• We are pleased to announce that the winner of our Feedback Prize Draw for the Winter 2024-25 session and winning £150 of gift vouchers is Zhao Liang Tay. Congratulations to Zhao Liang. If you fancy winning £150 worth of gift vouchers (from a major UK store) for the Summer 2025 exam sitting for just a few minutes of your time throughout the session, please see our website at https://www.acted.co.uk/further-info.html?pat=feedback#feedback-prize for more information on how you can make sure your name is included in the draw at the end of the session.
  • Please be advised that the SP1, SP5 and SP7 X1 deadline is the 14th July and not the 17th June as first stated. Please accept out apologies for any confusion caused.

September 2009 - Q10

R

Rebecca.Thomas

Member
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
 
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??
 
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)
 
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...
 
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!)
 
Back
Top