Lesson 5.4: Financial Mathematics

Introduction

In this lesson, we will explore the essential concepts of financial mathematics, specifically focusing on simple and compound interest, savings plans, loan repayments, and depreciation. By the end of this lesson, you, students, will be equipped with the knowledge to calculate various financial scenarios using sequences and series.

Learning Objectives

• Understand and apply simple and compound interest calculations.
• Analyze savings plans and loan repayments using geometric series.
• Compare different financial options using present and future value ideas.
• Calculate simple and compound interest over a stated period.
• Use geometric series to analyze savings or loan situations.

Understanding Simple Interest

Definition

Simple interest is calculated on the principal amount, or the initial amount of money. The formula for calculating simple interest is given by:

$$

I = P $\cdot$ r $\cdot$ t

$$

where:

• $I$ is the interest earned or paid,
• $P$ is the principal amount,
• $r$ is the annual interest rate (in decimal),
• $t$ is the time period in years.

Worked Example

Suppose you invest $1,000 at an annual interest rate of 5% for 3 years. We can calculate the simple interest as follows:

1. Identify the variables:

• $P = 1000$
• $r = 0.05$
• $t = 3$

1. Substitute the values into the formula:

$$

I = $1000 \cdot 0$.$05 \cdot 3$ = 150

$$

The total interest earned after 3 years is $150.

Common Misconception

A common misconception is that simple interest compounds. This is not true; simple interest remains constant for the entire period based solely on the principal.

Understanding Compound Interest

Definition

Compound interest is calculated on the initial principal, which also includes all of the accumulated interest from previous periods. The formula for compound interest is:

$$

A = P $\cdot$ $\left(1$ + $\frac{r}{n}$

$ight)^{nt}$

$$

where:

• $A$ is the amount of money accumulated after n years, including interest.
• $P$ is the principal amount.
• $r$ is the annual interest rate (in decimal).
• $n$ is the number of times that interest is compounded per year.
• $t$ is the time the money is invested or borrowed for, in years.

Worked Example

Consider an investment of $1,000 at an annual interest rate of 5% compounded annually for 3 years:

1. Identify the variables:

• $P = 1000$
• $r = 0.05$
• $n = 1$ (compounded annually)
• $t = 3$

1. Substitute the values into the formula:

$$

A = $1000 \cdot$ $\left(1$ + $\frac{0.05}{1}$

$ight)^{1 \cdot 3}$

$$

Simplifying gives:

$$

A = $1000 \cdot$ $\left(1$ + 0.05

$ight)^3 = 1000 \cdot (1.05)^3$

$$

Calculating $(1.05)^3$ gives approximately $1.157625$, so:

$$

A $\approx 1000$ $\cdot 1$.157625 = 1157.63

$$

The total amount after 3 years is approximately $1,157.63.

Effect of Compounding Frequency

The more frequently interest is compounded, the more interest will accumulate. For example, compare the following cases:

1. Quarterly compounding ($n = 4$):

$$A = 1000 \cdot \left(1 + \frac{0.05}{4}

ight)^{$4 \cdot 3$}$$

1. Monthly compounding ($n = 12$):

$$A = 1000 \cdot \left(1 + \frac{0.05}{12}

ight)^{$12 \cdot 3$}$$

Both will yield more than the annual compounding example. This highlights the importance of compounding frequency in financial decisions.

Savings Plans and Loan Repayments Using Geometric Series

Introduction to Geometric Series

Geometric series can be used to analyze situations involving constant growth or decay, such as savings plans and loan repayments.

Saving Plans Example

Suppose students decides to save $100 at the end of each month into a savings account that pays an annual interest rate of 6%, compounded monthly. We can analyze the total amount saved after $n$ months using a geometric series.

1. Identify the variables

• Monthly deposit = $100
• Monthly interest rate = $\frac{0.06}{12} = 0.005$
• Number of deposits = $n\$

1. Determine the future value of the savings

The future value after $n$ deposits can be calculated as:

$$

FV = $100 \cdot$ $\left($$\frac{(1 + 0.005)^n - 1}{0.005}$

ight)

$$

Here, we have a geometric series where the first term is $100$ and the common ratio is $(1 + 0.005)$.

This formula allows us to compute the future value of a growing savings plan efficiently.

Loan Repayment Example

For a loan of $10,000 at an interest rate of 5% per annum, compounded annually, and a loan period of 10 years, the monthly repayment can be described as a series.

1. Identify the variables:

• Loan amount = $10,000
• Annual interest rate = 0.05
• Number of payments = 120 (10 years x 12 months)

1. The monthly payment can be derived using the loan amortization formula:

$$

PMT = $\frac{P \cdot r(1 + r)^n}{(1 + r)^n - 1}$

$$

where $r$ is the monthly interest rate. After calculations, you can find that the total amount paid will form a geometric series that sums the paid amounts over time.

Comparing Financial Options: Present and Future Value

Present Value Concept

The present value is the current worth of a cash flow or series of cash flows that will be received in the future, discounting it back to the present using a specific interest rate. The formula is:

$$

$PV = \frac{FV}{(1 + r)^t}$

$$

where:

• $PV$ is the present value,
• $FV$ is the future value,
• $r$ is the interest rate,
• $t$ is the time period.

Future Value Concept

As previously discussed, the future value represents the amount of money that will be accumulated after a certain period at a specific interest rate. Understanding both is crucial for making informed financial choices.

Conclusion

In this lesson, students explored the significant concepts of financial mathematics. You learned to differentiate between simple and compound interest, how to use geometric series for savings plans and loan repayments, and the importance of present and future value concepts. Through examples and practical applications, you have gained the foundational tools necessary to navigate financial decisions effectively.

Study Notes

• Simple interest formula: $I = P \cdot r \cdot t$
• Compound interest formula: $A = P $\cdot$ $\left(1$ + $\frac{r}{n}

ight)^{nt}

• Financial decisions should consider the effect of compounding frequency.
• Geometric series can represent recurring financial actions like saving and loan repayment.
• Present value shows the worth of future cash flows discounted to today; the formula is $PV = \frac{FV}{(1 + r)^t}$.
• Future value calculates the total amount accumulated in a future period, important for assessing investment growth.