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Chapter 4 The Time Value Of Money

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Chapter 4 The Time Value Of Money

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    1. 1 Chapter 4 The Time Value Of Money

    2. Finance 311 2 Introduction This chapter introduces the concepts and skills necessary to understand the time value of money and its applications.

    3. Finance 311 3 TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important concept -- Almost everything from this point on in finance based upon understanding this concept. You must get this material down “cold.” You can do problems using formulas, calculators, and spreadsheets. We will primarily use financial calculators. No tables will be used. Please have your HP 10 b II financial calculators with you in class.

    4. Finance 311 4 Key Terms Future Value (FV) Future Value of an Annuity (FVAN) Future Value of an Annuity Due (FVAND) Present Value (PV) Present Value of an Annuity (PVAN) Present Value of an Annuity Due (PVAND)

    5. Finance 311 5 Examples of Problems Solved Sinking Fund Problems Capital Recovery Problems Loan Amortization Problems Retirement Planning Deferred Annuities Capital Budgeting

    6. Finance 311 6 Notation i (I/YR) to denote the interest rate per period n (N) to denote the number of periods PMT to denote cash payments (Annuity) PV to denote the present value dollar amount T to denote the tax rate t to denote time PV0 = principal amount at time 0 FVn = future value n time periods from time 0

    7. Finance 311 7 Simple Interest Interest = Principal X Rate X Time I = PV0 x i x t

    8. Finance 311 8 Future Value of a Cash Flow At the end of year n for a sum compounded at interest rate i is FVn = PV0 ( 1 + i )n Formula

    9. Finance 311 9 FINANCIAL CALCULATOR There are usually at least five input keys N I/yr PV PMT FV There are other keys that we will discuss later (for instance, P/YR or payments per year). Also, I strongly prefer that you use the HP 10b II calculator.

    10. Finance 311 10 Assume that you invest $1,000 at an annual interest rate of 6 percent, How much would you have at the end of 12 years?

    11. Finance 311 11 Present Value of a Cash Flow PV0 = FVn 1 Formula ( 1 + i ) n PVIF i , n = 1___ FVIFi,n

    12. Finance 311 12 Example Using Formula What is the PV of $100 one year from now if the i = 12%, compounded annually ? PV 0 = $100 x 1/ ( 1 + .12 ) 1 = $100 x 1/ ( 1.12) = $100 x ( .89286 ) = $ 89.286

    13. Finance 311 13 Example Using Financial Calculator P/YR =1 FV = 100 PMT = 0 n = 1 I = 12% PV = ?

    14. Finance 311 14 Example Using EXCEL

    15. Finance 311 15 Another Example Using the Formula What is the PV of $100 one year from now with 12 percent interest compounded monthly?

    16. Finance 311 16 Example using Financial Calculator Set P/YR to 12 by pressing 12 then pressing yellow then press P/YR N = 1 yellow XP/YR = 12 (or you can put 12 in the N function directly) FV = 100 PMT = 0 I/YR = 12 Then press PV to get (88.74) Then set the P/YR back to 1.

    17. Finance 311 17 Annuity A series of equal $ CF’s for a specified number of periods Ordinary annuity is where the CF’s occur at the end of each period Annuity due is where the CF’s occur at the beginning of each period - The easiest way to solve an annuity due problem is to use your begin (BEG) button on your calculator. But do not forget to set the calculator back to end of period after your calculation.

    18. Finance 311 18 Future Value of an Ordinary Annuity FVIFA i , n = ( 1 + i ) n - 1 Formula i Calculator - use your PMT key

    19. Finance 311 19 Present Value of an Ordinary Annuity 1- 1 ( 1+ i ) n Formula PVIFA i , n = i

    20. Finance 311 20 Annuity Due Future Value of an Annuity Due FVAND n = PMT ( FVIFA i , n ) ( 1 + i ) Useful for Retirement Planning Present Value of an Annuity Due PVAND 0 = PMT ( PVIFA i , n ) ( 1 + i ) Useful in leasing analysis. Please use your calculator and your BEG button. But do not forget to go back to the END mode after you complete the problem.

    21. Finance 311 21 Present Value of a Perpetuity PVPER 0 = PMT/ i Expect to receive $100 a year forever. Interest rate is 5%. PV =?

    22. Finance 311 22 FINDING INTEREST RATES OR GROWTH RATES You invest $300 and receive $483.15 five years later. What rate of return have you earned? 10.00% You invest $300 and receive $79.14 a year for each of five years. What rate of return have you earned? 10.00% The EPS of Tiger Incorporated grew from $.75 in 1997 to $1.85 in 2004. What was the annual rate of growth? 13.77%

    23. Finance 311 23 Simple and Compound Interest Simple Interest interest paid on the principal sum only Compound Interest interest paid on the principal and on prior interest which has not been paid or withdrawn

    24. Finance 311 24 Other than annual compounding You have $100 to invest: Bank A pays 6% compounded annually… $106.00 Bank B pays 6% compounded semi-annually…$106.09 Bank C pays 6% compounded quarterly…$106.14 Bank D pays 6% compounded monthly…$106.17 Bank E pays 6% compounded daily…$106.18 How much would you have at the end of one year in each bank?

    25. Finance 311 25 Present value PV 0 = FV n ( 1 +

    26. Finance 311 26 A Realistic Problem You want to purchase a car that costs $23,000. You have two financing alternatives. A. Receive a $2,000 rebate on the vehicle. But make (beginning of the month) monthly payments at a 5.4% annual rate for a period of three years. B. Obtain 0% (interest) financing for the three years. Make monthly payments at the beginning of the month. Your cost will be $23,000. What are the monthly payments under each alternative? Which alternative is cheaper?

    27. Finance 311 27 Effective annual rate of interest Effective rate of interest i eff = ( 1 + i nom/ m )m - 1 Bank X pays 5.60% compounded quarterly Bank Y pays 5.57% compounded daily Where do you invest for one year? 5.7187% for the quarterly compounding 5.7276% for the daily compounding

    28. Finance 311 28 Interest Compounded More Frequently Than Once Per Year

    29. Finance 311 29 Compounding and Effective Rates Effective annual rate of interest ieff = (1 + inom/m)m – 1 Rate of interest per compounding period im = (1 + ieff)1/m – 1

    30. Finance 311 30 LOAN AMORTIZATION TABLE Assume that you borrow $10,000 for five years at an annual interest rate of 12 percent. You make equal annual payments that include interest and principal. Prepare a loan amortization schedule or table.

    31. Finance 311 31 LOAN AMORTIZATION

    32. Finance 311 32 ASSUME THAT YOU BORROW $125,000 FOR THIRTY YEARS TO BUY A HOUSE AND THE INTEREST RATE IS 6.5%. What are the monthly payments? How much of the first payment goes to pay off the loan? How much do you pay back over the life of the loan? If the loan were 15 years at 6.25%, what would the monthly payments be? How much do you pay back over the life of the loan? Use your AMORT button to do these calculations.

    33. Finance 311 33 DEALING WITH AN UNEVEN CASH FLOW STREAM

    34. Finance 311 34 FINDING THE NET PRESENT VALUE OF AN INVESTMENT (NPV) NPV = PV of cash flows - cash flow in year 0 On calculator use CFj key and NPV key Use your Nj button for the same cash flows. IRR = interest rate that makes the NPV=0

    35. Finance 311 35 Example Using EXCEL

    36. Finance 311 36 Conclusion Future Value Future Value of an Annuity Future Value of an Annuity Due Present Value Present Value of an Annuity Present Value of an Annuity Due

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