Pretend I receive $50,000 cash returns at the end of each year for 3
years and then $75,000 cash returns at the end of each year for the
following 2 years.

At an 8 percent interest rate, what is the PRESENT VALUE of these cash returns?

NOTE: You will (probably) need a Present Value table for this one!

You must provide all the working and formulas (preferably with an
explantion / definition) as I would really like to learn to do this
myself.

It is VERY important to me that the answer is ONE HUNDRED PERCENT correct!

Thank you!

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Request for Question Clarification by omnivorous-ga on 22 Sep 2004 11:34 PDT

Are you comfortable using Microsoft Excel to see these calculations performed?

Best regards,

Omnivorous-GA

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Clarification of Question by thanksmate-ga on 22 Sep 2004 11:50 PDT

Sorry I would not like to use Excel.

Hi thanksmate!!


First of all I will start with some definitions:

The time value of money:
"... if we deposit money in a bank account we will receive interest.
Because of this, we prefer to receive money today rather than in the
future. Money we receive today is more valuable to us than money
received in the future by the amount of interest we can earn with the
money. This is referred to as the time value of money. It is the
change in purchasing power of money over time.
It also takes into account default risk and inflation. $100 today is a
sure thing and can be enjoyed now. In 5 years that money could be
worthless or not returned to the investor.
To adjust for this time value, we use two simple formulae. The present
value formula is used to discount future money streams: that is, to
convert future amounts to their equivalent present day amounts."
From "Time value of money - Wikipedia":
http://en.wikipedia.org/wiki/Time_value_of_money


Present Value:
"The present value of a future transaction is the nominal amount of
money to change hands, adjusted to account for the time value of
money. A given amount of money is almost always more valuable sooner
than later, so present values are generally smaller than corresponding
future values."
From "Present value - Wikipedia":
http://en.wikipedia.org/wiki/Present_value


The Present Value (PV) of a future cash flow (CF) received after N
years in a market where the annual interest rate is r can be
calculated as follows:

         CF
PV = -----------
      (1 + r)^N


To see a more detailed explanations about this topic please see:
"The value of time":
http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node3.html

"The Present Value":
http://garnet.acns.fsu.edu/~ppeters/fin3403/readings/tvm/tvm2.html


Now we have all that we need to find the total Present Value of the
five future cash returns:

We just need to calculate each Present Value and then add all to find the total.
So we will have:

        CF1          CF2         CF3         CF4         CF5
PV  = --------- + --------- + --------- + --------- + ---------
      (1 + r)^1   (1 + r)^2   (1 + r)^3   (1 + r)^4   (1 + r)^5

where:

CF1 = $50,000
CF2 = $50,000
CF3 = $50,000
CF4 = $75,000
CF5 = $75,000

and 

r = 0.08


Then:

PV = 50000/1.08 + 50000/1.1664 + 50000/1.259712 + 75000/1.36048896 +
     + 75000/1.4693280768 =

PV = $235,025.83

-----------------------------

Note: this result was tested in an Excel Spreadsheet using the NPV
function and the following online calculator (here I calculate each PV
and then added all in my calculator):
"Present Value": (see at the bottom of the page)
http://www.prenhall.com/divisions/bp/app/cfldemo/TVM/PresentValue.html

Note 2: NPV function in Excel calculate the Present value of a series
of future cash flows. NPV (Net Present Value) function was incorrectly
defined by someone in the past, and has never been corrected, so
please do not use the PV (Present Value) function of Excel for this
calculation. See the following page for reference:
"Calculating NPV in Excel":
In Excel to obtain the Present Value of Cash INFLOWS (like this
question problem) is just
=NPV(rate, value1, value2,....,valueN)
http://www2.hpu.edu/mlane/ExcelNPV.htm
 
---------------------------------


I hope that this helps you. Please feel free to request for an answer
clarification if you need further explanations or if you find
something unclear.


