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Internal rate of return and modified internal rate of return. Quark Industries has three potential projects,...
Internal rate of return and modified internal rate of return.
Quark Industries has three potential projects, all with an initial cost of $1,600,000.
Given the discount rate and the future cash flow of each project in the following table,
Cash Flow Project M Project N Project O
Year 1 $400,000 $500,000 $900,000
Year 2 $400,000 $500,000 $700,000
Year 3 $400,000 $500,000 $500,000
Year 4 $400,000 $500,000 $300,000
Year 5 $400,000 $500,000 $100,000
Discount rate 9% 14% 17%
Solutions
Expert Solution
Computation of IRR using trial and error method:
Project M:
Let’s compute NPV of project M using discount rate of 7 %:
|
Year |
Cash Flow (CM) |
PV Factor Computation |
PV Factor @ 7 % (F) |
PV (CM x F) |
|
0 |
-$1,600,000 |
1/ (1+0.07) ^0 |
1 |
-$1,600,000 |
|
1 |
$400,000 |
1/ (1+0.07) ^1 |
0.934579439 |
$373,831.77570 |
|
2 |
$400,000 |
1/ (1+0.07) ^2 |
0.873438728 |
$349,375.49131 |
|
3 |
$400,000 |
1/ (1+0.07) ^3 |
0.816297877 |
$326,519.15076 |
|
4 |
$400,000 |
1/ (1+0.07) ^4 |
0.762895212 |
$305,158.08482 |
|
5 |
$400,000 |
1/ (1+0.07) ^5 |
0.712986179 |
$285,194.47179 |
|
NPV M1 |
$40,078.97438 |
As NPV is positive let’s compute NPV at discount rate of 8 %.
|
Year |
Cash Flow (CM) |
PV Factor Computation |
PV Factor @ 8 % (F) |
PV (CM x F) |
|
0 |
-$1,600,000 |
1/ (1+0.08) ^0 |
1 |
-$1,600,000 |
|
1 |
$400,000 |
1/ (1+0.08) ^1 |
0.925925926 |
$370,370.37037 |
|
2 |
$400,000 |
1/ (1+0.08) ^2 |
0.85733882 |
$342,935.52812 |
|
3 |
$400,000 |
1/ (1+0.08) ^3 |
0.793832241 |
$317,532.89641 |
|
4 |
$400,000 |
1/ (1+0.08) ^4 |
0.735029853 |
$294,011.94112 |
|
5 |
$400,000 |
1/ (1+0.08) ^5 |
0.680583197 |
$272,233.27881 |
|
NPV M2 |
-$2,915.985169 |
IRR M = R1 + [NPV M1 x (R2 – R1)/ (NPV M1 – NPV M2)]
= 7 % + [$40,078.97438 x (8 % - 7 %)]/ [$40,078.97438 – (-$2,915.985169)]
= 7 % + ($40,078.97438 x 1 %)/ ($40,078.97438 + $2,915.985169)
= 7 % + ($400.7897438 / $42,994.95955)
= 7 % + 0.009321784
= 7 % + 0.9321784 %
= 7.93 %
Project N:
Let’s compute NPV of project N using discount rate of 16 %:
|
Year |
Cash Flow (CN) |
PV Factor Computation |
PV Factor @ 16 % (F) |
PV (CN x F) |
|
0 |
-$1,600,000 |
1/ (1+0.16) ^0 |
1 |
-$1,600,000 |
|
1 |
$500,000 |
1/ (1+0.16) ^1 |
0.8620689655172 |
431034.48276 |
|
2 |
$500,000 |
1/ (1+0.16) ^2 |
0.7431629013080 |
371581.45065 |
|
3 |
$500,000 |
1/ (1+0.16) ^3 |
0.6406576735414 |
320328.83677 |
|
4 |
$500,000 |
1/ (1+0.16) ^4 |
0.5522910978805 |
276145.54894 |
|
5 |
$500,000 |
1/ (1+0.16) ^5 |
0.4761130154142 |
238056.50771 |
|
NPV N1 |
$37,146.82683 |
As NPV is positive let’s compute NPV at discount rate of 17 %.
