Question

An annuity immediate has a first payment of 200 and increases by 100 each year until payments reach 600. there are 5 further payments of 600. find the present value at 5.5%. solution

Answer 1

I will be using present value notation of "v", where v is present value of 1 a year from now.
$v = \frac{1}{1+i}$
The series of payments looks like this:
$PV = 200 + 300v+400v^2+500v^3 +600v^4+600v^5 +...600v^9$
The first 5 payments form an increasing annuity, the last 5 form a standard constant annuity. The increasing part needs to be of the form 1,2,3...n. Since we have 2,3,4,5,6, subtract 1 from each term ---> (1,1,1,1,1) + (1,2,3,4,5).
Rearranging the equation gives:
$PV = 100(1+v+v^2+v^3+v^4) +100(1+2v+3v^2+4v^3+5v^4) \\ +600v^5(1+v+v^2+v^3+v^4)$

Now plugging in the sum formulas for geometric series and increasing series:
$PV = 100(\frac{1-v^5}{1-v}) +100(\frac{\frac{1-v^5}{1-v}-5v^5}{iv}) +600v^5(\frac{1-v^5}{1-v})$
Finally sub in values using i = 5.5% = .055
$PV = 3822.11$