Simple & Compound Interest – MIT‑Level Study Notes
Quantitative Aptitude – Lakshadweep TET
Author: ChatGPT, “expert teacher”
Audience: Advanced undergraduate students (MIT‑level)
Goal: Understand interest from first principles, derive the standard formulas, explore subtle edge cases, and solve a challenging problem.
1. Introduction
Interest quantifies the time value of money: a dollar today is worth more than a dollar tomorrow because it can earn a return. Two basic mechanisms exist:
| Mechanism | Short name | Key Feature |
|---|---|---|
| Simple Interest | SI | Linear growth in the principal. |
| Compound Interest | CI | Growth in principal and accrued interest. |
In the TET exam, questions often test the ability to manipulate and apply the formulas, but at an advanced level we must also understand why these formulas hold and how they behave under edge conditions.
2. First Principles
2.1. Notation
| Symbol | Meaning |
|---|---|
| (P) | Principal (initial deposit) |
| (r) | Annual nominal rate (as a decimal) |
| (t) | Time horizon in years |
| (n) | Compounding frequency per year |
| (F(t)) | Future value after (t) years |
| (I(t)) | Interest earned after (t) years |
2.2. Simple Interest: Linear Growth
Definition.
If a principal (P) accrues interest at a constant nominal rate (r) linearly over time, then after (t) years the interest earned is
[ I_{\text{SI}}(t) = P, r, t. ]
Proof (from linearity).
The rate (r) is defined as the ratio of interest earned per unit time to the principal:
[ r = \frac{\text{interest per year}}{P}. ]
Hence, over (t) years, the interest is simply (r) times (P) times the time:
[ I_{\text{SI}}(t) = r \times P \times t. ]
The future value follows:
[ F_{\text{SI}}(t) = P + I_{\text{SI}}(t) = P(1 + r t). ]
2.3. Compound Interest: Multiplicative Growth
Definition.
If interest is added to the principal at discrete intervals of length (\Delta t = \frac{1}{n}) years, and the rate per interval is (i = \frac{r}{n}), then after (t) years (an integer multiple of (\Delta t)) the future value is
[ F_{\text{CI}}(t) = P \left(1 + \frac{r}{n}\right)^{nt}. ]
Derivation.
Start with one compounding period:
[ P_1 = P(1 + i) = P\left(1 + \frac{r}{n}\right). ]
After two periods:
[ P_2 = P_1(1 + i) = P\left(1 + \frac{r}{n}\right)^2. ]
Continuing inductively, after (nt) periods:
[ F_{\text{CI}}(t) = P\left(1 + \frac{r}{n}\right)^{nt}. ]
2.4. Continuous Compounding
Let (n \to \infty) while keeping the annual nominal rate (r) fixed. By definition:
[ \lim_{n \to \infty} \left(1 + \frac{r}{n}\right)^n = e^r. ]
Thus the continuous future value is
[ F_{\text{cont}}(t) = P e^{r t}. ]
This limit is a standard result from the definition of the exponential function. It also shows that compound interest with infinite compounding frequency maximizes growth for a fixed nominal rate.
3. Comparative Properties
| Property | Simple Interest | Compound Interest |
|---|---|---|
| Formula for (F(t)) | (P(1 + r t)) | (P\left(1 + \frac{r}{n}\right)^{nt}) |
| Growth type | Linear | Exponential |
| Effective annual rate (EAR) | (r) | (\left(1 + \frac{r}{n}\right)^n - 1) |
| Time to double (Rule of 72) | (\frac{72}{r}) | (\frac{72}{\text{EAR}}) |
| Sensitivity to (n) | None | Increases with (n) (converges to (e^r)) |
| Applicability | Simple loans, short‑term | Savings, investments, loans with reinvestment |
4. Edge Cases & Pitfalls
4.1. Negative Rates
- Interpretation: Deflation or a penalty on deposits.
- SI: (F = P(1 + r t)) works; if (r < 0) the future value decreases linearly.
- CI: (