To find the present value of a series of cash flows (an annuity) that pays $1,716.00 every six months for 26 years at an interest rate of 14.00% per annum, we first have to calculate the present value of an annuity formula.
Given Data:
- Payment (PMT) = $1,716.00
- Number of years (n) = 26 years
- Interest rate (annual) = 14.00% (6.00% semi-annually)
- Number of payments per year = 2 (since payment is made every six months)
Steps to Calculate Present Value:
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Calculate the Number of Payments: \[ \text{Total Payments} = n \times \text{Payments per Year} = 26 \times 2 = 52 \text{ payments} \]
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Convert the Interest Rate for Each Period: \[ i = \frac{14.00%}{2} = 7.00% = 0.07 \text{ (per semi-annual period)} \]
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Use the Present Value of Annuity Formula: The present value \( PV \) of an annuity can be calculated using the formula: \[ PV = PMT \times \left(1 - (1 + i)^{-n}\right) / i \] where:
- \( PMT \) = payment per period
- \( i \) = interest rate per period
- \( n \) = total number of payments
Plugging in the values: \[ PV = 1716 \times \left(1 - (1 + 0.07)^{-52}\right) / 0.07 \]
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Calculate \( (1 + i)^{-n} \): \[ (1 + 0.07)^{-52} \approx (1.07)^{-52} \approx 0.026304 \]
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Calculate \( 1 - (1 + i)^{-n} \): \[ 1 - (1.07)^{-52} \approx 1 - 0.026304 \approx 0.973696 \]
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Calculate \( PV \): \[ PV = 1716 \times \frac{0.973696}{0.07} \] \[ PV = 1716 \times 13.19565714 \approx 22,648.65 \]
Conclusion:
The present value of a money machine that will pay $1,716.00 every six months for 26 years at an interest rate of 14.00% is approximately $22,648.65.