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Chapter 3 - Time Value of Money

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This document discusses the time value of money concepts of simple and compound interest, present and future value, and annuities. It provides formulas and examples for calculating future and present value of single deposits using tables or calculators. It also covers calculating the future value of annuities and using annuity tables. Key concepts covered include compound interest earning interest on interest, and the higher growth it provides over time compared to simple interest.

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1 Chapter 3Chapter 3 Time Value ofTime Value of MoneyMoney © 2001 Prentice-Hall, Inc. Fundamentals of Financial Management, 11/e Created by: Gregory A. Kuhlemeyer, Ph.D. Carroll College, Waukesha, WI
2 The Time Value of MoneyThe Time Value of Money The Interest Rate Simple Interest Compound Interest Amortizing a Loan
3 Obviously, $10,000 today$10,000 today. You already recognize that there is TIME VALUE TO MONEYTIME VALUE TO MONEY!! The Interest RateThe Interest Rate Which would you prefer -- $10,000$10,000 todaytoday or $10,000 in 5 years$10,000 in 5 years?
4 TIMETIME allows you the opportunity to postpone consumption and earn INTERESTINTEREST. Why TIME?Why TIME? Why is TIMETIME such an important element in your decision?
5 Types of InterestTypes of Interest Compound InterestCompound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). Simple InterestSimple Interest Interest paid (earned) on only the original amount, or principal borrowed (lent).
6 Simple Interest FormulaSimple Interest Formula FormulaFormula SI = P0(i)(n) SI: Simple Interest P0: Deposit today (t=0) i: Interest Rate per Period n: Number of Time Periods
7 SI = P0(i)(n) = $1,000(.07)(2) = $140$140 Simple Interest ExampleSimple 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?
8 FVFV = P0 + SI = $1,000 + $140 = $1,140$1,140 Future ValueFuture 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. Simple Interest (FV)Simple Interest (FV) What is the Future ValueFuture Value (FVFV) of the deposit?
9 The Present Value is simply the $1,000 you originally deposited. That is the value today! Present ValuePresent Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. Simple Interest (PV)Simple Interest (PV) What is the Present ValuePresent Value (PVPV) of the previous problem?
10 0 5000 10000 15000 20000 1st Year 10th Year 20th Year 30th Year Future Value of a Single $1,000 Deposit 10% Simple Interest 7% Compound Interest 10% Compound Interest Why Compound Interest?Why Compound Interest? FutureValue(U.S.Dollars)
11 Assume that you deposit $1,000$1,000 at a compound interest rate of 7% for 2 years2 years. Future ValueFuture Value Single Deposit (Graphic)Single Deposit (Graphic) 0 1 22 $1,000$1,000 FVFV22 7%
12 FVFV11 = PP00 (1+i)1 = $1,000$1,000 (1.07) = $1,070$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. Future ValueFuture Value Single Deposit (Formula)Single Deposit (Formula)
13 FVFV11 = PP00 (1+i)1 = $1,000$1,000 (1.07) = $1,070$1,070 FVFV22 = FV1 (1+i)1 = PP00 (1+i)(1+i) = $1,000$1,000(1.07)(1.07) = PP00 (1+i)2 = $1,000$1,000(1.07)2 = $1,144.90$1,144.90 You earned an EXTRA $4.90$4.90 in Year 2 with compound over simple interest. Future ValueFuture Value Single Deposit (Formula)Single Deposit (Formula)
14 FVFV11 = P0(1+i)1 FVFV22 = P0(1+i)2 General Future ValueFuture Value Formula: FVFVnn = P0 (1+i)n or FVFVnn = P0 (FVIFFVIFi,n) -- See Table ISee Table I General FutureGeneral Future Value FormulaValue Formula etc.
15 FVIFFVIFi,n is found on Table I at the end of the book or on the card insert. Valuation Using Table IValuation Using Table I Period 6% 7% 8% 1 1.060 1.070 1.080 2 1.124 1.145 1.166 3 1.191 1.225 1.260 4 1.262 1.311 1.360 5 1.338 1.403 1.469
16 FVFV22 = $1,000 (FVIFFVIF7%,2) = $1,000 (1.145) = $1,145$1,145 [Due to Rounding] Using Future Value TablesUsing Future Value Tables Period 6% 7% 8% 1 1.060 1.070 1.080 2 1.124 1.145 1.166 3 1.191 1.225 1.260 4 1.262 1.311 1.360 5 1.338 1.403 1.469
17 TVM on the CalculatorTVM on the Calculator Use the highlighted row of keys for solving any of the FV, PV, FVA, PVA, FVAD, and PVAD problems N: Number of periods I/Y: Interest rate per period PV: Present value PMT: Payment per period FV: Future value CLR TVM: Clears all of the inputs into the above TVM keys
18 Using The TI BAII+ CalculatorUsing The TI BAII+ Calculator N I/Y PV PMT FV Inputs Compute Focus on 3Focus on 3rdrd row of keys (will berow of keys (will be displayed in slides as shown above)displayed in slides as shown above)
19 Entering the FV ProblemEntering the FV Problem Press: 2nd CLR TVM 2 N 7 I/Y -1000 PV 0 PMT CPT FV
20 N: 2 periods (enter as 2) I/Y: 7% interest rate per period (enter as 7 NOT .07) PV: $1,000 (enter as negative as you have “less”) PMT: Not relevant in this situation (enter as 0) FV: Compute (Resulting answer is positive) Solving the FV ProblemSolving the FV Problem N I/Y PV PMT FV Inputs Compute 2 7 -1,000 0 1,144.90
21 Julie Miller wants to know how large her deposit of $10,000$10,000 today will become at a compound annual interest rate of 10% for 5 years5 years. Story Problem ExampleStory Problem Example 0 1 2 3 4 55 $10,000$10,000 FVFV55 10%
22 Calculation based on Table I: FVFV55 = $10,000 (FVIFFVIF10%, 5) = $10,000 (1.611) = $16,110$16,110 [Due to Rounding] Story Problem SolutionStory Problem Solution Calculation based on general formula: FVFVnn = P0 (1+i)n FVFV55 = $10,000 (1+ 0.10)5 = $16,105.10$16,105.10
23 Entering the FV ProblemEntering the FV Problem Press: 2nd CLR TVM 5 N 10 I/Y -10000 PV 0 PMT CPT FV
24 The result indicates that a $10,000 investment that earns 10% annually for 5 years will result in a future value of $16,105.10. Solving the FV ProblemSolving the FV Problem N I/Y PV PMT FV Inputs Compute 5 10 -10,000 0 16,105.10
25 We will use the ““Rule-of-72Rule-of-72””.. Double Your Money!!!Double Your Money!!! Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)?
26 Approx. Years to Double = 7272 / i% 7272 / 12% = 6 Years6 Years [Actual Time is 6.12 Years] The “Rule-of-72”The “Rule-of-72” Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)?
