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
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Identify the variables for the annuity formula:
- Payment (PMT) = $4,497.00
- Number of payments (n) = 29 years
- Interest rate (i) = 11% = 0.11
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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).
- 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
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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:
\[ (1 + 0.11)^{-29} \approx 0.042516 \] \[ PV_{Year8} = 4,497 \times \left( \frac{1 - 0.042516}{0.11} \right) \] \[ PV_{Year8} = 4,497 \times \left( \frac{0.957484}{0.11} \right) \approx 4,497 \times 8.6944 \approx 39,158.66 \]
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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.
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