Time Value of Money... Business Finance.pptx

time value of money

1. Time value of money describes the timing consideration associated with the value of money. TVM states that a present amount of money has the potential to grow over a period of time, if put into work . Being finance managers, we should always invest our free cash to earn more money, we should never keep our money idle. Time Value of Money (TVM).

2. TVM is based on a basic principle of financial decision making, which states; “ A dollar today worth more than a dollar tomorrow ” Reason for this is that if you get a dollar today and invest it, it would grow to more than a dollar later. It means you could earn interest/profit on the dollar you get and invest today, instead of waiting for the dollar promised at some future date. Time Value of Money (TVM).

3. Time Value of Money plays an important role in financial decision making. Most financial decisions, personal as well as business, involve time value of money considerations. It helps to answer two basic question related to financial decision making. To how much amount the current investment will grow over a specific period of time. (the concept of Future Value) How much a firm must invest now in order to earn an expected payoff later. (the concept of Present Value) Time Value of Money (TVM).

4. Why is TIME such an important element in your decision? Answer. TIME allows you the opportunity to postpone consumption and earn INTEREST/PROFIT. Time Value of Money (TVM).

5. In order to know about Time Value of Money we have to understand the concept of “Interest” , because interest provides the basis for Time Value of Money (TVM).

6. Interest is the money paid or earned for the use of money. Two types of Interest; Simple Interest Compound Interest. Interest

7. Simple interest is interest that is paid (earned) on only the original amount, or principal, borrowed (lent). The formula for calculating simple interest is SI = P ( i )(n) SI = simple interest in dollars P = principal, or original amount borrowed (lent) at time period 0 i = interest rate per time period n = number of time periods Simple Interest:

8. Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year? Solution; P0 = $1,000 i = 7% n = 2 years SI = ? SI = P ( i )(n) SI = $1,000 (.07) (2) SI = $140 Simple Interest Example

9. Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). Most situations in finance involving the time value of money do not rely on simple interest at all. Instead, compound interest is the norm. Compound Interest

10. Compound interest on $100 deposited at 8% interest rate for 3 years can be calculated as follow. Compound Interest Example Year Principal amount Principal + Previously Earned Interest Compound Interest 1 100 100 + 0 = 100 100(0.08) = 8 2 100 + 8 = 108 108 (0.08) = 8.64 3 100 + 8 + 8.64 = 116.64 116.64 (0.08)= 9.33 Total Compound Interest = 8+8.64+9.33 = 25.97

11. It is the amount to which the current amount of money grows after earning interest. It is the value of a present amount of money, or a series of payments, at some future time, evaluated at a given interest rate. Also known as the terminal value . Future Value

12. Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. It is the amount of money a firm must invest now in order to earn an expected amount later. Present Value

13. Future and Present values are calculated in following scenarios/cases; Case 1. Single Payment & Simple Interest. Case 2. Single Payment & Compound Interest. Case 3. Series of Payments & Compound Interest. Calculating Future & Present Values

14. Future value of a current single payment “ P ” made for a certain time period “n” evaluated with simple interest “ i ” is calculated with the following formula. Fv = P + SI Fv = P + (P . i . n) Fv = P (1 + i n) Calculating Future Value. Single Payment & Simple Interest

15. Present value “ Pv ” of a future amount of money “ Fv ” desired or promised at a certain time period “n” evaluated with simple interest “ i ” is calculated with the following formula. Pv = Fv . (1 + i n) Calculating Present Value. Single Payment & Simple Interest

16. Future value “Fv” of a current single amount “ P ” made for a certain time period “n” evaluated with compound interest “ i ” is calculated with the following formula. Fv = P (1 + i ) n Future Value is also know as “compound value”. Calculating future value of a present amount of money is known as Compounding . Calculating Future Value. Single Payment & Compound Interest

17. Example: consider a person deposits $100 into a savings account. If the interest rate is 8 percent, compounded annually, how much will the $100 be worth at the end of 3 year? Solution; P = $1,00 i = 8% n = 3 years FV = ? Fv = P (1 + i ) n Fv = 100(1+0.08) 3 Fv = 100(1.08) 3 Fv = 100(1.259) Fv = $125.9 Calculating Future Value. Single Payment & Compound Interest (Formula Method)

