NPV/IRR. A new computer system will require an initial outlay of
$20,000 but it will increase the firm?s cash flows by $4,000 a year
for each of the next 8 years. Is the system worth installing if the
required rate of return is 9 percent? What if it is 14 percent? How
high can the discount rate be before you would reject the project?

Buffcode --

Prof. Ramana Sonti notes in the presentation below that ?"an easy
way of thinking about IRR is to define it as the rate that forces the
NPV of the project to zero.?

Kent Sate University 
"Intermediate Financial Management," (Ramana Sonti)				
http://business.kent.edu/courses/spring02/Fin/36054/lecture_notes/chapter6.htm

The easiest way to do this is to set up a spreadsheet with your two
NPVs to see if both 9% and 14% are positive.  Note that normal
assumptions would include depreciation and tax effects but none are
assumed in this problem ?
http://www.mooneyevents.com/npvirr.xls

As you can see it?s positive at the lower rate of return (a GO at 9%)
and negative at 14% (a NO GO), so we have to seek a number at which
NPV is zero ? or very close to zero.  I?ve set up a test at 12% in the
spreadsheet and you can change the assumptions to get at the IRR.

At returns of 12% it?s still negative.?so you can try 11.9% (still
negative) and then 11.8% (positive, but only $4).  You?ve got your IRR
? unless you want to try more decimal places.  If you do, don?t forget
to copy the IRR test factor across the whole row 7 . . .

You can also use the Excel IRR functions -- but they work the same
way, with iterative calculations.


Google search strategy:
NPV + IRR + calculation

Best regards,

Omnivorous-GA

---

Request for Answer Clarification by buffcode-ga on 04 Jun 2005 07:19 PDT

It is possible to show me the steps without using Excel?

---

Clarification of Answer by omnivorous-ga on 04 Jun 2005 09:12 PDT

Buffcode ?

Here are the steps for calculating NPVs.  I?ll show the fundamental
calculations but will leave the details to the spreadsheet, as there
are at least 60 calculations (and that?s if you guess the IRR
correctly).

NPV 9%/14%
==============

1.  First set up your cash flows.  The convention is to use ?year 0?
as the up-front investments because they happen today, then returns
are typically calculated at the end of year 1 for simplicity.

Year 0: -$20,000
Year 1: $4,000
Year 2: $4,000
. 
.
.
Year 8: $4,000

2.  Next we figure out the NPV factor.  This is usually what confuses
early finance students ? but setting up the NPV factor is the easiest
way to check calculations when you?re doing a complex problem.  (Note
that there are other ways to calculate NPV that are mathematically
equivalent, such as dividing your cash flows by the compound interest
rate directly, but they mask any errors that you might make in the
process.)

Year 0 has no discount, so the NPV factor is one.  
It costs us 9% (or 14%) for year 1, so it?s 1/1.09 = 0.917431 or 1/1.14 = 0.877183.

Year 2 it?s 1/(1.09)^2 = 0.84168 or 1/(1.14)^2 = 0.769468
Year 3 it?s 1/(1.09)^3 = 0.772184 or 1/(1.14)^3 = 0.674972

 3.  Multiple each year?s cash flow by the NPV factor to get the
?discounted cash flow? for each year.  This is the line that I?ve
termed ?NPV@9%? or ?NPV@14%? or even ?IRR test return? on the
spreadsheet.

Check the line for errors ? by making sure that our $4,000 decreases
each year as the impact of discounting increases.

4.  Add the total of the discounted cash flows.  At 9% our positive
cash flows overwhelm the -$20,000 investment and produce an discounted
cash flow or NPV of $2,139.  That says the project is a ?Go.?

However, at 14% the cost of money is too high and our discounted cash
flows don?t produce a positive return.  Instead it?s negative, at
-$1,445.  It would take at least another 2 years of like returns to
turn the project positive at 14%.


IRR
====

We also know from the previous work that the point at which IRR is
zero is somewhere between 9% and 14%

I?ve set up a starting point of 12% for the calculations on the
spreadsheet.  Cash flows don?t change ? we simply have to try new
?rates of return? that get us to zero for the total NPV.

This is actually the hard part ? even Microsoft Excel does this
iteratively, trying rates until it get close to zero for the total
NPV.

At 12% you can see the NPV factors:
Year 0: 1.00
Year 1: 0.892857
Year 2: 0.797194
.
.
.
Year 8: 0.452349

At 12% the project still has a total of discounted cash flows of
-$129.  We?re getting close to zero but aren?t there yet.  You?ll want
to LOWER the rate of return until you get to the smallest possible
number that?s positive.

11.9% produces an NPV of -$63.
11.8% produces an NPV of $4 ? a positive number.
Can you get closer to zero?  Yes, but only if you go to more decimal
places ? 11.805% will deliver an NPV of zero with rounding.

Please let me know if there?s any part of this portion of the Answer
that?s unclear.

Best regards,

Omnivorous-GA