Geometric Series Question

[mhorng]

In ASM, the formula for the present value of payments in geometric progression (section 4j) has a denominator of (i-k). However, in some of the solutions it uses [1-(1+k)/(1+i)]. When is it appropriate to use either one of these?

[icon41gimp]

The first is dividing by (i-k) the second thing you have is dividing by (i-k)/(1+i) [after doing the algebra]

If you break the second thing into two steps it's dividing by (i-k) then multiplying by (1+i)

I don't have an ASM manual but it looks like they're accumulating whatever they calculating by another period; doing it a slightly concealed way

[Wedge^Product]

With (1-[(1+k)/(1+i)]^n)/(1-(1+k)/(1+i)) it's just an n term geometric series with r = (1+k)/(1+i) so you're summing 1 + v + v^2 + ... + v^n (Payments at the beginning of each period)

With (1-[(1+k)/(1+i)]^n)/(i-k) it's the above series multiplied by v = 1/(1+i). So you're summing v + v^2 + ... + v^n (Payments at the end of each period)

Personally, I'm more comfortable just figuring out what r is and using the formula for the geometric series. I feel like it gets you into less trouble because you never pick the wrong formula. Plus there have been some problems where the algebra simplifies neatly when you use v'=(1+k)/(1+i).

Is it odd that I think the hardest part about this exam is the little stuff? The stuff that really screws me up is something like Fund X accumulates from 1/1/02 to 12/31/33. I often don't take the time to think about how many periods it is and just use n=31. Super horrible mistakes ensue.

[alphatmw]

^ i didn't even notice your mistake in n=31 for about 30 seconds lol