27 The result indicates that a $1,000 investment that earns 12% annually will double to $2,000 in 6.12 years. Note: 72/12% = approx. 6 years Solving the Period ProblemSolving the Period Problem N I/Y PV PMT FV Inputs Compute 12 -1,000 0 +2,000 6.12 years
28 Assume that you need $1,000$1,000 in 2 years.2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually. 0 1 22 $1,000$1,000 7% PV1PVPV00 Present ValuePresent Value Single Deposit (Graphic)Single Deposit (Graphic)
29 PVPV00 = FVFV22 / (1+i)2 = $1,000$1,000 / (1.07)2 = FVFV22 / (1+i)2 = $873.44$873.44 Present ValuePresent Value Single Deposit (Formula)Single Deposit (Formula) 0 1 22 $1,000$1,000 7% PVPV00
30 PVPV00 = FVFV11 / (1+i)1 PVPV00 = FVFV22 / (1+i)2 General Present ValuePresent Value Formula: PVPV00 = FVFVnn / (1+i)n or PVPV00 = FVFVnn (PVIFPVIFi,n) -- See Table IISee Table II General PresentGeneral Present Value FormulaValue Formula etc.
31 PVIFPVIFi,n is found on Table II at the end of the book or on the card insert. Valuation Using Table IIValuation Using Table II Period 6% 7% 8% 1 .943 .935 .926 2 .890 .873 .857 3 .840 .816 .794 4 .792 .763 .735 5 .747 .713 .681
32 PVPV22 = $1,000$1,000 (PVIF7%,2) = $1,000$1,000 (.873) = $873$873 [Due to Rounding] Using Present Value TablesUsing Present Value Tables Period 6% 7% 8% 1 .943 .935 .926 2 .890 .873 .857 3 .840 .816 .794 4 .792 .763 .735 5 .747 .713 .681
33 N: 2 periods (enter as 2) I/Y: 7% interest rate per period (enter as 7 NOT .07) PV: Compute (Resulting answer is negative “deposit”) PMT: Not relevant in this situation (enter as 0) FV: $1,000 (enter as positive as you “receive $”) Solving the PV ProblemSolving the PV Problem N I/Y PV PMT FV Inputs Compute 2 7 0 +1,000 -873.44
34 Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000$10,000 in 5 years5 years at a discount rate of 10%. Story Problem ExampleStory Problem Example 0 1 2 3 4 55 $10,000$10,000 PVPV00 10%
35 Calculation based on general formula: PVPV00 = FVFVnn / (1+i)n PVPV00 = $10,000$10,000 / (1+ 0.10)5 = $6,209.21$6,209.21 Calculation based on Table I: PVPV00 = $10,000$10,000 (PVIFPVIF10%, 5) = $10,000$10,000 (.621) = $6,210.00$6,210.00 [Due to Rounding] Story Problem SolutionStory Problem Solution
36 Solving the PV ProblemSolving the PV Problem N I/Y PV PMT FV Inputs Compute 5 10 0 +10,000 -6,209.21 The result indicates that a $10,000 future value that will earn 10% annually for 5 years requires a $6,209.21 deposit today (present value).
37 Types of AnnuitiesTypes of Annuities Ordinary AnnuityOrdinary Annuity: Payments or receipts occur at the end of each period. Annuity DueAnnuity Due: Payments or receipts occur at the beginning of each period. An AnnuityAn Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.
38 Examples of AnnuitiesExamples of Annuities Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings
39 Parts of an AnnuityParts of an Annuity 0 1 2 3 $100 $100 $100 (Ordinary Annuity) EndEnd of Period 1 EndEnd of Period 2 Today EqualEqual Cash Flows Each 1 Period Apart EndEnd of Period 3
40 Parts of an AnnuityParts of an Annuity 0 1 2 3 $100 $100 $100 (Annuity Due) BeginningBeginning of Period 1 BeginningBeginning of Period 2 Today EqualEqual Cash Flows Each 1 Period Apart BeginningBeginning of Period 3
41 FVAFVAnn = R(1+i)n-1 + R(1+i)n-2 + ... + R(1+i)1 + R(1+i)0 Overview of anOverview of an Ordinary Annuity -- FVAOrdinary Annuity -- FVA R R R 0 1 2 nn n+1 FVAFVAnn R = Periodic Cash Flow Cash flows occur at the end of the period i% . . .
42 FVAFVA33 = $1,000(1.07)2 + $1,000(1.07)1 + $1,000(1.07)0 = $1,145 + $1,070 + $1,000 = $3,215$3,215 Example of anExample of an Ordinary Annuity -- FVAOrdinary Annuity -- FVA $1,000 $1,000 $1,000 0 1 2 33 4 $3,215 = FVA$3,215 = FVA33 7% $1,070 $1,145 Cash flows occur at the end of the period
43 Hint on Annuity ValuationHint on Annuity Valuation The future value of an ordinary annuity can be viewed as occurring at the endend of the last cash flow period, whereas the future value of an annuity due can be viewed as occurring at the beginningbeginning of the last cash flow period.
44 FVAFVAnn = R (FVIFAi%,n) FVAFVA33 = $1,000 (FVIFA7%,3) = $1,000 (3.215) = $3,215$3,215 Valuation Using Table IIIValuation Using Table III Period 6% 7% 8% 1 1.000 1.000 1.000 2 2.060 2.070 2.080 3 3.184 3.215 3.246 4 4.375 4.440 4.506 5 5.637 5.751 5.867
45 N: 3 periods (enter as 3 year-end deposits) I/Y: 7% interest rate per period (enter as 7 NOT .07) PV: Not relevant in this situation (no beg value) PMT: $1,000 (negative as you deposit annually) FV: Compute (Resulting answer is positive) Solving the FVA ProblemSolving the FVA Problem N I/Y PV PMT FV Inputs Compute 3 7 0 -1,000 3,214.90
46 FVADFVADnn = R(1+i)n + R(1+i)n-1 + ... + R(1+i)2 + R(1+i)1 = FVAFVAnn (1+i) Overview View of anOverview View of an Annuity Due -- FVADAnnuity Due -- FVAD R R R R R 0 1 2 3 n-1n-1 n FVADFVADnn i% . . . Cash flows occur at the beginning of the period
47 FVADFVAD33 = $1,000(1.07)3 + $1,000(1.07)2 + $1,000(1.07)1 = $1,225 + $1,145 + $1,070 = $3,440$3,440 Example of anExample of an Annuity Due -- FVADAnnuity Due -- FVAD $1,000 $1,000 $1,000 $1,070 0 1 2 33 4 $3,440 = FVAD$3,440 = FVAD33 7% $1,225 $1,145 Cash flows occur at the beginning of the period
48 FVADFVADnn = R (FVIFAi%,n)(1+i) FVADFVAD33 = $1,000 (FVIFA7%,3)(1.07) = $1,000 (3.215)(1.07) = $3,440$3,440 Valuation Using Table IIIValuation Using Table III Period 6% 7% 8% 1 1.000 1.000 1.000 2 2.060 2.070 2.080 3 3.184 3.215 3.246 4 4.375 4.440 4.506 5 5.637 5.751 5.867
49 Solving the FVAD ProblemSolving the FVAD Problem N I/Y PV PMT FV Inputs Compute 3 7 0 -1,000 3,439.94 Complete the problem the same as an “ordinary annuity” problem, except you must change the calculator setting to “BGN” first. Don’t forget to change back! Step 1: Press 2nd BGN keys Step 2: Press 2nd SET keys Step 3: Press 2nd QUIT keys
50 PVAPVAnn = R/(1+i)1 + R/(1+i)2 + ... + R/(1+i)n Overview of anOverview of an Ordinary Annuity -- PVAOrdinary Annuity -- PVA R R R 0 1 2 nn n+1 PVAPVAnn R = Periodic Cash Flow i% . . . Cash flows occur at the end of the period
51 PVAPVA33 = $1,000/(1.07)1 + $1,000/(1.07)2 + $1,000/(1.07)3 = $934.58 + $873.44 + $816.30 = $2,624.32$2,624.32 Example of anExample of an Ordinary Annuity -- PVAOrdinary Annuity -- PVA $1,000 $1,000 $1,000 0 1 2 33 4 $2,624.32 = PVA$2,624.32 = PVA33 7% $ 934.58 $ 873.44 $ 816.30 Cash flows occur at the end of the period
52 Hint on Annuity ValuationHint on Annuity Valuation The present value of an ordinary annuity can be viewed as occurring at the beginningbeginning of the first cash flow period, whereas the present value of an annuity due can be viewed as occurring at the endend of the first cash flow period.
