the money starts coming in 6 months.
the discount rate is 9.9% per year compounded quarterly.
you are talking about the present value of an annuity when the present value is 6 months in the future.
to bring that back to the present, you would first have to find the present value of the annuity and then find the present value of the present value of the annuity.
you can do it by formulas or you can use a financial calculator or you can treat it as a cash flow and find the present value of the cash flow.
using a financial calculator, i do the following:
present value = 0
future value = 0
interest rate per quarter = 9.9% per year / 4 = 2.475%
payments per quarter = 16,000
payments are made at the beginning of each quarter.
i then click on PV and the calculator tells me that the present value of the annuity is equal to $214,466.85
the first payment is 6 months out, so i have to bring that back 6 months to the current time period.
i use the calculator again with the following inputs.
present value = 0
future value = 214,466.85
interest rate per quarter is the same as before.
payments per quarter = 0
payments are made at the beginning of each quarter doesn't apply to the present value of a future value.
i then click on PV and the calculator tells me that the present value of the annuity brought back 6 months to the present time is $204,232.85.
here's what the results of the calculator look like.
first the present value of the annuity.
then the present value of the future value.
i then did a cash flow analysis using excel.
this is what it looks like.
time point 0 is the beginning of the first time period
time point 1 is the end of the first time period.
time point 2 is the end of the second time period.
etc.
each time period represents 1 quarter which is the same as 3 months.
the first payment of 16,000 is made at the end of time period 2, which is time point 2.
that's 6 months out.
the present value of 16,000 in time point 2 is brought back 6 months to time point 0, which is the current time period.
the cash slow and the financial formulas agree.
disregard the negative values.
they're a cash flow convention used by the calculator that i didn't follow too religiously because it didn't make a difference in the results, as long as you ignore the signs.