Level annuities

For the example question on page 15 chapter 6 on level annuities, the course notes find the PV using an annuity payable in advance, however I'm trying to look at using an annuity payable in arrear instead.

'Find the present value as at 01/01/2004 of a series of payments of £100 payable on the first day of each month during 2005, 2006 and 2007, assuming an effective rate of interest of 8% per annum.'

I'm interested to see how different people would approach this?

Many thanks

 

You could find it via an annuity in arrears, but to me this seems a bit circuitous - you have to start the annuity at 1/12/2004 and then discount it back 11 months.

 

Maybe the best way to tackle it is think what factors are readily available in the tables and go that route. That way there is less potential for extra calculation slip.

Not much in it really in this case.

Likewise in an exam if you don't do it the quickest way, and still get the right answer and its clear you knew what you were doing, its not an issue.

 

Another tip is that if you draw a quick yime line that can aid your thinking.

 

Thanks for the info.

Is it possible to find the annuity in arrears using annual terms?

 

Calum already answered this really.

In the example the payments are £100 on the first day of each month for 3 years. The first payment is on 01/01/05 and the last is on 01/12/07. The annuity in advance calc gives us the present value at 01/01/05 and then we can simply discount by V to get the present value at 01/01/04.

If we use a formula for an annuity in arrears (whether monthly or annual) your calculation start point on the timeline is 01/12/04. So for example using the annual arrears formula 1200 A_3(12 monthly) at 8% = [1200 * ((1-v^3) / i12] = [1200 * ((1-0.793832241) / 0.077208361)] = 3204.33 = the present value on 01/12/04. We have to multiply this by V^(11/12) to get the correct value of £2986 on 01/01/04.

Whether you use the arrears or advance formula you still have to calculate the pthly rate, so there's no speed advantage between the two in that regard. The advance method is quicker because you discount by v to get back to 01/01/04 and not some fraction of V.

 

Thanks, this has cleared up my confusion!

 

And just to throw a spanner in the works, another method that can often be useful when you have pre-computed tables to hand is to say the PV at 0 of a series of payments from time t to time s is the same as an annuity to s minus an annuity to t.