A person wishes to have $992354.70 after 40 years with 7% interest
compounded annually.

I believe this is done by taking the present value of an annuity. If
this is correct, please explain how this is done mathematically (IE
not using a financial calculator and not referring to tables for
needed values). If not, please explain the process keeping the above
in mind.

Please include necessary equations and work through them to show the
process. If fractions are used, write them using parentheses rather
than vertically.

Feel free to ask for clarification if you need more information to be
able to provide a helpful answer. Thanks..

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Request for Question Clarification by answerguru-ga on 23 Oct 2003 13:13 PDT

There seems to be some information missing here - it's not necessarily
an annuity unless the problem states that the person would like to
make equal payments each period over that 40 years. As it is stated,
it is possible to provide an amount that this person would need to
invest now in order to reach the given amount under these conditions.

Would you like me to show you this way of doing it?

answerguru-ga

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Clarification of Question by chrisn-ga on 23 Oct 2003 14:03 PDT

Equal payments (one per year) need to be made. I should have mentioned
that in the question text. Thanks for the question.

Hi chrisn-ga,

Here is your question again:

A person wishes to have $992354.70 after 40 years with 7% interest 
compounded annually. He needs to make equal payments each year.

Here is how you solve this type of problem:

This is a present value annuity problem - these types of annuities are
solved using the following formula:

PV annuity = C*((1 - (1/(1+r)^t))/r)

where:
C = Periodic cash payments
r = interest rate
t = number of periods

You are already given the PV annuity value, interest rate, and number
of periods, so we can solve for C:

$992354.70 = C*((1 - (1/(1+0.07)^40))/0.07)

C = $992354.70 / ((1 - (1/(1+0.07)^40))/0.07)
  = $992354.70 / (0.9332/0.07)
  = $992354.70 / 13.332
  = $74435.67

So this person will need to make annual payments of $$74435.67 in
order to have the desired amount in 40 years given an interest rate of
7%.

Hopefully this helps you understand the calculation behind this value.
If you have any problems understanding the information above please
let me know :)

Cheers!

answerguru-ga

Thank you. That is exactly what I was looking for.

I'm not a math whiz, but you might want to check the math again.

If you saved $74,435.67 for 40 years with 'no' interest, thats $2.97 million.

Did I miss something?

Yeah. The format, as far as how the information was presented, is
great. Something with the formula itself doesn't appear correct.
Consider this a request for clarification if you should come across
this question...otherwise it's my own fault for not checking the
results fully before closing the question.

You may want to try this online tool:
http://www.localok.com/savecalc.html

OK, I'm am feeling quite sheepish :)

I saw your mention of present value annuity in the question and
assumed that is what you wanted the calculation for - what you
actually need is the FUTURE value annuity.

The question answered in the orginal question was "how much should I
pay for an investment that gives an annual payment of 7% for the next
40 years". This is the inverse of the question you actually wanted
answered.

OK, so now for the future value annuity calculation:

FV annuity = C*(((1+r)^t - 1)/r)

C = FV annuity / (((1+r)^t - 1)/r)
  = $992354.70 / 199.635112
  = $4970.84

Hopefully that dissolves all the confusion :)

answerguru-ga