September 2009 - Q10

[Rebecca.Thomas]

Hi all,

I've had a poke around the forums although I gather that the general opinion is that this question was nasty, there doesn't seem to be anything explaining my area of confusion! So I was wondering if anyone could help

I'm generally unsure what's happening with the increasing annuity section of the payments - and how they jumped to a continuous annuity (in part iii, in part ii they basically said the increasing annuity didn't need to be discounted because increase rate = discount rate).

Does anyone have any idea what's going on here please?

Thanks x

 

[Rebecca.Thomas]

And also - I initially attempted to evalulate the increasing annuity in part ii using the formula (as given in the summary to chapter 7) with n = 50, i = 0.01 but this doesn't give me 50 as it should if I'm to obtain the 20 X 50 relating to this cost in the solutions.

Any ideas??

 

[didster]

The net rate you need to use is 0 not 0.01, since the increase and discount are at the same rate.
consider discrete payments first (annually)
If you have a payment of 100 now which increases at 4%pa. Next year's payment is 104.
If you discount this at 4% you get 100. Same for payment in two years, etc
And you can extend this to any discount rate and continuous payments.

As a general rule the formulae go belly up when you stick i=0 in. but you just need to forget about interest/discount, number of payments times rate.

The 20x50 is 20 years of 50 each (no discounting because increase = discount rate; they cancel out)

and continuous annuity because the questions says so (continuous increases, incurred continuously)

 

[Rebecca.Thomas]

This seems to cover ii, although I'm more confused about part iii...

From the wording of the question it's a continously increasing continously paid annuity and somehow they've got from that to a straight forward continously paid annuity (ignoring the continuously increasing portion??)

I would've expected to see a formula involved Ia bar (bar over both I and a) and they're using a bar...

 

[John Lee]

This seems to cover ii, although I'm more confused about part iii...

From the wording of the question it's a continuously increasing continuously paid annuity and somehow they've got from that to a straight forward continuously paid annuity (ignoring the continuously increasing portion??)

I would've expected to see a formula involved Ia bar (bar over both I and a) and they're using a bar...

The Ia "super" bar formula is used when there is a continuous linear increase in the payments.

Here we have a continuous compound increase - so as with all compound increases we need to go to first principles and work from there. In this case we need to use integrals:

integral (0 to 50) of (1.01^t × v^t)

We can then set V = 1.01v and note that this is a continuous level annuity with new improved rate (or we could just integrate it normally!)