18. We can also calculate Future values as follow; Fv = P (FVIF i,n ) Where ( FVIF i,n ) is the future value interest factor at i % for n periods . Tables, called future value interest factor tables have been constructed . These tables contain values of ( FVIF i,n ) or ( 1 + i ) n for wide ranges of i and n . ( 1 + i ) n is the future interest factor. These tables contain Interest Rates ( i ) on horizontal axis and Periods (n) on vertical axis. Similar to map coordinates they help us locate the appropriate future value interest factor. Calculating Future Value. Single Payment & Compound Interest (Table Method)

19. Sticking with our previous Example: Consider a person deposits $100 into a savings account. If the interest rate is 8 percent, compounded annually, how much will the $100 be worth at the end of 3 year? Solution; P = $1,00 i = 8% n = 3 years FV = ? Fv = P (FVIF i,n ) Fv = P (FVIF 8% , 3 ) Now we have to locate future value interest factor at 8 percent for 3 years (FVIF 8% , 3 ), in future value interest factor table. Calculating Future Value. Single Payment & Compound Interest (Table Method)

20. As we see in future value interest factor table, the value located at the intersection of the 8% column with the 3-period row is 1.260. Fv = P (FVIF 8% , 3 ) Fv = 100(1.260) Fv = $126 Note that the value is almost equal to the value that we have calculated with formula i.e. $ 125.9 . The slight difference in the values calculated through both methods may be due to rounding up of data. For more precise results stick to formula method, but for quicker calculations tables may be used.

21. With the increase in interest rate the future value also increases such that the proportionate increase in future value becomes greater as the interest rate rises . Future value and Interest rate

22. Present value “ Pv ” of a future single amount “ Fv ” desired or promised at a certain time period “n” evaluated with compound interest “ i ” is calculated with the following formula. Pv = Fv . (1 + i ) n Future Value is also know as “ Discounted Value”. Calculating present value of a future amount of money is known as D iscounting . Interest rate used to convert future values to present values is known as Discount rate or capitalization rate. Calculating Present Value. Single Payment & Compound Interest

23. Example: What would be the present value of $100 offered at the end of 3 year from now discounted at an interest rate of 8 percent? Solution; Fv = $1,00 i = 8% n = 3 years Pv = ? Pv = Fv . (1 + i ) n Pv = 100/(1+0.08) 3 Pv = 100/(1.08) 3 Pv = 100/1.259 Pv = $79.428 Calculating Present Value. Single Payment & Compound Interest (Formula Method)

24. We can also calculate Present values as follow; Pv = Fv (PVIF i,n ) Where ( P VIF i,n ) is the present value interest factor at i % for n periods . Tables, called present value interest factor tables have been constructed . These tables contain values of ( PVIF i,n ) or 1/ ( 1 + i ) n for wide ranges of i and n . 1/ ( 1 + i ) n is the present value interest factor. These tables contain Interest Rates ( i ) on horizontal axis and Periods (n) on vertical axis. Similar to map coordinates they help us locate the appropriate present value interest factor. Calculating Present Value. Single Payment & Compound Interest (Table Method)

25. Sticking with our previous Example: What would be the present value of $100 offered at the end of 3 year from now discounted at an interest rate of 8 percent? Solution; Fv = $1,00 i = 8% n = 3 years Pv = ? Pv = Fv (PVIF i,n ) Pv = Fv (PVIF 8% , 3 ) Now we have to locate present value interest factor at 8 percent for 3 years (PVIF 8% , 3 ), in present value interest factor table. Calculating Present Value. Single Payment & Compound Interest (Formula Method)

26. As we see in Present value interest factor table, the value located at the intersection of the 8% column with the 3-period row is 0.794. Pv = Fv (PVIF 8% , 3 ) Pv = 100(0.794) Pv = $79.4 Note that the value is almost equal to the value that we have calculated with formula i.e. $79.428 .

