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The dean of the School of Fine Arts is trying to decide whether to purchase a copy machine to place in the lobby of the building. The machine would add to student convenience, but the dean feels compelled to earn an 8 percent return on the investment of funds. Estimates of cash inflows from copy machines that have been placed in other university buildings indicate that the copy machine would probably produce incremental cash inflows of approximately $\$ 10,500$ per year. The machine is expected to have a three-year useful life with a zero salvage value.



Note: Use appropriate factor(s) from the tables provided.



Required:

AUse the Present Value Appendix (PV of \$1) to determine the maximum amount of cash the dean should be willing to pay for a copy machine.

Note: Round your intermediate calculations and final answer to 2 decimal places.


BUse the Present Value Appendix (PVA of \$1) to determine the maximum amount of cash the dean should be willing to pay for a copy machine.

Note: Round your final answer to 2 decimal places.



a. Maximum amount (PV)

b. Maximum amount (PVA)

Verified Answer :

  • Determine the annual PV factors for each of the three years: \(0.93\), \(0.86\), and \(0.79\).
  • Calculate the present value of each year's inflow and sum them to obtain \(\$27,090.00\).
  • Alternatively, compute the PVA factor of \(2.58\) and multiply by the annual inflow of \(\$10,500\) to reach the same result.
  • The maximum investment amount the dean should pay is \(\boxed{\$27,090.00}\).

Explanation

  1. Problem Analysis and Data Collection

To determine the maximum amount the dean should pay for the copy machine, we identify the key financial parameters: the annual cash inflow \(A = \$10,500\), the interest rate \(i = 8\%\) (or \(0.08\)), and the useful life of the investment \(n = 3\) years. We must calculate the present value of these future cash inflows using the requested discounting methods while rounding all intermediate and final figures to two decimal places.

  1. Calculation (a): Using PV of $1 Factors

First, we determine the Present Value (PV) of \$1 factors for each of the three years, rounded to two decimal places:

  • Year 1: \(PV_1 = \frac{1}{(1 + 0.08)^1} = \frac{1}{1.08} \approx 0.93\)
  • Year 2: \(PV_2 = \frac{1}{(1 + 0.08)^2} = \frac{1}{1.1664} \approx 0.86\)
  • Year 3: \(PV_3 = \frac{1}{(1 + 0.08)^3} = \frac{1}{1.2597} \approx 0.79\)

Next, we calculate the present value of each individual year's inflow:

  • PV Year 1: \(\$10,500 \times 0.93 = \$9,765.00\)
  • PV Year 2: \(\$10,500 \times 0.86 = \$9,030.00\)
  • PV Year 3: \(\$10,500 \times 0.79 = \$8,295.00\)

Summing these values gives the total maximum investment amount: \(\$9,765.00 + \$9,030.00 + \$8,295.00 = \boxed{\$27,090.00}\).

  1. Calculation (b): Using PVA of $1 Factor

Alternatively, we use the Present Value of an Annuity (PVA) factor, which represents the sum of the individual PV factors:

  • \(PVA\_factor = 0.93 + 0.86 + 0.79 = 2.58\)

We then multiply this factor by the constant annual cash inflow:

  • Maximum Investment = \(\$10,500 \times 2.58 = \boxed{\$27,090.00}\).

Both methods confirm that the maximum amount the dean should be willing to pay for the machine is \(\$27,090.00\).

Examples

Understanding the time value of money is essential for smart decision-making, whether for personal savings or large-scale business projects. By calculating the present value of future earnings, companies can determine if the immediate cost of an asset is justified by the benefits it will generate over time. For example, if a small business owner is considering buying a new coffee machine that generates extra daily profits, they can use the same PV formula to decide if the machine's price aligns with their expected financial goals. This approach ensures that investments are made only when they provide an adequate return on capital.

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