1 Chapter 4: Time Value of Money Copyright, 2000 Prentice Hall ©Author Nick Bagley, bdellaSoft, Inc. Objective Explain the concept of compounding and discounting and to provide examples of real life applications

2 Value of Investing $1 –Continuing in this manner you will find that the following amounts will be earnt:

3 Value of $5 Invested More generally, with an investment of $5 at 10% we obtainMore generally, with an investment of $5 at 10% we obtain

4 Future Value of a Lump Sum

5 Example: Future Value of a Lump Sum Your bank offers a CD with an interest rate of 3% for a 5 year investments. You wish to invest $1,500 for 5 years, how much will your investment be worth?

6 Present Value of a Lump Sum

7 Example: Present Value of a Lump Sum You have been offered $40,000 for your printing business, payable in 2 years. Given the risk, you require a return of 8%. What is the present value of the offer?You have been offered $40,000 for your printing business, payable in 2 years. Given the risk, you require a return of 8%. What is the present value of the offer?

8 Solving Lump Sum Cash Flow for Interest Rate

9 Example: Interest Rate on a Lump Sum Investment If you invest $15,000 for ten years, you receive $30,000. What is your annual return?If you invest $15,000 for ten years, you receive $30,000. What is your annual return?

10 Review of Logarithms The basic properties of logarithms that are used by finance are:The basic properties of logarithms that are used by finance are:

11 Review of Logarithms The following properties are easy to prove from the last ones, and are useful in financeThe following properties are easy to prove from the last ones, and are useful in finance

12 Solving Lump Sum Cash Flow for Number of Periods

13 Effective Annual Rates of an APR of 18%

14 The Frequency of Compounding Note that as the frequency of compounding increases, so does the annual effective rateNote that as the frequency of compounding increases, so does the annual effective rate What occurs as the frequency of compounding rises to infinity?What occurs as the frequency of compounding rises to infinity?

15 The Frequency of Compounding

16 The Frequency of Compounding

17 Derivation of PV of Annuity Formula: Algebra. 1 of 5

18 Derivation of PV of Annuity Formula: Algebra. 2 of 5

19 Derivation of PV of Annuity Formula: Algebra. 3 of 5

20 Derivation of PV of Annuity Formula: Algebra. 4 of 5

21 Derivation of PV of Annuity Formula: Algebra. 5 of 5

22 PV of Annuity Formula

23 PV Annuity Formula: Payment

24 PV Annuity Formula: Number of Payments

25 Annuity Formula: PV Annuity Due

26 Derivation of FV of Annuity Formula: Algebra

27 FV Annuity Formula: Payment

28 FV Annuity Formula: Number of Payments

29 Perpetual Annuities / Perpetuities Recall the annuity formula:Recall the annuity formula: Let n -> infinity with i > 0:

30 Mortgage: The payment We will examine this problem using a financial calculatorWe will examine this problem using a financial calculator The first quantity to determine is the amount of the loan and the pointsThe first quantity to determine is the amount of the loan and the points

31 Calculator Solution This is the monthly repayment

32 Calculator Solution 60 Months

33 Summary of Payments The family has made 60 payments = $ *12*5 = $161,878.64The family has made 60 payments = $ *12*5 = $161, Their mortgage repayment = 450, , = $31,255.39Their mortgage repayment = 450, , = $31, Interest = payments - principle reduction = 161, , = $130,623.25Interest = payments - principle reduction = 161, , = $130,623.25

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38 $10,000 $11,000 ¥ 1,000,000¥ 1,030,000¥ Time 10% $/$ (direct) 0.01 $/¥ 3% ¥ / ¥ ? $/¥ U.S.A.Japan

39 $10,000 $11,124 $11,000 ¥ 1,000,000¥ 1,030,000¥ Time 10% $/$ (direct) 0.01 $/¥ 3% ¥ / ¥ $/¥ U.S.A.Japan

40 $10,000 $10,918 ¥ $11,000 ¥ 1,000,000¥ 1,030,000¥ Time 10% $/$ (direct) 0.01 $/¥ 3% ¥ / ¥ $/¥ U.S.A.Japan

41 $10,000 $11,000 ¥ 1,000,000¥ 1,030,000¥ Time 10% $/$ (direct) 0.01 $/¥ 3% ¥ / ¥ $/¥ U.S.A.Japan