27. With the increase in interest rate the present value decreases by a decreasing rate. The greater the interest rate, the lower the present value. Present value and Interest rate

28. Sometimes we are faced with a time-value-of money situation in which we know both the future and present values, as well as the number of time periods involved. But we don’t know the compound interest rate ( i ) implicit in the situation. Unknown Interest Rate

29. Let’s assume that, if you invest $1,000 today, you will receive $3,000 in exactly 8 years. The compound interest (or discount) rate implicit in this situation can be found by rearranging either a basic future value or present value equation. Fv = P (FVIF i , 8 ) $3,000 = $1,000(FVIF i , 8 ) (FVIF i , 8 ) = $3,000/$1,000 = 3 Reading across the 8-period row in Table, we look for the future value interest factor (FVIF) that comes closest to our calculated value of 3. Unknown Interest Rate Example. (Table Method)

30. As we see, the interest factor that is closest to our calculated value of 3 in the table is 3.059 and is found in the 15% column. Because 3.059 is slightly larger than 3, we conclude that the interest rate implicit in the example situation is actually slightly less than 15 percent. For greater accuracy we should use formula method.

31. We know the formula for future value is Fv = P (1 + i ) n Rearranging the formula we can get the value of unknown interest rate. Fv = P (1 + i ) n Fv / P = (1 + i ) n $3,000/$1,000 = (1 + i ) 8 3 = (1 + i ) 8 3 1/8 = (1 + i ) 8 x 1/8 3 0.125 = (1 + i ) 1 1.1472 = 1 + i 1.1472 - 1 = i i = 0.147 OR i = 14.7 % Unknown Interest Rate Example. (Formula Method)

32. At times we don’t exactly know how long it will take for a dollar amount invested today to grow to a certain future value given a particular compound rate of interest. Unknown Number of Compounding (or Discounting) Periods, " n ".

33. For example, how long would it take for an investment of $1,000 to grow to $1,900 if we invested it at a compound annual interest rate of 10 percent? compounding periods ( n ) involved in this investment situation can be determined by rearranging either a basic future value or present value equation. Using future value Fv = P (FVIF i , n ) 1900 = 1,000(FVIF 10% , n ) (FVIF 10% , n ) = 1900/1,000 = 1.9 Reading down the 10% column in FVIF Table, we look for the future value interest factor ( FVIF ) in that column that is closest to our calculated value of 1.9. Unknown Number of Compounding Periods, " n ". Example. (Table Method)

34. We find that 1.949 comes closest to 1.9, and that this number corresponds to the 7-period row. Because 1.949 is a little larger than 1.9, we conclude that the compounding periods are slightly less than 7 annual periods. For greater accuracy we should use formula method.

35. We know the formula for future value is Fv = P (1 + i ) n Rearranging the formula we can get the value of unknown interest rate. Fv = P (1 + i ) n Fv / P = (1 + i ) n 1900/1,000 = (1 + 10% ) n 1.9 = (1 + 0.10 ) n 1.9 = (1 .10 ) n Taking the natural logarithm ( ln ) of both sides ln (1.9) = ln [(1 .10 ) n ] 0.64185 = n ( ln 1.10) 0.64185 = n (0.09531) n = 0.64185 / 0.09531 n = 6.73 years. Unknown Number of Compounding Periods, " n ". Example. (Formula Method)

36. Calculating Present & Future Values. In case of Series of Payments/Receipts & Compound Interest

37. An annuity is a series of equal payments or receipts occurring over a specified number of periods. For Example: Student Loan Payments, Car Loan Payments, Insurance Premiums, Mortgage Payments, Retirement Savings. There are two types of annuities 1. Ordinary Annuity: In an ordinary annuity, payments or receipts occur at the end of each period. 2. Annuity Due: In an annuity due , payments or receipts occur at the beginning of each period. A nnuity & its types.

38. Time line is a line that shows the cash-flow sequence for an annuity. A time line is calibrated, on which the compounding or discounting periods are shown above and cash flows (receipt or payments) are shown below where they actually occur. Time lines are very useful for solving complicated time value problems. Almost anytime you are having trouble with a present or future value problem, drawing a time line will help you to see what is happening. Time Line.

39. Time Line for Ordinary Annuity

40. Time Line for Annuity Due

41. Let’s assume that you deposit $1000 each annual receipt in a savings account earning 8 percent compound annual interest. How much money will you have at the end of three years? 0 1 2 3 1000 1000 1000 Futur Value of Ordinary Annuity

42. The formula for the future value of the above ordinary annuity is given as follow. FvAn = R (1 + i ) n-1 + R (1 + i ) n-2 + R (1 + i ) n-3 Where R represents the periodic Receipts or Payments. FvAn = 1000 (1 + 0.08 ) 3-1 + 1000 (1 + 0.08 ) 3-2 + 1000 (1 + 0.08 ) 3-3 FvAn = 1000 (1 .08 ) 2 + 1000 (1 .08 ) 1 + 1000 (1 .08 ) FvAn = 1000 (1.1664) + 1000 (1 .08 ) + 1000 (1) FvAn = 1166.4 + 1 080 + 1000 FvAn = 3246.4 Futur Value of Ordinary Annuity

