1. Besley Ch. 61 Time Value of Money

2. Besley Ch. 62 Cash Flow Time Lines CF Time Lines are a graphical representation of cash flows associated with a particular financial option. Time: 012 34 One Period 5% Interest Rate (per period) CF:-100 ? + indicates Cash Inflow - indicates Cash Outflow NOTE: Each tick mark denotes the end of one period.

3. Besley Ch. 63 Cash Flow Time Lines Outflow: A payment or disbursement of cash, such as for investment, or expenses. Inflow: A receipt of cash, can be in the form of dividends, principal, annuity payments, etc.

4. Besley Ch. 64 Future Value (FV) Future Value (FV): The ending value of a cash flow (or series of cash flows) over a given period of time, when compounded for a specified interest rate. Compounding: The process of calculating the amount of interest earned on interest. 012 3 5% -100 ? Present Value (PV) FV

5. Besley Ch. 65 FV Calculations Given: PV:$100 i:5% n: 1 INT:(PV x i) Solution: FV n = PV+INT = PV + (PV x i) = PV(1+i) 012 3 5% -100 ? Solution: FV n = 100+INT = 100 + (100 x 5%) = 100(1+ 0.05) = 105

6. Besley Ch. 66 FV Calculations FV 1 = PV(1+i) FV 2 = FV 1 (1+i) = [PV(1+i)](1+i) FV 3 = FV 2 (1+i) = {[PV(1+i)](1+i)}(1+i) FV n = PV(1+i) n 012 3 5% -100 ? INT 1 INT 2 INT 3 5.00 5.255.51  =15.76 Value at end of Period: 105.00 110.25115.76

7. Besley Ch. 67 FV Calculations Three ways to calculate Time Value of Money (TVM) solutions:  Numerical Solution: Calculate solution with formula  Tabular Solutions: Use Interest Factor tables to calculate  Financial Calculator Solutions: Use calculator

8. Besley Ch. 68 Numerical Solution Future Value Interest Factor for i and n (FVIF i,n ) is the factor by which the principal grows over a specified time period (n) and rate (i). FVIF i,n = (1+i) n Given:Solution: PV:$1FV n = PV(1+i) n = PV(FVIF i.n ) i:5%FV 5 = 1(1+.05) 5 n:5FV 5 = 1.2763

9. Besley Ch. 69 Tabular Solution FVIF i,n = (1 + i) n Given: i:5% n:5 FV n = PV(1+i) n = PV(FVIF i.n )

10. Besley Ch. 610 Financial Calculator Points to remember when using your Financial Calculator:  Check your settings:  END / BGN  P/Y  Clear TVM memory  Five Variables (N, I/Y, PV, PMT, FV) - with any 4 the 5th can be calculated Given: N:5 I/Y:5% PV:$1 PMT:0 FV:? Input: Output: NI/YPV PMT FV 550 1.2763

11. Besley Ch. 611 Present Value (PV) Present Value (PV): The current value of a future cash flow (or series of cash flows), when discounted for a specified period of time an rate. Discounting: The process of calculating the present value of a future cash flow or series of cash flows. 012 3 5% ?105 PV FV

12. Besley Ch. 612 PV Calculations Given: FV:105 i:5% n: 1 Solution: FV n = PV(1+i) n Solve for PV PV n = FV n / (1+i) n = FV n [1/(1+i) n ] = FV n (PVIF i,n ) 012 3 5% ?105 Solution: PV n = 105/(1+0.05) 1 = 100

13. Besley Ch. 613 PV Calculations FV 2 = FV 3 /(1+i) FV 1 = FV 2 /(1+i) = [FV 3 /(1+i)]/(1+i) PV = FV 1 /(1+i) = {[FV 1 /(1+i)]/(1+i)}/(1+i) PV n = FV n 012 3 5% ? Value at end of Period: 1 (1+i) n 115.7625  1.05 110.25  1.05 105.00 Given FV

