Andromeda
Note

Financial Models: Compound Interest

Definition

Compound interest is the interest calculated on the initial principal and also on the accumulated interest of previous periods. It is modeled by exponential functions where the growth happens at discrete intervals or continuously.

Why It Matters

Compound interest is the “gravity” of the financial world. If you understand it, you can harness it to build wealth over time; if you ignore it, it will eventually crush you under the weight of compounding debt. It is the ultimate example of the “snowball effect,” where small, consistent actions taken early have a vastly larger impact than heroic efforts taken later in life.

Core Concepts

  • Discrete Compounding A=P(1+rn)ntA = P \left(1 + \frac{r}{n}\right)^{nt}

    • How to read: “The amount A equals P times the quantity one plus r divided by n, all raised to the n t.”
    • Meaning: Principal grows by factor (1+r/n)(1 + r/n) each of ntnt compounding periods—interest earns interest at n times per year.

  • Continuous Compounding A=PertA = P e^{rt}

    • How to read: “The amount A equals P times e to the r t.”
    • Meaning: Limit as n→∞n \to \infty—money grows continuously at rate r with no discrete compounding intervals.

  • Annual Percentage Yield (APY) (Stewart): The simple interest rate equivalent to a given compound rate. For daily compounding at 6%, A=P(1+0.06/365)365=P(1.06183)A = P(1 + 0.06/365)^{365} = P(1.06183), so APY = 6.183%.

    • How to read: “The amount A equals P times the quantity one plus zero point zero six divided by three hundred sixty-five, all raised to the three hundred sixty-five.”
    • Meaning: Actual yearly growth factor after daily compounding—APY is the equivalent simple annual rate.

  • Effective Rate (rEr_E) rE=(1+r/n)n−1(discrete)rE=er−1(continuous)r_E = (1 + r/n)^n - 1 \quad \text{(discrete)} \qquad r_E = e^r - 1 \quad \text{(continuous)}

    • How to read: “The effective rate r E equals the quantity one plus r divided by n, all raised to the n, minus one; or e to the r minus one.”
    • Meaning: True annual percent gain after compounding—nominal rate r understates real growth when n > 1.

  • Present Value P=A(1+r/n)−ntP = A(1 + r/n)^{-nt}

    • How to read: “The present value P equals A times the quantity one plus r divided by n, all raised to the negative n t.”
    • Meaning / when to use: Reverse compound interest—how much to invest today to reach future amount A.

Connected Concepts

Connected notes