Continuous Compounding
Challenge

How to Play the Continuous Compounding Game
Here’s how to play:

1. Start: Each Quiz consists of 10 questions. Select one of the modes: Future Value (A), Find Principal (P), Find Rate (r) and Find Time (t).
   
2. Look at the Problem: Use the Continuous Compounding Formula to calculate the answer.
   
3. Select Your Answer: Select the correct answer.
   
4. Check Your Work: If you selected the right answer, it will be highlighted in green. If you are wrong, it will be highlighted in red and the correct answer will be highlighted in green.
   
5. Get a New Problem: Click “Next Calculation” for a new problem.
   Your score is tracked, showing how many you’ve gotten right.
   
6. Finish Game When you have completed 10 questions, click “Finish Game” to get your final score.
    
   

Continuous Compounding Formula (PERT Formula)
Continuous compounding is a mathematical concept that calculates interest assuming it is compounded infinitely many times over a given period. While not used for most bank accounts, it is essential for advanced financial modeling, especially in derivatives pricing.
This formula is often remembered by the acronym “PERT."

The Formula
The formula for the total Future Value ($A$) under continuous compounding is derived from the standard compound interest formula by applying limits as \(n \to \infty\):
\(A = P e^{rt}\)
Where:
A is the Future Value / Total Amount (The final balance after t years)
P is the Principal (The initial amount deposited or borrowed)
e is the Euler’s Number (An irrational constant, approximately 2.71828)
r is the Rate (The annual interest rate expressed as a decimal, e.g., 6% = 0.06).
t is the Time (The total duration of the investment or loan

 

How to Use the PERT Formula (Step-by-Step)
Step 1: Calculate the Product of Rate and Time (rt)
First, multiply the annual interest rate (as a decimal) by the total number of years. This determines the total growth factor needed for the exponent.
Step 2: Calculate the Growth Multiplier
Raise Euler’s number (e) to the power of the product found in Step 1.
Step 3: Calculate the Future Value (A)
Multiply the initial Principal (P) by the growth multiplier from Step 2.
\(A = P \times e^{rt}\)

Example Calculation
You invest $10,000 at an annual interest rate of 7%, compounded continuously, for 8 years. What will the total amount be after 8 years?
Given values:
P = $10,000
r = 7% = 0.07 (Decimal Rate)
t = 8 years
\(e \approx 2.71828\)

1. Plug the values into the formula:
   \(A = 10,000 \times e^{(0.07 \times 8)}\)
   
2. Calculate the exponent (rt):
   \(rt = 0.07 \times 8 = 0.56\)
   
3. Calculate the Growth Multiplier (\(e^{rt}\)):
   \(e^{0.56} \approx 1.75067\)
   
4. Calculate the Future Value (A):
   \(A = 10,000 \times 1.75067\)
   \(A \approx 17,506.70\)
   The total value of the investment after 8 years will be approximately $17,506.70.
   
   This video gives a clear, step-by-step approach to learn how to use the Continuous Compounding Formula.
   
   • Show Video
   
    
   

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