Subject A1 - September 1999 Exam Question 17 (iii)

I do not have a clue what this question is asking, nor do i understand anything in the examiners report.. Anyone care to help, or explain an easier alternative solution method?

Exam paper :

http://www.actuaries.org.uk/__data/assets/pdf_file/0004/32854/A1S99.PDF

 

The concept tested here is calculating accumulation factors for a given force of interest which varies with time.

Force of interest is interest that applies at an instant and you get the accumulation factor for a period as exp( - integral over period of delta(t) dt)

I can't imagine that this would have been removed from the course but it's possible. Have a look in the notes to see if it's there and if you understand the notes.

Then if you still need help post another question.

 

I havnt encountered anything of this sort before in the 20 exam papers i have done so im wondering where it actually fits into the course, it's still making 0 sense

 

Just realised you seem to have trouble with part (iii) only.
I'll assume you are familiar with parts (i) and (ii).
I'm not familiar with the course but in any event it is likely to be a combination of a few key ideas.

Part (iii) is asking you to show that the equivalent constant annual rate of interest is less than 5.5% if you invested the PV in (ii) and received the cashflows in (ii).

Put another way:
you invest 128.25 now
and receive (60-3t) for (5<=t<10)
Show the IRR is less than 5.5%

Ok so far?

One way to show that the IRR <5.5% is to find it explicitly but its not worth the time and complexity to do so, but as with finding it explicitly you start with an equation of value.

128.25 = v^5 (calculate pv at t=5 and discount to t-0)
*60 abar5 (payment of 60 part of {60-3t} paid cont for 5 yrs)
-(15abar5 + 3 Ibar_abar5)

This last part is the 3t payment. From t=5 to 10 the payment goes from 15 {ie 3*5} to 30. Put in terms of what we have notation for,
15 {abar5}
plus another payment starting at 0 increasing continuously at rate of 3 pa paid continuously for 5 years
3 {Ibar abar5}

Plug in 5.5 and show that RHS of the equation of value is too small, which means that you discounted too much so need a lower (discount/interest) rate. (note that this works here because you only have a positive income on the RHS and constant i, in different circumstances it may not always hold that too small a PV means a bigger i)

That seems to be the simplest way.
An alternative is to
128.25 = integral from 5 to 10 { v^t (60-3t)} dt at constant v
Then show that using i=5.5% (or equivalent v) makes RHS too small.

OK now?

 

i understand now, the inegral needs to be written interms of continuous annuities. I thought it would be easier to try and solve the integral directly but it involves integration by parts etc