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Word Problems

Word Problems. Organize the Data Given: Determine the objective and your strategy. Draw the Cash Flow Diagram. Write Equations and Solve. Reflect Back on What You Learned – What does your answer mean?. Reviewing…. Complex Cash Flows – Must split-up and recombine at the SAME point in time.

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Word Problems

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  1. Word Problems • Organize the Data Given: • Determine the objective and your strategy. • Draw the Cash Flow Diagram. • Write Equations and Solve. • Reflect Back on What You Learned – What does your answer mean?

  2. Reviewing… • Complex Cash Flows – Must split-up and recombine at the SAME point in time. • Linear gradient cash flows always begin at the end of year two. • Strategy – comparisons must occur at the same point in time. • Periods for i and n must match!

  3. Present Worth Analysis Net Present Worth of initial and future cash flows can be used to select among alternative projects. It is important to understand what Net Present Worth means, especially when the cash flows include both revenue and expenses.

  4. Present Worth Analysis • Project selection based on Present Worth Analysis: • If all expenses and revenues are included, select the largest NPW that is greater than zero. • Doing nothing is an option • (leaving the $ to earn “safe” interest) • If some or none of the revenues are included, select the largest NPW. • “Must Do situation”

  5. Terminology Salvage Value – the amount of money you can expect to receive by selling an asset when you are done with it. What value does it have when you are done with it? MARR – Minimum Attractive Rate of Return – I expect or need this return in order to be willing to invest my money.

  6. Example Problem Project A costs $10,000 and will last for 10 years. Annual, end of the year revenues will be $3,000, and expenses will be $1,000. There is no salvage value. Project B costs $10,000 and will also last for 10 years. Annual revenues will be $3,000 with annual expenses of $1,500. Salvage value is $5,000. Conduct an economic analysis to select the preferred project using a MARR of 10% per year, compounded annually.

  7. DIAGRAM: NPWA ? $3 000 10 0 1 2 3 4 $1 000 $10 000 Example Problem Project A costs $10,000 and will last for 10 years. Annual, end of the year revenues will be $3,000, and expenses will be $1,000. There is no salvage value. GIVEN: LIFETIME = 10 YRS MARR = 10%/YR, CPD ANNUALLY FIRST COST = $10 000 ANNUAL REVENUES = $3 000/YR ANNUAL COSTS = $1 000/YR SALVAGE VALUE = $0 FIND NPWA: NET ANNUAL = ANNUAL REVENUES – ANNUAL COSTS = $3000/YR–$1000/YR = $2000/YR NPWA = A(P|A,i,n) – 1ST COST = $2 000(P|A,10%,10)–$10 000 = $2 000(6.1446)–$10 000 = $2 289

  8. DIAGRAM: NPWB ? $5 000 $3 000 10 0 1 2 3 4 $1 500 $10 000 Example Problem Project B costs $10,000 and will also last for 10 years. Annual revenues will be $3,000 with annual expenses of $1,500. Salvage value is $5,000. GIVEN: LIFETIME = 10 YRS MARR = 10%/YR, CPD ANNUALLY FIRST COST = $10 000 ANNUAL REVENUES = $3 000/YR ANNUAL COSTS = $1 500/YR SALVAGE VALUE = $5 000 FIND NPWB: NET ANNUAL = ANNUAL REVENUES – ANNUAL COSTS = $3 000/YR–$1 500/YR = $1 500/YR NPWB = A(P|A,i,n) + SALVAGE(P|F,i,n) – 1ST COST = $1 500(P|A,10%,10)+ $5 000(P|F,10%,10)–$10 000 = $1 500(6.1446) + $5 000(0.3855)–$10 000 = $1 144 ►PREFER A

  9. What does this mean? NPWA = $2 289 NPWB = $1 144 We prefer project A over project B. Does NOT mean a $2 289 profit! Concept: We favor Project A by $2 289 over taking $10 000 and putting it in an account earning 10%.

  10. Thus… The project only costs $10 000, but at i = 10% it is equivalent to investing $10 000 + $2 289 for 10 years. Since PW > 0, you are actually earning more than 10% on investment.

  11. Present Worth Analysis • When applied correctly, NPW can be used to select among various alternative projects. • The larger the NPW the better. • Requires establishing MARR. • MARR is used as the (i) in the equations.

  12. Plotting NPW vs. i 10 000 NPW ($) IRR 2 289 0 10% 15.1% i (%) Why does NPW decrease as i increases?

  13. Problem 2 Project #1 costs $10,000 and has annual, end of the year revenues of $10,000 over its 5 year life. There is no salvage value. Project #2 costs $20,000 and has annual end of year revenues of $10,000 over its 10 year life. There is no salvage value. Conduct an economic analysis to select the preferred project using a MARR of 15% per year, compounded annually.

  14. Calculating NPWs… NPW1 = $23,522 NPW2 = $30,188 Why is it wrong to select Project 2 based on this analysis?

  15. (Net) Present Worth Analysis When comparing projects, it is necessary to compare alternatives with the same project life (i.e., over the same period of time).

  16. (Net) Present Worth Analysis Two possible approaches when project lives are different: Common Multiple Period: Projects are assumed to be repeated until a common multiple point in time is established. Study Period: Select a study period for both projects and estimate cash flows to conform to the study period.

  17. Problem 3 A firm is considering the purchase of one of two new machines. The data on each are as below: Machine A B Service Life 3 years 6 years Initial Cost $3,400 $6,500 Annual Net Operating Expense: $2,000 $1,800 Salvage Value $100 $500 Use a MARR of 12% compounded annually and the lowest common multiple assumption to determine the alternative to be selected.

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