Problem 45 - Calculus: Early Transcendental Functions

Question

Suppose that the value of a \(\$ 40,000\) asset decreases at a constant percentage rate of \(10 \% .\) Find its worth after (a) 10 years and (b) 20 years. Compare these values to a \(\$ 40,000\) asset that is depreciated to no value in 20 years using linear depreciation.

Step-by-Step Solution

Answer

With a 10% constant depreciation rate, the worth of the \$40,000 asset is approximately \$13,536.74 after 10 years, and \$1,831.56 after 20 years. The same asset is depreciated to \$20,000 after 10 years and to \$0 after 20 years using linear depreciation, i.e., \$2,000 depreciation annually. Thus, the asset loses value much slower under a constant rate of depreciation than under linear depreciation.

Step 1: Calculate value after 10 and 20 years with constant depreciation

First, let's find the value of the asset after 10 and 20 years with a constant percentage rate of \(10 \% \). We will use the formula A=P(1-r)^t. Our initial value (P) is \$40,000 and the rate (r) is 0.10. So the worth of the asset after 10 years would be A = \$40,000 * (1 - 0.10)^10 and after 20 years would be A = \$40,000 * (1 - 0.10)^20. This needs to be calculated.

Step 2: Calculate annual depreciation for linear depreciation

Next, to calculate the value of the asset with linear depreciation, we first need to find the amount depreciated each year. Since the asset is depreciated to no value in 20 years, the annual depreciation is simply the initial value divided by the number of years, which is \(\$40,000 / 20\).

Step 3: Calculate value after 10 and 20 years with linear depreciation

Now, we can find the value of the asset after 10 and 20 years using the annual depreciation calculated in the previous step. For each year, decrease the initial value of the asset by the annual depreciation. Therefore, the value of the asset after 10 years would be \$40,000 - 10 * annual depreciation, and after 20 years it would be \$40,000 - 20 * annual depreciation. As per the depreciation policy, after 20 years, the asset's value becomes 0.

Key Concepts

Constant Percentage Depreciation

Constant percentage depreciation, also known as exponential depreciation, occurs when an asset decreases in value by a fixed percentage each year. This method captures how certain assets lose value more rapidly over time.
It's calculated using the formula:

• \[ A = P(1 - r)^t \]

where:

• \( A \) = the asset's value after \( t \) years,
• \( P \) = the initial asset value,
• \( r \) = the constant depreciation rate, and
• \( t \) = time in years.

For example, a car with an initial value of $40,000 decreasing annually at 10% would be worth \( A = 40000(1 - 0.10)^{10} \) after 10 years. This reflects the nature of the asset's value decreasing exponentially rather than linearly.

Linear Depreciation

Linear depreciation, or straight-line depreciation, reduces an asset's value by a fixed amount each year over its useful life. It is one of the simplest methods to calculate depreciation, often used for accounting and tax purposes.

• The annual depreciation is calculated as:
  \[ \text{Annual Depreciation} = \frac{\text{Initial Value}}{\text{Number of Years}} \]

For example, if an asset is worth \(40,000 and will be depreciated completely over 20 years, the annual depreciation is \)2,000. Each year, the asset depreciates by this amount, reaching $0 at year 20.
This method assumes the asset loses an equal amount of value every year, making it very straightforward but sometimes less accurate in reflecting an asset's actual utilization and wear and tear.

Asset Value Calculation

Calculating an asset's value after certain years using different depreciation methods provides insights into financial planning and investment strategies.
The constant percentage depreciation method uses an exponential model where the asset's value decreases by a constant fraction annually, depicting a faster initial decline.
On the other hand, linear depreciation gives a uniform reduction in value over time. This helps businesses estimate budgets and taxes consistently.
By understanding these concepts, stakeholders can make informed decisions on asset management, providing clear forecasts for the asset's worth over time, and helping align financial strategies with realistic valuation methods.