Problem 15 - Accounting

Question

Western Rail Inc. is considering acquiring equipment at a cost of \(\$ 468,800\). The equipment has an estimated life of 10 years and no residual value. It is expected to provide yearly net cash flows of \(\$ 93,760\). The company's minimum desired rate of return for net present value analysis is \(12 \%\). Compute the following: a. The average rate of return, giving effect to straight-line depreciation on the investment. b. The cash payback period. c. The net present value. Use the table of the present value of an annuity of \(\$ 1\) appearing in this chapter. Round to the nearest dollar.

Step-by-Step Solution

Answer

a. Average Rate of Return: 20% b. Cash Payback Period: 5 years c. Net Present Value: $61,944

Step 1: Calculate Annual Depreciation

To find the annual depreciation using the straight-line method, divide the initial cost of the equipment by its useful life. This is given by \( \text{Annual Depreciation} = \frac{\\( 468,800}{10} = \\) 46,880 \).

Step 2: Calculate Average Income

The average income is obtained by subtracting the annual depreciation from the yearly net cash flows. That is, \( \text{Average Income} = \\( 93,760 - \\) 46,880 = \$ 46,880 \).

Step 3: Calculate Average Investment

The average investment is calculated as half of the initial cost, as the value depreciates linearly to zero. Thus, \( \text{Average Investment} = \frac{\\( 468,800}{2} = \\) 234,400 \).

Step 4: Calculate Average Rate of Return

The average rate of return can be calculated by dividing the average income by the average investment and multiplying by 100 to convert it into a percentage: \( \text{Average Rate of Return} = \left(\frac{\\( 46,880}{\\) 234,400}\right) \times 100 = 20\% \).

Step 5: Calculate Cash Payback Period

The cash payback period is computed by dividing the initial cost of the equipment by the annual net cash flow: \( \text{Cash Payback Period} = \frac{\\( 468,800}{\\) 93,760} = 5 \text{ years} \).

Step 6: Find Present Value of Annuity Factor

Consult the table of present value of annuity of \( \$ 1 \) for a rate of 12% over 10 years. Suppose the factor is approximately 5.65.

Step 7: Calculate Net Present Value

The net present value (NPV) is the total present value of cash flows minus the initial investment. Use the annuity factor: \( \text{NPV} = \\( 93,760 \times 5.65 - \\) 468,800 = \\( 530,744 - \\) 468,800 = \$ 61,944 \).

Key Concepts

Understanding the Average Rate of Return

The average rate of return (ARR) measures how much profit or "return" you will get from your investment, on average, every year. This is a useful concept when you're considering investments because it provides a clear percentage, which makes it easy to compare against other investment opportunities.

To calculate the average rate of return, follow these steps:

• First, determine the average annual income. In the context of our equipment purchase, this involves subtracting annual depreciation from the net cash flows. Depreciation is calculated using the straight-line method in this case.
• Next, calculate the average investment, which is typically half the original cost when the asset has a straight-line depreciation to zero value.
• Finally, divide the average annual income by the average investment, and multiply by 100 to get a percentage.

In our example, with an average income of $46,880 and an average investment of $234,400, the ARR comes out to 20%. This means that every year, on average, the company earns 20% returns on their investment.

Determining the Cash Payback Period

The cash payback period is a measure of how long it takes for an investment to "pay for itself." It answers the question: "How long will it take to recover the initial investment from the net returns it generates?"

Here's how you can calculate the cash payback period:

• Take the total initial cost of the investment, which is the purchase price of the asset.
• Divide this cost by the annual net cash flow that the asset generates.

If we refer back to our example, the equipment costs $468,800 and generates $93,760 annually.
By dividing these numbers, we find that it will take 5 years for the initial investment to be fully recovered.
This metric is straightforward and does not take into account the time value of money, but it does provide a simple snapshot of investment risk.

Exploring Straight-Line Depreciation

Straight-line depreciation is the simplest method to calculate the decrease in value of a tangible asset over time. It evenly spreads the depreciation cost over the useful life of the asset.

Calculating straight-line depreciation is simple:

• Find out the initial cost of the asset, which is often its purchase price.
• Divide this cost by the asset's useful life in terms of years.

In our scenario, the equipment cost is $468,800 with a useful life of 10 years. Using the straight-line depreciation method, each year, the equipment depreciates by $46,880.
This method is popular because of its ease and the fact that it allocates an equal expense each year, making financial planning and reporting tasks more predictable.