14. Besley Ch. 614 Numerical Solution Present Value Interest Factor for i and n (PVIF i,n ) is the discount factor applied to the FV in order to calculate the present value for a specific time period (n) and rate (i). PVIF i,n = 1/(1+i) n Given:Solution: FV:$1PV n = FV[1/(1+i) n ] = FV(PVIF i.n ) i:5%PV 5 = 1 [1/(1+.05) 5 ] n:5PV 5 = 0.7835

15. Besley Ch. 615 Tabular Solution PVIF i,n = (1 + i) n Given: i:5% n:5 PV n = FV[1/(1+i) n ] = FV(PVIF i.n )

16. Besley Ch. 616 Financial Calculator Given: N:5 I/Y:5% PV:? PMT:0 FV:-1 Input: Output: NI/YPV PMT FV 55.7835 0

17. Besley Ch. 617 Annuities Annuity: a series of equal payments made at specific intervals for a specified period. Types of Annuities: –Ordinary (Deferred) Annuity - is an annuity in which the payments occur at the end of each period. –Annuity Due - is an annuity in which the payments occur at the beginning of each period.

18. Besley Ch. 618 FV Ordinary Annuities Example: You decide that starting a year from now you will deposit $1,000 each year in a savings account earning 8% interest per year. How much will you have after 4 years? FV n =PV(1+i) n FVA n = PMT(1+i) 0 + PMT(1+i) 1 + PMT(1+i) 2 +... + PMT(1+i) n-1 01234 1,000 1,000.00 1,080.00 1,166.40 1,259.71 4,506.11  8%

19. Besley Ch. 619 FV Ordinary Annuities FVA n represents the future value of an ordinary annuity over n periods. FVA n = PMT(1+i) 0 + PMT(1+i) 1 + PMT(1+i) 2 +... + PMT(1+i) n-1 = PMT  (1+i) n-t = PMT  (1+i) t = PMT (1+i) n-1 = PMT  n t=1  n  n (1+i) n - 1 i

20. Besley Ch. 620 FV Ordinary Annuities Future Value Interest Factor for an Annuity (FVIFA i,n ) is the future value interest factor for an annuity (even series of cash flows) of n periods compounded at i percent. FVIFA i,n = (1+i) n-t = (1+i) n - 1 i  n t=1

21. Besley Ch. 621 Numerical Solution Given: PMT:$1,000 I:8% N:4 01234 1,000 1,000.00 8%(1+i) n - 1 i FVA n = PMT Solution: FVA n = PMT {[(1+i) n – 1]/i} = 1,000 {[(1+0.08) 4 – 1]/0.08] = 1,000 {4.5061} = $4,506.11

22. Besley Ch. 622 Tabular Solution Given: PMT:$1,000 I:8% N:4 FVA n = PMT(FVIFA i,n ) PVIFA i,n

23. Besley Ch. 623 Financial Calculator Given: N:4 I/Y:8% PV:0 PMT:1,000 FV:? Input: Output: NI/YPV PMT FV 48 4,506.11 1,000 0

24. Besley Ch. 624 FV Annuity Due Example: You decide that starting today you will deposit $1,000 each year in a savings account earning 8% interest per year. How much will you have after 4 years? FV n =PV(1+i) n 01234 1,000 1,166.40 1,259.71 1,360.49 4,866.60  8% 1,080.00 FVA(Due) n =PMT (1+i) t = PMT (1+i) n-t x (1+i)  n t=1  n (1+i) n - 1 i FVA(Due) n = PMT x (1+i)

25. Besley Ch. 625 FV Annuity Due FVA(Due) n represents the future value of an annuity due over n periods. FVA(Due) n = PMT (1+i) t  n t=1 = PMT x (1+i) (1+i) n - 1 i  n t=1 = PMT (1+i) n-t x (1+i)

