1. MONEY-TIME RELATIONSHIPS AND EQUIVALENCE
CHAPTER 3
MONEY-TIME RELATIONSHIPS AND EQUIVALENCE

2. WHAT IS MONEY? Medium of Exchange
Means of payment for goods or services
What sellers accept and buyers pay
Store of Value
A way to transport buying power from one time period to another
Unit of Account
A precise measurement of value or worth
Allows for tabulating debits and credits

3. WHAT IS CAPITAL?
Capital is wealth in the form of money or property that can be used to produce more wealth

4. KINDS OF CAPITAL
Equity capital is that owned by individuals who have invested their money or property in a business project or venture in the hope of receiving a profit. The value of equity capital is computed by estimating the current market value of everything the company owns MINUS all liabilities. On the balance sheet of the company, equity capital is listed as stockholder’ equity or owner’s equity.
Debt capital often called borrowed capital, is obtained from lenders for investment (bank loans, overdrafts, etc)

5. What is Bond?
A bond is a written and signed promise to pay a certain sum of money on a certain date, or on fulfillment of a specified condition.
All documented contracts and loan agreements are bonds.

6. What is Stock? What is Share?
Stock  is the equity capital that is raised through the sale of shares
A share is a unit of ownership that represents an equal proportion of a company’s capital. It entitles its holder (ie. the shareholder) to an equal claim on the company's profits and an equal obligation for the company’s debts and losses

7. Exchange money for shares of stock as proof of partial ownership

8. WHAT IS INTEREST?
Interest is the fee that a borrower pays to a lender for the use of her/his money. It is a rental amount charged for the use of money
WHAT IS INTEREST RATE?
Interest rate is the percentage of the money borrowed that is paid to the lender on some time basis. Money has time value because of interest

9. Interest from Religious Perspective
Through the mechanism of interest, unjust gains in trade is acquired through exploitation
Interest existed since 2000 B.C. in Babylon. Interest rates of 6-40% have been recorded
The Bible mentioned usury in Exodus. In 1536, John Calvin established the Protestant’s theory of usury

10. Interest from Religious Perspective
In Islam, interest is considered riba, and hence is strictly forbidden in Islamic economic jurisprudence (fiqh). It is a major sin
There are two types of riba - an increase in capital without any services provided and speculation (maisir), and commodity exchanges in unequal quantities. Both are prohibited in the Quran.

11. SIMPLE INTEREST
The total interest earned or charged is linearly proportional to the initial amount P of the loan (principal), the interest rate i and the number of interest periods N for which the principal is committed.
When applied, total interest “I” may be found by I = ( P ) ( N ) ( i ), where
P = principal amount lent or borrowed
N = number of interest periods ( e.g., years )
i = interest rate per interest period

12. Simple interest - example
You need RM1000 to pay for tuition fees. You want to borrow the money, and return it over 3 years. The interest rate is 10% (simple interest).
How much is the amount of interest you have to pay? How much is the overall amount that you have to pay?
I=( P ) ( N ) ( i ),
I= 1000*3*0.1=
Interest is RM300; overall to pay RM1,300

13. COMPOUND INTEREST
The interest charged for a given period is based on the remaining principal amount plus any accumulated interest charges up to the beginning of that period
Period Amount Owed Interest Amount Amount Owed at beginning of for Period at end of period 10% ) period
1 $1,000 $100 $1,100
2 $1,100 $110 $1,210
3 $1,210 $121 $1,331
« interest of the interest »

14. Compound interest - example
A loan of $1,000 is made at an interest of 12% for 5 years. The principal and interest are due at the end of the 5th year
Year
Amount at start of year, RM
Interest at end of year, RM
Amount owed at end of year, RM
Payment, RM 1
120.00 2
134.40 3
150.53 4
168.59 5
188.82

15. “ECONOMIC EQUIVALENCE”
Things are equivalent when they have the same effect
In engineering economics, we are usually not too concerned about the timing of a project's cash flow. Instead, we are concerned about the profitability of that project
This means we need to compare projects involving receipts and disbursements occurring at different times, with the goal of identifying an alternative having the largest eventual profitability

16. ECONOMIC EQUIVALENCE
Equivalence is established when we are indifferent about making a future payment, or a series of future payments, and a present sum of money
Equivalence is needed to compare alternative options or proposals, by reducing them to an equivalent basis, depending on:
interest rate
amounts of money involved
timing of the affected monetary receipts and/or expenditures
manner in which the interest or profit on invested capital is paid and the initial capital is recovered

