The approach you've taken is a good starting point, but it looks like there is a small mistake in not accounting for the fact that the first payment will be made 10 years from today. Let's go through the steps correctly.
You want to find the present value \(PV\) of a perpetuity that pays $1,128 per year, starting 10 years from now, at a discount rate of 6% (0.06).
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Calculate the Present Value of the Perpetuity at the Time of the First Payment:
The formula for the present value of a perpetuity is: \[ PV = \frac{PMT}{r} \] where \(PMT\) is the annual payment and \(r\) is the discount rate.
In your case: \[ PV = \frac{1128}{0.06} = 18800 \] This is the present value of the perpetuity at the time the first payment is received, which is 10 years from now.
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Discount this Amount Back to Today:
Now that you have the present value as of year 10, you need to discount it back to today (year 0).
The formula to discount it back is: \[ PV_{today} = \frac{PV_{future}}{(1 + r)^n} \] where \(n\) is the number of years until the cash flow starts.
In your situation: \[ PV_{today} = \frac{18800}{(1 + 0.06)^{10}} \]
First, calculate \( (1 + 0.06)^{10} \): \[ (1.06)^{10} \approx 1.79085 \]
Then compute the present value today: \[ PV_{today} = \frac{18800}{1.79085} \approx 10419.07 \]
So the value today of receiving $1,128.00 per year forever, with the first payment starting in 10 years, is approximately $10,419.07.
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