Functions, Graphs, and Models

Mathematical Models in Finance

Financial mathematics frequently uses functions to model the growth of investments and loans. This section introduces three primary interest models: simple interest, compound interest, and continuous compounding. Understanding these models is essential for analyzing various financial instruments such as bonds, savings accounts, and loans.

Simple Interest

• Definition: Simple interest is calculated only on the original principal over a period of time.

• Formula for Interest:

• Formula for Accumulated Amount:

• Variables:

  • P: Principal (initial amount invested or loaned)

  • R: Annual interest rate (as a decimal)

  • T: Time in years

  • I: Total interest accrued

  • A: Accumulated balance after T years

• Application: Simple interest is seldom used in modern financial products.

• Example: If , , , then and .

Compound Interest

• Definition: Compound interest is calculated on the principal and on the accumulated interest from previous periods.

• Formula:

• Variables:

  • P: Principal

  • r: Annual interest rate (as a decimal)

  • n: Number of compounding periods per year

  • t: Number of years

  • A: Accumulated balance after t years

• Application: Used in most savings accounts, certificates of deposit, and many loans.

• Example: If , , , , then .

Continuous Compounding

• Definition: Interest is compounded an infinite number of times per year, leading to exponential growth.

• Formula:

• Variables:

  • P: Principal

  • r: Annual interest rate (as a decimal)

  • t: Number of years

  • A: Accumulated balance after t years

• Application: Used in certain theoretical models and some financial products.

• Example: If , , , then .

Financial Instruments and Their Interest Models

Different financial products use different interest models and compounding frequencies. Below is a summary of common instruments and their characteristics.

Key Points

• Interest Rate (APR): The annual percentage rate, expressed as a decimal in formulas.

• Compounding Frequency: The number of times interest is added to the principal per year (e.g., monthly, semiannually).

• Face Value: The amount returned to the investor at the maturity of a bond.

• Inflation Protection: I bonds are designed to outpace inflation, while EE bonds have a fixed rate.

Additional Resources

• Government bonds and treasuries can be purchased at www.treasurydirect.gov.

Additional info: Understanding these models is foundational for later calculus topics, such as exponential growth and decay, and for solving real-world problems involving rates of change and accumulation.