1. Schweser CFA Level 1 Book 1 – Reading #5
The Time Value of Money
master time value of money mechanics
and crunch the numbers
Schweser CFA Level 1 Book 1 – Reading #5

2. #5 The time value of money
LOS 5.a - Interest rate interpretation
LOS 5.b - Real risk-free rate and premiums
LOS 5.c - Effective annual rate (EAR)
LOS 5.d - Different frequencies of compounding
LOS 5.e - Present value (PV); future value (FV); annuity
LOS 5.f - Solution by timeline demonstration
Concept checkers

3. Time Lines Draw time lines to better show the cash flows
Cash Inflow: positive; you receive money
Cash Outflow: negative; you pay money
Present value (PV): discount all future cash flows into today’s value
Future value (FV): compound all cash flows to the value of a future date
N = Number of compounding periods
I/Y = Interest rate of compounding periods
PMT = Payment, periodic cash flow
CPT = compute

4. Time Lines Time Line demonstration. T = 0, today, cash outflow = 1000
Time length : 8 years
At the end of each year : cash inflow = 600
Blue numeric numbers from 1 to 8: end of 1st year … end of 8th year
End of 1st year = beginning of 2nd year
End of year i = beginning of year i+1
+600
+600
+600
+600
+600
+600
+600
+600
-1000

5. LOS 5.a Interpret interest rates as required rates of return, discount rates, or opportunity costs

6. LOS 5.a Interpret interest rates as required rates of return, discount rates, or opportunity costs
Compound Interest or Interest on Interest: the interest earned
on the previous period's interest earnings.
Interest rate interpretation
Required rate of return: the required return at which investors will invest
Discount rates: borrow rate of investors from banks, as he will pay the interest
Opportunity cost of current consumption: This is value of the best alternative foregone, or in other words, this is the return of something else that you have to give up when consume in this project.

7. LOS 5.b Explain an interest rate as the sum of a real risk-free rate, and premiums that compensate investors for bearing distinct types of risk

8. Risk free rate: The interest rate without any potential risks
LOS 5.b Explain an interest rate as the sum of a real risk-free rate, and premiums that compensate investors for bearing distinct types of risk
Risk free rate: The interest rate without any potential risks
Nominal risk-free rate = real risk-free rate + expected inflation rate (e.g., U.S. Treasury bill (T-bills) )
Real risk-free rate: , if the risk-free rate of return is 3% and the inflation rate is 2%, the real risk-free rate of return is 1%.
Types of risks (risks of securities contain the first three types of risks):
Default risk: ex. Firms unable to pay back its debt (bonds)
Liquidity risk: ex. Sell securities for cash less than fair value in illiquid markets
Maturity risk: ex. Long term bonds are more volatile
Exchange rate risk: change in exchange rate if you buy or sell in foreign currency. ….
Required interest rate: the sum of nominal risk-free rate and risk premiums
Nominal risk-free rate +
Default risk premium +
Liquidity premium +
Maturity risk premium…
(premium: additional part as compensation for additional risk)
Real Risk-Free Rate of Return
The risk-free rate of return after taking inflation into account. For example, if the risk-free rate of return is 3% and the inflation rate is 2%, the real risk-free rate of return is 1%.

9. LOS 5.c Calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding

10. 𝐸𝐴𝑅= (1+𝑝𝑒𝑟𝑖𝑜𝑑𝑖𝑐 𝑟𝑎𝑡𝑒) 𝑚 −1
LOS 5.c Calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding
Stated annual interest rate: or quoted interest rate, is the interest rate announced by financial institutions yearly.
Effective annual rate (EAR): the annual rate of return actually being earned after adjustments have been made for different compounding periods
𝐸𝐴𝑅= (1+𝑝𝑒𝑟𝑖𝑜𝑑𝑖𝑐 𝑟𝑎𝑡𝑒) 𝑚 −1
Periodic rate: stated annual rate divided by m
m: the number of compounding periods per year

11. 𝐸𝐴𝑅= (1+𝑝𝑒𝑟𝑖𝑜𝑑𝑖𝑐 𝑟𝑎𝑡𝑒) 𝑚 −1
LOS 5.c Calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding
Effective annual rate (EAR): the annual rate of return actually being earned after adjustments have been made for different compounding periods
𝐸𝐴𝑅= (1+𝑝𝑒𝑟𝑖𝑜𝑑𝑖𝑐 𝑟𝑎𝑡𝑒) 𝑚 −1
Periodic rate: stated annual rate divided by m
m: the number of compounding periods per year

12. LOS 5.c Calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding

13. LOS 5.c Calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding

14. LOS 5.c Calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding
Additional mark: prove why continuous copmpounding ER = e^rt - 1

15. LOS 5.d Solve time value of money problems for different frequencies of compounding
We need to consider the case of compounding periods are other than annual, or the frequencies of compounding are different than annual.