43. FvAn = R [ _( 1 + i ) n -1 _ ] i Calculating for the same example. FvAn = 1000 [ _ ( 1 + 0.08 ) 3 -1 _ ] 0.08 FvAn = 1000 [ _ ( 1.08 ) 3 -1 _ ] 0.08 FvAn = 1000 ( 1.2597-1 ) 0.08 FvAn = 1000 ( _0.2597 ) 0.08 FvAn = 1000 (3.2463) FvAn = 3246.3 Futur Value of Ordinary Annuity

44. Let’s assume that you want to receive $1000 each year for 3 years. How much money would you have to deposit now in a bank account, if the bank offers an 8 percent compound annual interest? 0 1 2 3 1000 1000 1000 Present Value of Ordinary Annuity

45. The formula for the present value of the above ordinary annuity is given as follow. PvAn = R + R + R . (1 + i ) 1 ( 1 + i ) 2 (1 + i ) 3 Where R represents the periodic Receipts or Payments. PvAn = 1000 + 1000 + 1000 . (1 + 0.08) 1 (1 + 0.08) 2 (1 + 0.08) 3 PvAn = 1000 + 1000 + 1000 . (1.08) 1 (1.08) 2 (1.08) 3 PvAn = 1000 + 1000 + 1000 . (1.08) (1.1664) (1.2597) PvAn = 925.92 + 857.338 + 793.839 PvAn = 2577.1 Present Value of Ordinary Annuity

46. PvAn = R [ _ 1 - (1/ ( 1 + i ) n ) _ ] i Calculating for the same example. PvAn = 1000 [ _1 - (1/ ( 1+0.08) 3 )_ ] 0.08 PvAn = 1000 [ _1 - (1/ ( 1.08) 3 )_ ] 0.08 PvAn = 1000 [ _1 - (1/ 1.2597 )_ ] 0.08 PvAn = 1000 ( _1 - 0.7938_ ) 0.08 PvAn = 1000 ( _ 0.2062_ ) 0.08 PvAn = 1000 ( 2.5775 ) PvAn = 2577.5 Present Value of Ordinary Annuity

47. Let’s assume that you deposit $1000 each year in a savings account earning 8 percent compound annual interest. How much money will you have at the end of three years? FvAn = R(FVIFA i,n ) FvAn = 1000 (FVIFA 8%,3 ) For the value of (FVIFA 8%,3 ) we have to consult the Future Value Interest Factor of an (ordinary) Annuity (FVIFA) table . Futur Value of Ordinary Annuity ( Table Method)

48. The value of FVIFA for 8% compound annual interest and three years in the table is 3.246.

49. FvAn = 1000 (FVIFA 8%,3 ) FvAn = 1000 ( 3.246) FvAn = 3246. Note: Use of a table rather than a formula subjects us to some slight rounding error. Therefore, when extreme accuracy is called for, use formulas rather than tables.

50. Let’s assume that you want to receive $1000 each year for 3 years. How much money would you have to deposit now in a bank account, if the bank offers an 8 percent compound annual interest? PvAn = R(PVIFA i,n ) PvAn = 1000 (PVIFA 8%, 3 ) For the value of (PVIFA 8%, 3 ) we have to consult the Present Value Interest Factor of an (ordinary) Annuity (PVIFA) table , which is 2.577. PvAn = 1000 ( 2.577 ) PvAn = 2577 Present Value of Ordinary Annuity ( Table Method)

51. The unknown compound interest (discount) rate implicit in an annuity can be calculated if we know: The annuity’s future (present) value, The periodic payment or receipt, and The number of periods involved. Suppose that you need to have at least $9,500 at the end of 8 years in order to send your parents on a luxury cruise. To accumulate this sum, you have decided to deposit $1,000 at the end of each of the next 8 years in a bank savings account. If the bank compounds interest annually, what minimum compound annual interest rate must the bank offer for your savings plan to work? Unknown Interest (or Discount) Rate.

