📚 Advanced Study Notes – Simple & Compound Interest
Audience: Undergraduate (MIT‑level) students in finance, mathematics, or engineering
Focus: First‑principles derivation, rigorous reasoning, edge‑case analysis, and a challenging problem
1. Introduction
Interest is the cost of borrowing or the return on savings. Two canonical forms dominate most quantitative aptitude problems:
| Form | Symbol | Core Idea |
|---|---|---|
| Simple Interest | (SI) | Interest accrues linearly on the original principal. |
| Compound Interest | (CI) | Interest accrues on the current balance, i.e. on both principal and previously earned interest. |
Despite their ubiquity, the mathematics underlying these concepts has deep roots in calculus, differential equations, and exponential growth theory.
2. First‑Principles Derivation
2.1 Simple Interest (SI)
Assumptions
- Principal (P) is fixed.
- Annual nominal rate (r) (decimal form).
- Time (t) in years (can be fractional).
Derivation
Interest earned per unit time is a proportional function of the principal:
[ \frac{dI}{dt}=rP \quad \text{(constant)}. ]
Integrating from (0) to (t):
[ I(t) = \int_0^t rP , dt = rPt. ]
Hence:
[ \boxed{SI = P r, t} ]
Total Accumulated Amount
[ A_{\text{SI}} = P + SI = P(1+r t). ]
2.2 Compound Interest (CI)
Assumptions
- Principal (P) grows multiplicatively.
- Interest is compounded (n) times per year.
- Rate per compounding period (r_n = \frac{r}{n}).
Discrete Compounding
After one period:
[ P_1 = P(1+r_n). ]
After two periods:
[ P_2 = P_1(1+r_n) = P(1+r_n)^2. ]
Iterating (m) periods ((m = n t)):
[ A_{\text{CI}} = P (1+r_n)^{m} = P \left(1+\frac{r}{n}\right)^{nt}. ]
2.3 Continuous Compounding (Limit of (n \to \infty))
Let (n) be large. Then
[ A_{\text{cont}} = \lim_{n\to\infty} P\left(1+\frac{r}{n}\right)^{nt} = P e^{r t}, ]
using the standard limit (\displaystyle \lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n = e^x).
Differential Equation View
Define (A(t)) as the balance at time (t). For continuous compounding:
[ \frac{dA}{dt} = r A(t). ]
Solution with initial condition (A(0)=P):
[ A(t) = P e^{r t}. ]
3. Generalized Formulas
3.1 Variable Rates
If the nominal annual rate changes over disjoint time intervals (t_1,t_2,\dots,t_k) with rates (r_1,r_2,\dots,r_k), then:
[ A = P \prod_{i=1}^{k} \left(1+\frac{r_i}{n_i}\right)^{n_i t_i} ] for discrete compounding or [ A = P \exp!\left(\sum_{i=1}^{k} r_i t_i\right) ] for continuous compounding.
3.2 Multiple Cash Flows (Payments / Deposits)
For a series of deposits ({D_j}) at times ({t_j}), the accumulated value at time (T) is:
[ A(T) = \sum_j D_j \left(1+\frac{r}{n}\right)^{n(T-t_j)} ] (discrete) or [ A(T) = \sum_j D_j e^{r (T-t_j)} ] (continuous).
This is the foundation of annuity calculations.
4. Advanced Topics
4.1 Effective Annual Rate (EAR)
The nominal rate (r_{\text{nom}}) compounded (n) times per year yields an effective annual yield:
[ r_{\text{eff}} = \left(1+\frac{r_{\text{nom}}}{n}\right)^{n}-1. ]
As (n\to\infty), (r_{\text{eff}}\to e^{r_{\text{nom}}}-1).
4.2 Present Value (PV) and Discounting
Given a future cash flow (C) at time (t), its present value under rate