You have an investment worth $93,423 that is expected to make regular monthly payments of $1,710 for 40 months and a special payment of $X in 6months. The expected return for the investment is 0.71 percent per month and the first regular payment will be made in 1 month. What is X? Note: X is a positive number, round your answer to the nearest d

To solve this problem, we can break it down into the following steps using the concept of time value of money. Specifically, the present value of all future cash flows should equal the current value of the investment.

### Step 1: Present Value of the Regular Payments
We have 40 monthly payments of $1,710, and the expected return is 0.71% per month (which we interpret as a monthly discount rate).

The present value $ PV_{regular} $ of these monthly payments can be calculated using the formula for the present value of an annuity:

\[
PV_{regular} = P \times \left( 1 - \frac{1}{(1 + r)^n} \right) \div r
\]

Where:
- $ P = 1710 $ is the monthly payment,
- $ r = 0.0071 $ is the monthly interest rate (0.71% per month),
- $ n = 40 $ is the number of monthly payments.

### Step 2: Present Value of the Special Payment $ X $
The special payment $ X $ is made in 6 months, so the present value $ PV_X $ of this payment is:

\[
PV_X = \frac{X}{(1 + r)^6}
\]

### Step 3: Setting up the Equation
The total present value of the investment is the sum of the present value of the regular payments and the present value of the special payment $ X $. This should equal the initial investment of $93,423.

\[
93,423 = PV_{regular} + PV_X
\]

Substituting the equations for $ PV_{regular} $ and $ PV_X $, we get:

\[
93,423 = 1710 \times \left( 1 - \frac{1}{(1 + 0.0071)^{40}} \right) \div 0.0071 + \frac{X}{(1 + 0.0071)^6}
\]

Now we can calculate this equation step by step to find $ X $. I'll do the calculations.The value of the special payment $ X $ is approximately \$35,537 when rounded to the nearest dollar.

Would you like any further details or explanations on this calculation?

Here are 5 related questions for deeper understanding:
1. How would the result change if the interest rate was different?
2. What happens to the present value if the monthly payments are increased?
3. How is the formula for the present value of an annuity derived?
4. What would the investment value be if the special payment $ X $ were made after 12 months instead of 6?
5. How would the calculation change if the payments were made quarterly instead of monthly?

**Tip**: Always pay attention to the timing of cash flows when working with time value of money; small timing differences can significantly affect the outcome!

You have an investment worth $93,423 that is expected to make regular monthly payments of $1,710 for 40 months and a special payment of $X in 6 months. The expected return for the investment is 0.71 percent per month and the first regular payment will be made in 1 month. What is X?

Solve a complex time value of money problem where an investment of $93,423 makes monthly payments of $1,710 for 40 months, plus a special payment after 6 months. Learn how to use present value formulas for annuities and future cash flows to calculate X. Ideal for finance students and advanced high school math learners.

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