52. A rearrangement of the basic future value or present value of an annuity equation can be used to solve for the unknown compound interest (discount) rate. FvAn = R(FVIFA i,n ) 9500 = 1000 (FVIFA i,8 ) 9500/ 1000 = (FVIFA i,8 ) (FVIFA i,8 ) = 9500/ 1000 (FVIFA i,8 ) = 9.5 To find the unknown interest rate we read across the 8-period row in FVIFA table, we look for the future value interest factor of an annuity ( FVIFA ) that comes closest to our calculated value of 9.5 .

53. In our table, that interest factor is 9.549 and is found in the 5% column. Because 9.549 is slightly larger than 9.5 , we conclude that the interest rate implicit in the example situation is actually slightly less than 5 percent.

54. When dealing with annuities, one frequently encounters situations in which either the future (or present) value of the annuity, the interest rate, and the number of periodic payments (or receipts) are known. What needs to be determined, however, is the size of each equal payment or receipt. Rearrangement of either the basic present or future value annuity equation is necessary to solve for the periodic payment or receipt implicit in an annuity. Unknown Periodic Payment or Receipt

55. For Example; How much one must have to deposit at the end of each year in a savings account earning 5 percent compound annual interest to accumulate $10,000 at the end of 8 years? Here we have to compute the payment ( R ) going into the savings account each year with the help of future value of an annuity. FvAn = R(FVIFA i,n ) FvAn = R(FVIFA 5% ,8 ) 10,000 = R(FVIFA 5% ,8 ) The value of (FVIFA 5% ,8 ) in the table is 9.549. 10,000 = R( 9.549 ) R = 10,000/ 9.549 R = $ 1047.23

56. In a business setting, we most frequently encounter the need to determine periodic annuity payments in sinking fund (i.e., building up a fund through equal-dollar payments) and loan amortization (i.e., extinguishing a loan through equal-dollar payments) problems.

57. A perpetuity is an ordinary annuity whose payments or receipts continue forever. The ability to determine the present value of this special type of annuity is required in valuation of perpetual bonds and preferred stock. Perpetuity

58. PvA ∞ = R [ _ 1 - (1/( 1 + i ) ∞ ) _ ] i As 1/( 1 + i ) ∞ approaches to Zero, therefore; PvA ∞ = R [ _ 1 - 0 ] i PvA ∞ = R [ _ 1 ] i PvA ∞ = R / i Thus the present value of a perpetuity is simply the periodic receipt (payment) divided by the interest rate per period. Present Value of Perpetuity.

59. For example, if $100 is received each year forever and the interest rate is 8 percent, the present value of this perpetuity can be calculated as; PvA ∞ = R / i PvA ∞ = 100 / 0.08 PvA ∞ = $ 1250

60. In contrast to an ordinary annuity, where payments or receipts occur at the end of each period, an annuity due calls for a series of equal payments occurring at the beginning of each period. Luckily, only a slight modification to the procedures already outlined for the treatment of ordinary annuities will allow us to solve annuity due problems. Future Value of Annuity Due

61. 0 1 2 3 R1 R2 R3 0 1 2 3 R1 R2 R3 The real key to distinguishing between the future value of an ordinary annuity and an annuity due is the point at which the future value is calculated. For an ordinary annuity, future value is calculated as of the last cash flow. For an annuity due, future value is calculated as of one period after the last cash flow. Note that the future value of the three-year annuity due is simply equal to the future value of a comparable three-year ordinary annuity, compounded for one more period.

62. Formula for Future Value of Annuity Due Formula for future value of an Ordinary Annuity is FvAn = R [ _( 1 + i ) n -1 _ ] i Compounding this formula for one more period, it becomes the formula for future value of annuity due. Formula for future value of an Annuity Due. FvAD = R [ _( 1 + i ) n -1 _ ] X (1+ i ) 1 i

63. Let’s assume that you deposit $1000 at the beginning of each year in a savings account, earning 8 percent compound annual interest. How much money will you have at the end of three years? FvAD = R [ _( 1 + i ) n -1 _ ] X (1+ i ) 1 i FvAD = 1000 [ ( 1 + 0.08 ) 3 -1 ] X (1+0.08) 1 0.08 FvAD = 1000 [ ( 1.08 ) 3 -1 ] X (1.08) 1 0.08 FvAD = 1000 ( 1.2597-1 ) X (1.08) 0.08 FvAD = 1000 ( 0.2597 ) X (1.08) 0.08 FvAD = [ 1000 (3.2463) ] X (1.08) FvAD = 3246.3 X (1.08) FvAD = $ 3506.

