From time to time I still get confused and have a temporary mind block about the timings of these annuities. For example taking an annuity payable in advance for 25 years payable half yearly is it written out as 0.5[1 + v^0.5 +v^1 +...+v^24.5]? If it is right, I always make the mistake of thinking the last term's v^25. What's the explanation for why the last term is 24.5? Is it because the annuity is in advance of half a year so the payment at 25 would be paid on 24.5? If I were then to put this as a geometric progression would it be: 0.5[1-(v^0.5)^25]/1-v^0.5 i.e for 25 terms? Sometimes I get confused because when writing it out the last term is at 24.5 but it's actually for 25 terms if you get what I mean Now moving onto annuity payable in arrears for 25 years payable half yearly. Is this written out as 0.5[v+v^0.5+v^1+...+v^25]? So then the geometric progression would be 0.5v[1-(v^0.5)^25]/1-v^0.5? for x^n n would still be for 25 terms right? I know it's such a basic thing but it still confuses me. If I am right I just need to remember to watch out for annuities payable in advance. Thanks!
It's actually for 50 terms if you write it that way, two payments each year for 25 years, for both annuities. It's 24.5 because it's the start of every six months for 25 years. The last six months starts at 24.5 years from now. One way to look at it is to remember one way and realise the other is either six months earlier, or six months later, so a simple timing adjustment will convert one to the other. If six months later, discount the immediate annuity for six months.
