1. APPLICATIONS OF MONEY-TIME RELATIONSHIPS
CHAPTER 4
APPLICATIONS OF MONEY-TIME RELATIONSHIPS

2. MINIMUM ATTRACTIVE RATE OF RETURN ( MARR )
An interest rate used to convert cash flows into equivalent worth at some point(s) in time
Usually a policy issue based on:
- amount, source and cost of money available for investment
- number and purpose of good projects available for investment
- amount of perceived risk of investment opportunities and estimated cost of administering projects over short and long run
- type of organization involved
MARR is sometimes referred to as hurdle rate

3. CAPITAL RATIONING MARR approach involving opportunity cost viewpoint
Exists when management decides to restrict the total amount of capital invested, by desire or limit of available capital
Select only those projects which provide annual rate of return in excess of MARR
As amount of investment capital and opportunities available change over time, a firm’s MARR will also change

4. PRESENT WORTH METHOD ( PW )
Based on concept of equivalent worth of all cash flows relative to the present as a base
All cash inflows and outflows discounted to present at interest -- generally MARR
PW is a measure of how much money can be afforded for investment in excess of cost
PW is positive if dollar amount received for investment exceeds minimum required by investors

5. FINDING PRESENT WORTH
Discount future amounts to the present by using the interest rate over the appropriate study period

6. FINDING PRESENT WORTH PW = Fk ( 1 + i ) - k
Discount future amounts to the present by using the interest rate over the appropriate study period
PW = Fk ( 1 + i ) - k
i = effective interest rate, or MARR per compounding period
k = index for each compounding period
Fk = future cash flow at the end of period k
N = number of compounding periods in study period N
k = 0

7. FINDING PRESENT WORTH PW = Fk ( 1 + i ) - k
Discount future amounts to the present by using the interest rate over the appropriate study period
PW = Fk ( 1 + i ) - k
i = effective interest rate, or MARR per compounding period
k = index for each compounding period
Fk = future cash flow at the end of period k
N = number of compounding periods in study period
interest rate is assumed constant through project N
k = 0

8. FINDING PRESENT WORTH PW = Fk ( 1 + i ) - k
Discount future amounts to the present by using the interest rate over the appropriate study period
PW = Fk ( 1 + i ) - k
i = effective interest rate, or MARR per compounding period
k = index for each compounding period
Fk = future cash flow at the end of period k
N = number of compounding periods in study period
interest rate is assumed constant through project
The higher the interest rate and further into future a cash flow occurs, the lower its PW N
k = 0

9. BOND AS EXAMPLE OF PRESENT WORTH
The value of a bond, at any time, is the present worth of future cash receipts from the bond
The bond owner receives two types of payments from the borrower:
-- periodic interest payments until the bond is retired ( based on r );
-- redemption or disposal payment when the bond is retired ( based on i );
The present worth of the bond is the sum of the present values of these two payments at the bond’s yield rate

10. VN = C ( P / F, i%, N ) + rZ ( P / A, i%, N )
PRESENT WORTH OF A BOND
For a bond, let
Z = face, or par value
C = redemption or disposal price (usually Z )
r = bond rate (nominal interest) per interest period
N = number of periods before redemption
i = bond yield (redemption ) rate per period
VN = value (price) of the bond N interest periods prior to redemption -- PW measure of merit
VN = C ( P / F, i%, N ) + rZ ( P / A, i%, N )
Periodic interest payments to owner = rZ for N periods -- an annuity of N payments
When bond is sold, receive single payment (C), based on the price and the bond yield rate ( i )

11. FUTURE WORTH METHOD (FW )
FW is based on the equivalent worth of all cash inflows and outflows at the end of the planning horizon at an interest rate that is generally MARR

12. FUTURE WORTH METHOD (FW )
FW is based on the equivalent worth of all cash inflows and outflows at the end of the planning horizon at an interest rate that is generally MARR
The FW of a project is equivalent to PW
FW = PW ( F / P, i%, N )

13. FUTURE WORTH METHOD (FW )
FW is based on the equivalent worth of all cash inflows and outflows at the end of the planning horizon at an interest rate that is generally MARR
The FW of a project is equivalent to PW
FW = PW ( F / P, i%, N )
If FW > 0, it is economically justified

14. FUTURE WORTH METHOD (FW )
FW is based on the equivalent worth of all cash inflows and outflows at the end of the planning horizon at an interest rate that is generally MARR
The FW of a project is equivalent to PW
FW = PW ( F / P, i%, N )
If FW > 0, it is economically justified
FW ( i % ) =  Fk ( 1 + i ) N - k N
k = 0

15. FUTURE WORTH METHOD (FW )
FW is based on the equivalent worth of all cash inflows and outflows at the end of the planning horizon at an interest rate that is generally MARR
The FW of a project is equivalent to PW
FW = PW ( F / P, i%, N )
If FW > 0, it is economically justified
FW ( i % ) =  Fk ( 1 + i ) N - k N
k = 0
i = effective interest rate
k = index for each compounding period
Fk = future cash flow at the end of period k
N = number of compounding periods in study period

16. ANNUAL WORTH METHOD ( AW )
AW is an equal annual series of dollar amounts, over a stated period ( N ), equivalent to the cash inflows and outflows at interest rate that is generally MARR
AW is annual equivalent revenues ( R ) minus annual equivalent expenses ( E ), less the annual equivalent capital recovery (CR)
AW ( i % ) = R - E - CR ( i % )
AW = PW ( A / P, i %, N )
AW = FW ( A / F, i %, N )
If AW > 0, project is economically attractive
AW = 0 : annual return = MARR earned

17. CR ( i % ) = I ( A / P, i %, N ) - S ( A / F, i %, N )
CAPITAL RECOVERY ( CR )
CR is the equivalent uniform annual cost of the capital invested
CR is an annual amount that covers:
Loss in value of the asset
Interest on invested capital ( i.e., at the MARR )
CR ( i % ) = I ( A / P, i %, N ) - S ( A / F, i %, N )
I = initial investment for the project
S = salvage ( market ) value at the end of the study period
N = project study period

18. CAPITAL RECOVERY ( CR)
CR is also calculated by adding sinking fund amount (i.e., deposit) to interest on original investment
CR ( i % ) = ( I - S ) ( A / F, i %, N ) + I ( i % )
CR is also calculated by adding the equivalent annual cost of the uniform loss in value of the investment to the interest on the salvage value
CR ( i % ) = ( I - S ) ( A / P, i %, N ) + S ( i % )

19. INTERNAL RATE OF RETURN METHOD ( IRR )
IRR solves for the interest rate that equates the equivalent worth of an alternative’s cash inflows (receipts or savings) to the equivalent worth of cash outflows (expenditures)
Also referred to as:
investor’s method
discounted cash flow method
profitability index
IRR is positive for a single alternative only if:
both receipts and expenses are present in the cash flow pattern
the sum of receipts exceeds sum of cash outflows

20.  R k ( P / F, i’ %, k ) =  E k ( P / F, i’ %, k )
INTERNAL RATE OF RETURN METHOD ( IRR )
IRR is i’ %, using the following PW formula:
 R k ( P / F, i’ %, k ) =  E k ( P / F, i’ %, k ) N N
k = 0
k = 0

21.  R k ( P / F, i’ %, k ) =  E k ( P / F, i’ %, k )
INTERNAL RATE OF RETURN METHOD ( IRR )
IRR is i’ %, using the following PW formula:
 R k ( P / F, i’ %, k ) =  E k ( P / F, i’ %, k )
R k = net revenues or savings for the kth year N N
k = 0
k = 0

22.  R k ( P / F, i’ %, k ) =  E k ( P / F, i’ %, k )
INTERNAL RATE OF RETURN METHOD ( IRR )
IRR is i’ %, using the following PW formula:
 R k ( P / F, i’ %, k ) =  E k ( P / F, i’ %, k )
R k = net revenues or savings for the kth year
E k = net expenditures including investment costs for the kth year N N
k = 0
k = 0

23.  R k ( P / F, i’ %, k ) =  E k ( P / F, i’ %, k )
INTERNAL RATE OF RETURN METHOD ( IRR )
IRR is i’ %, using the following PW formula:
 R k ( P / F, i’ %, k ) =  E k ( P / F, i’ %, k )
R k = net revenues or savings for the kth year
E k = net expenditures including investment costs for the kth year
N = project life ( or study period ) N N
k = 0
k = 0

24.  R k ( P / F, i’ %, k ) =  E k ( P / F, i’ %, k )
INTERNAL RATE OF RETURN METHOD ( IRR )
IRR is i’ %, using the following PW formula:
 R k ( P / F, i’ %, k ) =  E k ( P / F, i’ %, k )
R k = net revenues or savings for the kth year
E k = net expenditures including investment costs for the kth year
N = project life ( or study period )
If i’ > MARR, the alternative is acceptable N N
k = 0
k = 0

25. INTERNAL RATE OF RETURN METHOD ( IRR )
IRR is i’ %, using the following PW formula:
 R k ( P / F, i’ %, k ) =  E k ( P / F, i’ %, k )
R k = net revenues or savings for the kth year
E k = net expenditures including investment costs for the kth year
N = project life ( or study period )
If i’ > MARR, the alternative is acceptable
To compute IRR for alternative, set net PW = 0
PW =  R k ( P / F, i’ %, k ) -  E k ( P / F, i’ %, k ) = 0
i’ is calculated on the beginning-of-year unrecovered investment through the life of a project N N
k = 0
k = 0 N N
k = 0
k = 0

26. INTERNAL RATE OF RETURN PROBLEMS
The IRR method assumes recovered funds, if not consumed each time period, are reinvested at i ‘ %, rather than at MARR
The computation of IRR may be unmanageable
Multiple IRR’s may be calculated for the same problem
The IRR method must be carefully applied and interpreted in the analysis of two or more alternatives, where only one is acceptable

27. THE EXTERNAL RATE OF RETURN METHOD ( ERR )
ERR directly takes into account the interest rate (  ) external to a project at which net cash flows generated over the project life can be reinvested (or borrowed ).
If the external reinvestment rate, usually the firm’s MARR, equals the IRR, then ERR method produces same results as IRR method

28. CALCULATING EXTERNAL RATE OF RETURN ( ERR )
1. All net cash outflows are discounted to the present (time 0) at  % per compounding period.
2. All net cash inflows are discounted to period N at  %.
3. ERR -- the equivalence between the discounted cash inflows and cash outflows -- is determined.
The absolute value of the present equivalent worth of the net cash outflows at  % is used in step 3.
A project is acceptable when i ‘ % of the ERR method is greater than or equal to the firm’s MARR

29. CALCULATING EXTERNAL RATE OF RETURN ( ERR )
 Ek ( P / F,  %, k )( F / P, i ‘ %, N ) =
 Rk ( F / P,  %, N - k )
k = 0 N
k = 0

30. CALCULATING EXTERNAL RATE OF RETURN ( ERR )
 Ek ( P / F,  %, k )( F / P, i ‘ %, N ) =
 Rk ( F / P,  %, N - k )
Rk = excess of receipts over expenses in period k
k = 0 N
k = 0

31. CALCULATING EXTERNAL RATE OF RETURN ( ERR )
 Ek ( P / F,  %, k )( F / P, i ‘ %, N ) =
 Rk ( F / P,  %, N - k )
Rk = excess of receipts over expenses in period k
Ek = excess of expenses over receipts in period k
k = 0 N
k = 0

32. CALCULATING EXTERNAL RATE OF RETURN ( ERR )
 Ek ( P / F,  %, k )( F / P, i ‘ %, N ) =
 Rk ( F / P,  %, N - k )
Rk = excess of receipts over expenses in period k
Ek = excess of expenses over receipts in period k
N = project life or period of study
k = 0 N
k = 0

33. CALCULATING EXTERNAL RATE OF RETURN ( ERR )
 Ek ( P / F,  %, k )( F / P, i ‘ %, N ) =
 Rk ( F / P,  %, N - k )
Rk = excess of receipts over expenses in period k
Ek = excess of expenses over receipts in period k
N = project life or period of study
 = external reinvestment rate per period
k = 0 N
k = 0

34. CALCULATING EXTERNAL RATE OF RETURN ( ERR )
 Ek ( P / F,  %, k )( F / P, i ‘ %, N ) =
 Rk ( F / P,  %, N - k )
Rk = excess of receipts over expenses in period k
Ek = excess of expenses over receipts in period k
N = project life or period of study
 = external reinvestment rate per period
k = 0 N
k = 0 N
 Rk ( F / P,  %, N - k )
k = 0
i ‘ %= ?
Time N N
 Ek ( P / F,  %, k )( F / P, i ‘ %, N )
k = 0

35. ERR ADVANTAGES ERR has two advantages over IRR:
1. It can usually be solved for directly, rather than by trial and error.
2. It is not subject to multiple rates of return.

36. PAYBACK PERIOD METHOD
Sometimes referred to as simple payout method

37. PAYBACK PERIOD METHOD Sometimes referred to as simple payout method
Indicates liquidity (riskiness) rather than profitability

38. PAYBACK PERIOD METHOD Sometimes referred to as simple payout method
Indicates liquidity (riskiness) rather than profitability
Calculates smallest number of years (  ) needed for cash inflows to equal cash outflows -- break-even life

39. PAYBACK PERIOD METHOD Sometimes referred to as simple payout method
Indicates liquidity (riskiness) rather than profitability
Calculates smallest number of years (  ) needed for cash inflows to equal cash outflows -- break-even life
 ignores the time value of money and all cash flows which occur after 

40. PAYBACK PERIOD METHOD ( Rk -Ek) - I > 0
Sometimes referred to as simple payout method
Indicates liquidity (riskiness) rather than profitability
Calculates smallest number of years (  ) needed for cash inflows to equal cash outflows -- break-even life
 ignores the time value of money and all cash flows which occur after 
( Rk -Ek) - I > 0 
k = 1

41. PAYBACK PERIOD METHOD ( Rk -Ek) - I > 0
Sometimes referred to as simple payout method
Indicates liquidity (riskiness) rather than profitability
Calculates smallest number of years (  ) needed for cash inflows to equal cash outflows -- break-even life
 ignores the time value of money and all cash flows which occur after 
( Rk -Ek) - I > 0
If  is calculated to include some fraction of a year, it is rounded to the next highest year 
k = 1

42. PAYBACK PERIOD METHOD
The payback period can produce misleading results, and should only be used with one of the other methods of determining profitability

43. PAYBACK PERIOD METHOD
The payback period can produce misleading results, and should only be used with one of the other methods of determining profitability
A discounted payback period ‘ ( where ‘ < N ) may be calculated so that the time value of money is considered

44. ( Rk - Ek) ( P / F, i %, k ) - I > 0
PAYBACK PERIOD METHOD
The payback period can produce misleading results, and should only be used with one of the other methods of determining profitability
A discounted payback period ‘ ( where ‘ < N ) may be calculated so that the time value of money is considered 
( Rk - Ek) ( P / F, i %, k ) - I > 0
k = 1

45. ( Rk - Ek) ( P / F, i %, k ) - I > 0
PAYBACK PERIOD METHOD
The payback period can produce misleading results, and should only be used with one of the other methods of determining profitability
A discounted payback period ‘ ( where ‘ < N ) may be calculated so that the time value of money is considered
i‘ is the MARR 
( Rk - Ek) ( P / F, i %, k ) - I > 0
k = 1

46. ( Rk - Ek) ( P / F, i %, k ) - I > 0
PAYBACK PERIOD METHOD
The payback period can produce misleading results, and should only be used with one of the other methods of determining profitability
A discounted payback period ‘ ( where ‘ < N ) may be calculated so that the time value of money is considered
i‘ is the MARR
I is the capital investment made at the present time 
( Rk - Ek) ( P / F, i %, k ) - I > 0
k = 1

47. ( Rk - Ek) ( P / F, i %, k ) - I > 0
PAYBACK PERIOD METHOD
The payback period can produce misleading results, and should only be used with one of the other methods of determining profitability
A discounted payback period ‘ ( where ‘ < N ) may be calculated so that the time value of money is considered
i‘ is the MARR
I is the capital investment made at the present time
( k = 0 ) is the present time 
( Rk - Ek) ( P / F, i %, k ) - I > 0
k = 1

48. ( Rk - Ek) ( P / F, i %, k ) - I > 0
PAYBACK PERIOD METHOD
The payback period can produce misleading results, and should only be used with one of the other methods of determining profitability
A discounted payback period ‘ ( where ‘ < N ) may be calculated so that the time value of money is considered
i‘ is the MARR
I is the capital investment made at the present time
( k = 0 ) is the present time
‘ is the smallest value that satisfies the equation ’
( Rk - Ek) ( P / F, i %, k ) - I > 0
k = 1

49. INVESTMENT-BALANCE DIAGRAM
Describes how much money is tied up in a project and how the recovery of funds behaves over its estimated life.

50. INTERPRETING IRR USING INVESTMENT-BALANCE DIAGRAM
P (1 + i‘)
Unrecovered
Investment
Balance, $
[ P (1 + i‘) - (R1 - E1) ] (1 +i‘)
1 + i‘
1 + i‘
(R1 - E1)
1 + i‘
(R2 - E2)
(R3 - E3)
Initial investment
= P
1 + i‘
(RN-1 - EN-1)
(RN - EN) $0 1 2 3 N
downward arrows represent annual returns (Rk - Ek) : 1 < k < N
dashed lines represent opportunity cost of interest, or interest on BOY investment balance
IRR is value i ‘ that causes unrecovered investment balance to equal 0 at the end of the investment period.

51. INVESTMENT-BALANCE DIAGRAM EXAMPLE
Capital Investment ( I ) = $10,000
Uniform annual revenue = $5,310
Annual expenses = $3,000
Salvage value = $2,000
MARR = 5% per year

52. ’ Investment Balance, $ MARR = 5% Years Area of Negative Investment
5,000
MARR = 5%
$2,001 ( = FW ) ’
+ $4,310
Years 5 1 2 3 4
Area of Negative
Investment
Balance
- $2,199
- $2,310
- $2,310
- 5,000
- $4,294
- $4,509
- $6,290
- $2,310
- $8,190
- $2,310
- $6,604
- $2,310
- $8,600
- 10,000
-$10,500

53. WHAT INVESTMENT-BALANCE DIAGRAM PROVIDES
Discounted payback period ( ‘) is 5 years
FW is $2,001
Investment has negative investment balance until the fifth year
Investment-balance diagram provides additional insight into worthiness of proposed capital investment opportunity and helps communicate important economic information