26. Besley Ch. 626 FV Annuity Due Future Value Interest Factor for an Annuity Due (FVIFA(Due) i,n ) is the future value interest factor for an annuity due of n periods compounded at i percent. FVIFA(Due) i,n = x (1+i) (1+i) n - 1 i

27. Besley Ch. 627 Numerical Solution Given: PMT:$1,000 - BGN I:8% N:4 01234 1,000 8% Solution: FVA(Due) n = PMT [{((1+i) n – 1)/i}x (1+i)} = 1,000 [{((1+0.08) 4 – 1)/0.08}x (1+0.08)} = 1,000 {4.8666} = $4,866.60 FVA(Due) i,n = PMT x (1+i) (1+i) n - 1 i

28. Besley Ch. 628 Tabular Solution Given: PMT:$1,000 - BGN I:8% N:4 FVA(Due) n = PMT[(FVIFA i,n )(1+i)] PVIFA i,n

29. Besley Ch. 629 Financial Calculator Given: N:4 I/Y:8% PV:0 PMT:1,000 - BGN FV:? Input: Output: NI/YPV PMT FV 48 -4,866.60 1,000 0 BGN

30. Besley Ch. 630 PV Ordinary Annuities Example: You decide that starting a year from now you will withdraw $1,000 each year for the next 4 years from a savings account which earns 8% interest per year. How much do you need to deposit today? PVA n = PMT[1/(1+i) 1 ] + PMT[1/(1+i) 2 ] +... + PMT[1/(1+i) n ] The present value of an annuity is calculated by adding the PV of the individually discounted/compounded cash flows. 01234 1,000 8% (925.93) (857.34) (793.83) (735.03) (3,312.13)

31. Besley Ch. 631 PV Ordinary Annuities PVA n represents the present value of an ordinary annuity over n periods. PVA n = PMT[1/(1+i) 1 ] + PMT[1/(1+i) 2 ] +... + PMT[1/(1+i) n ] = PMT (1+i) t = PMT  n t=1 1 1 - (1+i) n i 1

32. Besley Ch. 632 PV Ordinary Annuities Present Value Interest Factor for an Annuity (PVIFA i,n ) is the present value interest factor for an annuity (even series of cash flows) of n periods compounded at i percent. PVIFA i,n = 1 - (1+i) n i 1

33. Besley Ch. 633 Numerical Solution Given: PMT:$1,000 I:8% N:4 01234 1,000 8% Solution: PVA n = PMT {1-[1/(1+i) n ]/i} = 1,000 {1-[1/(1+0.08) 4 ]/0,08} = 1,000 {3.3121} = $3,312.13 PVA n = PMT 1 - (1+i) n i 1

34. Besley Ch. 634 Tabular Solution Given: PMT:$1,000 I:8% N:4 PVA n = PMT(PVIFA i,n ) PVIFA i,n Periods7%8%9% 10.93460.92590.9174 21.80801.78331.7591 32.62432.57712.5313 43.38723.31213.2397 54.10023.99273.8897

35. Besley Ch. 635 Financial Calculator Given: N:4 I/Y:8% PV:? PMT:1,000 FV:0 Input: Output: NI/YPV PMT FV 48 -3,3121.13 1,000 0

36. Besley Ch. 636 PV Annuity Due Example: You decide that starting today you will withdraw $1,000 each year for the next four years from a savings account earning 8% interest per year. How much do you need today? 01234 1,000 8% (925.93) (857.34) (793.83) (1,000.00) (3,577.10)

37. Besley Ch. 637 PV Annuity Due PVA(Due) n represents the future value of an annuity due over n periods. PVA(Due) n = PMT  n-1 t=0 (1+i) t 1 = PMT x (1+i)  n t=1 (1+i) t 1 = PMT x (1+i) 1 - (1+i) n i 1