53 PVAPVAnn = R (PVIFAi%,n) PVAPVA33 = $1,000 (PVIFA7%,3) = $1,000 (2.624) = $2,624$2,624 Valuation Using Table IVValuation Using Table IV Period 6% 7% 8% 1 0.943 0.935 0.926 2 1.833 1.808 1.783 3 2.673 2.624 2.577 4 3.465 3.387 3.312 5 4.212 4.100 3.993
54 N: 3 periods (enter as 3 year-end deposits) I/Y: 7% interest rate per period (enter as 7 NOT .07) PV: Compute (Resulting answer is positive) PMT: $1,000 (negative as you deposit annually) FV: Not relevant in this situation (no ending value) Solving the PVA ProblemSolving the PVA Problem N I/Y PV PMT FV Inputs Compute 3 7 -1,000 0 2,624.32
55 PVADPVADnn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1 = PVAPVAnn (1+i) Overview of anOverview of an Annuity Due -- PVADAnnuity Due -- PVAD R R R R 0 1 2 n-1n-1 n PVADPVADnn R: Periodic Cash Flow i% . . . Cash flows occur at the beginning of the period
56 PVADPVADnn = $1,000/(1.07)0 + $1,000/(1.07)1 + $1,000/(1.07)2 = $2,808.02$2,808.02 Example of anExample of an Annuity Due -- PVADAnnuity Due -- PVAD $1,000.00 $1,000 $1,000 0 1 2 33 4 $2,808.02$2,808.02 = PVADPVADnn 7% $ 934.58 $ 873.44 Cash flows occur at the beginning of the period
57 PVADPVADnn = R (PVIFAi%,n)(1+i) PVADPVAD33 = $1,000 (PVIFA7%,3)(1.07) = $1,000 (2.624)(1.07) = $2,808$2,808 Valuation Using Table IVValuation Using Table IV Period 6% 7% 8% 1 0.943 0.935 0.926 2 1.833 1.808 1.783 3 2.673 2.624 2.577 4 3.465 3.387 3.312 5 4.212 4.100 3.993
58 Solving the PVAD ProblemSolving the PVAD Problem N I/Y PV PMT FV Inputs Compute 3 7 -1,000 0 2,808.02 Complete the problem the same as an “ordinary annuity” problem, except you must change the calculator setting to “BGN” first. Don’t forget to change back! Step 1: Press 2nd BGN keys Step 2: Press 2nd SET keys Step 3: Press 2nd QUIT keys
59 1. Read problem thoroughly 2. Determine if it is a PV or FV problem 3. Create a time line 4. Put cash flows and arrows on time line 5. Determine if solution involves a single CF, annuity stream(s), or mixed flow 6. Solve the problem 7. Check with financial calculator (optional) Steps to Solve Time ValueSteps to Solve Time Value of Money Problemsof Money Problems
60 Julie Miller will receive the set of cash flows below. What is the Present ValuePresent Value at a discount rate of 10%10%? Mixed Flows ExampleMixed Flows Example 0 1 2 3 4 55 $600 $600 $400 $400 $100$600 $600 $400 $400 $100 PVPV00 10%10%
61 1. Solve a “piece-at-a-timepiece-at-a-time” by discounting each piecepiece back to t=0. 2. Solve a “group-at-a-timegroup-at-a-time” by first breaking problem into groups of annuity streams and any single cash flow group. Then discount each groupgroup back to t=0. How to Solve?How to Solve?
62 ““Piece-At-A-Time”Piece-At-A-Time” 0 1 2 3 4 55 $600 $600 $400 $400 $100$600 $600 $400 $400 $100 10% $545.45$545.45 $495.87$495.87 $300.53$300.53 $273.21$273.21 $ 62.09$ 62.09 $1677.15$1677.15 == PVPV00 of the Mixed Flowof the Mixed Flow
63 ““Group-At-A-Time” (#1)Group-At-A-Time” (#1) 0 1 2 3 4 55 $600 $600 $400 $400 $100$600 $600 $400 $400 $100 10% $1,041.60$1,041.60 $ 573.57$ 573.57 $ 62.10$ 62.10 $1,677.27$1,677.27 == PVPV00 of Mixed Flowof Mixed Flow [Using Tables][Using Tables] $600(PVIFA10%,2) = $600(1.736) = $1,041.60 $400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57 $100 (PVIF10%,5) = $100 (0.621) = $62.10
64 ““Group-At-A-Time” (#2)Group-At-A-Time” (#2) 0 1 2 3 4 $400 $400 $400 $400$400 $400 $400 $400 PVPV00 equals $1677.30.$1677.30. 0 1 2 $200 $200$200 $200 0 1 2 3 4 5 $100$100 $1,268.00$1,268.00 $347.20$347.20 $62.10$62.10 PlusPlus PlusPlus
65 Use the highlighted key for starting the process of solving a mixed cash flow problem Press the CF key and down arrow key through a few of the keys as you look at the definitions on the next slide Solving the Mixed FlowsSolving the Mixed Flows Problem using CF RegistryProblem using CF Registry
66 Defining the calculator variables: For CF0: This is ALWAYS the cash flow occurring at time t=0 (usually 0 for these problems) For Cnn:* This is the cash flow SIZE of the nth group of cash flows. Note that a “group” may only contain a single cash flow (e.g., $351.76). For Fnn:* This is the cash flow FREQUENCY of the nth group of cash flows. Note that this is always a positive whole number (e.g., 1, 2, 20, etc.). Solving the Mixed FlowsSolving the Mixed Flows Problem using CF RegistryProblem using CF Registry * nn represents the nth cash flow or frequency. Thus, the first cash flow is C01, while the tenth cash flow is C10.
