mceanswers,
The present worth can be found by solving this problem as a geometric
series
The basic equation is this:
Present Worth = a + ar + ar^2 + ar^3 + ... + ar^(n1) + ...
Where:
a = Initial Sum, such as the $10,000 in your example.
r = (1  Discount Rate) > 10% discount becomes r = 90%
n = number of payments, including the initial one
For a value of "r" less than one, there is a simple formula that can
tell you what the sum (the Discounted Present Worth) is:
Sn = a * ( (1r^n) / (1r) )
Going with your example.... we start out with $10,000, have a total
term of 3 payments, and use a discount rate of 10%. Using the above
equation, the value is:
Sn = $10,000 * 2.71 = $27,100
Using common logic, like the initial example you used.
Present Worth = $10,000 + 0.9*$10,000 + 0.9*$9000 = $27,100
I'm not sure what you are using this for, but this formula is useful
for determining whether to make an investment based upon it's present
earning power ("a" in this equation) and an external force like
inflation (represented by "1r").
Assuming that you have an infinite period of time in which to collect
this income, the total present worth can be found by the simple
formula:
S = a / (1  r)
For the same numbers described above ($10,000; 10%), the DPW is:
$100,000
However, if you are clever, you will notice that in the above equation
you can simply substitute the raw discount rate for the value of "1 
r". So, just take the present income value ($10,000) and divide it by
the discount rate (10%) and you will get $100,000. To tell you what
you will have after a specified number of periods, use the first
equation I wrote (Sn).
Don't hesitate to ask any questions!
krobertga 
Request for Answer Clarification by
mceanswersga
on
15 Sep 2003 09:29 PDT
KRobert,
I don't we've yet arrived at the answer. This should be an equation
with 4 variables. We currently have just 3:
a = Initial Sum, such as the $10,000 in your example.
r = (1  Discount Rate) > 10% discount becomes r = 90%
n = number of payments, including the initial one
We need to add a variable for the decline rate:
d = annual percentage decline
So, just to set up another example question, let's say that I am
trying to determine the DPW for the following situation:
a = $30,000
r = 12%
n = 17
d = 6.5%
such that year 1 payment is $30,000, yr 2 = 28,500, yr 3 = 26,227, etc
out to 17 years. What is the 125DPW of this series of payments?
thanks, MCE

Request for Answer Clarification by
mceanswersga
on
15 Sep 2003 09:31 PDT
KRobert,
Please excuse the typos in the previous question. It should have read:
I don't think we've yet arrived at the answer. This should be an equation
with 4 variables. We currently have just 3:
a = Initial Sum, such as the $10,000 in your example.
r = (1  Discount Rate) > 10% discount becomes r = 90%
n = number of payments, including the initial one
We need to add a variable for the decline rate:
d = annual percentage decline
So, just to set up another example question, let's say that I am
trying to determine the DPW for the following situation:
a = $30,000
r = 12%
n = 17
d = 6.5%
such that year 1 payment is $30,000, yr 2 = 28,500, yr 3 = 26,227, etc
out to 17 years. What is the 12%DPW of this series of payments?
thanks, MCE

Clarification of Answer by
krobertga
on
15 Sep 2003 11:04 PDT
mceanswersga,
I want to make sure that I understand you....
This income stream is declining at 6.5% per year, but is also earning
12% per year? Is that right?
krobertga

Request for Answer Clarification by
mceanswersga
on
15 Sep 2003 12:21 PDT
To clarify:
This income stream is declining at 6.5% per year.
What is the X DPW of this stream of income? (where X is a variable, in
this case we shall say 12% per year)
thanks, MCE

Request for Answer Clarification by
mceanswersga
on
15 Sep 2003 12:24 PDT
krobertga,
Sorry, but to clarify again:
This income stream is declining at 6.5% per year.
What would I pay for this stream of income if I want to realize a 12%
annualized rate of return, which, according to my understanding is the
same as asking "what is the 12% DPW of this stream of income?
thanks, MCE

Clarification of Answer by
krobertga
on
15 Sep 2003 16:10 PDT
mceanswersga,
To be honest, I've never heard it quite put that way. One more
question though, and we should have an answer:
Let's take the one and two year models:
If we have $30,000 in year one, and you wanted to purchase this for an
annual return of 12%, you would pay $26,786. Does that agree with your
calculation?
For two years, we have income of $30,000 in year one and $28050 in
year two (a 6.5% decline).
So, we have the original amount, $26,786, and a new amount $22,361.
So, you would be willing to pay $49,147 for income of $30,000 and
$28,050 ($58,050 total), spaced one year apart, for a return of 12%
per annum on your investment. (This assumes that you pay for the
initial $30,000 a year in advance).
Is this correct, can you follow the logic?
krobertga
