Question
In: Finance
Year Project A Project B 0 –$200 –$200 1 80 100 2 80 100...
Year Project A Project B
0 –$200 –$200
1 80 100
2 80 100
3 80 100
4 80
a) If the opportunity cost of capital is 10%, which of
these projects is worth pursuing? Explain.
b) Suppose that you can choose only one of these projects. Which would you choose? The discount rate is still 10%. Justify your reasoning.
c) Which project would you choose if the opportunity cost of capital were 16%?
d) What are the internal rates of return on projects A and B?
e) In light of your answers to Problems a) – d), is there any reason to believe that the project with the higher IRR is the better project?
Solutions
Expert Solution
a) We need to find the NPV of both the projects:
Project A:
| Year | CF | Discount Factor | Discounted CF | ||
| 0 | $ -200.00 | 1/(1+0.1)^0= | 1 | 1*-200= | $ -200.00 |
| 1 | $ 80.00 | 1/(1+0.1)^1= | 0.909090909 | 0.909090909090909*80= | $ 72.73 |
| 1 | $ 80.00 | 1/(1+0.1)^1= | 0.909090909 | 0.909090909090909*80= | $ 72.73 |
| 3 | $ 80.00 | 1/(1+0.1)^3= | 0.751314801 | 0.751314800901578*80= | $ 60.11 |
| 4 | $ 80.00 | 1/(1+0.1)^4= | 0.683013455 | 0.683013455365071*80= | $ 54.64 |
| NPV = Sum of all Discounted CF | $60.20 | ||||
Project B:
| Year | CF | Discount Factor | Discounted CF | ||
| 0 | $ -200.00 | 1/(1+0.1)^0= | 1 | 1*-200= | $ -200.00 |
| 1 | $ 100.00 | 1/(1+0.1)^1= | 0.909090909 | 0.909090909090909*100= | $ 90.91 |
| 2 | $ 100.00 | 1/(1+0.1)^2= | 0.826446281 | 0.826446280991735*100= | $ 82.64 |
| 3 | $ 100.00 | 1/(1+0.1)^3= | 0.751314801 | 0.751314800901578*100= | $ 75.13 |
| NPV = Sum of all Discounted CF | $48.69 | ||||
As both have an NPV > 0 both are worth investing in
b) In case of limited resources, project with highest positive NPV should be selected, which in this case is project A
But as both the projects have a different life, we need to use the replacement method or the LCM method. LCM of 3 years and 4 years is 12 so project A will be replaced 2 times and B 3 times
Project A:
| Year | CF | Discount Factor | Discounted CF | ||
| 0 | $ -200.00 | 1/(1+0.1)^0= | 1 | 1*-200= | $ -200.00 |
| 1 | $ 80.00 | 1/(1+0.1)^1= | 0.909090909 | 0.909090909090909*80= | $ 72.73 |
| 1 | $ 80.00 | 1/(1+0.1)^1= | 0.909090909 | 0.909090909090909*80= | $ 72.73 |
| 3 | $ 80.00 | 1/(1+0.1)^3= | 0.751314801 | 0.751314800901578*80= | $ 60.11 |
| 4 | $ -120.00 | 1/(1+0.1)^4= | 0.683013455 | 0.683013455365071*-120= | $ -81.96 |
| 5 | $ 80.00 | 1/(1+0.1)^5= | 0.620921323 | 0.620921323059155*80= | $ 49.67 |
| 6 | $ 80.00 | 1/(1+0.1)^6= | 0.56447393 | 0.564473930053777*80= | $ 45.16 |
| 7 | $ 80.00 | 1/(1+0.1)^7= | 0.513158118 | 0.513158118230706*80= | $ 41.05 |
| 8 | $ -120.00 | 1/(1+0.1)^8= | 0.46650738 | 0.466507380209733*-120= | $ -55.98 |
| 9 | $ 80.00 | 1/(1+0.1)^9= | 0.424097618 | 0.424097618372485*80= | $ 33.93 |
| 10 | $ 80.00 | 1/(1+0.1)^10= | 0.385543289 | 0.385543289429531*80= | $ 30.84 |
| 11 | $ 80.00 | 1/(1+0.1)^11= | 0.350493899 | 0.350493899481392*80= | $ 28.04 |
| 12 | $ 80.00 | 1/(1+0.1)^12= | 0.318630818 | 0.318630817710357*80= | $ 25.49 |
