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# Time Value of Money

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1. Time Value of Money Lecture No.3

2. Chapter 2Time 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

3. What Do We Need to Know? • To make such comparisons (the lottery decision problem), we must be able to compare the value of money at different point in time. • To do this, we need to develop a method for reducing a sequence of benefits and costs to a single point in time. Then, we will make our comparisons on that basis.

4. Time Value of Money • Money has a time value because it can earn more money over time (earning power). • Money has a time value because its purchasing power changes over time (inflation). • Time value of money is measured in terms of interest rate. • Interest is the cost of money—a cost to the borrower and an earning to the lender

5. Delaying Consumption

6. Which Repayment Plan?

7. Cash Flow Diagram

8. End-of-Period Convention Interest Period 0 1 End of interest period Beginning of Interest period 0 1

9. Methods of Calculating Interest • Simple interest: the practice of charging an interest rate only to an initial sum (principal amount). • Compound interest: the practice of charging an interest rate to an initial sum and to any previously accumulated interest that has not been withdrawn.

10. Simple Interest • P = Principal amount • i= Interest rate • N = Number of interest periods • Example: • P = \$1,000 • i = 8% • N = 3 years

11. Simple Interest Formula F = P+(iP)N where P: principal amount i: simple interest rate N: number of interest periods F: total amount accumulated at the end of period N

12. Compound Interest • Compound interest: the practice of charging an interest rate to an initial sum and to any previously accumulated interest that has not been withdrawn.

13. Compound Interest • P = Principal amount • i= Interest rate • N = Number of interest periods • Example: • P = \$1,000 • i = 8% • N = 3 years

14. Compounding Process \$1,080 \$1,166.40 0 \$1,259.71 1 \$1,000 2 3 \$1,080 \$1,166.40

15. \$1,259.71 2 1 0 3 \$1,000

16. Compound Interest Formula

17. Some Fundamental Laws The Fundamental Law of Engineering Economy

18. Practice Problem: Warren Buffett’s Berkshire Hathaway • Went public in 1965: \$18 per share • Worth today (August 22, 2003): \$76,200 • Annual compound growth: 24.58% • Current market value: \$100.36 Billion • If he lives till 100 (current age: 73 years as of 2003), his company’s total market value will be ?

19. Market Value • Assume that the company’s stock will continue to appreciate at an annual rate of 24.58% for the next 27 years.

20. Example • In 1626 the Indians sold Manhattan Island to Peter Minuit • of the Dutch West Company for \$24. • If they saved just \$1 from the proceeds in a bank account • that paid 8% interest, how much would their descendents • have now? • As of Year 2003, the total US population would be close to • 275 millions. If the total sum would be distributed equally • among the population, how much would each person receive?

21. Excel Solution =FV(8%,377,0,1) = \$3,988,006,142,690

22. Practice Problem • Problem Statement If you deposit \$100 now (n = 0) and \$200 two years from now (n = 2) in a savings account that pays 10% interest, how much would you have at the end of year 10?

23. Solution F 0 1 2 3 4 5 6 7 8 9 10 \$100 \$200

24. Practice problem • Problem Statement Consider the following sequence of deposits and withdrawals over a period of 4 years. If you earn 10% interest, what would be the balance at the end of 4 years? ? \$1,210 1 4 0 2 3 \$1,500 \$1,000 \$1,000

25. ? \$1,210 0 1 3 2 4 \$1,000 \$1,000 \$1,500 \$1,100 \$1,000 \$2,981 \$1,210 \$2,100 \$2,310 + \$1,500 -\$1,210 \$1,100 \$2,710

26. Solution