Calculating the Present Value of a Flat After Depreciation

This question asks us to determine the present value of a flat whose value has decreased (depreciated) over a period of three years at different rates each year.

Depreciation is the decrease in the value of an asset over time due to wear and tear, obsolescence, or other factors. When the depreciation rate is applied annually, the value for the next year is calculated based on the value at the start of the current year.

Here's the information given:

• Initial Value of the flat (3 years ago) = Rs. 65,00,000
• Depreciation Rate in the 1st year = 5%
• Depreciation Rate in the 2nd year = 4%
• Depreciation Rate in the 3rd year = 3%

We need to calculate the value of the flat at the end of the third year, which is its present value.

Step-by-Step Flat Value Depreciation Calculation

We will calculate the value of the flat year by year, considering the depreciation rate for each specific year.

Year 1 Depreciation:

The depreciation in the first year is 5% of the initial value.

Value after 1st year = Initial Value $\times$ (1 - Depreciation Rate Year 1)

Value after 1st year = Rs. $65,00,000 \times (1 - 0.05)$

Value after 1st year = Rs. $65,00,000 \times 0.95$

Value after 1st year = Rs. $61,75,000$

Year 2 Depreciation:

The depreciation in the second year is 4% of the value at the beginning of the second year (which is the value after the first year).

Value after 2nd year = Value after 1st year $\times$ (1 - Depreciation Rate Year 2)

Value after 2nd year = Rs. $61,75,000 \times (1 - 0.04)$

Value after 2nd year = Rs. $61,75,000 \times 0.96$

Value after 2nd year = Rs. $59,28,000$

Year 3 Depreciation:

The depreciation in the third year is 3% of the value at the beginning of the third year (which is the value after the second year).

Value after 3rd year = Value after 2nd year $\times$ (1 - Depreciation Rate Year 3)

Value after 3rd year = Rs. $59,28,000 \times (1 - 0.03)$

Value after 3rd year = Rs. $59,28,000 \times 0.97$

Value after 3rd year = Rs. $57,50,160$

The value of the flat at the end of the third year is Rs. 57,50,160. This is the present value of the flat.

Summary of Flat Value Calculation

Alternatively, the present value can be calculated directly:

Present Value = Initial Value $\times$ (1 - Rate$_1$) $\times$ (1 - Rate$_2$) $\times$ (1 - Rate$_3$)

Present Value = $65,00,000 \times (1 - 0.05) \times (1 - 0.04) \times (1 - 0.03)$

Present Value = $65,00,000 \times 0.95 \times 0.96 \times 0.97$

Present Value = $65,00,000 \times 0.88404$

Present Value = $57,56,260$

Let's re-check the step-by-step calculation:

• $65,00,000 \times 0.95 = 61,75,000$
• $61,75,000 \times 0.96 = 59,28,000$
• $59,28,000 \times 0.97 = 57,50,160$

The step-by-step calculation is correct. The direct calculation requires multiplying the factors first: $0.95 \times 0.96 \times 0.97$.

• $0.95 \times 0.96 = 0.912$
• $0.912 \times 0.97 = 0.88464$

Present Value = $65,00,000 \times 0.88464 = 57,50,160$.

Both methods yield the same result.

Revision Table: Key Concepts in Depreciation

Additional Information on Asset Value and Depreciation

Understanding how asset values change over time is important in finance and accounting. Depreciation is a common concept for physical assets like buildings, machinery, and vehicles.

Different assets depreciate at different rates and using different methods (e.g., straight-line depreciation, declining balance depreciation). In this problem, we used a declining balance approach implicitly because the percentage was applied to the remaining value each year, although the rate itself changed annually.

When dealing with successive percentage changes, remember that the order of multiplication does not matter, i.e., $V_0 \times (1-r_1) \times (1-r_2)$ is the same as $V_0 \times (1-r_2) \times (1-r_1)$. However, the rates must be applied based on the value at the *beginning* of the period the rate applies to.

This type of calculation is also relevant when dealing with compound interest or growth, where the value increases rather than decreases. The formula would involve $(1 + \text{rate})$ instead of $(1 - \text{rate})$.