Outline 2: Time Value of Money & Introduction to Discount Rates & Rate of Return

1. Outline 2: Time Value of Money & Introduction to Discount Rates & Rate of Return 2.1 Future Values 2.2 Present Values 2.3 Multiple Cash Flows 2.4 Perpetuities and Annuities 2.5 Effective Annual Interest Rate 2.6 Loan Amortization Appendix on Time Value of Money

2. Future Values Future Value - Amount to which an investment will grow after earning interest. Compound Interest - Interest earned on interest. Simple Interest - Interest earned only on the original investment.

3. Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Interest Earned Per Year = 100 x .06 = $ 6

4. Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100.

5. Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years 12345 Interest Earned Value 100

6. Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years 12345 Interest Earned 6 Value 100 106

7. Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years 12345 Interest Earned 6 6 Value 100 106 112

8. Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years 12345 Interest Earned 6 6 6 Value 100 106 112 118

9. Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years 12345 Interest Earned 6 6 6 6 Value 100 106 112 118 124

10. Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today Future Years 12345 Interest Earned 6 6 6 6 6 Value 100 106 112 118 124 130 Value at the end of Year 5 = $130

11. Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance.

12. Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Interest Earned Per Year =Prior Year Balance x .06

13. Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Future Years 1 2 34 5 Interest Earned Value 100

14. Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Future Years 1 2 34 5 Interest Earned 6.00 Value 100 106.00

15. Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Future Years 1 2 34 5 Interest Earned 6.00 6.36 Value 100 106.00 112.36

16. Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Future Years 1 2 34 5 Interest Earned 6.00 6.36 6.74 Value 100 106.00 112.36 119.10

17. Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Future Years 1 2 34 5 Interest Earned 6.00 6.36 6.74 7.15 Value 100 106.00 112.36 119.10 126.25

18. Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today Future Years 1 2 34 5 Interest Earned 6.00 6.36 6.74 7.15 7.57 Value 100 106.00 112.36 119.10 126.25 133.82 Value at the end of Year 5 = $133.82

19. Future Values Future Value of $100 = FV

20. Future Values Future Value of any Present Value = FV where t= number of time periods r=is the discount rate

21. Future Values if t=4: FV = PV(1+r)(1+r) (1+r)(1+r) = PV(1+r)4 if t=10: FV = PV(1+r)(1+r)(1+r)(1+r)(1+r)(1+r)(1+r) (1+r)(1+r)(1+r) = PV(1+r)10

22. Future Values if t=n: FV = PV(1+r)(1+r) (1+r)(1+r)…(1+r) = PV(1+r)n if t=0: FV = PV(1+r) = PV(1+r)0 = PV

23. Future Values Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for five years?

24. Future Values Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for five years?

25. Future Values: FV with Compounding Interest Rates

26. Future Value: Manhattan Island Sale Peter Minuit bought Manhattan Island for $24 in 1626. Was this a good deal? To answer, determine $24 is worth in the year 2008 (correct from 2006), compounded at 12.5% (long-term average annual return on S&P 500): FYI - The value of Manhattan Island land is a very small fraction of this number.

27. Present Values Present Value Value today of a future cash flow. Discount Factor Present value of a $1 future payment. Discount Rate Interest rate used to compute present values of future cash flows.

28. Present Values

29. Present Values Since FV = PV (1+r) then solve for PV by dividing both sides by (1+r):

30. Present Values Example You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years?

31. Present Values Example You are twenty years old and want to have $1 million in cash when you are 80 years old (you can expect to live to one-hundred or more). If you expect to earn the long-term average 12.4% in the stock market how much do you need to invest now?

32. Present Values Discount Factor = DF = PV of $1 • Discount Factors can be used to compute the present value of any cash flow. • r is the discount rate (of return)

33. Present Value • The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable.

34. Present Value: PV of Multiple Cash Flows Example Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer?

35. Present Value: PV of Multiple Cash Flows • PVs can be added together to evaluate multiple cash flows.

36. Present Value: Perpetuities & Annuities Perpetuity A stream of level cash payments that never ends. Annuity Equally spaced level stream of cash flows for a limited period of time.

37. Present Value: Perpetuities & Annuities PV of Perpetuity Formula C = constant cash payment r = interest rate or rate of return

38. Present Value: Perpetuities & Annuities Example - Perpetuity In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%?

39. Present Value: Perpetuities & Annuities Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?

40. Present Value: Perpetuities & Annuities PV of Annuity Formula C = cash payment r = interest rate t = Number of years (periods) cash payment is received

41. Present Value: Perpetuities & Annuities If PV of Annuity Formula is: Then formula for annuity payment is:

42. Present Value: Perpetuities & Annuities Formula for annuity payment can be used to find loan payments. Just think of C as Payment, PV as loan amount, t as the number of months, and r must be the periodic loan r to coincide with the frequency of payments:

43. Present Value: Perpetuities & Annuities PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years.

44. Present Value: Perpetuities & Annuities Example - Annuity You are purchasing a car. You are scheduled to make 60 month installments of $500 for a $25,000 auto. Given an annual market rate of interest of 5% for a car loan, what is the price you are paying for the car (i.e. what is the PV)?

45. Present Value: Perpetuities & Annuities Example - Annuity You have just won the NJ lottery for $2 million over 25 years. How much is the “$2 million” NJ Lottery really worth at an opportunity cost rate of return of 12.4% - long-run annual stock market rate of return (ignoring income taxes)?

46. Present Value: Perpetuities & Annuities Example - Annuity Now what if you took the lump-sum based on a 5% discount rate by the State of New Jersey?

47. Perpetuities & Annuities Example - Future Value of annual payments You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account?

48. Perpetuities & Annuities Future Value of Ordinary Annuity:

49. Perpetuities & Annuities Present Value of Ordinary Annuity:

50. Effective Interest Rates Effective Annual Interest Rate - Interest rate that is annualized using compound interest. r = annual or nominal rate of interest or return m= number of compounding periods per year rnom/m=also known as the periodic interest rate