1. Present Value Present value is the current value of a future sum

2. If you receive $100 one year from now, what is the present value (PV) of that $100 if your opportunity cost is 6%?

3. Int = 6 N = 1 FV = 100 PV = 94.34 0 1 0 1 PV = FV = 100

4. If you receive $100 five years from now, what is the present value (PV) of that $100 if your opportunity cost is 6%?

5. PV = FV / (1 + i) n PV = 100 / (1.06) 5 = $ 74.73 0 5 0 5 PV = FV = 100

6. Example What is the present value (PV) of $500 to be received 10 years from now if the opportunity cost is 6%?

7. PV = FV / (1 + i) n PV = 500 / (1.06) 10 = 500(0.558) = $279 0 10 0 10 PV = FV = 500

8. Periods per year = 12FV = 500 I = 9.6PV = -100 N = ? months Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? 0 ? 0 ? PV = -100 FV = 500

9. Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? PV = FV / (1 + i) n 100 = 500 / (1+.008) N 5 = (1.008) N ln 5 = ln (1.008) N ln 5 = N ln (1.008) 1.60944 =.007968 N N = 202 months

10. The present value (PV) of $500 to be received 10 years from now is $279

11. Investment that has two cash flows in different time periods What is the present value (PV) of an investment that results in $500 to be received in 5 years and $1000 to be received in 10 years if the opportunity cost is 4%?

12. Present value (PV) is the sum of the following: 500 / (1.04) 5 and 1000 / (1.04) 10 = 500(0.822) + 1000(0.676) = 411 + 676 = $1087

13. Annuities

14. Annuity A sequence of equal cash flows, occurring at the end of each period

15. Annuity A sequence of equal cash flows, occurring at the end of each period 01 234 Annuities

16. Examples of Annuities: If you buy a bond with interest paid out semi-annually, you will receive equal semi-annual coupon interest payments over the life of the bond If you borrow money to buy a house or a car, you will pay a stream of equal payments

17. Ordinary Annuity Cash flows of an ordinary annuity occur at the end of each period Annuity Due Cash flows of an annuity due occur at the beginning of each period

18. Compound Annuity Compound annuity involves depositing or investing an equal sum of money at the end of each year for a certain number of years

19. If we invest $1,000 at the end of each year at 8% for purchasing a car. How much would you have after 3 years ?

20. Periods / year = 1I = 8N = 3 PMT = -1,000 FV = $3246.40 If you invest $1,000 each year at 8%, how much would you have after 3 years? 0 1 2 3 1000 1000 1000 1000 1000 1000

21. Time line A horizontal line on which the present time period (t=0) is at the left most end Future time periods are shown along the line moving from left to right

22. Time line The amounts of cash flow are shown below the line Positive values represent cash inflows and negative values represent cash outflows at various time periods

23. Future value of compound annuity The future value of a compound annuity is as follows:

24. The future-value of the annuity at the end of the year n is denoted by The amount of annuity payment deposited at the end of each year is denoted by PMT i is the annual rate of interest n is the number of years for which the annuity will last

25. Future value of compound annuity The future value of a compound annuity can be expressed as follows:

26. Example We need an amount of $10,000 for university education in 8 years. How much do we need to deposit at the end of each year in the bank at 6% interest to have the money ready for payment to the University?

28. Present value of an annuity The present value of an annuity is as follows:

29. The present value of the annuity The amount of annuity received at the end of each year is denoted by PMT i is the discount rate n is the number of years for which the annuity will last

30. Present value of an annuity The present value of an annuity can be expressed as follows:

31. Example What is the present value of a 10 year $1000 annuity discounted back to the present at 5 percent?

33. Annuity Due Cash flows of an annuity due occur at the beginning of each period

34. Future value of an annuity due As an annuity due, shifts the payments from the end of the year to the beginning of the year, we compound the cash flows for an another year

35. Future value of an annuity due Future value of an annuity due is computed as follows:

36. Periods / year = 1I = 8N = 3 PMT = -1,000 FV = 3246.40(1.08) If you invest $1,000 each year in an annuity due at 8%, how much would you have after 3 years? 0 1 2 3 1000 1000 1000 1000 1000 1000

37. Future value of an annuity due Future value of this annuity due is $3,506.11 In an annuity due, compounding is for 1 additional year

38. Present value of an annuity due Present value of an ordinary annuity is multiplied by (1+i), which cancels out 1 year’s discounting:

39. Example What is the present value of a 10 year $1000 annuity due discounted back to the present at 5 percent?

41. In the discounting of an annuity due, the cash flows are discounted for 1 less year

42. Amortized Loan An amortized loan is a loan that is repaid in equal installments over time

43. Example We borrow US$ 20,000 for renovation of a house. The loan has to be repaid in 4 equal installments at the end of each of the 4 years. The lender charges an interest rate of 15% on the outstanding loan. What is the installment associated with repayment of the loan?

44. The installment for repaying the principal and interest on the loan in 4 years is US$7005.25

45. Uneven cash flows A project involving uneven cash flows over several years For example, investments in fixed assets

46. We discount each of the cash flows back to the present. We add the positive cash flows and subtract the negative cash flows

47. Perpetuity A perpetuity is an investment that pays a constant permanent currency amount every year

48. The equation representing a perpetuity is as follows:

49. Example We have invested in a US$ 1000 perpetuity. Our discount rate is 9 percent. What is the present value of the perpetuity?

50. The present value of the perpetuity is as follows: