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

2. PW, FW, AW, IRR, ERR, payback period
We want to know if a proposed capital investment and its associated expenditures can be recovered by revenue (or savings) over time in addition to a sufficiently attractive return on the capital.
Six methods for evaluating the economic profitability of a proposed problem solution:-
PW, FW, AW, IRR, ERR, payback period
PW, FW, and AW convert cash flows from the alternative into equivalent worth using MARR (minimum attractive rate of return)
Present worth, future worth, annual worth, internal rate of return, external rate of return

3. MINIMUM ATTRACTIVE RATE OF RETURN (MARR )
MARR is a reasonable rate of return established for the evaluation and selection of alternatives. A project is not considered to be economically viable if its expected returns is less than the MARR.
The MARR is used to convert cash flows into equivalent worth at some point(s) in time
How to determine MARR? MARR is based on:
- amount, source and cost of money available for investment
- number and purpose of good projects available for investment (are the projects essential or are they elective?)
- amount of perceived risk of investment opportunities and the estimated cost of administering projects over short and long runs
- type of organization involved (government? private?)
MARR (sometimes called ‘hurdle rate’) must be chosen to MAXIMIZE the economic well-being of the organization

4. CAPITAL RATIONING
Usually, MARR is established using the ‘opportunity cost’ viewpoint, which results from ‘capital rationing’
Capital rationing happens when management restricts the total amount of capital invested, due to insufficient funds to sponsor all good investment opportunities
Management would select projects that give annual rate of return (profit) that is higher than MARR
(see figure for explanation)
As amount of investment capital and opportunities available change over time, a firm’s MARR will also change

6. PRESENT WORTH METHOD ( PW )
PW is based on the concept of equivalent worth of all cash flows relative to the present as the base
All cash inflows and outflows are discounted to the present at an interest rate of the MARR
PW is a measure of how much money a firm can afford for investment in excess of cost
PW is positive if the dollar amount received for investment exceeds the minimum amount required by the investor (ie. the firm makes a profit)

7. FINDING PRESENT WORTH PW = PW of cash inflows – PW of cash outflows;
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

8. 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 nominal interest rate, r );
-- redemption or disposal payment when the bond is retired ( based on effective interest rate, i );
The present worth of the bond is the sum of the present values of these two payments at the bond’s yield rate

9. 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 bond at N interest periods prior to redemption
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 )

10. 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
FW = PW ( F / P, i%, N )
If FW > 0,
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

11. 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 an interest rate that is generally the 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

12. 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

13. 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

14. INTERNAL RATE OF RETURN METHOD ( IRR, i’)
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

15. IRR CHARACTERISTICS
The IRR method assumes that recovered funds, if not consumed at 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

16. 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
If the external reinvestment rate, usually the firm’s MARR, equals the IRR, then ERR method produces same results as IRR method

17.  Ek ( P / F,  %, k )( F / P, i ‘ %, N )
CALCULATING ERR N
 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

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

19. 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

20. ( 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

21. SUMMARY

22. N FW ( i % ) =  Fk ( 1 + i ) N - k FW = PW (1 + i ) N
PW is based on the concept of equivalent worth of all cash flows relative to the present as the base
PW = PW of cash inflows – PW of cash outflows;
PW = Fk ( 1 + i ) – k
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 N
FW ( i % ) =  Fk ( 1 + i ) N - k
k = 0
FW = PW (1 + i ) N

23. 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 ( i % ) = R - E - CR ( i % )
Where R is equivalent revenues, E is annual equivalent expenses, CR is the annual equivalent capital recovery

24. Rate of Return
ROR – the gain or loss on an investment over a specified period, expressed as a percentage over the initial investment cost
IRR – internal ROR. Most widely used. Also called ‘investor’s method’, ‘discounted cash flow method’, ‘profitability index’. Uses i that equates the equivalent worth of an alternative’s cash inflows to the equivalent worth of cash outflows.
ERR – external ROR. 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). ERR compensates for IRR’s weakness.

25. Example 1
An investment of RM10,000 will give a uniform annual revenue of RM5310 for 5 years. The project has a salvage value (market value) of RM2,000. Maintenance is RM3,000/year. The company accepts any project which will earn at least 10%/year. Is this project worth taking? Use the PW method

27. 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

28. Cash outflows, RM
Cash inflows, RM
Initial investment
10,000
Annual revenue
RM5310 (P/A, 10%, 5)
20,129
Market (salvage) value
RM2000 (P/F, 10%, 5)
1,242
Annual expenses
RM3000 (P/A, 10%, 5)
11,372
TOTAL
21,372
21,371
Total PW
PW(10%) is ~ RM0. So, the project is marginally acceptable

29. Example 2
You want to buy a bond that has a face value of RM5,000 and pays 8% interest/year. The bond will be redeemed in 20 years’ time. The first interest payment will be 1 year from now. (a) In order to receive 10%/year on this investment, how much should be paid now? (b) If you pay RM4,600 for the bond now, how much would you receive on an annual basis?

30. (a) VN = C ( P / F, i%, N ) + r Z ( P / A, i%, N )
For a bond, 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 bond at N interest periods prior to redemption
(a) VN = C ( P / F, i%, N ) + r Z ( P / A, i%, N )
= (RM5,000)(P/F, 10%, 20) + (0.08)(RM5000)(P/A, 10%, 20)
= (RM5000)(0.1486) + (0.08)(RM5000)(8.5136)
= RM743 + RM = RM
(b) VN = C ( P / F, i%, N ) + r Z ( P / A, i%, N )
RM4600 = (RM5000)(P/F, i’%, 20) + (RM5000)(0.08)(P/A,i’%,20)
(Go through the tables to find the closest value. This is the trial and error procedure. Answer – i’= 8.9%))

31. Example 3
You have been offered an investment, whereby, if you put in RM60,000 today, you will be guaranteed that the money will grow to be RM 105,000 in 3 years’ time. What is the effective interest rate?

32. FW = PW (1 + i ) N
105,000 = 60,000 ( 1 + i ) 3

33. Example 4
A new equipment that costs RM25,000, if used, will increase the productivity of a manufacturing line. After 5 years, the machine will have a market value of RM5,000. It has been calculated that the manufacturing line will have an increased productivity that amounts to RM8,000 (net). The company’s MARR is 20%/year.
(a) Draw a cash-flow diagram for this investment.
(b) Evaluate if this is a viable investment using the FW method
(c) Now, use the PW method, and compare your results

36. FW method
FW = FW of cash inflows – FW of cash outflows
FW (20%) = (-RM25000) (F/P, 20%, 5) + (RM8000)(F/A, 20%, 5) + RM5000 = RM 2,324.80
This is a good investment, as FW> 0
PW method
PW = PW of cash inflows – PW of cash outflows
PW (20%) = (RM8000)(P/A, 20%,5) + (RM5000)(P/F, 20%,5) – RM25000 = RM934.29
This is a good investment, as PW>0
Both methods show that this project is a viable one, and hence should be taken up