1. Evaluating a Single Project
Chapter 5
Evaluating a Single Project
The objective of Chapter 5 is to discuss contemporary methods for determining project profitability.

2. Proposed capital projects can be evaluated in several ways.
Present worth (PW)
Future worth (FW)
Annual worth (AW)
Internal rate of return (IRR)
External rate of return (ERR)
Payback period

3. To be attractive, a capital project must provide a return that exceeds a minimum level established by the organization.
This minimum level is reflected in a firm’s Minimum Attractive Rate of Return (MARR).

4. The present worth method.
The present worth (PW) is found by discounting all cash inflows and outflows to the present time at an interest rate that is generally the MARR.
A positive PW for an investment project means that the project is acceptable (it satisfies the MARR).

5. PW = $4,671.40  This is a good investment!
Example:
Consider a project that has an initial investment of $50,000 and that returns $18,000 per year for the next four years. If the MARR is 12%, is this a good investment?
PW = -50, ,000 (P/A, 12%, 4)
PW = -50, ,000 (3.0373)
PW = $4,  This is a good investment!

9. Assumptions of the PW method
It is assumed we know the future with certainty.
It is assumed we can borrow or lend money at the same interest rate.

10. Bond value is a good example of present worth.
Two type of payments:
A series of periodic interest payments will be received until the bond is retired.
A single payment when the bond is retired or sold. (Bond’s maturity date is reached )
Face, Par, or value = Bond value (received after N periods )
Redemption or disposal price = bond value
Bond rate = nominal interest per interest period
N = number of periods before redemption
i = bond yield rate per period

11. Bond value = 10,000$
Bond rate = 8% / 4 = 2% per 3 months (quarter)
i = 10% / 4 = 2.5% per 3 months (quarter)
N = 8×4 = 32 quarters.
Uniform received payments = Bond value × Bond rate
= 0.02 ×10,000 = 200$ per quarter
P = 10,000 (P/F, 2.5%, 32) (P/A, 2.5%, 32) = $

12. Bond value = 5,000$. Bond rate = 8% per year. i = 10% per year.
N = 20 quarters.
Uniform received payments = Bond value × Bond rate
= 0.08 ×5,000 = 400$ per year
P = 5,000 (P/F, 10%, 20) (P/A, 10%, 20) = $
(b) P = 4600$ find i?
4600 = 5,000 (P/F, i%, 20) (P/A, i%, 20)
By trial and error, i = 8.9%

13. Capitalized worth CW
Capitalized worth is the present worth of all revenues or expenses over an infinite length of time.
If only expenses are considered this is sometimes referred to as capitalized cost.
The capitalized worth method is especially useful in problems involving endowments and public projects with indefinite lives.
As N becomes very large, (P/A) = 1/i. So, CW = A(1/i).

14. A = 30, ,000 (A/F, 8%, 4) = 34,438$
CW = -100,000 – 34,438 (P/A, 8%, infinity )
= -100,000 – 34,438/.08 = -530,475$

15. Future Worth (FW)
FW is based on the equivalent worth of all cash inflows and outflows at the end of the study period at an interest rate that is generally the MARR.
Decisions made using FW and PW will be the same.
A positive FW for an investment project means that the project is acceptable (it satisfies the MARR).

17. Annual Worth (AW)
Annual worth is an equal periodic series of dollar amounts that is equivalent to the cash inflows and outflows, at an interest rate that is generally the MARR.
The AW of a project is annual equivalent revenue or savings (R) minus annual equivalent expenses (E), minus annual capital recovery (CR) amount.

18. Capital Recovery CR is the annual equivalent cost of the capital invested.
The CR covers the following items.
Loss in value of the asset.
Interest on invested capital (at the MARR).
The CR distributes the initial cost (I) and the salvage value (S) across the life of the asset.
Equivalent Uniform Annual Cost (EUAC) = CR + E

19. Example:
A corporate jet costs 1,350,000$ and will incur 200,000$ per year in fixed cost (maintenance,… ) and 277$ per hour variable cost (fuel,…). The jet will be operated 1200 hours per year for five years and then sold for 650,000$. The jet revenues 1000$ per hour. The MARR is 15% per year.
Revenue, R = 1000$ × 1200 = 1,200,000$ per year
Expense, E = 200,000$ + 277$ ×1200 = 532,400$ per year
Capital Recovery,
CR = 1,350,000(A/P, 15%, 5) – 650,000(A/F, 15%, 5)
= 306,310$ per year
4. Annual Worth, AW = R – E – CR = 361,290$ per year
5. Equivalent Uniform Annual Cost,
EUAC = E + CR = 838,710$ per year

20. 1. Revenue, R = 300,000 × 0.1 = 30,000$ per year
2. Expense, E = 0 $
3. Capital Recovery,
CR = 110,000(A/P, 15%, 6) – 8,000(A/F, 15%, 5)
= 28,148.4$ per year
4. Annual Worth, AW = R – E – CR = 1,851.6$ per year
5. Equivalent Uniform Annual Cost,
EUAC = E + CR = 28,148.4$ per year

21. Internal Rate of Return
The internal rate of return (IRR) method is the most widely used rate of return method for performing engineering economic analysis.
It is also called the investor’s method, the discounted cash flow method, and the profitability index.
If the IRR for a project is greater than the MARR, then the project is acceptable.
The IRR is the interest rate that equates the equivalent worth of an alternative’s cash inflows (revenue, R) to the equivalent worth of cash outflows (expenses, E).
The IRR is sometimes referred to as the breakeven interest rate.

22. The IRR is the interest i'% at which
The method of solving for the i'% that equates revenues and expenses normally involves trial-and-error calculations, or solving numerically using mathematical software.

23. -345, ,000(P/F, i, 6) + 98,000(P/A, i, 6) = 0
OR ,000(F/P, i, 6) + 115, ,000(F/A, i, 6) = 0
OR ,000(A/P, i, 6) + 115,000(A/F, i, 6) + 98,000 = 0

24. -345, ,000(P/F, i, 6) + 98,000(P/A, i, 6) = 0
Try i = MARR = 20%
-345, ,000× ,000× = 19,412.5
Try i = 25%
-345, ,000× ,000× = -25,621
20% ,412.5 i%
25% ,621
i = IRR = 22.1% per year

25. -25, ,000(P/F, i, 5) + 8,000(P/A, i, 5) = 0
Try i = 20% , ,000× ,000× = 934.3$
Try i = 25% , ,000× ,000× =
20% i%
25%
i = IRR = 21.68% per year

26. i = IRR = 6.75% per three months
A = 0.07 × 7000 = 490$
(P/A, i, 50) = (P/A, i, 50) =
i = 6% (P/A, 6%, 50) =
i = 7% (P/A, 7%, 50) =
i = IRR = 6.75% per three months

28. (P/A, i, 24) = (P/A, i, 50) =
i = 1% (P/A, 1%, 24) =
i = 0.75% (P/A, 0.75%, 24) =
i = IRR = 0.93% per month