Chapter 4 – Introduction to Valuation: The Time Value of Money

1. Chapter 4 – Introduction to Valuation: The Time Value of Money

2. Time value of money • When money is invested, the amount of money grows over time. • Although money will lose value over time due to inflation, the amount of money in an account that earns a positive RoR will be greater in the future than what it is today. • Thus the money in the accounts has different values at different points in time • This is what “Time Value of Money” refers to.

3. Time value of money • The RoR should compensate investors for opportunity cost, inflation and risk • The increased money should make up for the value lost due to inflation due to the Inflation Premium • The Real Risk Free Rate (r*) and Risk Premium (RP) increase the amount of money more than is necessary to compensate for inflation and thus produce a profit for the investor.

4. Time Lines / Cash Flow Diagrams • For the rest of this course, we will deal with cash flows that occur over some period of time. • It is helpful (recommended) that you depict these cash flows graphically • To do this we will use time lines. This will: • Visually show the cash flows, both positive and negative, so you have a clear idea of what is happening • It shows what you know and what you don’t know about a problem • It helps us decide what we need to find and do in order to solve the problem.

5. Time Lines / Cash Flow Diagrams • Your basic time line will look something like this. 0 2 3 1 The Future  Today • Note: The time units can be what ever they need to be; days, weeks, months, years, etc.

6. Future Value • Future Value - The amount an investment grown when it earns a positive RoR. • Compounding - The process of going from today's values (present value) to values at some point in the future. (future value) • Compounded Interest – Interest earned on interest In general: FVn = PV(1+i)n

7. FVn = PV(1+i)n PV = Present Value of the principle or investment i = Interest Rate n = number of periods FV = Future Value of the investment This includes the amount invested plus the return / profit

8. Two ways to Find Future Values • Solve the equation using a regular Calculator • Use a financial calculator.

9. Example: Suppose you deposit $100 in a bank account that pays 10% interest each year. How much do you have at the end of the third year?

10. First Year Starting Balance = Interest = Ending Balance = Second Year Starting Balance = Interest = Ending Balance = Third Year Starting Balance = Interest = Ending balance = Or

11. So for our example 0 2 3 1 The Future  Today Beginning Balance Interest Earned Ending Balance

12. Solve by calculator: • Our Example: n = PV = i = So FV3 =

13. Using a financial calculator: • Important! • Set P/Y (payments per year) to 1 • Clear the register • Enter 3, then press n • Enter 10 (calculator interprets this as 10%), then press i • Enter 100, then press PV • We have no “pmt”, so enter 0, then press pmt • Press FV, and voila! FV = ___________. • Why is FV Negative? • FV = PV(1+i)n

14. Example What is the FV of $5000 @ 6% after 5years?

15. Present Value • Present Value – The value today of money to be received in the future. • Discounting – The process of going from Future Values to Present Values (the reverse of compounding) • Opportunity Cost Rate – The rate of return on the best available alternative. In general: PV = FVn / (1+i)n

16. Two ways to Find Present Values • Solve the equation using a regular Calculator • Use a financial calculator.

17. Example: • What is the present value of $100 due you in 3 years if you assume the opportunity cost rate of 10% Draw a time line _________________________________ PV = FVn/ (1+i)n

18. PV = FVn / (1+i)n Using a regular calculator for our example n = FVn = i = PV =

19. Using a financial calculator: • Important! • Set P/Y (payments per year) to 1 • Clear the register • Enter 3, then press n • Enter 10 (calculator interprets this as 10%), then press i • Enter 100, then press FV • We have no “pmt”, so enter 0, then press pmt • Press PV, and voila! PV = ___________. • Why is PV Negative? • FV = PV(1+i)n

20. Example: • What is the PV of $5000 discounted for 3 years @ 12%?

21. Example reword: • You are considering purchasing a security that promises to pay $5,000 three years from now. What is this securities value today if your opportunity cost rate is 12%?

22. Using the Financial Valuation to make a decision • You would like to buy a new automobile. You have $50,000, but the car costs $68,000. If you can earn 9 percent, how much do you have to invest today to buy the car in two years? • Need to know the present value of $68,000 • PV = 68,500 / 1.092 • PV = $57,655.08 • You are still short $7,655.08

23. Solving for Time and Interest • Example: Your broker proposed an investment that will pay you $1,000 one year from now for an initial cost of $900 today. What is the annual return on this investment? Draw a time line 1 0 r = ?% FV = $1,000 PV = $900

24. Solving for Time and Interest • Solve for r using the PV or FV formula FVn = PV(1+i)n

25. Solving for Time and Interest Using a financial Calculator • Important: • Set P/Y (payments per year) to 1 • Clear register • Enter, 1, then press n • Enter, -900 then press PV • Enter, 1,000 then press FV • Enter, 0, then press pmt • Press i, and viola i = ______________ “Time Value of Money” Menu n i PV pmt FV

26. Solving for Time and Interest • Example: Your broker proposed an investment that will pay you $1,000 two years from now for an initial cost of $900 today. What is the annual return on this investment? Draw a time line 1 2 0 r = ?% FV = $1,000 PV = $900

27. Solving for Time and Interest • Solve for r using the PV or FV formula FVn = PV(1+i)n

28. Solving for Time and Interest Using a financial Calculator • Important: • Set P/Y (payments per year) to 1 • Clear register • Enter, 2, then press n • Enter, -900 then press PV • Enter, 1,000 then press FV • Enter, 0, then press pmt • Press i, and viola i = ______________ “Time Value of Money” Menu n i PV pmt FV

29. Solving for Time and Interest • Example: Suppose you invest in a 5 year Certificate of Deposit that cost you $7,500 today and pays you $10,000 in 5 years. What is the return (each year) that you will earn on this investment? Draw a time line r = ?% 5 0 FV = $10,000 PV = $7,500

30. Solving for Time and Interest • Solve for r using the PV or FV formula FVn = PV(1+i)n

31. Solving for Time and Interest Using a financial Calculator • Important: • Set to “End” Mode • Set P/Y (payments per year) to 1 • Clear register • Enter, 5, then press n • Enter, -7,500 then press PV • Enter, 10,000 then press FV • Enter, 0, then press pmt • Press i, and viola i = ______________ “Time Value of Money” Menu n i PV pmt FV

32. Solving for Time and Interest • Example: How long will it take to double an investment of $1000 if you can invest it at a rate of 6%? Draw a time line r = 6% n 0 FV = $2,000 PV = $1,000

33. Solving for Time and Interest Using a financial Calculator • Important: • Set to “End” Mode • Set P/Y (payments per year) to 1 • Clear register • Enter -1,000, then press PV • Enter -2,000, then press FV • Enter 6, then press i • Enter, 0, then press pmt • Press n, and viola n = ______________ “Time Value of Money” Menu n i PV pmt FV

34. Other Key Points • Compounding means we are making the number bigger, we are growing it. • FV are bigger than PV’s. Check your answers • Discounting means we are shrinking the number • PV are smaller than FV. Check your answers.