Applications of Compound Interest: Depreciation, Growth & CI-SI Differences

1. Commercial Applications of the Compounding Principle

The mathematical formula of compound interest extends far beyond bank loans. It underpins financial accounting and managerial planning across three core areas:

2. Asset Depreciation (Written Down Value Method)

Under the Written Down Value (WDV) or Reducing Balance method prescribed by the Companies Act 2013 and the Income Tax Act 1961, depreciation is charged as a constant percentage on the diminishing book value of the asset:

Book Value after n years (V_n) = V₀ × [ 1 - (r ÷ 100) ]ⁿ

Where:

• V₀ = Initial acquisition cost or scrap value baseline
• r = Annual rate of depreciation (%)
• n = Operational life in years
• V_n = Net book value at the end of year n

Total Cumulative Depreciation charged over n years:

Total Depreciation = Initial Cost - Scrap Value = V₀ - V_n

3. Algebraic Differences Between CI and SI

In university examinations, finding the principal or interest rate from the difference between compound interest and simple interest is a standard question:

4. Worked Numerical Examples

Problem 1: Machinery Depreciation

Question: A textile mill purchases spinning machinery for ₹5,00,000. Depreciation is charged at 10% per annum under the diminishing balance method. What will be its book value after 3 years?

Step-by-Step Solution:

1. Identify parameters:
   • Initial Cost (V₀) = ₹5,00,000
   • Depreciation Rate (r) = 10% p.a.
   • Time (n) = 3 years
2. Apply WDV Depreciation Formula:
   • V₃ = V₀ × [ 1 - (r ÷ 100) ]³
   • V₃ = 5,00,000 × [ 1 - 0.10 ]³ = 5,00,000 × (0.90)³
3. Calculate value:
   • (0.90)³ = 0.9 × 0.9 × 0.9 = 0.729
   • V₃ = 5,00,000 × 0.729 = ₹3,64,500
4. Cumulative Depreciation:
   • Total Depreciation = 5,00,000 - 3,64,500 = ₹1,35,500
   • Conclusion: Book value after 3 years is ₹3,64,500.

Problem 2: Finding Principal from 2-Year CI-SI Difference

Question: The difference between compound interest (compounded annually) and simple interest on a certain sum for 2 years at 8% per annum is ₹192. Find the principal sum.

Step-by-Step Solution:

1. Use the 2-Year Difference Formula:
   • CI - SI = P × (R ÷ 100)²
2. Substitute given values:
   • 192 = P × (8 ÷ 100)²
   • 192 = P × (64 ÷ 10,000)
3. Solve for P:
   • P = (192 × 10,000) ÷ 64
   • 192 ÷ 64 = 3
   • P = 3 × 10,000 = ₹30,000
   • Conclusion: The principal sum is ₹30,000.

Problem 3: Demographic Population Expansion

Question: The population of an industrial town is currently 2,50,000. It increases at the rate of 4% per annum. What will be its population after 2 years?

Step-by-Step Solution:

1. Identify parameters:
   • Base Population (P₀) = 2,50,000
   • Growth Rate (r) = 4% p.a.
   • Time (n) = 2 years
2. Compute future population:
   • P₂ = P₀ × [ 1 + (r ÷ 100) ]²
   • P₂ = 2,50,000 × (1 + 0.04)²
   • (1.04)² = 1.0816
   • P₂ = 2,50,000 × 1.0816 = 2,70,400
   • Conclusion: Town population after 2 years will be 2,70,400.