Geometric Progressions in Finance: Loans & Depreciation

Learn how to apply geometric progression to real-world finance, including loan repayment schemes, asset depreciation, and business decision-making.

#financial-mathematics#loan-repayment#depreciation#geometric-progression#business-finance#math-education

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Practical Financial Applications of Geometric Progression

We will apply geometric progression to:

* Loan repayments

* Depreciation

* Business decision-making

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Loan Repayments: Equal Monthly Payments

KCL Ltd borrows £331,000 for 3 months at 10% interest/month. Let monthly payment = x.

Step-by-step balances:

After month 1: 331000(1.1) - x

After month 2: (331000(1.1) - x)(1.1) - x

After month 3 = 0: ((331000(1.1)-x)(1.1)-x)(1.1)-x = 0

Simplify:

331000(1.1)^3 - x(1.1^2 + 1.1 + 1) = 0

x = 331000(1.1)^3 / (1 + 1.1 + 1.1^2)

Calculation:

x = 331000 x 1.331 / (1 + 1.1 + 1.21)

x = 440561 / 3.31

x = £133,100 per month

Total repaid = 3 x £133,100

Total repaid = £399,300

Note: geometric series sum appears in denominator

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Loan Repayments: Differentiated Payments

Scheme 2: debt reduces by equal amount each month

Equal monthly reduction: £331,000 / 3 = £110,333.33

Month 1:

Debt after interest: 331000 × 1.1 = £364,100

Payment: 364,100 − 220,666.67 = £143,433.33

Month 2:

Debt after interest: 220,666.67 × 1.1 = £242,733.34

Payment: 242,733.34 − 110,333.34 = £132,400

Month 3:

Debt after interest: 110,333.34 × 1.1 = £121,366.67

Payment: 121,366.67 − 0 = £121,366.67

Total repaid:

143,433.33 + 132,400 + 121,366.67

= £397,200

Comparison:

Scheme 1 (Equal): £399,300

Scheme 2 (Differentiated): £397,200

399,300 − 397,200 = £2,100

Differentiated scheme is cheaper by £2,100

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Depreciation

A machine costs £50,000 and loses 15% of its value each year.

Setting up the model:

100% − 15% = 85% → r = 0.85

Vₙ = V₀ × (0.85)ⁿ

Calculation for year 3:

V₃ = 50,000 × (0.85)³

V₃ = 50,000 × 0.614125

V₃ = £30,706.25

Business Use:

* Asset valuation

* Budgeting

* Replacement planning

Limitation:

Assumes a constant depreciation rate — real-world rates may vary

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Key Takeaways

Core Principles:

* Geometric progression models repeated percentage change

* Loan repayment models use compounding

* Depreciation models repeated decline over time

Results Summary:

Equal-payment scheme: £399,300

Differentiated scheme: £397,200

Saving: £2,100

Conclusion:

Geometric progression supports practical financial decisions - from loan structuring to asset management.

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Practical Financial Applications of Geometric Progression

Loan Repayments: Equal Monthly Payments

Loan Repayments: Differentiated Payments

Depreciation

Key Takeaways

DESIGNER-MADE PRESENTATION, GENERATED FROM YOUR PROMPT

Geometric Progressions in Finance: Loans & Depreciation

DESIGNER-MADE
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