67 Solving the Mixed FlowsSolving the Mixed Flows Problem using CF RegistryProblem using CF Registry Steps in the Process Step 1: Press CF key Step 2: Press 2nd CLR Work keys Step 3: For CF0 Press 0 Enter ↓ keys Step 4: For C01 Press 600 Enter ↓ keys Step 5: For F01 Press 2 Enter ↓ keys Step 6: For C02 Press 400 Enter ↓ keys Step 7: For F02 Press 2 Enter ↓ keys
68 Solving the Mixed FlowsSolving the Mixed Flows Problem using CF RegistryProblem using CF Registry Steps in the Process Step 8: For C03 Press 100 Enter ↓ keys Step 9: For F03 Press 1 Enter ↓ keys Step 10: Press ↓ ↓ keys Step 11: Press NPV key Step 12: For I=, Enter 10 Enter ↓ keys Step 13: Press CPT key Result: Present Value = $1,677.15
69 General Formula: FVn = PVPV00(1 + [i/m])mn n: Number of Years m: Compounding Periods per Year i: Annual Interest Rate FVn,m: FV at the end of Year n PVPV00: PV of the Cash Flow today Frequency ofFrequency of CompoundingCompounding
70 Julie Miller has $1,000$1,000 to invest for 2 years at an annual interest rate of 12%. Annual FV2 = 1,0001,000(1+ [.12/1])(1)(2) = 1,254.401,254.40 Semi FV2 = 1,0001,000(1+ [.12/2])(2)(2) = 1,262.481,262.48 Impact of FrequencyImpact of Frequency
71 Qrtly FV2 = 1,0001,000(1+ [.12/4])(4)(2) = 1,266.771,266.77 Monthly FV2 = 1,0001,000(1+ [.12/12])(12)(2) = 1,269.731,269.73 Daily FV2 = 1,0001,000(1+[.12/365])(365)(2) = 1,271.201,271.20 Impact of FrequencyImpact of Frequency
72 The result indicates that a $1,000 investment that earns a 12% annual rate compounded quarterly for 2 years will earn a future value of $1,266.77. Solving the FrequencySolving the Frequency Problem (Quarterly)Problem (Quarterly) N I/Y PV PMT FV Inputs Compute 2(4) 12/4 -1,000 0 1266.77
73 Solving the FrequencySolving the Frequency Problem (Quarterly Altern.)Problem (Quarterly Altern.) Press: 2nd P/Y 4 ENTER 2nd QUIT 12 I/Y -1000 PV 0 PMT 2 2nd xP/Y N CPT FV
74 The result indicates that a $1,000 investment that earns a 12% annual rate compounded daily for 2 years will earn a future value of $1,271.20. Solving the FrequencySolving the Frequency Problem (Daily)Problem (Daily) N I/Y PV PMT FV Inputs Compute 2(365) 12/365 -1,000 0 1271.20
75 Solving the FrequencySolving the Frequency Problem (Daily Alternative)Problem (Daily Alternative) Press: 2nd P/Y 365 ENTER 2nd QUIT 12 I/Y -1000 PV 0 PMT 2 2nd xP/Y N CPT FV
76 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. (1 + [ i / m ] )m - 1 Effective AnnualEffective Annual Interest RateInterest Rate
77 Basket Wonders (BW) has a $1,000 CD at the bank. The interest rate is 6% compounded quarterly for 1 year. What is the Effective Annual Interest Rate (EAREAR)? EAREAR = ( 1 + 6% / 4 )4 - 1 = 1.0614 - 1 = .0614 or 6.14%!6.14%! BW’s EffectiveBW’s Effective Annual Interest RateAnnual Interest Rate
78 Converting to an EARConverting to an EAR Press: 2nd I Conv 6 ENTER ↓ ↓ 4 ENTER ↑ CPT 2nd QUIT
79 1. Calculate the payment per period. 2. Determine the interest in Period t. (Loan balance at t-1) x (i% / m) 3. Compute principal paymentprincipal payment in Period t. (Payment - interest from Step 2) 4. Determine ending balance in Period t. (Balance - principal paymentprincipal payment from Step 3) 5. Start again at Step 2 and repeat. Steps to Amortizing a LoanSteps to Amortizing a Loan
80 Julie Miller is borrowing $10,000$10,000 at a compound annual interest rate of 12%. Amortize the loan if annual payments are made for 5 years. Step 1: Payment PVPV00 = R (PVIFA i%,n) $10,000$10,000 = R (PVIFA 12%,5) $10,000$10,000 = R (3.605) RR = $10,000$10,000 / 3.605 = $2,774$2,774 Amortizing a Loan ExampleAmortizing a Loan Example
81 Amortizing a Loan ExampleAmortizing a Loan Example End of Year Payment Interest Principal Ending Balance 0 --- --- --- $10,000 1 $2,774 $1,200 $1,574 8,426 2 2,774 1,011 1,763 6,663 3 2,774 800 1,974 4,689 4 2,774 563 2,211 2,478 5 2,775 297 2,478 0 $13,871 $3,871 $10,000 [Last Payment Slightly Higher Due to Rounding]
82 The result indicates that a $10,000 loan that costs 12% annually for 5 years and will be completely paid off at that time will require $2,774.10 in annual payments. Solving for the PaymentSolving for the Payment N I/Y PV PMT FV Inputs Compute 5 12 10,000 0 -2774.10
83 Using the AmortizationUsing the Amortization Functions of the CalculatorFunctions of the Calculator Press: 2nd Amort 1 ENTER 1 ENTER Results: BAL = 8,425.90 ↓ PRN = -1,574.10 ↓ INT = -1,200.00 ↓ Year 1 information only
84 Using the AmortizationUsing the Amortization Functions of the CalculatorFunctions of the Calculator Press: 2nd Amort 2 ENTER 2 ENTER Results: BAL = 6,662.91 ↓ PRN = -1,763.99 ↓ INT = -1,011.11 ↓ Year 2 information only
85 Using the AmortizationUsing the Amortization Functions of the CalculatorFunctions of the Calculator Press: 2nd Amort 1 ENTER 5 ENTER Results: BAL = 0.00 ↓ PRN =-10,000.00 ↓ INT = -3,870.49 ↓ Entire 5 Years of loan information
86 Usefulness of AmortizationUsefulness of Amortization 2.2. Calculate Debt OutstandingCalculate Debt Outstanding -- The quantity of outstanding debt may be used in financing the day-to-day activities of the firm. 1.1. Determine Interest ExpenseDetermine Interest Expense -- Interest expenses may reduce taxable income of the firm.

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Chapter 3 - Time Value of Money

  • 1. 1 Chapter 3Chapter 3 Time Value ofTime Value of MoneyMoney © 2001 Prentice-Hall, Inc. Fundamentals of Financial Management, 11/e Created by: Gregory A. Kuhlemeyer, Ph.D. Carroll College, Waukesha, WI
  • 2. 2 The Time Value of MoneyThe Time Value of Money The Interest Rate Simple Interest Compound Interest Amortizing a Loan
  • 3. 3 Obviously, $10,000 today$10,000 today. You already recognize that there is TIME VALUE TO MONEYTIME VALUE TO MONEY!! The Interest RateThe Interest Rate Which would you prefer -- $10,000$10,000 todaytoday or $10,000 in 5 years$10,000 in 5 years?
