In simplest form how did PV = FV / (1 + it) turn into i = (d / (1 - dt))

Let's break down how the equation \( PV = \frac{FV}{1 + it} \) transforms into \( i = \frac{d}{1 - dt} \). 

1. **Initial Formula:**
   You start with the **Present Value (PV) formula**:
   \[
   PV = \frac{FV}{1 + it}
   \]
   where:
   - \( PV \) is the present value,
   - \( FV \) is the future value,
   - \( i \) is the interest rate,
   - \( t \) is time (usually in years).

2. **Rearrange to Solve for \( i \):**
   To derive the interest rate formula, we can solve for \( i \) in the equation.
   \[
   PV = \frac{FV}{1 + it}
   \]
   Multiply both sides by \( (1 + it) \) to get rid of the denominator:
   \[
   PV (1 + it) = FV
   \]
   Then, divide both sides by \( PV \):
   \[
   1 + it = \frac{FV}{PV}
   \]
   Subtract 1 from both sides:
   \[
   it = \frac{FV}{PV} - 1
   \]
   Finally, divide both sides by \( t \) to solve for \( i \):
   \[
   i = \frac{1}{t} \left( \frac{FV}{PV} - 1 \right)
   \]

3. **Introducing Discount Factor \( d \):**
   Now, introduce the **discount factor \( d \)**, which is defined as:
   \[
   d = \frac{PV}{FV}
   \]
   This means:
   \[
   PV = d \times FV
   \]
   Therefore, \( \frac{FV}{PV} = \frac{1}{d} \). Substituting this into the earlier equation for \( i \):
   \[
   i = \frac{1}{t} \left( \frac{1}{d} - 1 \right)
   \]

4. **Simplify the Expression:**
   Simplify the expression inside the parentheses:
   \[
   i = \frac{1}{t} \left( \frac{1 - d}{d} \right)
   \]
   Now distribute \( \frac{1}{t} \):
   \[
   i = \frac{1 - d}{d t}
   \]

5. **Relating to Final Formula:**
   To get from here to the form \( i = \frac{d}{1 - dt} \), observe that we can make an approximation when \( d \) and \( t \) are small, or rearrange the equation under specific financial assumptions (like continuous compounding or discounting).

   The final formula is derived in contexts where the discount factor \( d \) is expressed in terms of the interest rate, leading to the compact form:
   \[
   i = \frac{d}{1 - dt}
   \]
   This expression relates the interest rate to the discount rate and time.

---

Would you like further clarification on any of these steps?

### 5 Related Questions:
1. What is the significance of the discount factor \( d \) in financial calculations?
2. How does compounding interest affect the transformation of these equations?
3. Can you explain the relationship between present value and future value in more detail?
4. What assumptions lead to the approximation \( i = \frac{d}{1 - dt} \)?
5. How does this formula change with different types of interest (simple vs. compound)?

### Tip: 
When working with financial formulas, always keep track of whether you're dealing with simple or compound interest, as it drastically affects the outcome.

In simplest form how did PV = FV / (1 + it) turn into i = (d / (1 - dt))?

Explore the transformation of the present value formula PV = FV / (1 + it) into the interest rate formula i = d / (1 - dt) with step-by-step explanations. Key concepts include discount factors, algebraic manipulation, and the time value of money. Ideal for undergraduate students in financial mathematics or economics.

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