📚 Advanced Study Notes: Simple & Compound Interest
Subject: Quantitative Aptitude
Audience: MIT‑level students, graduate courses in Financial Mathematics or Actuarial Science
Goal: Build a rigorous, first‑principles understanding of interest, derive all standard formulas, examine edge cases, and tackle a challenging problem that integrates theory with application.
1. Foundations
| Concept | Definition | Intuition |
|---|---|---|
| Principal | Initial amount of money, (P). | The base on which interest accrues. |
| Rate | Annual nominal rate (r) (decimal). | The price of borrowing or the return on investment. |
| Time | Number of years, (t). | How long money is outstanding. |
| Interest | Incremental payment due to time and rate. | Either simple (linear) or compound (exponential). |
Remark:
In quantitative aptitude, we usually assume constant rate and fixed compounding schedule unless stated otherwise. Real‑world problems may deviate, and we’ll cover those later.
2. Simple Interest (SI)
2.1 Derivation from First Principles
Assume a linear growth: for each infinitesimal time interval (dt), interest accrues proportionally to the current principal.
[ dA_{\text{SI}} = P \cdot r , dt ]
Integrate over (t) years:
[ A_{\text{SI}} = P + \int_0^t P r , dt = P(1 + rt) ]
- Total accumulated amount (A_{\text{SI}})
- Interest earned (I_{\text{SI}} = A_{\text{SI}} - P = P r t)
2.2 Assumptions & Limitations
| Assumption | Validity | Edge Cases |
|---|---|---|
| Rate (r) constant over (t) | ✔ | If rate changes, we must break into intervals. |
| No compounding → interest not reinvested | ✔ | For long periods, this underestimates growth. |
| Time measured in whole years | ✔ | For non‑integer (t), formula still works (e.g., 1.5 years). |
2.3 Effective Annual Rate (EAR) for SI
Since SI does not compound, EAR = nominal rate (r).
3. Compound Interest (CI)
3.1 Discrete Compounding
Let (n) be the number of compounding periods per year (e.g., (n=12) for monthly).
During each period of length (1/n) year, the amount multiplies by (1 + \frac{r}{n}).
After (N = nt) periods:
[ A_{\text{CI}} = P \left(1 + \frac{r}{n}\right)^{nt} ]
- Interest earned: (I_{\text{CI}} = A_{\text{CI}} - P)
3.2 Continuous Compounding
Take the limit as (n \to \infty):
[ \lim_{n\to\infty} \left(1 + \frac{r}{n}\right)^{nt} = \exp(r t) ]
Thus:
[ A_{\text{cont}} = P e^{rt} ]
- Interpretation: Interest is reinvested instantaneously.
3.3 Effective Annual Rate (EAR)
For nominal rate (r) compounded (n) times per year:
[ \boxed{\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1} ]
- Monotonicity: Increasing (n) increases EAR for (r>0).
- Limit: (\lim_{n\to\infty} \text{EAR} = e^r - 1).
3.4 Derivation via Differential Equations
Let (A(t)) denote the amount at time (t). For continuous compounding:
[ \frac{dA}{dt} = r A(t) ]
Solve:
[ \frac{dA}{A} = r , dt ;\Longrightarrow; \ln A = rt + C ] [ A(t) = A(0) e^{rt} = P e^{rt} ]
This confirms the earlier limit result.
4. Advanced Topics
4.1 Sensitivity Analysis (Gateaux Derivatives)
We may ask how sensitive the accumulated amount is to changes in (r) or (t).
4.1.1 Partial Derivative w.r.t. Rate
[ \frac{\partial A_{\text{CI}}}{\partial r} = P \cdot nt \cdot \frac{1}{n} \left(1 + \frac{r}{n}\right)^{nt-1} = P t \left(1 + \frac{r}{n}\right)^{nt-1} ]
For continuous compounding:
[ \frac{\partial A_{\text{cont}}}{\partial r} = P t e^{rt} ]
4.1.2 Partial Derivative w.r.t. Time
For CI:
[ \frac{\partial A_{\text{CI}}}{\partial t} = P \left(