 $400 $100 PV0 IEG2H2-w2 49

50 How to Solve? 1. Solve a piece piece--at at--a-time time by discounting each piece back to t=0. 2. Solve a group group--at at--a-time time by first breaking problem into groups of annuity streams and any single cash flow groups. Then discount each group back to t=0. IEG2H2-w2 50

51 Piece-At-A-Time 0 10% $ $ $ $ $ $ $600 $400 $400 $100 $ = PV0 of the Mixed Flow IEG2H2-w2 51

52 Group-At-A-Time (#1) 0 10% $1, $ $ $600 $600 $400 $400 $100 $1, = PV0 of Mixed Flow [Using Tables] $600(P/A,10%,2) = $600(1.736) = $1, $400(P/A,10%,2).(P/F, 10%, 2) = $400(1.736)(0.826) = $ $100 (P/F, 10%,2) = $100 (0.621) = $62.10 IEG2H2-w2 52

53 Group-At-A-Time 0 $1, Plus $ Plus $ $400 $ $200 $ (#2) 4 $400 $400 PV0 equals $ $100 IEG2H2-w2 53

54 Frequency of Compounding General Formula: Fn = PV0(1 + [i/m])mn n: m: i : F n: P0: Number ofyears Compounding Periods peryear Annual Interest Rate FV at the end ofyear n PV of the Cash Flow today IEG2H2-w2 54

55 Impact of Frequency Julie Miller has $1,000 to invest for 2Years at an annual interest rate of 12%. Annual F2 = 1,000 1,000(1+ [.12/1])(1)(2) = 1, Semi F2 = 1,000 1,000(1+ [.12/2])(2)(2) = 1, IEG2H2-w2 55

56 Impact of Frequency Qrtly F2 Monthly F2 Daily F2 = 1,000 1,000(1+ [.12/4])(4)(2) = 1, = 1,000 1,000(1+ [.12/12])(12)(2) = 1, = 1,000 1,000(1+[.12/365])(365)(2) = 1, IEG2H2-w2 56

57 Effective Annual Interest Rate Effective Annual Interest Rate The actual rate of interest earned (paid) after adjusting the nominal rate for factors such as the number of compounding periods per year. EAR = (1 + [ i / m ] )m - 1 IEG2H2-w2 57

58 Effective Annual Interest Rate Welly has a $1,000 Cash Deposit at the bank. The interest rate is 6% compounded quarterly for 1 year. What is the Effective Annual Interest Rate (EAR EAR)? EAR = ( 1 + 6% / 4 )4-1 = =.0614 or 6.14% IEG2H2-w2 58