17. ECONOMIC EQUIVALENCE FOR FOUR REPAYMENT PLANS OF AN $8,000 LOAN
Plan #1: $2,000 of loan principal plus 10% at BOY paid at end of year. Notice - interest paid at the end of each year is reduced by $200 (i.e., 10% of remaining principal)
Year Amount Owed Interest Accrued Total Principal Total end at beginning for Year Money Payment of Year of Year owed at Payment ( BOY ) end of Year
1 $8,000 $ $8,800 $2,000 $2,800
2 $6,000 $ $6,600 $2,000 $2,600
3 $4,000 $ $4,400 $2,000 $2,400
4 $2,000 $ $2,200 $2,000 $2,200
10,000

18. ECONOMIC EQUIVALENCE FOR FOUR REPAYMENT PLANS OF AN $8,000 LOAN, 10% interest rate
Plan #2: $0 of loan principal paid at end of fourth year; $800 interest paid at the end of each year
Year Amount Owed Interest Accrued Total Principal Total end at beginning for Year Money Payment of Year of Year owed at Payment ( BOY ) end of Year
1 $8,000 $ $8,800 $ $800
2 $8,000 $ $8,800 $0 $800
3 $8,000 $ $8,800 $ $800
4 $8,000 $ $8,800 $8,000 $8,800
11,200`

19. ECONOMIC EQUIVALENCE FOR FOUR REPAYMENT PLANS OF AN $8,000 LOAN, 10% interest rate
Plan #3: Pay interest at end of each year, plus some premium. A total of $2,524 paid at the end of each year
Year Amount Owed Interest Accrued Total Principal Total end at beginning for Year Money Payment of Year of Year owed at Payment ( BOY ) end of Year
1 $8,000 $ $8,800 $1,724 $2,524
2 $6,276 $ $6,904 $1,896 $2,524
3 $4,380 $ $4,818 $2,086 $2,524
4 $2,294 $ $2,524 $2,294 $2,524
1724(principal) = 2524.
= 6276.
=6904
10,096

20. ECONOMIC EQUIVALENCE FOR FOUR REPAYMENT PLANS OF AN $8,000 LOAN, 10% interest rate
Plan #4: No interest and no principal paid for first three years. At the end of the fourth year, the original principal plus accumulated (compounded) interest is paid.
Year Amount Owed Interest Accrued Total Principal Total end at beginning for Year Money Payment of Year of Year owed at Payment ( BOY ) end of Year
1 $8,000 $ $8,800 $0 $0
2 $8,800 $ $9,680 $0 $0
3 $9,680 $ $10,648 $0 $0
4 $10,648 $1, $11,713 $8,000 $11,713

21. What is Cash Flow?
Cash flow is the movement of money into or out of a business, project, or financial product
It is usually measured during a specified, limited period of time
Measurement of cash flow can be used for calculating other parameters that give information about a company's value and situation.

22. What is Cash Flow?
In an ideal business cycle, there will be more cash flowing in than flowing out.
In reality, however, most businesses have to produce or deliver goods/services to their customers while also paying their staff and suppliers before they get paid themselves
This lag in payments-in and payments-out is a major challenge for businesses. How well it is managed is critical to the business' immediate financial health and long term sustainability.

23. Cash Inflows
Cash inflows are any receipts of cash to a business and can include:
Payment for goods or services from customers
Receipt of a bank loan
Interest on savings and investments
Shareholder investments
Tax returns

24. Cash Outflows Purchase of stock, raw materials or equipment
Cash outflows are any cash outgoings and can include:
Purchase of stock, raw materials or equipment
Wages, rents, daily operating expenses
Loan repayments, maintenance cost payments
Income tax, payroll tax, other taxes
Asset purchases

25. What is Cash Flow Diagram?
It is a tool to represent cash transactions in a given time duration
It includes information on cash inflows and cash outflows
Outflow is placed UNDER the line, while inflow is placed OVER the line

26. CASH FLOW DIAGRAM NOTATION1 1 2 3 4
5 = N 1
Time scale with progression of time moving from left to right
The numbers represent time periods (e.g., years, months, quarters, etc...)