38. Besley Ch. 638 PV Annuity Due Present Value Interest Factor for an Annuity Due (PVIFA(Due) i,n ) is the present value interest factor for an annuity due of n periods compounded at i percent. PVIFA(Due) i,n = PMT x (1+i) 1 - (1+i) n i 1

39. Besley Ch. 639 Numerical Solution Given: PMT:$1,000 - BGN I:8% N:4 01234 1,000 8% Solution: PVA(Due) n = PMT [{(1-1/(1+i) n )/i}x (1+i)] = 1,000 [{(1-1/(1+0.08) 4 )/0.08}x (1+0.08)] = 1,000 {3.5771} = $3,577.10 PVIFA(Due) i,n = PMT x (1+i) 1 - (1+i) n i 1

40. Besley Ch. 640 Tabular Solution Given: PMT:$1,000 - BGN I:8% N:4 PVA(Due) n = PMT[(PVIFA i,n )(1+i)] PVIFA i,n Periods7%8%9% 10.93460.92590.9174 21.80801.78331.7591 32.62432.57712.5313 43.38723.31213.2397 54.10023.99273.8897

41. Besley Ch. 641 Financial Calculator Given: N:4 I/Y:8% PV:? PMT:1,000 - BGN FV:0 Input: Output: NI/YPV PMT FV 48 -3,577.10 1,000 0 BGN

42. Besley Ch. 642 Solving for Interest Rates with Annuities PVA n =PMT(PVIFA i,n ) -3,239.72 = 1,000(PVIFA i,n ) -3.2397 = PVIFA i,n Numerical Solution: Trial & Error  Solve for PVIFA 01234 1,000 ?% -3,239.72 PVIFA i,n = 1 - (1+i) n i 1

43. Besley Ch. 643 Solving for Interest Rates with Annuities Tabular Solution: -3.2397 = PVIFA i,n 01234 1,000 ?% -3,239.72 PVIFA i,n Periods7%8%9% 10.93460.92590.9174 21.80801.78331.7591 32.62432.57712.5313 43.38723.31213.2397 54.10023.99273.8897

44. Besley Ch. 644 Solving for Interest Rates with Annuities Financial Calculator: N:4 I/Y:? PV:-3,239.72 PMT:1,000 FV:0 01234 1,000 ?% -3,239.72 Input: Output: NI/YPV PMT FV 4 9 -3, 239.72 1,000 0

45. Besley Ch. 645 Perpetuities Perpetuity: A perpetual annuity, an annuity which continues forever. Consol A perpetual bond issued by the British government where the proceeds were used to consolidate past debts. PVP = PMT / i

46. Besley Ch. 646 Perpetuities PVA 5%,100 = $19,848

47. Besley Ch. 647 Uneven Cash Flow Streams 01234 250 750 8% (231.48) (643.00) (595.37) (2,204.88) 1,000 (735.03) PV n = FV[1/(1+i) n ] = FV(PVIF i.n )

48. Besley Ch. 648 Semiannual and Other Compounding Periods Simple Interest Rate: The interest rate used to compute the interest rate per period; the quoted interest rate is always in annual terms. Effective Annual Rate (EAR): The actual interest rate being earned during a year when compounded interest is considered.

49. Besley Ch. 649 Semiannual and Other Compounding Periods Types of Compounding: –Annual Compounding –Semiannual Compounding (Bonds) –Quarterly (Stock Dividends) –Daily (Bank Accounts/Credit Cards) EAR Formula EAR = 1+ -1 i simple m m

50. Besley Ch. 650 Semiannual and Other Compounding Periods Annual Percentage Rate (APR): the periodic rate multiplied by the number of period per year.

51. Besley Ch. 651 Fractional Time Periods F Use current formulas and convert time (n) into a fraction.

52. Besley Ch. 652 Amortized Loans Amortized Loan: a loan that is repaid in equal payments (an annuity) over the life of the loan. Amortization Schedule: A financial schedule illustrating each payment in the loan, and further breaking that down between principal and interest.