If you find your method more intuitive, it is perfectly fine. Keep in mind that, in the solutions, for b) it says PV(month#1) and for c) it says PV because the first one is the present value one month from now, and the second is the present value today. You are using PV in b) for the present value next month and in c) for the present value the month before today.
If you ask my favorite method, I would do
b) PV=(200/0,4%)*(1-1/(1+0,4%))^11*1)/(1+0.4%) --> This way it is today's value because the annuity formula gives me the value one month before the first payment (month 1)
FV= PV*(1+0,4%)^12 -> because it is 12 months from today
c) PV=(200/0,4%)*(1-1/(1+0,4%))^11*1)*(1+0.4%) --> This way it is today's value because the annuity formula gives me the value one month before the first payment (month -1)
FV= PV*(1+0,4%)^12 -> because it is 12 months from today
Instead of moving the PV to today every time and capitalizing 12 months, you preferred leaving the value where the annuity formula computes it and capitalizing accordingly. It will work every time as long as you keep in mind that the annuity formula always provides the value in the month before
If you ask my favorite method, I would do
b) PV=(200/0,4%)*(1-1/(1+0,4%))^11*1)/(1+0.4%) --> This way it is today's value because the annuity formula gives me the value one month before the first payment (month 1)
FV= PV*(1+0,4%)^12 -> because it is 12 months from today
c) PV=(200/0,4%)*(1-1/(1+0,4%))^11*1)*(1+0.4%) --> This way it is today's value because the annuity formula gives me the value one month before the first payment (month -1)
FV= PV*(1+0,4%)^12 -> because it is 12 months from today
Instead of moving the PV to today every time and capitalizing 12 months, you preferred leaving the value where the annuity formula computes it and capitalizing accordingly. It will work every time as long as you keep in mind that the annuity formula always provides the value in the month before