27. CASH FLOW DIAGRAM NOTATION1 1 2 3 4
5 = N 2
P =$8,000 2
Present expense (cash outflow) of $8,000

28. CASH FLOW DIAGRAM NOTATION 3 1 1 2 3 4 5 = N
P =$8,000 3
Annual income (cash inflow) of $2,524

29. Interest rate of loan CASH FLOW DIAGRAM NOTATION 3 1 1 2 3 4 5 = N 2 4
P =$8,000
i = 10% per year
Interest rate of loan 4

30. CASH FLOW DIAGRAM NOTATION 5 3 1 1 2 3 4 5 = N
P =$8,000
i = 10% per year
Dashed-arrow line indicates
amount to be determined 5

31. Problem to Solve
The heating, ventilation, and air conditioning system (HAVC) used in the shop floor needs to be upgraded. There are 2 alternatives –
Alternative A: rebuild the existing HAVC system
(2) Alternative B: Install a new HAVC system that utilizes existing ductwork
Which alternative would you choose? Why?

32. Alternative A – rebuild existing HAVC system
Equipment, labor, materials - RM18,000
Annual electricity cost – RM32,000
Annual maintenance – RM2,400
Alternative B – Install new HAVC system
Equipment, labor, materials – RM60,000
Annual electricity cost – RM9,000
Annual maintenance – RM16,000
Replacement of major component at the end
of 4 years – RM9,400
You have been told that at the end of 8 years, the estimated market value of Alternative A is RM2,000 and for Alternative B, the value is RM8,000. Which alternative would you choose? Why?
At the end of the 8th year, u would have spent RM 291,200 if you chose Alternative 1
At the end of the 8th year, u would have spent RM 261,400 if you chose Alternative 2
Hence, a difference of RM29,800 between the two.

33. End of year…
Alternative A (RM)
Alternative B (RM)
0 (now)
18,000
60,000 1
34,400
25,000 2 3 4 5 6 7 8
34,400 – 2,000 = 32,400
25,000 – 8,000 = 17,000
TOTAL
291,200
261,400

34. NOTATION i = effective interest rate per interest period
N = number of compounding periods (e.g., years)
P = present sum of money; the equivalent value of one or more cash flows at the present time
F = future sum of money; the equivalent value of one or more cash flows at a future time
Cash flow is the movement of money into or out of a business, project, or financial product. It is usually measured during a specified, limited period of time. Measurement of cash flow can be used for calculating other parameters that give information on a company's value and situation.

35. RELATING PRESENT AND FUTURE EQUIVALENT VALUES OF SINGLE CASH FLOWS
(a) Find F (future value) when given P (present value)
F = P ( 1+i ) N
(1+i)N single payment compound amount factor
functionally expressed as F = P (F / P, i%, N )
predetermined values of this can be found in standard interest factor tables P
N =
F = ?

36. RELATING PRESENT AND FUTURE EQUIVALENT VALUES OF SINGLE CASH FLOWS
(b) Find P when given F:
P = F [1 / (1 + i ) ] N
(1+i)-N single payment present worth factor
functionally expressed as P = F ( P / F, i%, N )
predetermined values of this can be found in standard interest factor tables F
N =
P = ?

37. Example – (a) find F, given P (single cash flow)
Your company borrows RM1000 for 8 years. How much must it repay in a lump sum at the end of the 8th year? (use interest rate 10% per year)
The question above can be written this way: What is the future equivalence at the end of 8 years of RM1000 at the beginning of the 8 years?
F = P (F / P, i%, N )
= RM1000 (2.1436)
= RM 2,143.60

39. Example – (b) find P, given F (single cash flow)
Your company would like to have RM2, eight years from now. What amount should be deposited now to get that amount in 8 years’ time? (10% annual interest rate)
What is the present equivalence of RM2, that will be received 8 years from now?
P = F ( P / F, i%, N ) (try it!)
P = F ( P / F, i%, N )
RM * = RM 1000

40. NOTATION
A = end-of-period cash flows (or equivalent end-of-period values ) in a uniform series continuing for a specified number of periods, starting at the end of the first period and continuing through the last period Ordinary annuity – the first cash flow is made at the end of the first period Deferred annuity – the cash flow begins at some later date

41. uniform series compound amount factor in [ ]
RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES
(1) Find F given A
Finding future equivalent income (inflow) value given a series of uniform equal payments
( 1 + i ) N - 1
F = A i
uniform series compound amount factor in [ ]
functionally expressed as F = A ( F / A, i%, N )
predetermined values are given in standard interest factor tables
F = ? 1 2 3 4 5 6 7 8
A =

42. uniform series present worth factor in [ ]
RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES
(2) Find P given A
Finding present equivalent value given a series of uniform equal receipts
( 1 + i ) N - 1
P = A
i ( 1 + i ) N
uniform series present worth factor in [ ]
functionally expressed as P = A ( P / A, i%, N )
predetermined values given in standard interest factor tables
A = 1 2 3 4 5 6 7 8
P = ?

43. sinking fund factor in [ ]
RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES
(3) Find A given F
Finding amount A of a uniform series when given the equivalent future value i
A = F
( 1 + i ) N -1
sinking fund factor in [ ]
functionally expressed as A = F ( A / F, i%, N )
predetermined values are given in standard interest factor tables
F = 1 2 3 4 5 6 7 8
A =?

44. capital recovery factor in [ ]
RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES
(4) Find A given P
Finding amount A of a uniform series when given the equivalent present value
i ( 1+i )N
A = P
( 1 + i ) N -1
capital recovery factor in [ ]
functionally expressed as A = P ( A / P, i%, N )
predetermined values are in standard interest factor tables
P = 1 2 3 4 5 6 7 8
A =?

45. Example – (1) find F, given A (uniform series, ordinary annuity)
If 8 annual deposits of RM each are placed in an account, how much money will be accumulated immediately after the last deposit? (use10% annual interest rate)
What amount, at the end of the 8th year, is equivalent to 8 end-of-the-year payments of RM each?
F = A ( F / A, i%, N ) (try it!)
RM * = RM2,143.60

47. Example – (2) find P, given A (uniform series, ordinary annuity)
How much should be deposited in a fund now so that at the end of every year, for 8 years, a sum of RM can be withdrawn?
What is the present equivalence of 8 end-of-the-year payments of RM each?
P = A ( P / A, i%, N ) (try it!)
P = A ( P / A, i%, N ) (try it!)
P= * = RM

48. Example – (3) find A, given F (uniform series, ordinary annuity)
What uniform annual amount should be deposited every year so that RM can be accumulated at the time of the 8th annual deposit?
What uniform payment, made at the end of every year, is equivalent to RM at the end of the 8th year?
A = F ( A / F, i%, N ) (try it!)
A = F ( A / F, i%, N )
A= * = RM

49. Example – (4) find A, given P (uniform series, ordinary annuity)
What is the size of 8 equal annual payments to repay a loan of RM1000? The first payment is due one year after receiving the loan. (Interest 10% per year)
What is the amount of the uniform payment that must be made, such that at the end of 8 successive years, the payment would be equivalent to RM1000 at the beginning of the first year?
A = P ( A / P, i%, N ) (try it!)
A = P ( A / P, i%, N )
A=1000 * = RM187.4

50. Annuity is deferred for j periods, where j < N Find P given A
RELATING UNIFORM SERIES (DEFERRED ANNUITY) TO PRESENT / FUTURE EQUIVALENT VALUES
Annuity is deferred for j periods, where j < N
Find P given A
For ordinary annuity: P = A ( P / A, i%, N )
For deferred annuity:
P= A ( P / A, i%, N - j ) at end of period j
P= A ( P / A, i%, N - j ) ( P / F, i%, j ) at time 0 (time present)

51. Example – deferred annuity
A mother, on the day her daughter is born, wants to put aside a lump sum of money in an account bearing a 12%/year interest. The mother wishes that her daughter withdraws RM2000 on her 18th, 19th, 20th, and 21st birthday. How much should the lump sum be?
P = A ( P / A, i%, N - j ) ( P / F, i%, j )
(j = 17, N = 21)
P = A ( P / A, i%, N - j ) ( P / F, i%, j ) = 2000*3.0373*0.1456=RM884.46

53. F = P (F / P, i%, N )
P = F ( P / F, i%, N )
F = A ( F / A, i%, N )
P = A ( P / A, i%, N )
A = F ( A / F, i%, N )
A = P ( A / P, i%, N )
P= A ( P / A, i%, N - j ) at end of period j
P= A ( P / A, i%, N - j ) ( P / F, i%, j ) at time 0

54. EQUIVALENCE CALCULATIONS INVOLVING MULTIPLE PARAMETERS
So far, we have seen that all compounding interest takes place once per time period (e.g., a year), and to this point, cash flows also occur once per time period.
However, there are cases where a series of cash outflows occur over a number of years, and that the value of the outflows is unique for each of the years (for example, the first three years). It could be that the value of outflows is the same for the last four years.
You might be asked to find a) the present equivalent expenditure; b) the future equivalent expenditure; and c) the annual equivalent expenditure