64. In ordinary annuity we consider the cash flows as occurring at the end of periods. The present value of an ordinary annuity, is therefore, calculated as of one period before the first cash flow. 0 1 2 3 R1 R2 R 3 The cash flows in annuity due occur at the beginning of periods. The present value of an annuity due, is therefore, calculated as of the first cash flow. 0 1 2 3 R1 R2 R 3 Present Value of Annuity Due

65. Formula for Present value of an Ordinary Annuity is PvAn = R [ _ 1 - (1/ ( 1 + i ) n ) _ ] i Compounding this formula for one more period, it becomes the formula for Present value of annuity due. Formula for present value of an Annuity Due. PVAD = R [ _ 1 - (1/ ( 1 + i ) n ) _ ] x (1+ i ) 1 i Formula for Present Value of Annuity Due

66. Let’s assume that you want to receive $1000 at the beginning of each year, for 3 years. How much money would you have to deposit now in a bank account, if the bank offers an 8 percent compound annual interest? PVAD = R [ 1 - (1/ ( 1 + i ) n ) ] x (1+ i ) 1 I PVAD = 1000 [ 1 - (1/ ( 1+0.08) 3 ) ] x (1.08) 1 0.08 PVAD = 1000 [ 1 - (1/ ( 1.08) 3 ) ] x (1.08) 0.08 PVAD = 1000 [ 1 - (1/ 1.2597 ) ] x (1.08) 0.08 PVAD = 1000 ( 1 - 0.7938 ) x (1.08) 0.08 PVAD = 1000 ( 0.2062 ) x (1.08) 0.08 PVAD = 1000 ( 2.5775 ) x (1.08) PVAD = 2577.5 x (1.08) PVAD = $ 2783.7

67. FvAD = R(FVIFA i,n ) X (1+ i ) 1 FvAD = 1000 (FVIFA 8%,3 ) X (1.08) 1 FvAD = 1000 ( 3.246 ) X (1.08) FvAD = $ 3505.7 Future Value of Annutiy Due Table method.

68. PVAD = R(PVIFA i,n ) X (1+ i ) 1 PVAD = 1000 (PVIFA 8%,3 ) X (1.08) 1 PVAD = 1000(2.577 ) X (1.08) PVAD = $ 2783.16 Present Value of Annutiy Due Table method.

69. We know that in annuities the cash flows are of equal amounts. But many times we face a scenario where the cash flows are not of equal amounts i.e. we encounter a mixed (or uneven) pattern of cash flows. Mixed flow problems can always be solved by adjusting each flow individually and then summing the results . The first step in solving the question above, or any similar problem, is to draw a time line, position the cash flows, and draw arrows indicating the direction and position to which you are going to adjust the flows. Second , make the necessary calculations as indicated by your diagram. Mixed Flows

70. What is the present value of $5,000 to be received annually at the end of years 1 and 2, followed by $6,000 annually at the end of years 3 and 4, and concluding with a final payment of $1,000 at the end of year 5, all discounted at 5 percent ? Solution PV0 = 5000 + 5000 + 6000 + 6000 + 1000 (1 + i ) 1 ( 1 + i ) 2 ( 1 + i ) 3 ( 1 + i ) 4 (1 + i ) 5 PV0 = 5000 + 5000 + 6000 + 6000 + 1000 ( 1 + 0.05 ) 1 (1 + 0.05 ) 2 (1 + 0.05 ) 3 (1 + 0.05 ) 4 (1 + 0.05 ) 5 PV0 = 5000 + 5000 + 6000 + 6000 + 1000 ( 1 .05 ) 1 (1 .05 ) 2 (1 .05 ) 3 (1 .05 ) 4 (1 .05 ) 5 PV0 = 5000 + 5000 + 6000 + 6000 + 1000 1 .05 1 .1025 1 .1576 1 .2155 1 .2762 PVo = 4761.9 + 4535.1 + 5183.1 + 4936.2 + 783.6 PVo = $ 20200

71. We can recognize certain patterns within mixed cash flows that allow us to take some calculation shortcuts . Patterns in mixed cash flows.