  • 4. 4 TIMETIME allows you the opportunity to postpone consumption and earn INTERESTINTEREST. Why TIME?Why TIME? Why is TIMETIME such an important element in your decision?
  • 5. 5 Types of InterestTypes of Interest Compound InterestCompound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). Simple InterestSimple Interest Interest paid (earned) on only the original amount, or principal borrowed (lent).
  • 6. 6 Simple Interest FormulaSimple Interest Formula FormulaFormula SI = P0(i)(n) SI: Simple Interest P0: Deposit today (t=0) i: Interest Rate per Period n: Number of Time Periods
  • 7. 7 SI = P0(i)(n) = $1,000(.07)(2) = $140$140 Simple Interest ExampleSimple 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?
  • 8. 8 FVFV = P0 + SI = $1,000 + $140 = $1,140$1,140 Future ValueFuture 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. Simple Interest (FV)Simple Interest (FV) What is the Future ValueFuture Value (FVFV) of the deposit?
  • 9. 9 The Present Value is simply the $1,000 you originally deposited. That is the value today! Present ValuePresent Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. Simple Interest (PV)Simple Interest (PV) What is the Present ValuePresent Value (PVPV) of the previous problem?
  • 10. 10 0 5000 10000 15000 20000 1st Year 10th Year 20th Year 30th Year Future Value of a Single $1,000 Deposit 10% Simple Interest 7% Compound Interest 10% Compound Interest Why Compound Interest?Why Compound Interest? FutureValue(U.S.Dollars)
  • 11. 11 Assume that you deposit $1,000$1,000 at a compound interest rate of 7% for 2 years2 years. Future ValueFuture Value Single Deposit (Graphic)Single Deposit (Graphic) 0 1 22 $1,000$1,000 FVFV22 7%
  • 12. 12 FVFV11 = PP00 (1+i)1 = $1,000$1,000 (1.07) = $1,070$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. Future ValueFuture Value Single Deposit (Formula)Single Deposit (Formula)
  • 13. 13 FVFV11 = PP00 (1+i)1 = $1,000$1,000 (1.07) = $1,070$1,070 FVFV22 = FV1 (1+i)1 = PP00 (1+i)(1+i) = $1,000$1,000(1.07)(1.07) = PP00 (1+i)2 = $1,000$1,000(1.07)2 = $1,144.90$1,144.90 You earned an EXTRA $4.90$4.90 in Year 2 with compound over simple interest. Future ValueFuture Value Single Deposit (Formula)Single Deposit (Formula)
  • 14. 14 FVFV11 = P0(1+i)1 FVFV22 = P0(1+i)2 General Future ValueFuture Value Formula: FVFVnn = P0 (1+i)n or FVFVnn = P0 (FVIFFVIFi,n) -- See Table ISee Table I General FutureGeneral Future Value FormulaValue Formula etc.
  • 15. 15 FVIFFVIFi,n is found on Table I at the end of the book or on the card insert. Valuation Using Table IValuation Using Table I Period 6% 7% 8% 1 1.060 1.070 1.080 2 1.124 1.145 1.166 3 1.191 1.225 1.260 4 1.262 1.311 1.360 5 1.338 1.403 1.469
  • 16. 16 FVFV22 = $1,000 (FVIFFVIF7%,2) = $1,000 (1.145) = $1,145$1,145 [Due to Rounding] Using Future Value TablesUsing Future Value Tables Period 6% 7% 8% 1 1.060 1.070 1.080 2 1.124 1.145 1.166 3 1.191 1.225 1.260 4 1.262 1.311 1.360 5 1.338 1.403 1.469
  • 17. 17 TVM on the CalculatorTVM on the Calculator Use the highlighted row of keys for solving any of the FV, PV, FVA, PVA, FVAD, and PVAD problems N: Number of periods I/Y: Interest rate per period PV: Present value PMT: Payment per period FV: Future value CLR TVM: Clears all of the inputs into the above TVM keys
  • 18. 18 Using The TI BAII+ CalculatorUsing The TI BAII+ Calculator N I/Y PV PMT FV Inputs Compute Focus on 3Focus on 3rdrd row of keys (will berow of keys (will be displayed in slides as shown above)displayed in slides as shown above)
  • 19. 19 Entering the FV ProblemEntering the FV Problem Press: 2nd CLR TVM 2 N 7 I/Y -1000 PV 0 PMT CPT FV
  • 20. 20 N: 2 periods (enter as 2) I/Y: 7% interest rate per period (enter as 7 NOT .07) PV: $1,000 (enter as negative as you have “less”) PMT: Not relevant in this situation (enter as 0) FV: Compute (Resulting answer is positive) Solving the FV ProblemSolving the FV Problem N I/Y PV PMT FV Inputs Compute 2 7 -1,000 0 1,144.90
  • 21. 21 Julie Miller wants to know how large her deposit of $10,000$10,000 today will become at a compound annual interest rate of 10% for 5 years5 years. Story Problem ExampleStory Problem Example 0 1 2 3 4 55 $10,000$10,000 FVFV55 10%
  • 22. 22 Calculation based on Table I: FVFV55 = $10,000 (FVIFFVIF10%, 5) = $10,000 (1.611) = $16,110$16,110 [Due to Rounding] Story Problem SolutionStory Problem Solution Calculation based on general formula: FVFVnn = P0 (1+i)n FVFV55 = $10,000 (1+ 0.10)5 = $16,105.10$16,105.10
  • 23. 23 Entering the FV ProblemEntering the FV Problem Press: 2nd CLR TVM 5 N 10 I/Y -10000 PV 0 PMT CPT FV
  • 24. 24 The result indicates that a $10,000 investment that earns 10% annually for 5 years will result in a future value of $16,105.10. Solving the FV ProblemSolving the FV Problem N I/Y PV PMT FV Inputs Compute 5 10 -10,000 0 16,105.10
  • 25. 25 We will use the ““Rule-of-72Rule-of-72””.. Double Your Money!!!Double Your Money!!! Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)?
  • 26. 26 Approx. Years to Double = 7272 / i% 7272 / 12% = 6 Years6 Years [Actual Time is 6.12 Years] The “Rule-of-72”The “Rule-of-72” Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)?