55. Example - problem involving multiple parameters
Your company needs to make a series of year-end payments of equipment maintenance over 8 years – RM100 on the 1st year, RM200 on the 2nd year, RM500 on 3rd year, RM400 each year on the 4th – 8th year. Find a) the present equivalent expenditure; b) the future equivalent expenditure; and c) the annual equivalent expenditure of these cash flows, given annual interest rate of 20%

56. Present EQUIVALENT EXPENDITURE CALCULATIONS
P0 = F( P / F, i%, N ) for each of the unique years
-- F is a series of unique outflow for year 1 through year 3
-- i is common for each calculation
-- N is the year in which the outflow occurred
-- Multiply the outflow with the associated table value
-- Add the three products together
Use A ( P / A, i%, N - j ) ( P / F, i%, j ) -- deferred annuity -- for the remaining (common outflow) years:
-- A is common for years 4 through 7
-- i remains the same
-- N is the final year
-- j is the last year the UNIQUE outflow occurred
-- multiply the common outflow value with the table values
-- add this to the previous total for the present equivalent expenditure.
J=3, N=8

58. Present equivalent expenditure--
Po = F1 (P/F, 20%, 1) + F2 (P/F, 20%, 2) + F3 (P/F, 20%, 3) + A (P/A, 20%, 5) * (P/F, 20%, 3)
Po = RM RM RM RM = RM
Future equivalent expenditure—
F8 = Po (F / P, 20%, 8) = RM * = RM
Annual equivalent expenditure—
A = F ( A / F, 20%, 8 ) = RM * = RM313.7
A = P ( A / P, 20%, 8 ) = RM * = RM313.7
Deferred annuity – 8years minus 3years = 5;

59. Find F given G: (G/ i) (F/A, i%, N) - (NG/ i)
Relating Uniform Gradient of Cash Flow to its Future, Annual and Present Equivalents
G = uniform gradient amounts -- used if cash flows increase by a constant (uniform) amount in each period
Find F given G: (G/ i) (F/A, i%, N) - (NG/ i)
Find A given G: A = G ( A / G, i%, N )
Find P given G: P = G ( P / G, i%, N )
Direct use of gradient conversion factors applies when there is NO cash flow at the end of period one.

60. Example-gradient problem
Payment for a certain technical service is given as follows: RM1000 in the 2nd year of service, RM2000 in the 3rd year of service, RM3000 in the 4th year of service. Interest rate is 15% per year. Find (a) present equivalence at the beginning of the 1st year, (b) annual equivalent values at the end of each of the 4 years.
Direct use of gradient conversion factors applies when there is NO cash flow at the end of period one.

62. (or, use A = P ( A / P, 15%, 4 ) = 3789*0.3503 =RM1,327 )
Present equivalence
P = G ( P / G, 15%, 4 )
= 1000*3.786 = RM3789
Annual equivalence
A = G ( A / G, 15%, 4 )
= 1000* = RM1,326
(or, use A = P ( A / P, 15%, 4 ) = 3789* =RM1,327 )
Direct use of gradient conversion factors applies when there is NO cash flow at the end of period one.

63. VARYING INTEREST RATES
Sometimes, the interest rate on a loan varies with time. This must be addressed when calculating the equivalent values of the loan
Find P given F; with interest rates that vary over N
Find F given P; with interest rates that vary over N

64. NOMINAL AND EFFECTIVE INTEREST RATES
Sometimes, the interest period is less than a year
Nominal Interest Rate - r - rates compounded more than once a year; the stated annual interest rate
Annual Percentage Rate - APR – percentage rate per period times number of compounding periods. APR = r x M
Effective Interest Rate - i - rates compounded more than once a year, the actual amount of interest paid. The actual rate is higher than the nominal rate because compounding happens more than once a year
i = ( 1 + r / M )M - 1 = ( F / P, r / M, M ) -1
M is the number of compounding periods per year

65. Example – nominal, APR, effective interest rates
A credit card company charges an interest rate of 1.375% per month on the unpaid balance of all accounts. What is the nominal interest rate? What is the APR? What is the effective interest rate per year?
Nominal interest rate = 1.375% per month
APR = (12)(1.375) = 16.5%/year
Effective interest rate = i = ( 1 + r / M )M - 1
= ( 1 + (0.165/ 12 )12 - 1
= or 17.81%/year

