Financial Discrete Models

Mathematical Modeling · Axiom Academy

EXAMPLE Financial Discrete Models Worked examples applying difference equations to personal finance You deposit 10,000 into a savings account that pays 5% annual interest , compounded yearly. You want to understand how your investment grows over time without making additional deposits. What type of growth does compound interest exhibit? Sensitivity Analysis: Interest Rate Impact Explore how changes in the interest rate affect your investment growth over 20 years. Key Takeaways: Compound Interest Model: is a first-order linear homogeneous difference equation Solution: Geometric growth with explicit formula Rule of 72: Divide 72 by the interest rate percentage to estimate doubling time Sensitivity: Small changes in interest rate compound dramatically over long periods You take out a 20,000 car loan at 6% annual interest , making 400 monthly payments . How long until the loan is paid off? In the loan equation B_n+1 = (1+r)B_n - P, what does the term (1+r)B_n represent? Sensitivity Analysis: Payment Amount Impact See how increasing your monthly payment affects payoff time and total interest. Key Takeaways: Loan Amortization Model: is a first-order linear non-homogeneous equation Equilibrium: When P/r > B_0, the loan will eventually be paid off Condition: For loan payoff, payment P must exceed interest on balance: P > rB_0 Strategy: Extra payments early have the biggest impact due to compounding

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