  • 27. 27 The result indicates that a $1,000 investment that earns 12% annually will double to $2,000 in 6.12 years. Note: 72/12% = approx. 6 years Solving the Period ProblemSolving the Period Problem N I/Y PV PMT FV Inputs Compute 12 -1,000 0 +2,000 6.12 years
  • 28. 28 Assume that you need $1,000$1,000 in 2 years.2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually. 0 1 22 $1,000$1,000 7% PV1PVPV00 Present ValuePresent Value Single Deposit (Graphic)Single Deposit (Graphic)
  • 29. 29 PVPV00 = FVFV22 / (1+i)2 = $1,000$1,000 / (1.07)2 = FVFV22 / (1+i)2 = $873.44$873.44 Present ValuePresent Value Single Deposit (Formula)Single Deposit (Formula) 0 1 22 $1,000$1,000 7% PVPV00
  • 30. 30 PVPV00 = FVFV11 / (1+i)1 PVPV00 = FVFV22 / (1+i)2 General Present ValuePresent Value Formula: PVPV00 = FVFVnn / (1+i)n or PVPV00 = FVFVnn (PVIFPVIFi,n) -- See Table IISee Table II General PresentGeneral Present Value FormulaValue Formula etc.
  • 31. 31 PVIFPVIFi,n is found on Table II at the end of the book or on the card insert. Valuation Using Table IIValuation Using Table II Period 6% 7% 8% 1 .943 .935 .926 2 .890 .873 .857 3 .840 .816 .794 4 .792 .763 .735 5 .747 .713 .681
  • 32. 32 PVPV22 = $1,000$1,000 (PVIF7%,2) = $1,000$1,000 (.873) = $873$873 [Due to Rounding] Using Present Value TablesUsing Present Value Tables Period 6% 7% 8% 1 .943 .935 .926 2 .890 .873 .857 3 .840 .816 .794 4 .792 .763 .735 5 .747 .713 .681
  • 33. 33 N: 2 periods (enter as 2) I/Y: 7% interest rate per period (enter as 7 NOT .07) PV: Compute (Resulting answer is negative “deposit”) PMT: Not relevant in this situation (enter as 0) FV: $1,000 (enter as positive as you “receive $”) Solving the PV ProblemSolving the PV Problem N I/Y PV PMT FV Inputs Compute 2 7 0 +1,000 -873.44
  • 34. 34 Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000$10,000 in 5 years5 years at a discount rate of 10%. Story Problem ExampleStory Problem Example 0 1 2 3 4 55 $10,000$10,000 PVPV00 10%
  • 35. 35 Calculation based on general formula: PVPV00 = FVFVnn / (1+i)n PVPV00 = $10,000$10,000 / (1+ 0.10)5 = $6,209.21$6,209.21 Calculation based on Table I: PVPV00 = $10,000$10,000 (PVIFPVIF10%, 5) = $10,000$10,000 (.621) = $6,210.00$6,210.00 [Due to Rounding] Story Problem SolutionStory Problem Solution
  • 36. 36 Solving the PV ProblemSolving the PV Problem N I/Y PV PMT FV Inputs Compute 5 10 0 +10,000 -6,209.21 The result indicates that a $10,000 future value that will earn 10% annually for 5 years requires a $6,209.21 deposit today (present value).
  • 37. 37 Types of AnnuitiesTypes of Annuities Ordinary AnnuityOrdinary Annuity: Payments or receipts occur at the end of each period. Annuity DueAnnuity Due: Payments or receipts occur at the beginning of each period. An AnnuityAn Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.
  • 38. 38 Examples of AnnuitiesExamples of Annuities Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings
  • 39. 39 Parts of an AnnuityParts of an Annuity 0 1 2 3 $100 $100 $100 (Ordinary Annuity) EndEnd of Period 1 EndEnd of Period 2 Today EqualEqual Cash Flows Each 1 Period Apart EndEnd of Period 3
  • 40. 40 Parts of an AnnuityParts of an Annuity 0 1 2 3 $100 $100 $100 (Annuity Due) BeginningBeginning of Period 1 BeginningBeginning of Period 2 Today EqualEqual Cash Flows Each 1 Period Apart BeginningBeginning of Period 3
  • 41. 41 FVAFVAnn = R(1+i)n-1 + R(1+i)n-2 + ... + R(1+i)1 + R(1+i)0 Overview of anOverview of an Ordinary Annuity -- FVAOrdinary Annuity -- FVA R R R 0 1 2 nn n+1 FVAFVAnn R = Periodic Cash Flow Cash flows occur at the end of the period i% . . .
  • 42. 42 FVAFVA33 = $1,000(1.07)2 + $1,000(1.07)1 + $1,000(1.07)0 = $1,145 + $1,070 + $1,000 = $3,215$3,215 Example of anExample of an Ordinary Annuity -- FVAOrdinary Annuity -- FVA $1,000 $1,000 $1,000 0 1 2 33 4 $3,215 = FVA$3,215 = FVA33 7% $1,070 $1,145 Cash flows occur at the end of the period
  • 43. 43 Hint on Annuity ValuationHint on Annuity Valuation The future value of an ordinary annuity can be viewed as occurring at the endend of the last cash flow period, whereas the future value of an annuity due can be viewed as occurring at the beginningbeginning of the last cash flow period.
  • 44. 44 FVAFVAnn = R (FVIFAi%,n) FVAFVA33 = $1,000 (FVIFA7%,3) = $1,000 (3.215) = $3,215$3,215 Valuation Using Table IIIValuation Using Table III Period 6% 7% 8% 1 1.000 1.000 1.000 2 2.060 2.070 2.080 3 3.184 3.215 3.246 4 4.375 4.440 4.506 5 5.637 5.751 5.867
  • 45. 45 N: 3 periods (enter as 3 year-end deposits) I/Y: 7% interest rate per period (enter as 7 NOT .07) PV: Not relevant in this situation (no beg value) PMT: $1,000 (negative as you deposit annually) FV: Compute (Resulting answer is positive) Solving the FVA ProblemSolving the FVA Problem N I/Y PV PMT FV Inputs Compute 3 7 0 -1,000 3,214.90
  • 46. 46 FVADFVADnn = R(1+i)n + R(1+i)n-1 + ... + R(1+i)2 + R(1+i)1 = FVAFVAnn (1+i) Overview View of anOverview View of an Annuity Due -- FVADAnnuity Due -- FVAD R R R R R 0 1 2 3 n-1n-1 n FVADFVADnn i% . . . Cash flows occur at the beginning of the period
  • 47. 47 FVADFVAD33 = $1,000(1.07)3 + $1,000(1.07)2 + $1,000(1.07)1 = $1,225 + $1,145 + $1,070 = $3,440$3,440 Example of anExample of an Annuity Due -- FVADAnnuity Due -- FVAD $1,000 $1,000 $1,000 $1,070 0 1 2 33 4 $3,440 = FVAD$3,440 = FVAD33 7% $1,225 $1,145 Cash flows occur at the beginning of the period
  • 48. 48 FVADFVADnn = R (FVIFAi%,n)(1+i) FVADFVAD33 = $1,000 (FVIFA7%,3)(1.07) = $1,000 (3.215)(1.07) = $3,440$3,440 Valuation Using Table IIIValuation Using Table III Period 6% 7% 8% 1 1.000 1.000 1.000 2 2.060 2.070 2.080 3 3.184 3.215 3.246 4 4.375 4.440 4.506 5 5.637 5.751 5.867
  • 49. 49 Solving the FVAD ProblemSolving the FVAD Problem N I/Y PV PMT FV Inputs Compute 3 7 0 -1,000 3,439.94 Complete the problem the same as an “ordinary annuity” problem, except you must change the calculator setting to “BGN” first. Don’t forget to change back! Step 1: Press 2nd BGN keys Step 2: Press 2nd SET keys Step 3: Press 2nd QUIT keys
  • 50. 50 PVAPVAnn = R/(1+i)1 + R/(1+i)2 + ... + R/(1+i)n Overview of anOverview of an Ordinary Annuity -- PVAOrdinary Annuity -- PVA R R R 0 1 2 nn n+1 PVAPVAnn R = Periodic Cash Flow i% . . . Cash flows occur at the end of the period
  • 51. 51 PVAPVA33 = $1,000/(1.07)1 + $1,000/(1.07)2 + $1,000/(1.07)3 = $934.58 + $873.44 + $816.30 = $2,624.32$2,624.32 Example of anExample of an Ordinary Annuity -- PVAOrdinary Annuity -- PVA $1,000 $1,000 $1,000 0 1 2 33 4 $2,624.32 = PVA$2,624.32 = PVA33 7% $ 934.58 $ 873.44 $ 816.30 Cash flows occur at the end of the period
  • 52. 52 Hint on Annuity ValuationHint on Annuity Valuation The present value of an ordinary annuity can be viewed as occurring at the beginningbeginning of the first cash flow period, whereas the present value of an annuity due can be viewed as occurring at the endend of the first cash flow period.
