Amount desired at the end $17,600, length 7 years, rate 2%, quarterly. What’s the period used, rate used, PV factor used, PV desired at the end?

Let's break down the elements and solve for the required values step-by-step.

### Given Data:
- **Future Value (FV)**: $17,600
- **Time (n)**: 7 years
- **Annual interest rate (r)**: 2% (or 0.02 as a decimal)
- **Compounding frequency**: Quarterly (4 times a year)

### Step 1: **Determine the number of periods (n)**
Since the interest is compounded quarterly, we need to calculate the total number of compounding periods:

\[
n = \text{Number of years} \times \text{Compounding periods per year} = 7 \times 4 = 28 \text{ periods}
\]

### Step 2: **Determine the periodic interest rate (i)**
The annual interest rate of 2% needs to be adjusted for quarterly compounding. The periodic interest rate (i) is:

\[
i = \frac{\text{Annual interest rate}}{\text{Compounding periods per year}} = \frac{2\%}{4} = 0.5\% = 0.005 \text{ (decimal)}
\]

### Step 3: **Calculate the Present Value (PV) factor**
The present value factor can be found using the formula for the present value of a lump sum:

\[
\text{PV factor} = \frac{1}{(1 + i)^n} = \frac{1}{(1 + 0.005)^{28}}
\]

Let’s calculate that:

\[
\text{PV factor} = \frac{1}{(1.005)^{28}} \approx \frac{1}{1.148883} \approx 0.8703
\]

### Step 4: **Determine the Present Value (PV)**
To find the present value (the amount of money you need to invest today), multiply the future value (FV) by the PV factor:

\[
PV = \text{FV} \times \text{PV factor} = 17,600 \times 0.8703 \approx 15,320.79
\]

### Summary of Results:
1. **Period used**: 28 quarters (7 years × 4 quarters/year)
2. **Rate used**: 0.5% per quarter
3. **PV factor used**: 0.8703
4. **PV desired at the end**: $15,320.79

Would you like more details on any of these steps or have further questions? 

Here are 5 related questions to deepen understanding:
1. What happens if the interest rate changes to 3%?
2. How does the present value change with monthly compounding?
3. What is the future value if the initial investment was $15,320.79 at 2%?
4. How do different compounding frequencies (e.g., annually, semiannually) affect the PV factor?
5. How can you calculate the amount of interest earned over the 7-year period?

**Tip:** Always adjust the interest rate and number of periods to match the compounding frequency when solving time value of money problems.

Discover how to calculate the present value of $17,600 with 2% annual interest compounded quarterly over 7 years. Learn step-by-step about periods, interest rates, the PV factor, and the required investment today. Suitable for high school and college-level finance or math students.

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