16. LOS 5.d Solve time value of money problems for different frequencies of compounding

17. LOS 5.d Solve time value of money problems for different frequencies of compounding

18. LOS 5.d Solve time value of money problems for different frequencies of compounding

19. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows

20. I|Y = rate of return per compounding period
LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows
Future value of a single sum: this value determines the value of an investment at the end of project with n compounding periods.
𝐹𝑉=𝑃𝑉× (1+𝐼|𝑌) 𝑁
PV = present value
I|Y = rate of return per compounding period
N = total number of compounding periods
(this expression only considers one cash outflow of PV at t=0; FV is the value of this investment at the end of period N)

21. Future value of a single sum 𝐹𝑉=𝑃𝑉× (1+𝐼|𝑌) 𝑁 PV = present value
LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows
Future value of a single sum
𝐹𝑉=𝑃𝑉× (1+𝐼|𝑌) 𝑁
PV = present value
I|Y = rate of return per compounding period
N = total number of compounding periods
(this expression only considers one cash outflow of PV at t=0; FV is the value of this investment at the end of period N)
or use 𝐹𝑉=300× (1+0.08) 10 =$647.68

22. 𝑃𝑉= 𝐹𝑉 (1+𝐼|𝑌) 𝑁 Present value of a single sum
LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows
Present value of a single sum
𝑃𝑉= 𝐹𝑉 (1+𝐼|𝑌) 𝑁
(1+𝐼|𝑌) 𝑁 is referred as future value factor or future value interest factor

23. 𝑃𝑉= 𝐹𝑉 (1+𝐼|𝑌) 𝑁 Present value of a single sum
LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows
Present value of a single sum
𝑃𝑉= 𝐹𝑉 (1+𝐼|𝑌) 𝑁

24. 𝑃𝑉= 𝐹𝑉 (1+𝐼|𝑌) 𝑁 Present value of a single sum
LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows
Present value of a single sum
𝑃𝑉= 𝐹𝑉 (1+𝐼|𝑌) 𝑁
Ex. PV of a zero-coupon bond

25. Ex. Receiving $1000 per year at the end of next 8 years
LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows
Annuities: An annuity is a series of equal dollar payments that are made at the end of equidistant points in time such as monthly, quarterly, or annually over a finite period of time.
Ex. Receiving $1000 per year at the end of next 8 years
Ordinary annuities: If payments are made at the end of each period, the annuity is referred to as ordinary annuity.

26. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows
You can use the following formula to calculate FV of an ordinary Annuity

27. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows
You can use the formula in previous slide or use calculator to crunch the result as follows:

28. Mathematic formula to calculate the Present value of
LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows
Mathematic formula to calculate the Present value of

29. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows

30. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows

31. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows

32. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows

33. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows
Annuity Due: is an annuity in which all the cash flows occur at the beginning of the period.
For example, rent payments on apartments are typically annuity due as rent is paid at the beginning of the month.
Computation of future/present value of an annuity due requires compounding the cash flows for one additional period, beyond an ordinary annuity.

34. Future value of annuity due (first payment at t=0)
LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows
Future value of annuity due (first payment at t=0)
Method 1: Set to BGN mode (button press [2ND] [PMT], [2ND] [ENTER]; screen upper right corner will appear BGN)
Method 2: By default END mode, compute as normal annuity, then convert to annuity due by the formula below:
𝐹𝑉𝐴 𝐷 = 𝐹𝑉𝐴 𝑂 ×(1+𝐼|𝑌)
FVA(D)
FVA(O)

35. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows

36. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows

37. Present value of annuity due (first payment at t=0)
LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows
Present value of annuity due (first payment at t=0)
Method 1: Set to BGN mode (button press [2ND] [PMT], [2ND] [ENTER]; screen upper right corner will appear BGN)
Method 2: By default END mode, compute as normal annuity, then convert to annuity due by the formula below:
𝑃𝑉𝐴 𝐷 = 𝑃𝑉𝐴 𝑂 ×(1+𝐼|𝑌)
PVA(D)
PVA(O)

38. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows

39. Present value of a perpetuity 𝑃𝑉 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 = 𝑃𝑀𝑇 𝐼|𝑌
LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows
Perpetuity: When a constant cash flow (C) will occur at regular intervals forever it is called a perpetuity. Perpetuity is continuous and has no end, therefore, there is no future value of perpetuity.
Present value of a perpetuity
𝑃𝑉 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 = 𝑃𝑀𝑇 𝐼|𝑌
To find a future value of a perpetuity, we force to set an end for this flows. In this case, it will be an ordinary till this end-point.

