Annuities Question

Hi,

I've having a little problem understanding a question that I am working on.

The part of the question that I am having trouble with is outlined below:

The project is expected to provide a continuous income at a rate of $80,000 in th first year, $83,200 in the second year, and so on, with income increasing each year by 4%p.a. compound. The income is recieved for 25 years and calculate the present value of the project assuming a interest rate of 11%p.a

Am I correct in assuming that there are essentially 2 annuities present in this?

1. for the continuous payment of $80,000​

2. Annuity for increasing amounts​

Therefore we would calculate the annuity for a continuous annuity for 1 year, and multiply this by an annuity due for the 25 year period.

The above is an assumption making based on the worked solution and it was the only explanation I could come up with. However reading the question it looks more like the payments are continuous for the 25 year period and increasing by 4%.

Are these sorts of questions common in the exam?

 

Continuous increase annuity.
try (Ia bar)n

 

Here the payments are continuous and the increases occur discretely, at the end of each year. As the increases are compound, we cannot use an (Ia-bar) annuity, as that applies where there are simple increases (not compound).

The best thing to do with an unusual case is to write down an expression from first principles:

80,000*(a-bar_1) (1+1.04*v+1.04^2*v^2+....+1.04^24*v^24)

The summation can be converted into an annuity due, but it's probably quicker to evaluate it using the sum of the first 25 terms of a geometric progression with first term 1 and common ratio 1.04*v.

These questions can appear on the exam - though not hugely frequently in this format. Compound increasing annuities are quite common.

 

Hi Mark,

Thanks for your reply, I was looking through the first principals explanation and was wondering why it becomes 80,000*(a-bar_1)?

Is this due to the continuous payment occurring for each year. So we use that to signify the continuous payment annuity?

I.e. 80,000*(a-bar_1) = the present value of 80,000 as a continuous payment?

That is what was confusing me.

Regards
Noel

 

Yaa you are right , a-bar 1 is used because the payments are made continously throught the year in each year , if payments were made at the end of year then we would have used the discount factor ' v '

 

divyam's right.

a-bar 1 is the present value at the start of the year, of continuous payments at a rate of 1 made over the year.

So the present value of the first year's payment in this example is:

80,000*a_bar 1.