end of year 2 is end of semi-annual period 4.
end of year 4 is semi-annual period 8.
end of year 8 is semi-annual period 16.
interest rate of 6% per year becomes interest rate of 6/2 = 3% per semi-annual period.
present value of 20,000 at end of period 4 = 20,000 / (1.03)^4 = 17,769.74096.
present value of 40,000 at end of period 8 = 40,000 / (1.03)^8 = 31,576.36937.
present value of 80,000 atend of period 16 = 80,000 / (1.03)^16 = 49,853.35514.
total present value is equal to 99,199.46547.
you need to find the semi-annual payment required at the end of each semi-annual period for 16 semi-annual periods that will satisfy this present value.
use of a financial calculator helps, although you can also solve with a manual formula that is constructed to handle it.
using the financial calculator, you get a semi-annual payment of $7897.35.
the financial calculator i used can be found at https://arachnoid.com/finance/
here are the results from using that calculator.
inputs were:
preseent value = 99199.46547
future value = 0
number of time periods = 16
interest rate percent per time period = 3
payments made at end of each time period.
click on PMT and the calculator tells you that the monthly payment required is 7897.35.
if you find the payments required by manual formula, then the formula to use is:
a = (p*r)/(1-(1/(1+r)^n))
a is the annuity.
p is the present amount.
r is the interest rate per time period.
n is the number of time periods.
set p = 99199.46547
set r = .03
set n = 16
formula becomes a = (99199.46547*.03)/(1-(1/(1+.03)^16))
result is a = 7897.353693 which equals 7897.35 when rounded to 2 decimal places.
fyi,
this problem was previous solved using excel and cash flow analysis.
the solution was the same, as you can see by checking out the following link.
https://www.algebra.com/algebra/homework/word/misc/Miscellaneous_Word_Problems.faq.question.1119013.html