What is the difference between simple and compound interest?

Simple interest is earned only on the initial principal (e.g., $100 earning 5% simple interest gets $5 every year). Compound interest is earned on the principal plus all interest accumulated previously, meaning your balances grow exponentially.

What happens to APY as compounding frequency increases?

As compounding frequency increases, the APY (effective return rate) increases. However, it approaches a mathematical ceiling known as continuous compounding, defined by the constant e.

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Core Study Guide

The Mathematics of Compounding Interest

Understanding the force that turns small deposits into exponential wealth.

Compounding interest represents interest calculated on the initial principal of a deposit or loan, plus all prior accumulated interest. Described by Albert Einstein as the "eighth wonder of the world," compounding interest transforms linear growth into exponential growth over time.

In commerce and corporate finance, understanding compounding is vital for evaluating bonds, capital expenditure projects, retirement portfolios, and mortgage amortizations. By reinvesting interest earnings rather than withdrawing them, the compounding base grows larger each period, causing earnings to grow faster and faster.

Key Takeaways

•Simple interest earns a fixed percentage on the starting principal only; compound interest earns interest on interest.
•Compounding frequency (daily, monthly, quarterly, yearly) determines how often interest is calculated and added.
•A higher compounding frequency yields a higher effective annual rate, known as Annual Percentage Yield (APY).

Core Concepts & Definitions

1Principal and Future Value

Principal (P) is the starting baseline amount of money invested or borrowed. Future Value (A) is the nominal value that the principal grows to after earning interest over a specific duration of years (t).

•The interest rate (r) is expressed as a decimal (e.g., 5% = 0.05).

•Time (t) is almost always measured in annual increments.

2The Impact of Compounding Frequencies

The variable n represents the number of compounding periods per year. If interest compounds monthly, n = 12; if quarterly, n = 4; if daily, n = 365.

•More frequent compounding leads to higher future returns because interest starts earning interest sooner.

•Continuous compounding represents the mathematical limit of infinitely small periods.

3Annual Percentage Yield (APY)

APY, or the Effective Annual Rate, represents the actual rate of return earned on an investment in one year, taking into account the effects of compounding.

•APY = (1 + r/n)^n - 1.

•For yearly compounding, APY equals the nominal interest rate (r).

Equations & Calculation Methods

Compound Interest Future Value

A = P * (1 + r/n)^(nt)

Computes the final accumulated amount (A) given principal (P), annual interest rate (r), compounding periods per year (n), and total years (t).

Annual Percentage Yield (APY)

APY = (1 + r/n)^n - 1

Converts the nominal nominal rate (r) to the effective rate that accounts for compounding cycles within a year.

Rule of 72 (Doubling Time)

Years ≈ 72 / (Interest Rate * 100)

A quick, simple shortcut to estimate how many years it will take for an investment to double in value at a constant compounding rate.