Simple & compound interest

📚 Advanced Study Notes – Simple & Compound Interest

(Designed for SSC CPO aspirants with a university‑level depth)

Why go beyond the “formula‑book”?
SSC CPO often tests intuition, edge‑case handling, and the ability to adapt a formula to an atypical situation.
A solid grasp of the mathematical foundation lets you solve any variation—whether the exam throws a “negative rate” or asks for “continuous compounding” in disguise.

1. Foundations

Notation convention – All formulas below assume annual compounding unless otherwise specified.

2. Simple Interest (SI)

2.1 First‑principle derivation

1. Linear growth – Over one year, the amount grows by a fixed percentage of the principal.
2. Over (t) years, the growth is proportional to time:

[ A_{\text{SI}} = P + (P \cdot r \cdot t) = P(1 + rt) ]

3. Interest earned:

[ I_{\text{SI}} = A_{\text{SI}}-P = P,r,t ]

Key insight: Simple interest is a linear function of time; no compounding.

2.2 Edge Cases

3. Compound Interest (CI)

3.1 First‑principle derivation

1. Discrete compounding – Interest is added at the end of each period; the new principal is the old principal plus the interest earned during that period.
2. Let (n) be the number of compounding periods per year (e.g., (n=12) for monthly).
3. Each period lasts (\frac{1}{n}) yr, so the period rate is (\frac{r}{n}).

Recurrence relation for each period (k):

[ P_{k+1} = P_k \left(1 + \frac{r}{n}\right) ]

4. After (N = nt) periods:

[ A_{\text{CI}} = P \left(1 + \frac{r}{n}\right)^{nt} ]

5. Interest earned:

[ I_{\text{CI}} = A_{\text{CI}} - P ]

Take‑away: The compound factor ( \left(1 + \frac{r}{n}\right)^{nt} ) is an exponential growth with base (1 + \frac{r}{n}).

3.2 Continuous Compounding

When the compounding frequency goes to infinity ((n \to \infty)), the discrete factor approaches the exponential constant (e):

[ \lim_{n\to\infty} \left(1 + \frac{r}{n}\right)^{n} = e^r ]

Thus

[ A_{\text{cont}} = P,e^{rt} ]

Proof sketch (using the definition of (e)):

[ \begin{aligned} A_{\text{cont}} &= P \lim_{n\to\infty}\left(1 + \frac{r}{n}\right)^{nt} = P \Big[ \lim_{n\to\infty}\left(1 + \frac{r}{n}\right)^{n}\Big]^t \ &= P, (e^r)^t = P,e^{rt} \end{aligned} ]

Intuition: Continuous compounding yields the maximum amount obtainable for a given (P,r,t).

3.3 Differential‑Equation Perspective

Let (A(t)) be the account value at time (t). For continuous compounding,

[ \frac{dA}{dt} = r A(t) ]

Solving:

[ A(t) = P e^{rt} ]

Remark – This differential equation is the continuous analog of the discrete recurrence.

4. Related Concepts