given an intial investment B, and interest rate r per annum, what's the equation of the return after T years?

The equation for the return after \( T \) years, given an initial investment \( B \) and an interest rate \( r \) per annum, depends on whether the interest is **simple** or **compound**.

### 1. **Simple Interest**:
The return after \( T \) years, where the interest is added linearly based on the initial principal \( B \), is given by:
\[
A = B(1 + rT)
\]
Where:
- \( A \) is the total amount after \( T \) years.
- \( r \) is the annual interest rate (expressed as a decimal).
- \( T \) is the number of years.

### 2. **Compound Interest**:
If the interest is compounded annually, the equation for the return is:
\[
A = B(1 + r)^T
\]
Where:
- \( A \) is the total amount after \( T \) years.
- \( r \) is the annual interest rate (expressed as a decimal).
- \( T \) is the number of years.

For more frequent compounding (like monthly, quarterly, etc.), the formula changes slightly:
\[
A = B\left(1 + \frac{r}{n}\right)^{nT}
\]
Where:
- \( n \) is the number of compounding periods per year (e.g., 12 for monthly, 4 for quarterly).

Would you like more details on one of these formulas or a specific example calculation?

### Related Questions:
1. How does continuous compounding differ from annual compounding?
2. What happens if the interest is compounded monthly instead of annually?
3. How does the total return change if the interest rate is compounded more frequently?
4. What’s the difference in returns between simple and compound interest over 10 years?
5. How do you derive the compound interest formula from first principles?

### Tip:
Always make sure to express the interest rate \( r \) as a decimal when plugging it into the formulas (e.g., 5% → \( r = 0.05 \)).

Given an initial investment B, and interest rate r per annum, what's the equation of the return after T years?

Learn how to calculate returns on an initial investment using simple and compound interest formulas. This guide covers key financial mathematics concepts, including the equations for simple and compound interest, and how compounding frequency affects the total return. Suitable for students in Grades 9-12.

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