5️⃣ Application of Financial Mathematics — Slides

Mathematical Fundamentals · MS01-001-G

Learning Objectives

• Compute simple and compound interest
• Understand how capitalisation frequency affects future value
• Choose the correct interest model for a given financial situation

1. Simple Interest

Formula

\[C_n = C_0(1 + nt)\]

where \(C_0\) = initial capital, \(t\) = periodic rate, \(n\) = number of periods.

Interest earned per period: \(I = C_0 \times t\) (constant).

Growth is linear — each period adds the same absolute amount.

Application: short-term loans, trade credit.

2. Compound Interest

Formula

\[C_n = C_0(1 + t)^n\]

Interest is reinvested each period — growth is exponential.

The key difference: with compound interest, the base grows, so each period’s interest is larger than the last.

Application: savings accounts, long-term investments, loan amortisation.

Capitalisation Frequency

Annual rate \(r\). If compounded \(k\) times per year:

Tip

Effective rate per sub-period: \(t = \dfrac{r}{k}\)

Total periods: \(n = k \times T\)

Example: 6 % p.a. compounded monthly → \(t = 0.5\%\)/month.

The higher \(k\), the larger the final amount.

Worked Example — Two Investments

Compare €5,000 invested for 4 years at 6 % p.a.

Simple interest: \[C = 5000(1 + 4 \times 0.06) = 5000 \times 1.24 = €6{,}200\]

Compound interest (annual): \[C = 5000 \times 1.06^4 \approx 5000 \times 1.2625 \approx €6{,}312.38\]

Compound yields €112.38 more — the difference widens with time and rate.

Summary

1. Simple interest: linear growth — the same amount is added each period.
2. Compound interest: exponential growth — each period’s interest is larger because the base grows.
3. For long horizons or high rates, compound interest significantly outperforms simple interest.

📖 Full Reference

→ Module 5: Application of Financial Mathematics