Calculating periodic rate of interest - i
• The formula is derived from FV = PV(1 + i)n
1 + 𝑖 𝑛 = 𝐹𝑉
𝑃𝑉
1 + 𝑖 = 𝐹𝑉
𝑃𝑉
1 𝑛
𝒊 = 𝑭𝑽
𝑷𝑽
𝟏 𝒏
− 𝟏 use this formula

Sure! Let's use the formula you provided to calculate the periodic rate of interest $ i $.

### Given:
- **Future Value (FV)**: $589.20
- **Present Value (PV)**: $300.00
- **Number of periods (n)**: 12 (since the interest is compounded quarterly for 3 years, $ n = 4 \times 3 = 12 $)

### Formula:
The periodic rate of interest $ i $ is given by:

\[
i = \left(\frac{FV}{PV}\right)^{\frac{1}{n}} - 1
\]

### Step 1: Plug in the values

\[
i = \left(\frac{589.20}{300.00}\right)^{\frac{1}{12}} - 1
\]

### Step 2: Calculate the value inside the parentheses

\[
\frac{589.20}{300.00} = 1.964
\]

### Step 3: Calculate the 12th root

\[
i = 1.964^{\frac{1}{12}} - 1
\]

Using a calculator:

\[
i \approx 1.059947 - 1
\]

\[
i \approx 0.059947
\]

### Step 4: Convert to a percentage

Since $ i $ is the periodic rate of interest for one quarter, it is approximately:

\[
i \approx 5.9947\%
\]

### Step 5: Nominal Annual Rate
To find the nominal annual rate $ r $, multiply the periodic rate by the number of periods per year (4 for quarterly compounding):

\[
r = 4 \times i \approx 4 \times 0.059947 = 0.239788
\]

\[
r \approx 23.9788\%
\]

### Final Answer:
The nominal annual interest rate is approximately **23.9788%** per annum, compounded quarterly.

Would you like any further details or have any other questions?

### Related Questions:
1. How does the periodic rate differ when compounded semi-annually versus quarterly?
2. What is the periodic interest rate if the future value is $1,000 and the present value is $600 over 5 years compounded annually?
3. How can we derive the formula for the nominal annual interest rate from the periodic rate?
4. What would be the nominal annual rate if the future value was $800 instead of $589.20?
5. How does changing the number of compounding periods per year affect the nominal rate?
6. What is the relationship between the periodic rate and the effective annual rate?
7. How do we calculate the nominal rate if only the present value and future value are known?
8. How do different interest rates compare when compounding is done monthly, quarterly, and annually?

### Tip:
Always ensure to convert the periodic interest rate back to the nominal annual rate correctly, especially when dealing with different compounding frequencies.

Learn how to calculate the periodic rate of interest (i) using the formula FV = PV(1 + i)^n. Follow a step-by-step explanation with specific Future Value (FV) $589.20, Present Value (PV) $300.00, and 12 periods (n). Understand the process to find both the periodic rate and the nominal annual rate.

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