72. In the above example we can also consider the last 3 cash flows of $6000 as an ordinary annuity, and calculate its present value as of one period before the first $6000 cash flow using formula for present value of ordinary annuity, and then discount it further for two periods to bring it to the present. PvAn = R [ _ 1 - (1/ ( 1 + i ) n ) _ ] i PvAn = 6000 [ _ 1 - (1/ ( 1.05 ) 3 ) _ ] 0.05 PvAn = 6000 [ _ 1 - (1/ 1.1576 ) _ ] = 6000 [ 1 - (0.8639 ) ] 0.05 0.05 PvAn = 6000 [0.1361 ] = 6000 ( 2.7229 ) 0.05 PvAn = 16337.2

73. We can also consider the two equal amounts of $5000 as one ordinary annuity and the other equal amounts of $6000 as another ordinary annuity, and calculate their present value respectively, using the following formula for present value of ordinary annuity. PvAn = R [ _ 1 - (1/ ( 1 + i ) n ) _ ] i

74. Up to now, we have assumed that interest is paid annually i.e. once in a year. But in many cases the interest is earned and paid more than once in a year i.e. semiannually, quarterly, monthly or continuously. In this case the general formula for solving for the future value at the end of n years where interest is paid m times a year is; Fv = PV (1 + i /m ) mn Where m is the number of times the interest is compounded annually. In case of semiannually compounding m = 2 , in quarterly compounding m = 4 and in case of monthly compounding m = 12 . The more times during the year that interest is paid/compounded, the greater the future value at the end of a given year . Compounding More Than Once a Year

75. When interest is compounded more than once a year, the formula for calculating present value must be revised too. Pv = Fv . (1 + i /m ) mn The fewer times a year that the discount rate is compounded, the greater the present value. This relationship is just the opposite of that for future values.

76. The Future and Present Values of $100 invested for 3 years, with a stated interest rate of 8%, compounded quarterly, can be calculated as follow; Future Value Fv = PV (1 + i /m ) mn Fv = 100 (1 + 0.08/4) 3x4 Fv = 100 (1 + 0.02) 12 Fv = 100 (1.02) 12 Fv = 100 (1.268) Fv = $126.8 Present Value Pv = Fv . (1 + i/m ) mn Pv = 100 . (1 + 0.08/4 ) 3x4 Pv = 100 . (1 + 0.02 ) 12 Pv = 100 . (1 .02 ) 12 Pv = 100 . 1.268 Pv = $78.86

77. In practice, interest is sometimes compounded continuously. In continuous compounding the number of times the interest is compounded in a year, approaches infinity. The term (1 + [ i / m ]) mn approaches “ e in ” , where “ e” is approximately 2.71828. Therefore the future value at the end of n years of an initial deposit of PV 0 where interest is compounded continuously at a rate of i percent is FV n = PV ( e ) in Continuous Compounding

78. The present value PV of a future amount FV n at the end of n years where interest is compounded continuously at a rate of i percent is PV = FV n / ( e ) in The Future and Present Values of $100 invested for 3 years, with a stated interest rate of 8%, compounded continuously , can be calculated as follow ; Future Value FVn = PV0(e) in FVn = 100 (2.71828) (0.08)(3) FVn = 100 (2.71828) (0.24) FVn = 100 (1.2712) FVn = 127.12 Present Value PV0 = FVn / (e) in PV0 = 100 / (2.71828) (0.08)(3) PV0 = 100 / (2.71828) (0.24) PV0 = 100 / (1.2712) PV0 = 78.66

79. Nominal (stated) interest rate is a rate of interest quoted for a year that has not been adjusted for frequency of compounding. Effective annual interest rate is the actual rate of interest earned (paid) after adjusting the nominal rate for the number of com pounding periods per year. When we want to compare alternative investments that have different compounding periods, we need to state their interest on some common, or standardized, basis . The effective annual interest rate is the interest rate compounded annually that provides the same annual interest as the nominal rate does when compounded m times per year. Effective annual interest rate = [1 + ( i / m)] m − 1 Effective Annual Interest Rate

80. Inflation - Rate at which prices as a whole are increasing. Nominal Interest Rate - Rate at which money invested grows. Real Interest Rate - Rate at which the purchasing power of an investment increases. 1 + real interest rate = 1+nominal interest rate 1+inflation rate Real Interest Rate

81. The End