📚 Study Notes – Simple & Compound Interest
Target Audience: Odisha Patwari / Lekhpal (Land Record Officers)
Subject: Quantitative Aptitude – Advanced University Level (MIT‑style)
Why It Matters
Patwaris/Lekhpal often have to compute interest on unpaid land dues, mortgaged land, or advance payments. A rigorous understanding of interest theory guarantees accuracy, fairness, and legal compliance in record‑keeping and settlement.
1. Foundations – From First Principles
1.1 What Is Interest?
- Interest is the extra amount paid (or earned) for using money over time.
- In mathematical terms, interest reflects the time‑value of money (TVM).
1.2 Basic Variables
| Symbol | Meaning | Units |
|---|---|---|
| (P) | Principal (initial amount) | ₹ |
| (r) | Annual nominal rate (decimal) | – |
| (t) | Time in years | yr |
| (n) | Number of compounding periods per year | – |
| (A) | Accumulated amount after (t) years | ₹ |
| (S) | Simple interest earned | ₹ |
| (C) | Compound interest earned | ₹ |
2. Simple Interest (SI)
2.1 Definition
Interest earned once on the principal for the whole duration.
[ \boxed{S = P , r , t} ] [ \boxed{A_{\text{SI}} = P + S = P(1 + rt)} ]
2.2 Derivation from First Principles
Assume the rate (r) is the annual rate of return per unit time.
- Annual increment: ( \Delta P_{\text{annual}} = P \cdot r ).
- Over (t) years: linear accumulation, no compounding → simply multiply.
2.3 Assumptions & Edge Cases
| Assumption | Edge Case | Remedy |
|---|---|---|
| Rate is constant | Negative rate (e.g., deflationary environment) | Use ( r < 0 ) → ( A_{\text{SI}} = P(1 + rt) ) still valid |
| Time measured in complete years | Partial years (e.g., 4 months) | Convert to fraction: ( t = \frac{4}{12} = \frac{1}{3} ) |
| No additional deposits | Additional deposits/withdrawals | Break into separate SI calculations and sum |
| No compounding | – | – |
3. Compound Interest (CI)
3.1 Discrete Compounding
Interest is added to the principal at each period, then earns interest in subsequent periods.
[ \boxed{A_{\text{CI}} = P\left(1 + \frac{r}{n}\right)^{nt}} ] [ \boxed{C = A_{\text{CI}} - P} ]
3.2 Derivation (Iterative & Limit)
Iterative Approach
- After first period ((1/n) of a year): ( P_1 = P\left(1 + \frac{r}{n}\right) ).
- After second period: ( P_2 = P_1\left(1 + \frac{r}{n}\right) = P\left(1 + \frac{r}{n}\right)^2 ).
- After (nt) periods: ( P_{nt} = P\left(1 + \frac{r}{n}\right)^{nt} ).
Continuous Limit
Let ( n \to \infty ).
[
\lim_{n\to\infty}\left(1 + \frac{r}{n}\right)^{nt} = e^{rt}
]
Hence continuous compounding:
[ \boxed{A_{\text{cont}} = P e^{rt}} ]
3.3 Effective Annual Rate (EAR)
Given nominal rate (r) compounded (n) times per year, the effective annual yield is:
[ \boxed{\text{EAR} = \left(1 + \frac{r}{n}\right)^{n} - 1} ]
3.4 Edge Cases & Practical Variations
| Scenario | Formula | Notes |
|---|---|---|
| Zero compounding periods ((n=0)) | Not defined – treat as SI (no compounding) | (A = P(1+rt)) |
| Very large (n) (daily, hourly) | Use continuous formula | (A \approx Pe^{rt}) |
| Negative rate | Same formulas, result may be below principal | e.g., (r=-0.02) → (A_{\text{CI}} = P(1 - 0.02/n)^{nt}) |
| Partial compounding periods | Use fractional exponent | (A = P\left(1 + \frac{r}{n}\right)^{n t}) where (t) may be non‑integer |