Math Problem Statement

Bottoms Up Diaper Service is considering the purchase of a new industrial washer. It can purchase the washer for $6,900 and sell its old washer for $2,100. The new washer will last for 6 years and save $1,900 a year in expenses. The opportunity cost of capital is 18%, and the firm’s tax rate is 21%.

If the firm uses straight-line depreciation over a 6-year life, what are the cash flows of the project in years 0 to 6? The new washer will have zero salvage value after 6 years, and the old washer is fully depreciated.

Note: Negative amounts should be indicated by a minus sign.

Solution

To calculate the cash flows for the project over the 6-year life of the washer, we need to analyze the initial investment, annual savings, depreciation, and tax savings. Here’s a step-by-step breakdown of each component:

1. Initial Investment (Year 0)

  • Cost of New Washer: $6,900
  • Sale of Old Washer: $2,100
  • Net Initial Investment: Net Initial Investment=Cost of New WasherSale of Old Washer=6,9002,100=4,800\text{Net Initial Investment} = \text{Cost of New Washer} - \text{Sale of Old Washer} = 6,900 - 2,100 = 4,800

So, the cash flow for Year 0 is -$4,800.

2. Annual Savings

The new washer saves the company $1,900 per year in expenses.

3. Depreciation

Since the new washer is depreciated using straight-line depreciation over 6 years: Annual Depreciation=Cost of New WasherUseful Life=6,9006=1,150\text{Annual Depreciation} = \frac{\text{Cost of New Washer}}{\text{Useful Life}} = \frac{6,900}{6} = 1,150

4. Tax Savings from Depreciation

The tax savings from depreciation each year is calculated as: Tax Savings from Depreciation=Annual Depreciation×Tax Rate=1,150×0.21=241.5\text{Tax Savings from Depreciation} = \text{Annual Depreciation} \times \text{Tax Rate} = 1,150 \times 0.21 = 241.5

5. Total Annual Cash Flow (Years 1 to 6)

Each year, the total cash flow is calculated as follows: Annual Cash Flow=Annual Savings+Tax Savings from Depreciation\text{Annual Cash Flow} = \text{Annual Savings} + \text{Tax Savings from Depreciation} Substituting the values: Annual Cash Flow=1,900+241.5=2,141.5\text{Annual Cash Flow} = 1,900 + 241.5 = 2,141.5

Summary of Cash Flows

The cash flows over the 6-year period are as follows:

  • Year 0: -$4,800 (initial investment)
  • Year 1 to Year 6: $2,141.5 each year

Final Cash Flow Table

YearCash Flow ($)
0-4,800
12,141.5
22,141.5
32,141.5
42,141.5
52,141.5
62,141.5

Let me know if you’d like more details on the calculations or have further questions!


Here are some related questions you might find useful to explore:

  1. How would the cash flows change if the tax rate were different?
  2. What impact would a residual (salvage) value have on the final year's cash flow?
  3. How would a different depreciation method, like double-declining balance, affect the annual cash flows?
  4. What would be the net present value (NPV) of the project with a 10% discount rate instead of 18%?
  5. How would extending the life of the washer beyond 6 years affect the financial assessment?

Tip: Calculating the tax impact of depreciation helps capture the full tax shield benefit for capital investments.

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Math Problem Analysis

Mathematical Concepts

Finance
Investment Analysis
Depreciation
Cash Flow Analysis

Formulas

Net Initial Investment = Cost of New Washer - Sale of Old Washer
Annual Depreciation = Cost of New Washer / Useful Life
Tax Savings from Depreciation = Annual Depreciation * Tax Rate
Annual Cash Flow = Annual Savings + Tax Savings from Depreciation

Theorems

Time Value of Money
Straight-line Depreciation
Tax Shield from Depreciation

Suitable Grade Level

Grades 11-12 (High School), College Level Finance