Chapter: Interest and the Time Value of Money (Applied Mathematics 2)

Hello everyone! Welcome to one of the most "relatable" chapters in mathematics: Interest and the Time Value of Money. This chapter isn't just for exams; the knowledge here will help you manage your money better in the future, whether it's saving, borrowing, or making installment payments.

If you feel like math is tough, "don't worry!" This chapter focuses on understanding the concepts and applying the formulas. I’ll walk you through it step-by-step!

1. Simple Interest

Simple Interest is interest calculated solely on the "initial principal" throughout the entire duration of the deposit or loan. It’s like having an initial sum of money that generates the same amount of profit every year.

Calculation Formula:

\( A = P(1 + rt) \)

  • \( A \) is the total amount (Principal + Interest)
  • \( P \) is the Principal
  • \( r \) is the annual interest rate (always convert to decimal, e.g., 5% becomes 0.05)
  • \( t \) is the time period (in years)

Important Point: Don't forget that \( r \) and \( t \) must be consistent. If the problem gives time in months, you must divide by 12 to convert it into years first!

Example: Deposit 1,000 Baht at a simple interest rate of 5% per year for 3 years.
We get \( A = 1,000(1 + (0.05 \times 3)) = 1,000(1.15) = 1,150 \) Baht.

Summary Keyword: Simple interest = Same amount of interest earned every year.

2. Compound Interest

This is the "8th wonder of the world" because the interest earned in the first period is added to the principal to calculate interest for the next period. Simply put, it's "interest earning interest."

Calculation Formula (Interest compounded annually):

\( A = P(1 + i)^n \)

Calculation Formula (Interest compounded \( k \) times per year):

\( A = P(1 + \frac{r}{k})^{kn} \)

  • \( i \) or \( \frac{r}{k} \) is the interest rate per period.
  • \( n \) or \( kn \) is the total number of periods the interest is compounded.

Comparison Example: If the interest rate is 12% per year:
- Compounded annually: Interest per period is 12% (0.12)
- Compounded every 6 months (2 times/year): Interest per period is 12/2 = 6% (0.06)
- Compounded every 3 months (4 times/year): Interest per period is 12/4 = 3% (0.03)

Common Mistakes: Students often forget to divide the annual rate by the number of compounding periods (\( k \)) and forget to multiply the number of years by \( k \). Be careful with these!

Did you know?: The more frequently you compound interest (e.g., monthly vs. annually), the more total money you will end up with.

3. Present Value and Future Value

Have you ever heard the saying, "100 Baht today is worth more than 100 Baht 10 years from now"? This is the core principle of the time value of money.

  • Future Value (\( S \)): The amount of money you will receive in the future, including interest.
  • Present Value (\( P \)): The value of money today that will grow into a specific amount in the future.
Relationship Formula:

\( P = S(1 + i)^{-n} \)

Simple Technique: To find the future value (\( S \)), multiply by the interest factor. To find the current principal (\( P \)), divide (or use a negative exponent).

Example: If you want to have 10,000 Baht in 2 years, with the bank offering 4% interest per year (compounded annually), how much must you deposit today?
\( P = 10,000(1 + 0.04)^{-2} \)

Summary Keyword: Present value is "stripping away" the interest to see the starting principal.

4. Annuities

This part might look difficult, but it's actually an application of "geometric series." An annuity is a series of equal payments or receipts, such as home loans or monthly savings.

Two types of annuities:

1. Ordinary Annuity (End of period):
e.g., Receiving a salary after working a full month, or paying a credit card bill.
Total amount formula: \( R + R(1+i) + R(1+i)^2 + ... + R(1+i)^{n-1} \)
Using the geometric series formula: \( S_n = R \frac{(1+i)^n - 1}{i} \)

2. Annuity Due (Beginning of period):
e.g., Paying dorm rent (must pay before living), or saving money at the start of every month.
Total amount formula: \( R(1+i) + R(1+i)^2 + ... + R(1+i)^n \)
The total amount will always be one interest period higher than an ordinary annuity!

  • \( R \) is the payment per period.
  • \( i \) is the interest rate per period.
  • \( n \) is the total number of periods.

Important Point: In A-Level 2 exams, questions often ask you to "set up" the answer in an exponential form rather than calculating the final numerical result. You must practice identifying the geometric series and determining what \( a_1 \) and \( r \) (common ratio) are.

Summary for Exam Preparation

1. Read the question carefully: Is it simple interest or compound interest?
2. Check time units: Is the interest rate annual, but compounded monthly? (If yes, don't forget to divide by 12).
3. Draw a Timeline: For annuities, drawing a timeline helps you see how many interest periods each payment accumulates, effectively reducing confusion between beginning-of-period and end-of-period payments.
4. Memorize this: The geometric series formula \( S_n = \frac{a_1(r^n - 1)}{r - 1} \) is the heart of the annuity topic.

"If you understand where each payment comes from, this chapter will be a great source of easy marks. Good luck!"