April 2007 Q9

[vgvu99]

The answer given in ASET and Revision Notes requires a trial & error method (or an iterative process) to find the value of n. i.e. n = 100 ..... n = 110, n = 111 etc...

The following method solves the exact value of n (DPP). Can some1 please verify if it is correct. Thanks.

Please note the notation used: a|n](p) = PV of immediate annuity certain payable pthly for a period of n years.
Also working in £million


Costs: 40 + 36.a|0.5](12) + 2.a|t](12).v^(1/2) @10%
Incomes: 1.v^(5/12) + 12.a|t](12).v^(1/2)

<the idea here is to separate the first payment of the rental income>

And therefore we have the equation of value:
0 = 12.a|t](12).v^(6/12) - 57.50812 - 2.a|t](12).v^(1/2) + v^(5/12)

==> (12.v^(6/12) - 2.v^(1/2)).a|t](12) - 57.50812 + v^(5/12)
==> a|t](12) = 5.9307
==> t = 8.794 years or 105.53 months
==> DPP >= 105.53 + 6 = 111.53 or 112 months.

 

[vgvu99]

A similar method can be found in the examiners' report:

http://www.actuaries.org.uk/__data/assets/pdf_file/0010/32788/ct1r_a07.pdf


I suggest ActEd to implement this method to the Revision Notes & ASET

 

[Mark Mitchell]

You're right - this can be solved by an exact method, as you spotted and as is given in the examiners' report. We will update ASET and the Revision Notes booklets accordingly.

A couple of points:

- there's one small error in your solution, I think. Where you evaluate the present value of the income, the first term should be 1.v^(6/12) rather than 1.v^(5/12) as the first income payment is received at the start of the seventh month i.e. time 6. It doesn't change the final answer for the DPP, though.

- note that either the exact method or the trial and error method would be suitable to use in an exam. The exact method depends on having spotted the "trick" of splitting the first of the income payments off when writing down the present value. If you missed this, then trial and error is the sensible thing to do.

Thanks again,

Mark.

 

[johnpe21]

could you please explain how we could try the trial and error method for this problem ? we should use the equations described below and change n ? or use different equations?

I tried an equation like the one below, but for payments I used 12.a(..)|t](12).v^(6/12) and came up with a different result. Can you please tell me where is the mistake?

Thanks a lot

 

[John Lee]

could you please explain how we could try the trial and error method for this problem ? we should use the equations described above and change n ?

Yes

I tried an equation like the one above, but for payments I used 12.a(..)|t](12).v^(6/12) and came up with a different result. Can you please tell me where is the mistake?

This is exactly the same as above - the only difference is in the term. The t is from when these payments start and not from time 0. Hence, you'll get a different answer.

 

[johnpe21]

i am still a bit confused. So the equation I have used is not correct? do I need to change sth in it to get the correct value? Or you mean that in the value that I calculate I should add 6 months to get the correct one?

 

[John Lee]

i am still a bit confused. So the equation I have used is not correct? do I need to change sth in it to get the correct value? Or you mean that in the value that I calculate I should add 6 months to get the correct one?

I'm thinking the latter - you should add 6 months

 

[Montgomery]

Help Help Help¬¬¬

CAn anyone please explain why we have takes 1v^(6.12) taken into account. Why is one months income is considered when when we have an annuity function for the income payements.

Thanks
Montgomery

You're right - this can be solved by an exact method, as you spotted and as is given in the examiners' report. We will update ASET and the Revision Notes booklets accordingly.

A couple of points:

- there's one small error in your solution, I think. Where you evaluate the present value of the income, the first term should be 1.v^(6/12) rather than 1.v^(5/12) as the first income payment is received at the start of the seventh month i.e. time 6. It doesn't change the final answer for the DPP, though.

- note that either the exact method or the trial and error method would be suitable to use in an exam. The exact method depends on having spotted the "trick" of splitting the first of the income payments off when writing down the present value. If you missed this, then trial and error is the sensible thing to do.

Thanks again,

Mark.

 

[Mark Mitchell]

The first income payment is split off from the annuity representing the rest of the income payments, so that we can write the expressions for the PV of income and PV of costs in terms of the same the annuity symbol.

It simplifies the algrebra that follows.