|
Year |
Cash Flow (CN) |
PV Factor Computation |
PV Factor @ 17 % (F) |
PV (CN x F) |
|
0 |
-$1,600,000 |
1/ (1+0.17) ^0 |
1 |
-$1,600,000 |
|
1 |
$500,000 |
1/ (1+0.17) ^1 |
0.93457943925234 |
427,350.42735 |
|
2 |
$500,000 |
1/ (1+0.17) ^2 |
0.87343872827321 |
365,256.77551 |
|
3 |
$500,000 |
1/ (1+0.17) ^3 |
0.81629787689085 |
312,185.27822 |
|
4 |
$500,000 |
1/ (1+0.17) ^4 |
0.76289521204753 |
266,825.02412 |
|
5 |
$500,000 |
1/ (1+0.17) ^5 |
0.71298617948367 |
228,055.57617 |
|
NPV N2 |
-$326.91864 |
IRR N = R1 + [NPV N1 x (R2 – R1)/ (NPV N1 – NPV N2)]
= 16 % + [$37,146.82683 x (17 % - 16 %)]/ [$37,146.82683 – (-$326.91864)]
= 16% + ($37,146.82683 x 1 %)/ ($37,146.82683+ $326.91864)
= 16 % + ($371.4682683/ $37,473.74547)
= 16 % + 0.009912761
= 16% + 0.9912761%
= 16.93 %
Project O:
Let’s compute NPV of project O using discount rate of 24 %:
|
Year |
Cash Flow (CO) |
PV Factor Computation |
PV Factor @ 24 % (F) |
PV (CO x F) |
|
0 |
-$1,600,000 |
1/ (1+0.24) ^0 |
1 |
-$1,600,000.00 |
|
1 |
900,000 |
1/ (1+0.24) ^1 |
0.8064516129032 |
725,806.45161 |
|
2 |
700,000 |
1/ (1+0.24) ^2 |
0.6503642039542 |
455,254.94277 |
|
3 |
500,000 |
1/ (1+0.24) ^3 |
0.5244872612534 |
262,243.63063 |
|
4 |
300,000 |
1/ (1+0.24) ^4 |
0.4229735977850 |
126,892.07934 |
|
5 |
100,000 |
1/ (1+0.24) ^5 |
0.3411077401492 |
34,110.77401 |
|
NPV O1 |
$4,307.87836 |
As NPV is positive let’s compute NPV at discount rate of 25 %.
|
Year |
Cash Flow (CO) |
PV Factor Computation |
PV Factor @ 25 % (F) |
PV (CO x F) |
|
0 |
-$1,600,000 |
1/ (1+0.25) ^0 |
1 |
-$1,600,000 |
|
1 |
900,000 |
1/ (1+0.25) ^1 |
0.800000 |
720,000 |
|
2 |
700,000 |
1/ (1+0.25) ^2 |
0.640000 |
448,000 |
|
3 |
500,000 |
1/ (1+0.25) ^3 |
0.512000 |
256,000 |
|
4 |
300,000 |
1/ (1+0.25) ^4 |
0.409600 |
122,880 |
|
5 |
100,000 |
1/ (1+0.25) ^5 |
0.327680 |
32,768 |
|
NPV O2 |
-$20,352 |
IRR O = R1 + [NPV O1 x (R2 – R1)/ (NPV O1 – NPV O2)]
= 24 % + [$4,307.87836 x (25 % - 24 %)]/ [$4,307.87836– (-$20,352)]
= 24 % + ($4,307.87836 x 1 %)/ ($4,307.87836 + $20,352)
= 24 % + ($43.0787836/ $24659.87836)
= 24 % + 0.001746918
= 24 % + 0.1746918 %
= 24.17 %
Computation of MIRR:
MIRR = n√ (Future value of positive cash flow/Present value of negative cash flow) – 1
Project M:
|
Year |
Cash Flow (CM) |
Computation of FV Factor |
FV Factor @ 9 % (F) |
FV (CM x F) |
|
1 |
$400,000 |
(1+0.09) ^4 |
1.41158161 |
564632.644 |
|
2 |
$400,000 |
(1+0.09) ^3 |
1.295029 |
518011.600 |
|
3 |
$400,000 |
(1+0.09) ^2 |
1.1881 |
475240.000 |
|
4 |
$400,000 |
(1+0.09) ^1 |
1.09 |
436000.000 |
|
5 |
$400,000 |
(1+0.09) ^0 |
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