Question

given data: key center chocolates, inc. cost of new machine $24,000 life expectancy of new machine, in years 10 parts replacement costs at the end of year 5 $3,000 salvage value of new machine at the end of year 10 $2,000 annual operating costs of new machine $10,500 annual operating costs of current method $12,000 increased production (boxes) per year $2,300 contribution margin per box $1.90 required return on investment 12% the syntax of the pv function is: pv(rate, nper, pmt, fv, type) rate = interest rate, in this problem, .12 12% nper = number of periods, in years pmt = this argument is used only for the total net annual cash inflows, which is computed in part 1. leave this argument blank for all other pv computations. fv = future value. in this problem, the future value is the cost of the replacement of parts at the end of year 6 and the salvage value of the machine at the end of year 10. type = leave this field blank. doing so indicates payments are made at the end of the period. problem information: key center chocolates, inc. is considering buying a new machine that will automatically dip chocolates. (the dipping operation is currently done primarily by hand.) the annual costs of the current method, relevant costs if the new machine is purchased, and other important information are provided on the given data worksheet. key center chocolates, inc. net annual cash inflows provided by the new dipping machine reduction in annual operating costs: less: annual operating costs using current hand method less: annual operating costs of new machine = annual savings in operating costs plus: increased annual contribution margin due to increased production with the new machine total net annual cash inflows net present value of the new machine item amount of cash flows pv of cash flows years(s) cost of the new machine parts replacement costs end of 5th year net annual cash inflows salvage value of new machine net present value using the general decision rule for the net present value method, should the management of the company accept this project? type yes or no in the cell to the right. no explanation is needed.

Explanation:

Step1: Calculate Annual Savings in Operating Costs

Annual operating costs of current method: $12,000
Annual operating costs of new machine: $10,500
Annual savings = \( 12,000 - 10,500 = 1,500 \)

Step2: Add Increased Contribution Margin

Increased production contribution margin: $2,300
Total annual cash inflows (before salvage and parts) = \( 1,500 + 2,300 = 3,800 \)

Step3: Calculate PV of Annual Cash Inflows (Years 1 - 10)

Using \( PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} \), where \( PMT = 3,800 \), \( r = 0.12 \), \( n = 10 \)
\( PV_{annual} = 3,800 \times \frac{1 - (1 + 0.12)^{-10}}{0.12} \)
\( (1 + 0.12)^{-10} \approx 0.3220 \), \( \frac{1 - 0.3220}{0.12} \approx 5.6502 \)
\( PV_{annual} \approx 3,800 \times 5.6502 \approx 21,470.76 \)

Step4: Calculate PV of Parts Replacement Cost (Year 5)

Parts replacement cost: $3,000 (outflow, so negative)
\( PV_{parts} = -3,000 \times (1 + 0.12)^{-5} \)
\( (1 + 0.12)^{-5} \approx 0.5674 \)
\( PV_{parts} \approx -3,000 \times 0.5674 \approx -1,702.20 \)

Step5: Calculate PV of Salvage Value (Year 10)

Salvage value: $2,000 (inflow)
\( PV_{salvage} = 2,000 \times (1 + 0.12)^{-10} \)
\( PV_{salvage} \approx 2,000 \times 0.3220 \approx 644.00 \)

Step6: Calculate Initial Cost (Year 0)

Cost of new machine: $24,000 (outflow, negative)
\( PV_{initial} = -24,000 \)

Step7: Total Net Present Value (NPV)

\( NPV = PV_{initial} + PV_{annual} + PV_{parts} + PV_{salvage} \)
\( NPV = -24,000 + 21,470.76 - 1,702.20 + 644.00 \)
\( NPV = -24,000 + 20,412.56 = -3,587.44 \)? Wait, correction: Wait, maybe I missed the contribution margin per box? Wait, the contribution margin per box is $1.90, but maybe the increased production is in boxes? Wait, original data: "Increased production (boxes) per year" is 2,300? Wait, no, the "Increased production (boxes) per year" and "Contribution margin per box" is $1.90. Oh! I made a mistake. Step2 should be: Increased contribution margin = 2,300 boxes × $1.90/box = $4,370. Then annual savings is $1,500, so total annual cash inflows = $1,500 + $4,370 = $5,870. Let's recalculate:

Step2 (Corrected): Increased Contribution Margin

Increased production (boxes): 2,300
Contribution margin per box: $1.90
Increased contribution margin = \( 2,300 \times 1.90 = 4,370 \)
Annual savings: $1,500
Total annual cash inflows = \( 1,500 + 4,370 = 5,870 \)

Step3 (Corrected): PV of Annual Cash Inflows (Years 1 - 10)

\( PV_{annual} = 5,870 \times \frac{1 - (1 + 0.12)^{-10}}{0.12} \)
\( \frac{1 - 0.3220}{0.12} \approx 5.6502 \)
\( PV_{annual} \approx 5,870 \times 5.6502 \approx 33,166.67 \)

Step4 (Corrected): PV of Parts Replacement Cost (Year 5)

\( PV_{parts} = -3,000 \times 0.5674 \approx -1,702.20 \) (same as before)

Step5 (Corrected): PV of Salvage Value (Year 10)

\( PV_{salvage} = 2,000 \times 0.3220 \approx 644.00 \) (same as before)

Step6 (Corrected): Initial Cost

\( PV_{initial} = -24,000 \) (same as before)

Step7 (Corrected): Total NPV

\( NPV = -24,000 + 33,166.67 - 1,702.20 + 644.00 \)
\( NPV = -24,000 + 32,108.47 = 8,108.47 \)

Wait, now positive. So the net present value is positive? Wait, let's check the formula again. The PV function in Excel: \( PV(rate, nper, pmt, [fv], [type]) \). Let's use Excel-like calculation:

For annual cash inflows (pmt = 5,870, nper = 10, rate = 12%):
\( PV_{annual} = PV(0.12, 10, -5870, 0, 0) \approx 33,166.67 \) (since pmt is outflow in Excel, but we have inflow, so sign matters)

Parts replacement: \( PV(0.12, 5, 0, -3000, 0) \approx -1,702.20 \) (fv = -3000, so PV is positive? Wait, no: PV of a future outflow is negative. Wait, \( PV = FV / (1 + r)^n \), so \( 3000 / (1.12)^5 \approx 1,702.20 \), so as an outflow, it's -1,702.20.

Salvage value: \( PV(0.12, 10, 0, 2000, 0) \approx 644.00 \) (inflow, positive)

Initial cost: -24,000

Total NPV: \( -24,000 + 33,166.67 - 1,702.20 + 644.00 = 8,108.47 \)

Now, the decision rule: If NPV > 0, accept the project. So the answer to "should the management of the company accept this project? Type Yes or No in the cell to the right." is Yes? Wait, but my first calculation was wrong because I misread the increased production as 2,300 (cash) instead of 2,300 boxes. So the correct NPV is positive, so accept.

Answer:

Yes (assuming the NPV is positive, so the management should accept the project)