Mathematical Models in Finance: Simple, Compound, and Continuous Interest

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: $I = PRT$

• Formula for Accumulated Amount: $A = P(1 + RT)$

• 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 $P = 1000$, $R = 0.05$, $T = 3$, then $I = 1000 \times 0.05 \times 3 = 150$ and $A = 1000 + 150 = 1150$.

Compound Interest

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

• Formula: $A = P \left(1 + \frac{r}{n}\right)^{nt}$

• 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 $P = 1000$, $r = 0.06$, $n = 12$, $t = 2$, then $A = 1000 \left(1 + \frac{0.06}{12}\right)^{24}$.

Continuous Compounding

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

• Formula: $A = Pe^{rt}$

• 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 $P = 500$, $r = 0.04$, $t = 5$, then $A = 500e^{0.2}$.

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.

Instrument	Type	Interest Payment	Compounding Frequency	Notes
Municipal Bond	Loan to local government	Semiannual	Semiannual	Face value returned at maturity
Corporate Bond	Loan to corporation	Semiannual	Semiannual	Face value returned at maturity
T-bonds	Loan to federal government	Semiannual	Semiannual	Term: 20 or 30 years
T-notes	Loan to federal government	Semiannual	Semiannual	Term: 2, 3, 5, 7, or 10 years
Savings Account	Loan to bank	Varies	Monthly	Generally compounded monthly
Certificate of Deposit (CD)	Loan to bank	Varies	Varies	Early withdrawal may incur a fee
Savings Bonds (EE/I)	Loan to government	Varies	Semiannual	I bonds adjust for inflation; EE bonds have fixed rate
Credit Card Loan	Loan from bank	Varies	Varies	High interest rates

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.