Best regards.
livioflores-ga

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Request for Answer Clarification by thanksmate-ga on 23 Sep 2004 06:10 PDT

Thank you everyone!

Livioflores-ga, there is 3 / 4 different answers but only one can be
correct. Can you please confirm which answer is correct and why the
other answers are wrong?

Thanks again!

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Clarification of Answer by livioflores-ga on 23 Sep 2004 08:59 PDT

Hi!!

As I told you my answer was be double tested, by an Excel function and
by an online calculator, so I am pretty sure that my answer is
correct.

I used directly the definition of present value and the formula showed
in all texts and pages for cases like this.

Note that the first comment start at the right way for the $50,000
payments, but for the two $75,000 payments the calculation is for cash
flows in years 1 and 2 not years 4 and 5 and then do an strange use of
the PV formula trying to find the correct value of PV for this
payments. The correct second part is:
75,000/(1.08)^4 + 75,000/(1.08)^5 = 106,170.98
The sum is: $128,854.85 + $106,170.98 = $235,025.83


For the second comment I do not understand which method he/she is
trying to use, but he/she do calculations for years 5 and 6!! Year 6
not exist for this problem.


If you want to check this solution by yourself using Excel do the following:
Input the cash flows series:
A1: 50000    B1: =NPV(0.08;A1:A5)
A2: 50000
A3: 50000
A4: 75000
A5: 75000

The cell B1 will show the Present Value of all cash returns. 

I hope that this clarifies the problem. If not please let me know and
I will gladly give you further assistance.


Regards.
livioflores-ga

Thank you! Well done :-)

The present value of that series of cash flows at 8% is $227,161.39. 
I calculated that by first determining the present value of three
$50,000 payments discounted back three years at 8% using the "Present
Worth Value of Cash Flow Series" formula at
http://www.wheatworks.com/formula.htm.  So you get 50,000/(1.08)^1 +
50,000/(1.08)^2 + 50,000/(1.08)^3.  The summation of that series gives
$128,854.85 as the PV for the first three cash flows at 8%.

I then found the PV at the start of year four for the next two $75K
flows using the same formula, i.e, 75,000/(1.08)^1 + 75,000/(1.08)^ to
get $133,744.86, which I then discounted back four years as a lump sum
at 8% using the "Present Value of a Single Sum" formula at
http://www.wheatworks.com/formula.htm.  So you take $133,744.86 (the
FV in year four) and divide by (1.08)^4 to get $98,306.46.  Then you
add the two PV's ($128,854.85 + $98,306.54) and get $227,161.39.

I invite a Researcher to check me, but I am pretty sure that's how you do it.

The answer is quite simple = the present value is $240,204.37

Let's begin with the last half of the equation, 2 payments of $75,000
in year 5 and 6. Simply bring $156,000 dollars (75,000 x 8% + 75,000)
back 6 years less the 8% per year. $156,000(.92)(.92)(.92)(.92)(.92)=
$102,816.72 - This is the present value of the last 2 75,000 payments.

Now look at the first half of the equation-

Year 1 payment= $50,000(1.08)(1.08)= $58,320
Year 2 payment= $50,000(1.08)= $54,000
Year 3 payment= $50,000 = $50,000

Total is $162,320 at year 3 end, now bring this total back to year 1
less interest = $162,320(.92)(.92) = $137,38765

Now toal both equations- $137,38765 + $102,816.72 = $240,204.37

The solution purposed by the first respondent is incorrect - try
entering $50,000 - 8% = $46,000

Just to explain my answer, I thought that the cash flows all had to be
equal in amount to use the formula that livioflores used (and that I
used for the first three cash flows).  Thus, I found the PV of the
last two flows at the beginning of year four, then discounted that
amount back as a lump sum to the beginning of year one.  The "Present
Worth Value of Cash Flow Series" formula at
http://www.wheatworks.com/formula.htm is the correct one to use,
however.  Just use whatever the amount is for that particular cash
flow for "CF" in that formula.