Notes › MATH 5441: Financial Mathematics Lecture 3

Effective Interest Rate

calendar_today August 27, 2025 article 668 words access_time 8 min replay Modified October 1, 2025

Table of Contents

EIR

The current definition of interest (amount value minus principal) is not good for analyzing profitability
- We need a better definition that uses $a(t)$

Definition 03.1 (Effective Interest Rate).

The amount that one unit invested at the beginning of a period will earn during the period, with interest paid at the end of the period.

$$i = a(1) - a(0) = a(1) - 1$$

Written in terms of $A(t)$:

$$\frac{A(1)-A(0)}{A(0)} = \frac{I}{A(0)}$$

Alternative definition:
- The amount of interest gained during a period divided by the principal at the beginning of the period
For the $n$th time period:
- Note: $I_n$ is the amount of growth in the period whereas $i_n$ is the rate of growth

$$i_{n} = \frac{A(n)-A(n-1)}{A(n-1)} = \frac{I_{n}}{A(n-1)}$$

In terms of $a$:

$$i_{n} = \frac{a(n)-a(n-1)}{a(n-1)}$$

Example 03.2.

(Example 3.1 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 26])

03 Effective Interest Rate 2025-08-27 20.49.09

The fund at the end of the $n$th period equals the fund at the beginning of the period plus the interest earned during the period

$$A(n) = A(n-1) + i_{n}A(n-1)=(1+i_{n})A(n-1)$$

This leads to:

$$A(n) = (1 + i_{1})(1+i_{2})\dots(1+i_{n})A(0)$$

Example 03.3.

(Example 3.2 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 26])

03 Effective Interest Rate 2025-10-01 09.30.26

Example 03.4.

(Example 3.3 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 27])

03 Effective Interest Rate 2025-10-01 09.34.38

Example 03.5.

(Example 3.4 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 27])

03 Effective Interest Rate 2025-10-01 09.42.26

Negative Interest Rates

$i$ can be $< 0$ if $a$ is decreasing (which also means $A$ is decreasing)

Example 03.6.

(Example 3.5 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 27])

03 Effective Interest Rate 2025-10-01 09.49.35

Practice

Example 03.7.

(Problem 3.1 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 29])

03 Effective Interest Rate 2025-10-01 09.51.00

Example 03.8.

(Problem 3.2 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 29])

03 Effective Interest Rate 2025-10-01 10.10.29

Example 03.9.

(Problem 3.3 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 29])

03 Effective Interest Rate 2025-10-01 10.15.33

Example 03.10.

(Problem 3.4 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 29])

03 Effective Interest Rate 2025-10-01 10.19.51

Example 03.11.

(Problem 3.5 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 29])

03 Effective Interest Rate 2025-10-01 10.23.31

Example 03.12.

(Problem 3.6 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 29])

03 Effective Interest Rate 2025-10-01 10.25.34

Example 03.13.

(Problem 3.7 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 29])

03 Effective Interest Rate 2025-10-01 10.29.48

Example 03.14.

(Problem 3.8 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 29])

03 Effective Interest Rate 2025-10-01 10.37.12

Example 03.15.

(Problem 3.9 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 30])

03 Effective Interest Rate 2025-10-01 10.44.28

Example 03.16.

(Problem 3.10 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 30])

03 Effective Interest Rate 2025-10-01 10.48.56

Example 03.17.

(Problem 3.11 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 30])

03 Effective Interest Rate 2025-10-01 10.50.43

Example 03.18.

(Problem 3.12 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 30])

03 Effective Interest Rate 2025-10-01 10.52.36

Example 03.19.

(Problem 3.13 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 30])

03 Effective Interest Rate 2025-10-01 11.17.55

Example 03.20.

(Problem 3.14 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 30])

03 Effective Interest Rate 2025-10-01 11.25.27

Example 03.21.

(Problem 3.15 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 31])

03 Effective Interest Rate 2025-10-01 11.27.38

Example 03.22.

(Problem 3.16 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 31])

03 Effective Interest Rate 2025-10-01 11.47.06

Example 03.23.

(Problem 3.17 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 31])

03 Effective Interest Rate 2025-10-01 11.49.18

Example 03.24.

(Problem 3.18 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 31])

03 Effective Interest Rate 2025-10-01 11.50.39

Example 03.25.

(Problem 3.19 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 31])

03 Effective Interest Rate 2025-10-01 11.52.39

Example 03.26.

(Problem 3.20 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 31])

03 Effective Interest Rate 2025-10-01 11.54.41

Example 03.27.

(Problem 3.21 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 31])

03 Effective Interest Rate 2025-10-01 11.56.25

Example 03.28.

(Problem 3.22 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 31])

03 Effective Interest Rate 2025-10-01 12.07.49

Sources

Finan, A Basic Course in the Theory of Interest and Derivatives Markets

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