Question

how do I solve the problem on a finance calculator.
What is the value today of a money machine that will pay $4,497.00 per year for 29.00 years? Assume the first payment is made 8.00 years from today and the interest rate is 11.00%.

Answer 1

To solve this problem using a finance calculator, you need to calculate the present value (PV) of a series of cash flows (the payments from the money machine) that occur in the future. Since the first payment occurs 8 years from today, and the payments continue for 29 years, you will need to break this into two parts: the present value of the future cash flows and then discount that amount back to today.

### Step 1: Calculate the Present Value of Future Payments

1. **Identify the variables for the annuity formula:**
- Payment (PMT) = $4,497.00
- Number of payments (n) = 29 years
- Interest rate (i) = 11% = 0.11

2. **Calculate the Present Value (PV) of the annuity at the time of the first payment (which is at Year 8):**
To calculate the present value of an annuity, you can use the formula:
\[
PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]
where:
- PMT = $4,497.00
- r = interest rate = 0.11
- n = total number of payments = 29

Plugging in the values:
\[
PV = 4,497 \times \left( \frac{1 - (1 + 0.11)^{-29}}{0.11} \right)
\]

### Step 2: Calculate the Present Value at Year 0

Since the calculated PV above will be the value at Year 8, we now need to discount that value back to today (Year 0).

3. **Discount the annuity PV back to Year 0 (using the formula):**
\[
PV_{today} = PV \div (1 + r)^t
\]
where:
- t = number of years until the first payment = 8 years.

Putting it all together, the calculations would look like this:

### Calculation Steps

1. **Calculate the PV of annuity at Year 8**:
\[
PV_{Year8} = 4,497 \times \left( \frac{1 - (1 + 0.11)^{-29}}{0.11} \right)
\]

Let's calculate this by using a calculator:

\[
(1 + 0.11)^{-29} \approx 0.042516
\]
\[
PV_{Year8} = 4,497 \times \left( \frac{1 - 0.042516}{0.11} \right)
\]
\[
PV_{Year8} = 4,497 \times \left( \frac{0.957484}{0.11} \right) \approx 4,497 \times 8.6944 \approx 39,158.66
\]

2. **Discount the PV to Year 0**:
\[
PV_{today} = \frac{39,158.66}{(1 + 0.11)^8}
\]
Calculate \((1 + 0.11)^8\):
\[
(1 + 0.11)^8 \approx 2.3996
\]
Now calculate:
\[
PV_{today} = \frac{39,158.66}{2.3996} \approx 16,303.50
\]

### Final Answer
The value today of the money machine is approximately **$16,303.50**.