To determine the present value of receiving $1,470 per year forever (a perpetuity) with the first payment starting in 7 years, we can follow these steps:
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Calculate the present value of the perpetuity at the time the payments start.
The formula for the present value of a perpetuity is: \[ PV = \frac{C}{r} \] Where:
- \(C\) is the annual cash flow,
- \(r\) is the discount rate.
Here:
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\(C = 1,470\)
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\(r = 0.12\)
\[ PV = \frac{1,470}{0.12} = 12,250 \]
This value of $12,250 is the present value of the perpetuity at the time the first payment is received (which is 7 years from today).
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Discount this value back to today.
Now, we need to discount this present value back to today (which is 7 years earlier). The formula for the present value (PV) is: \[ PV = \frac{FV}{(1 + r)^n} \] Where:
- \(FV\) is the future value (which is $12,250),
- \(r\) is the discount rate (0.12),
- \(n\) is the number of years (7).
\[ PV = \frac{12,250}{(1 + 0.12)^7} \]
First, calculate \((1 + 0.12)^7\): \[ (1 + 0.12)^7 \approx 2.21068 \]
Now, calculate the present value: \[ PV \approx \frac{12,250}{2.21068} \approx 5,537.25 \]
So, the value today of receiving $1,470.00 per year forever, starting 7 years from today, at a discount rate of 12.00% is approximately $5,537.25.
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