Engineering Economic Analysis

(^234) RATEOF RETURNANALYSIS
"1
SOLUTION.
The first step is to draw the cash flow diagram. Then counting the three sign changes is easy.
$30K $32K $34K $36K $40K $42K $44K $46K $48K
t t t t t t t t t
0-1-2-3-4-5-6-7-8-9-10
! -$221{~ JoK~ $38K
-$120K

With three sign changes, there may be 0, 1, 2, or 3 positive roots for the PW =0 equation.

Evaluati ng How Many Roots There Are

The number of sign changes tells us how many roots are possible-not how many roots

there are. Rather than covering the many mathematical approaches thatmaytell us if the

root will be unique, it is more useful to employ the power of the spreadsheet. A spreadsheet

can show us if a root is unique and the value of each root that exists.

The approach is simple. For any set of cash flows, graph the PW as a function of the

interest rate. We are interested in positive interest rates, so the graph usually starts ati=O.

SinceDescartes' ruleis based onx > 0, andx =(1+i)-1, sometimes the graph is started

close toi= -1. This will identify any negative values ofithat solve the equation.

If the root is unique, we're done. We've found the internal rate of return. If there are

multiple roots, then we know the project's PW at all interest rates-including the one our

organizationuses. Wecan alsouse the approach of the next section to findamodifiedinternal

rate of return.

The easiest way to build the spreadsheet is to use the NPV(i,values) function in Excel.

Remember that this function applies to cash flows in periods 1 toN, so that the cash flow

at time 0 must be added in separately.The following examplesuse a spreadsheet to answer

the question of how many roots each cash flow diagram in the last section has.

This project is representative of ones with a salvage cost. How many roots for the PW equation

exist?

$50K

t t t t t t

0-1~2-3-4-5-6-7

j '. .. $1

_ .$180K-- - = -=

• : .. :;::


• 
  

  ---