  • 53. 53 PVAPVAnn = R (PVIFAi%,n) PVAPVA33 = $1,000 (PVIFA7%,3) = $1,000 (2.624) = $2,624$2,624 Valuation Using Table IVValuation Using Table IV Period 6% 7% 8% 1 0.943 0.935 0.926 2 1.833 1.808 1.783 3 2.673 2.624 2.577 4 3.465 3.387 3.312 5 4.212 4.100 3.993
  • 54. 54 N: 3 periods (enter as 3 year-end deposits) I/Y: 7% interest rate per period (enter as 7 NOT .07) PV: Compute (Resulting answer is positive) PMT: $1,000 (negative as you deposit annually) FV: Not relevant in this situation (no ending value) Solving the PVA ProblemSolving the PVA Problem N I/Y PV PMT FV Inputs Compute 3 7 -1,000 0 2,624.32
  • 55. 55 PVADPVADnn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1 = PVAPVAnn (1+i) Overview of anOverview of an Annuity Due -- PVADAnnuity Due -- PVAD R R R R 0 1 2 n-1n-1 n PVADPVADnn R: Periodic Cash Flow i% . . . Cash flows occur at the beginning of the period
  • 56. 56 PVADPVADnn = $1,000/(1.07)0 + $1,000/(1.07)1 + $1,000/(1.07)2 = $2,808.02$2,808.02 Example of anExample of an Annuity Due -- PVADAnnuity Due -- PVAD $1,000.00 $1,000 $1,000 0 1 2 33 4 $2,808.02$2,808.02 = PVADPVADnn 7% $ 934.58 $ 873.44 Cash flows occur at the beginning of the period
  • 57. 57 PVADPVADnn = R (PVIFAi%,n)(1+i) PVADPVAD33 = $1,000 (PVIFA7%,3)(1.07) = $1,000 (2.624)(1.07) = $2,808$2,808 Valuation Using Table IVValuation Using Table IV Period 6% 7% 8% 1 0.943 0.935 0.926 2 1.833 1.808 1.783 3 2.673 2.624 2.577 4 3.465 3.387 3.312 5 4.212 4.100 3.993
  • 58. 58 Solving the PVAD ProblemSolving the PVAD Problem N I/Y PV PMT FV Inputs Compute 3 7 -1,000 0 2,808.02 Complete the problem the same as an “ordinary annuity” problem, except you must change the calculator setting to “BGN” first. Don’t forget to change back! Step 1: Press 2nd BGN keys Step 2: Press 2nd SET keys Step 3: Press 2nd QUIT keys
  • 59. 59 1. Read problem thoroughly 2. Determine if it is a PV or FV problem 3. Create a time line 4. Put cash flows and arrows on time line 5. Determine if solution involves a single CF, annuity stream(s), or mixed flow 6. Solve the problem 7. Check with financial calculator (optional) Steps to Solve Time ValueSteps to Solve Time Value of Money Problemsof Money Problems
  • 60. 60 Julie Miller will receive the set of cash flows below. What is the Present ValuePresent Value at a discount rate of 10%10%? Mixed Flows ExampleMixed Flows Example 0 1 2 3 4 55 $600 $600 $400 $400 $100$600 $600 $400 $400 $100 PVPV00 10%10%
  • 61. 61 1. Solve a “piece-at-a-timepiece-at-a-time” by discounting each piecepiece back to t=0. 2. Solve a “group-at-a-timegroup-at-a-time” by first breaking problem into groups of annuity streams and any single cash flow group. Then discount each groupgroup back to t=0. How to Solve?How to Solve?
  • 62. 62 ““Piece-At-A-Time”Piece-At-A-Time” 0 1 2 3 4 55 $600 $600 $400 $400 $100$600 $600 $400 $400 $100 10% $545.45$545.45 $495.87$495.87 $300.53$300.53 $273.21$273.21 $ 62.09$ 62.09 $1677.15$1677.15 == PVPV00 of the Mixed Flowof the Mixed Flow
  • 63. 63 ““Group-At-A-Time” (#1)Group-At-A-Time” (#1) 0 1 2 3 4 55 $600 $600 $400 $400 $100$600 $600 $400 $400 $100 10% $1,041.60$1,041.60 $ 573.57$ 573.57 $ 62.10$ 62.10 $1,677.27$1,677.27 == PVPV00 of Mixed Flowof Mixed Flow [Using Tables][Using Tables] $600(PVIFA10%,2) = $600(1.736) = $1,041.60 $400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57 $100 (PVIF10%,5) = $100 (0.621) = $62.10
  • 64. 64 ““Group-At-A-Time” (#2)Group-At-A-Time” (#2) 0 1 2 3 4 $400 $400 $400 $400$400 $400 $400 $400 PVPV00 equals $1677.30.$1677.30. 0 1 2 $200 $200$200 $200 0 1 2 3 4 5 $100$100 $1,268.00$1,268.00 $347.20$347.20 $62.10$62.10 PlusPlus PlusPlus
  • 65. 65 Use the highlighted key for starting the process of solving a mixed cash flow problem Press the CF key and down arrow key through a few of the keys as you look at the definitions on the next slide Solving the Mixed FlowsSolving the Mixed Flows Problem using CF RegistryProblem using CF Registry
  • 66. 66 Defining the calculator variables: For CF0: This is ALWAYS the cash flow occurring at time t=0 (usually 0 for these problems) For Cnn:* This is the cash flow SIZE of the nth group of cash flows. Note that a “group” may only contain a single cash flow (e.g., $351.76). For Fnn:* This is the cash flow FREQUENCY of the nth group of cash flows. Note that this is always a positive whole number (e.g., 1, 2, 20, etc.). Solving the Mixed FlowsSolving the Mixed Flows Problem using CF RegistryProblem using CF Registry * nn represents the nth cash flow or frequency. Thus, the first cash flow is C01, while the tenth cash flow is C10.