66. COMPOUNDING MORE THAN ONCE A YEAR
Single Amounts
Find P, F, A, given r, M
F = P ( F / P, i%, N )
i% = ( 1 + r / M ) M – 1
Uniform and / or Gradient Series
Find P, F, A, given r, M, and the existence of a cash flow at the end of each period - use formulas and tables for uniform annual series and uniform gradient series

67. CASH FLOWS LESS OFTEN THAN COMPOUNDING PERIODS
Find A, given i, k and X, where:
i is the effective interest rate per interest period
k is the period at the end of which cash flow occurs
X is the uniform cash flow amount
Use: A = X (A / F, i%, k )
k is the period at the beginning of which cash flow occurs
Use: A = X ( A / P, i%, k )

68. CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS
Continuous compounding assumes cash flows occur at discrete intervals, but compounding is continuous throughout the interval.
Given nominal per year interest rate -- r, compounding per year -- M one unit of principal = [ 1 + (r / M ) ] M
Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp
Given lim [ 1 + (1 / p) ] p = e1 =
( F / P, r%, N ) = e rN
i = e r - 1
pv

69. CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Single Cash Flow
Find F given P
F = P (e rN)
Functionally expressed as ( F / P, r%, N )
e rN is continuous compounding compound amount
Predetermined values are in standard tables.

70. CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Single Cash Flow
Find P given F
P = F (e -rN)
Functionally expressed as ( P / F, r%, N )
e -rN is continuous compounding present equivalent
Predetermined values are given in standard interest tables

71. CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Uniform Series
Find F given A
F = A (e rN- 1)/(e r- 1)
Functionally expressed as ( F / A, r%, N )
(e rN- 1)/(e r- 1) is continuous compounding compound amount
Predetermined values are in standard interest tables

72. CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Uniform Series
Find P given A
Finding present equivalent value given a series of uniform equal receipts
P = A (e rN- 1) / (e rN ) (e r- 1)
Functionally expressed as ( P / A, r%, N )
(e rN- 1) / (e rN ) (e r- 1) is continuous compounding present equivalent
Predetermined values are in column 5 of appendix D of text

73. CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Uniform Series
Find A given F
Finding a uniform series given a future value
A = F (e r- 1) / (e rN - 1)
Functionally expressed as ( A / F, r%, N )
(e r- 1) / (e rN - 1) is continuous compounding sinking fund
Predetermined values are in column 6 of appendix D of text

74. CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Uniform Series
Find A given P
Finding a series of uniform equal receipts given present equivalent value
A = P [e rN (e r- 1) / (e rN - 1) ]
Functionally expressed as ( A / P, r%, N )
[e rN (e r- 1) / (e rN - 1) ] is continuous compounding capital recovery
Predetermined values are in column 7 of appendix D of text

75. CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS
Continuous flow of funds suggests a series of cash flows occurring at infinitesimally short intervals of time
Given:
a nominal interest rate or r
p is payments per year
[ 1 + (r / p ) ] p - 1
P =
r [ 1 + ( r / p ) ] p
Given Lim [ 1 + ( r / p ) ] p = e r
For one year ( P / A, r%, 1 ) = ( e r - 1 ) / re r
p --> oo

76. CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS
Find F given A
Finding the future equivalent given the continuous funds flow
F = A [ ( erN - 1 ) / r ]
Functionally expressed as ( F / A, r%, N )
( erN - 1 ) / r is continuous compounding compound amount
Predetermined values are found in column 6 of appendix D of text.

77. CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS
Find P given A
Finding the present equivalent given the continuous funds flow
P = A [ ( erN - 1 ) / rerN ]
Functionally expressed as ( P / A, r%, N )
( erN - 1 ) / rerN is continuous compounding present equivalent
Predetermined values are found in column 7 of appendix D of text.

78. CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS
Find A given F
Finding the continuous funds flow given the future equivalent
A = F [ r / ( erN - 1 )]
Functionally expressed as ( A / F, r%, N )
r / ( erN - 1 ) is continuous compounding sinking fund

79. CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS
Find A given P
Finding the continuous funds flow given the present equivalent
A = F [ rerN / ( erN - 1 )]
Functionally expressed as ( A / P, r%, N )
rerN / ( erN - 1 ) is continuous compounding capital recovery