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An extract from College Teaching Methods & Styles Journal – September 2008, Volume 4, Number 9, Finding Multiple Internal Rates Of Return For A Project With Non-Conventional Cash Flows: Utilizing Popular Financial/Graphing Calculators And Spreadsheet Software {pgs. 36-37}

3. UTILIZING FINANCIAL CALCULATORS TO FIND MULTIPLE IRRS

3.1. HP 17bII+ Financial Calculator


There are two ways to find the multiple IRRs.

Method I: Using the function keys

First, input the cash flows stream. Turn on HP 17bII+ and choose FIN and then choose CFLO.

FLOW(0)=? Input 580 (the initial cost) and press +/- to be -580. Press INPUT. FLOW(1)=? Input 530 (the first stage cash flows) and press INPUT.
#TIMES(1)=1 Input 3 (there are three years for the first stage) and press INPUT.
FLOW(2)=? Input 1080 (the second stage cash flow) and press +/- to be -1,080. Press INPUT.
#TIMES(2)=1 Just leave it there (there is only one year for the second stage). Choose EXIT and CALC and IRR% and then the following result appears.

MANY/NO SOLUTIONS; KEY
IN GUESS; [STO] {IRR%}

The calculator perceives there is more than one IRR. Do the following steps to find two IRRs.

Input 10 (or input a number that is closer to the first IRR compared to the second IRR5) and press STO and choose IRR%. Then after a few seconds of trial-and-error process, IRR%=9.8966 shows up if four decimal places are set. So, the first IRR = 9.8966%. To find the second IRR, repeat the same procedure again.
Choose IRR% and the following result appears on the screen again:

MANY/NO SOLUTIONS; KEY
IN GUESS; [STO] {IRR%}

Input 32 (or input a number that is closer to the second IRR compared to the first IRR6) and press STO and choose IRR%. Then after a few seconds of trial-and-error process, IRR%=32.1866 is displayed. So, the second IRR = 32.1866%.

Method II: Using equation solver

HP 17bII+ has the equation solver function. Input the whole equation first and use the solver function to solve for the equation. Select SOLVE and then select NEW. After that, TYPE EQUATION; [INPUT] is displayed. Type the following equation.

NPV=-580+530÷(1+K)+530÷((1+K)^2)+530÷((1+K)^3)-1080÷((1+K)^4)

Press EXIT twice, then “SAVE THIS EQUATION” is shown. Select YES and select CALC. Based on the graph, two IRRs are around 10% (0.1) and 32% (0.32). To solve for K that makes NPV = 0, input a number for K to compute NPV first. Then, change NPV to 0 and compute K again. The calculator will base on the previously inputted number for K and use trial-and-error process to get the accurate value of K, which makes NPV = 0.

To find the first IRR, input 0.1 and select K (K=0.1). Then select NPV and “NPV=0.377023” is displayed (Assume 6 decimal places are used). Input 0 and select NPV(to make NPV=0) and then select K. After a few seconds, K=0.098966 is displayed. The first IRR is 9.8966%.

To find the second IRR, input 0.32 and select K (K=0.32). Then select NPV and “NPV=0.395111” is displayed. Input 0 and select NPV(to make NPV=0) and then select K. After a few seconds, K=0.321866 is displayed. The second IRR is 32.1866%.

BEST!
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