  • 67. 67 Solving the Mixed FlowsSolving the Mixed Flows Problem using CF RegistryProblem using CF Registry Steps in the Process Step 1: Press CF key Step 2: Press 2nd CLR Work keys Step 3: For CF0 Press 0 Enter ↓ keys Step 4: For C01 Press 600 Enter ↓ keys Step 5: For F01 Press 2 Enter ↓ keys Step 6: For C02 Press 400 Enter ↓ keys Step 7: For F02 Press 2 Enter ↓ keys
  • 68. 68 Solving the Mixed FlowsSolving the Mixed Flows Problem using CF RegistryProblem using CF Registry Steps in the Process Step 8: For C03 Press 100 Enter ↓ keys Step 9: For F03 Press 1 Enter ↓ keys Step 10: Press ↓ ↓ keys Step 11: Press NPV key Step 12: For I=, Enter 10 Enter ↓ keys Step 13: Press CPT key Result: Present Value = $1,677.15
  • 69. 69 General Formula: FVn = PVPV00(1 + [i/m])mn n: Number of Years m: Compounding Periods per Year i: Annual Interest Rate FVn,m: FV at the end of Year n PVPV00: PV of the Cash Flow today Frequency ofFrequency of CompoundingCompounding
  • 70. 70 Julie Miller has $1,000$1,000 to invest for 2 years at an annual interest rate of 12%. Annual FV2 = 1,0001,000(1+ [.12/1])(1)(2) = 1,254.401,254.40 Semi FV2 = 1,0001,000(1+ [.12/2])(2)(2) = 1,262.481,262.48 Impact of FrequencyImpact of Frequency
  • 71. 71 Qrtly FV2 = 1,0001,000(1+ [.12/4])(4)(2) = 1,266.771,266.77 Monthly FV2 = 1,0001,000(1+ [.12/12])(12)(2) = 1,269.731,269.73 Daily FV2 = 1,0001,000(1+[.12/365])(365)(2) = 1,271.201,271.20 Impact of FrequencyImpact of Frequency
  • 72. 72 The result indicates that a $1,000 investment that earns a 12% annual rate compounded quarterly for 2 years will earn a future value of $1,266.77. Solving the FrequencySolving the Frequency Problem (Quarterly)Problem (Quarterly) N I/Y PV PMT FV Inputs Compute 2(4) 12/4 -1,000 0 1266.77
  • 73. 73 Solving the FrequencySolving the Frequency Problem (Quarterly Altern.)Problem (Quarterly Altern.) Press: 2nd P/Y 4 ENTER 2nd QUIT 12 I/Y -1000 PV 0 PMT 2 2nd xP/Y N CPT FV
  • 74. 74 The result indicates that a $1,000 investment that earns a 12% annual rate compounded daily for 2 years will earn a future value of $1,271.20. Solving the FrequencySolving the Frequency Problem (Daily)Problem (Daily) N I/Y PV PMT FV Inputs Compute 2(365) 12/365 -1,000 0 1271.20
  • 75. 75 Solving the FrequencySolving the Frequency Problem (Daily Alternative)Problem (Daily Alternative) Press: 2nd P/Y 365 ENTER 2nd QUIT 12 I/Y -1000 PV 0 PMT 2 2nd xP/Y N CPT FV
  • 76. 76 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. (1 + [ i / m ] )m - 1 Effective AnnualEffective Annual Interest RateInterest Rate
  • 77. 77 Basket Wonders (BW) has a $1,000 CD at the bank. The interest rate is 6% compounded quarterly for 1 year. What is the Effective Annual Interest Rate (EAREAR)? EAREAR = ( 1 + 6% / 4 )4 - 1 = 1.0614 - 1 = .0614 or 6.14%!6.14%! BW’s EffectiveBW’s Effective Annual Interest RateAnnual Interest Rate
  • 78. 78 Converting to an EARConverting to an EAR Press: 2nd I Conv 6 ENTER ↓ ↓ 4 ENTER ↑ CPT 2nd QUIT
  • 79. 79 1. Calculate the payment per period. 2. Determine the interest in Period t. (Loan balance at t-1) x (i% / m) 3. Compute principal paymentprincipal payment in Period t. (Payment - interest from Step 2) 4. Determine ending balance in Period t. (Balance - principal paymentprincipal payment from Step 3) 5. Start again at Step 2 and repeat. Steps to Amortizing a LoanSteps to Amortizing a Loan
  • 80. 80 Julie Miller is borrowing $10,000$10,000 at a compound annual interest rate of 12%. Amortize the loan if annual payments are made for 5 years. Step 1: Payment PVPV00 = R (PVIFA i%,n) $10,000$10,000 = R (PVIFA 12%,5) $10,000$10,000 = R (3.605) RR = $10,000$10,000 / 3.605 = $2,774$2,774 Amortizing a Loan ExampleAmortizing a Loan Example
  • 81. 81 Amortizing a Loan ExampleAmortizing a Loan Example End of Year Payment Interest Principal Ending Balance 0 --- --- --- $10,000 1 $2,774 $1,200 $1,574 8,426 2 2,774 1,011 1,763 6,663 3 2,774 800 1,974 4,689 4 2,774 563 2,211 2,478 5 2,775 297 2,478 0 $13,871 $3,871 $10,000 [Last Payment Slightly Higher Due to Rounding]
  • 82. 82 The result indicates that a $10,000 loan that costs 12% annually for 5 years and will be completely paid off at that time will require $2,774.10 in annual payments. Solving for the PaymentSolving for the Payment N I/Y PV PMT FV Inputs Compute 5 12 10,000 0 -2774.10
  • 83. 83 Using the AmortizationUsing the Amortization Functions of the CalculatorFunctions of the Calculator Press: 2nd Amort 1 ENTER 1 ENTER Results: BAL = 8,425.90 ↓ PRN = -1,574.10 ↓ INT = -1,200.00 ↓ Year 1 information only
  • 84. 84 Using the AmortizationUsing the Amortization Functions of the CalculatorFunctions of the Calculator Press: 2nd Amort 2 ENTER 2 ENTER Results: BAL = 6,662.91 ↓ PRN = -1,763.99 ↓ INT = -1,011.11 ↓ Year 2 information only
  • 85. 85 Using the AmortizationUsing the Amortization Functions of the CalculatorFunctions of the Calculator Press: 2nd Amort 1 ENTER 5 ENTER Results: BAL = 0.00 ↓ PRN =-10,000.00 ↓ INT = -3,870.49 ↓ Entire 5 Years of loan information
  • 86. 86 Usefulness of AmortizationUsefulness of Amortization 2.2. Calculate Debt OutstandingCalculate Debt Outstanding -- The quantity of outstanding debt may be used in financing the day-to-day activities of the firm. 1.1. Determine Interest ExpenseDetermine Interest Expense -- Interest expenses may reduce taxable income of the firm.