40. Similarly, we can set N as large as possible to get this $56.25
LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows
Similarly, we can set N as large as possible to get this $56.25
N= 9999; I|Y = 8; PMT = 4.5; FV = 0; [CPT][PV] =

41. Similarly, we can set N as large as possible to get this 6%
LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows
Present value of a perpetuity
𝑃𝑉 𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 = 𝑃𝑀𝑇 𝐼|𝑌
Similarly, we can set N as large as possible to get this 6%
N= 9999; PV = -75; PMT = 4.5; FV = 0; [CPT][I|Y] = 6

42. PV and FV of Uneven Cash Flow series
LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows
PV and FV of Uneven Cash Flow series
Year 1 and 2: negative cash flows
Year 3: 0 cash flow
Year 4, 5 and 6: positive cash flows
Compute the FV for each cash flow: compound each flow to year 6
Sum up them
𝑖=1 𝑛 𝐶𝑜𝑚𝑝𝑜𝑢𝑛𝑑 𝐶𝐹 𝑖
+4000
+3500
+2000
-500
-1000

43. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows

44. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows

45. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows

46. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows

47. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows

48. LOS 5.e Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows

49. Applications
What is the balance in an account at the end of 10 years if $2,500 is deposited today and the account earns 4%, compounded annually? Quarterly?
Annual compounding:
FV = $2,500 ( )^10
= $2,500 (1.4802) =
$3,700.61
Quarterly compounding:
FV = $2,500 ( )^40
= $2,500 (1.4889) =
$3,722.16

50. Applications
If you deposit $10 in an account that pays 5% interest, compounded annually, how much will you have at the end of 10 years ? 50 years? 100 years?
10 years: FV = $10 (1+0.05)^ 10 = $10 (1.6289) = $16.29
50 years: FV = $10 ( )^50= $10 ( ) = $114.67
100 years: FV = $10 ( )^100 = $10 (131.50) = $1,315.01

51. Applications
How much interest on interest is earned in an account by the end of 5 years if $100,000 is deposited and interest is 4% per year, compounded continuously?

52. Applications
How much will be in an account at the end of five years the amount deposited today is $10,000 and interest is 8% per year, compounded semi-annually ?

53. Applications
Suppose you want to have 0.5$ million saved by the time you reach age 30 and suppose that you are 20 years old today. If you can earn 5% on your funds, how much would you have to invest today to reach your goal?

54. Applications
How much would I have to deposit in an account today that pays 12% interest, compounded quarterly, so that I have a balance of $20,000 in the account at the end of 10 years?

55. Applications
Suppose I want to be able to withdraw $5000 at the end of five years and withdraw $6,000 at the end of six years, leaving a zero balance in the account after the last withdrawal. If I can earn 5% on my balances, how much must I deposit today to satisfy my withdrawals needs?

56. Applications
Suppose you deposit $100,000 in an account today that pays 6% interest, compounded annually. How long does it take before the balance in your account is $500,000?

57. Applications
The lucky Loan Company will lend you $100,000 today with terms that require you to pay off the loan in thirty-six monthly installments of $500 each. What is the effective annual rate of interest that the Lucky Loan Company is charging you?

58. Applications
How much does it take for your money to grow to ten times its original value if the interest rate of 5% per year?

59. Applications
Under what conditions does the effective annual rate of interest(EAR) differ from the annual percentage rate (APR)?

60. Applications
As the frequency of compounding increases within the annual period, what happens to the relation between the EAR and the APR?

61. Applications
If interest is paid at a rate of 5% per year, compounded quarterly, what is the : a) annual percentage rate? b) effective annual rate?

62. Applications
Consider an annuity consisting of three cash flows of $2,000 each. Assume a 4% interest rate. What is the present value of the annuity if the first cash flow occurs: a) today; b) one year from today; c) two years from today; d) three years from today; e) four years from today

63. Applications
Suppose you wish to retire forty years from today. You determine that you need $50,000 per year once you retire, with the first retirement funds withdrawn once year from the day you retire. You estimate that you will earn 6% per year on your retirement funds and that you will need funds up to and including your 25th birthday after retirement.
How much must you deposit in an account today so that you have enough funds for retirement?
How much must you deposit each year in an account, starting on year from today, so that you have enough funds for retirement?

64. Applications
Jim needs $800,000 to retire in 15 years. He will save $20,000 at the end of each of the next five years, and $40,000 at the end of the years If his investment account returns 11% per year, what equal payments must he make into the account at the end of years 6 thru 10?

65. LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems
Loan Amortization
Amortized loan: a special type of loan that is paid off by making a series of regular & equal payments
Part of each payment goes towards paying off the simple interest from the unpaid balance while the rest goes towards paying off the principal of the loan
This differs from installment loans where the interest over the lifetime of the loan is computed at purchase
Interest for an amortized loan is computed on the unpaid balance
The amount of loan and interest payment do not remain fixed over the term of loan.
Examples: House Mortgaged Loans, Auto Loans
In an amortized loan, the present value can be thought of as the amount borrowed, n is the number of periods the loan lasts for, i is the interest rate per period, and payment is the loan payment that is made.

66. LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems
Loan Amortization
Interest Component = Beginning Balance * Periodic Interest Rate
Principal Component = Payment - Interest
Sum of Principal Repayments = Original Amount of Loan
Sum of Interest Payments =
Sum of Total Payments – Sum of Principal Repayments

67. LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems
Loan Amortization
For Example: Beginning balance of a loan is $100000, interest rate is 10% and loan term is 10 years. What is the installment payment? Construct amortization schedule.
Amortization Schedule:
Noted: You need to input FV equal to 0 to make sure at the end of year Ten, you pay an amount of 0. If you don’t put FV=0, that means at year ten you still have to pay a given amount.

68. LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems
Loan Amortization
For Example: Beginning balance of a loan is $100000, interest rate is 10% and loan term is 10 years.
Amortization Schedule:

69. LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems

70. LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems

71. LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems

72. LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems
Note: Once you have solved for the payment, $ , the remaining principal on any payment date can be calculated by entering N = number of remaining payments and solving for the PV
For example:
N=4, PMT = , I|Y = 10, FV = 0, [CPT] -> PV =

73. Faster method PMT = 802.43 N = 18 I|Y = 5 FV = 0
LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems
Faster method
PMT =
N = 18
I|Y = 5
FV = 0
[CPT] -> PV =

74. LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems
Solving for payment:

75. Solving for number of periods:
LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems
Solving for number of periods:

76. LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems

77. Solving for rate of return/discount rate:
LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems
Solving for rate of return/discount rate:

78. The formula for computation of geometric mean in constant time is:
LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems
Geometric Mean:
The formula for computation of geometric mean in constant time is:

79. LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems

80. LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems

81. LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems

82. LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems

83. The connection Between PV, FV and Series of Cash Flows
LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems
The connection Between PV, FV and Series of Cash Flows
Alternative interpretation of present value:
How much money you put in the bank today
In order to make future withdraws
The final withdraw will exhaust the account
PV of 100, 200, 300 for the following 3 years will be , the assumed interest rate is 10%.
This is similar to invest now, and withdraw 100, 200, 300 for each year of the following 3 years, the money in account will generate interest but finally just be offset by the final with draw of 300
Another way is to look at future values of this 3 cash flows for 641 PV CF 1 2 3
100
200
300
Comp ratio
1.1
1.21
1.331
Sum(PV)
CF1
CF2
CF3
Invest
300
Withdraw
100
200
Balance

84. We compute the PV, and what do we observe?
LOS 5.f Demonstrate the use of a time line in modeling and solving time value of money problems
Cash Flow additivity principle: It refers to the present value of any stream of cash flows equal the sum of the present values of the cash flows.
The sum of the two series of cash flows is same as the present values of the two series taken together.
Example: If we have 3 projects 1 2 and 3, each project’s cash flows are indicated in the table below
We compute the PV, and what do we observe?
Indeed, the P3 CF = P1 CF + P2 CF if we add each period Project 1 and Project 2 cash flows, we get Project 3
The PV show the same result that PV3 = PV1 + PV2 = =
This is a demonstration of the additivity principle
rate = 10% 1 2 3 4 PV
Project 1
100
Project 2
300
Project 3
400

85. End of Chapter
Reading #5: the time value of money
(1) Nominal risk-free rate = real risk-free rate + expected inflation rate
(2) Required return on a security = real risk-free rate + expected inflation + default risk premium + liquidity premium + maturity risk premium
(3) Effective annual rate (EAR)
𝐸𝐴𝑅= (1+ 𝑠𝑡𝑎𝑡𝑒𝑑 𝑎𝑛𝑛𝑢𝑎𝑙 𝑟𝑎𝑡𝑒 𝑚 ) 𝑚 −1
(4) 𝐹𝑉 = 𝑃𝑉(1+𝐼/𝑌) 𝑁
(5) PV = 𝐹𝑉 (1+𝐼/𝑌) 𝑁
(6) Annuity due: cash flows occur at the beginning of each time period
(7) Ordinary annuity: cash flows occur at the end of each time period
(8) Perpetual annuities: 𝑃𝑉= 𝑃𝑀𝑇 𝐼/𝑌
Concept Checkers: Page 131, to be done by yourself including challenge problems. Check the answers to find out the mistake you make and try to comprehend the answers of each question.