Problem

[Anu Dhanuka]

As we all know exam has started, and we are not supposed to ask any particular question here....but to do if we face with some problem, and need to clarify it....
Please let me know, i am facing problem in some questiona...

[dmbfan41]

i'm pretty sure you can still ask questions, we just can't discuss the exam from this sitting until it's over.

[Anu Dhanuka]

thats cool...
my problem is in ques 14 of nov 05 paper...

Payments of X are made at the beginning of each year for 20 years. These payments
earn interest at the end of each year at an annual effective rate of 8%. The interest is
immediately reinvested at an annual effective rate of 6%. At the end of 20 years, the
accumulated value of the 20 payments and the reinvested interest is 5600.
Calculate X.

I dont knw what they have done with the reinvested amount...according to me it should be simple, like Xi invested for 20 yrs at 6%.
please if possible let me know the solution....
I am writing my paper on monday....

Thanks a lot...

[greeneggs]

You are investing x at the start of EACH year for 20 years. so the interest payments are increasing and reinvested. so you would have an accumulated increasing annuity, plus 20x at the end of 20 years.

[AganaBelea]

At the end of the first year, you have earned 0.08X in interest on your principal of X. You take out that interest, and then add another X to the principal, so at the end of year two you take out 0.08(2X) in interest, etc, until at the end of year 20, you take out 0.08(20X) in interest.

You take out all of these portions of interest and reinvest them at 6%. So:
At t = 1, you invest 0.08X
At t = 2, you invest 0.08(2X)
...
At t = 20, you invest 0.08(20X)
...at 6%.

The accumulated amounts of these portions of interest would accumulate to:

0.08(20X) + 0.08(19X)(1.06) + ... + 0.08(2X)(1.06^18) + 0.08(X)(1.06^19)

This accumulated amount, along with your 20 original contributions of X (and remember, you've removed ALL of the interest those contributions generated), combine to be:

0.08(20X) + 0.08(19X)(1.06) + ... + 0.08(2X)(1.06^18) + 0.08(X)(1.06^19) + 20X = 5600

[Anu Dhanuka]

I got it now, the mistake i was doin, is that wondering if we have used Xi of the first payment in the reinvestment, why we are using it again....but now i got, as the payments of X are done regularly, so at time 2 we have interest on 2X and not on only X....

Thanks a lot for the help....

At the end of the first year, you have earned 0.08X in interest on your principal of X. You take out that interest, and then add another X to the principal, so at the end of year two you take out 0.08(2X) in interest, etc, until at the end of year 20, you take out 0.08(20X) in interest.

You take out all of these portions of interest and reinvest them at 6%. So:
At t = 1, you invest 0.08X
At t = 2, you invest 0.08(2X)
...
At t = 20, you invest 0.08(20X)
...at 6%.

The accumulated amounts of these portions of interest would accumulate to:

0.08(20X) + 0.08(19X)(1.06) + ... + 0.08(2X)(1.06^18) + 0.08(X)(1.06^19)

This accumulated amount, along with your 20 original contributions of X (and remember, you've removed ALL of the interest those contributions generated), combine to be:

0.08(20X) + 0.08(19X)(1.06) + ... + 0.08(2X)(1.06^18) + 0.08(X)(1.06^19) + 20X = 5600