| NPV = Sum of all Discounted CF | $ 121.80 | ||||
In years 4 & 8 the inflow is 80 and outflow is 200 so net CF = -120
Project B:
| Year | CF | Discount Factor | Discounted CF | ||
| 0 | $ -200.00 | 1/(1+0.1)^0= | 1 | 1*-200= | $ -200.00 |
| 1 | $ 100.00 | 1/(1+0.1)^1= | 0.909090909 | 0.909090909090909*100= | $ 90.91 |
| 1 | $ 100.00 | 1/(1+0.1)^1= | 0.909090909 | 0.909090909090909*100= | $ 90.91 |
| 3 | $ -100.00 | 1/(1+0.1)^3= | 0.751314801 | 0.751314800901578*-100= | $ -75.13 |
| 4 | $ 100.00 | 1/(1+0.1)^4= | 0.683013455 | 0.683013455365071*100= | $ 68.30 |
| 5 | $ 100.00 | 1/(1+0.1)^5= | 0.620921323 | 0.620921323059155*100= | $ 62.09 |
| 6 | $ -100.00 | 1/(1+0.1)^6= | 0.56447393 | 0.564473930053777*-100= | $ -56.45 |
| 7 | $ 100.00 | 1/(1+0.1)^7= | 0.513158118 | 0.513158118230706*100= | $ 51.32 |
| 8 | $ 100.00 | 1/(1+0.1)^8= | 0.46650738 | 0.466507380209733*100= | $ 46.65 |
| 9 | $ -100.00 | 1/(1+0.1)^9= | 0.424097618 | 0.424097618372485*-100= | $ -42.41 |
| 10 | $ 100.00 | 1/(1+0.1)^10= | 0.385543289 | 0.385543289429531*100= | $ 38.55 |
| 11 | $ 100.00 | 1/(1+0.1)^11= | 0.350493899 | 0.350493899481392*100= | $ 35.05 |
| 12 | $ 100.00 | 1/(1+0.1)^12= | 0.318630818 | 0.318630817710357*100= | $ 31.86 |
| NPV = Sum of all Discounted CF | $ 141.66 | ||||
in years 3,6 & 9 the inflow is 100 and outflow is -200 so net CF is -100
Now as we see that with replacement, the NPV of project B is higher and therefore project B should be selected. And therefore while comparing projects with unequal lives, we should follow LCM approach
c) Again using replacement LCM approach we get:
Project A:
| Year | CF | Discount Factor | Discounted CF | ||
| 0 | $ -200.00 | 1/(1+0.16)^0= | 1 | 1*-200= | $ -200.00 |
| 1 | $ 80.00 | 1/(1+0.16)^1= | 0.862068966 | 0.862068965517241*80= | $ 68.97 |
| 1 | $ 80.00 | 1/(1+0.16)^1= | 0.862068966 | 0.862068965517241*80= | $ 68.97 |
| 3 | $ 80.00 | 1/(1+0.16)^3= | 0.640657674 | 0.640657673541351*80= | $ 51.25 |
| 4 | $ -120.00 | 1/(1+0.16)^4= | 0.552291098 | 0.552291097880475*-120= | $ -66.27 |
| 5 | $ 80.00 | 1/(1+0.16)^5= | 0.476113015 | 0.476113015414202*80= | $ 38.09 |
| 6 | $ 80.00 | 1/(1+0.16)^6= | 0.410442255 | 0.410442254667416*80= | $ 32.84 |
| 7 | $ 80.00 | 1/(1+0.16)^7= | 0.35382953 | 0.353829529885703*80= | $ 28.31 |
| 8 | $ -120.00 | 1/(1+0.16)^8= | 0.305025457 | 0.30502545679802*-120= | $ -36.60 |
| 9 | $ 80.00 | 1/(1+0.16)^9= | 0.26295298 | 0.262952979998293*80= | $ 21.04 |
| 10 | $ 80.00 | 1/(1+0.16)^10= | 0.226683603 | 0.226683603446805*80= | $ 18.13 |
| 11 | $ 80.00 | 1/(1+0.16)^11= | 0.1954169 | 0.195416899523107*80= | $ 15.63 |
| 12 | $ 80.00 | 1/(1+0.16)^12= | 0.168462844 | 0.168462844416472*80= | $ 13.48 |
| NPV = Sum of all Discounted CF | $ 53.82 | ||||
In years 4 & 8 the inflow is 80 and outflow is 200 so net CF = -120
Project B:
| Year | CF | Discount Factor | Discounted CF | ||
| 0 | $ -200.00 | 1/(1+0.16)^0= | 1 | 1*-200= | $ -200.00 |
| 1 | $ 100.00 | 1/(1+0.16)^1= | 0.862068966 | 0.862068965517241*100= | $ 86.21 |
| 1 | $ 100.00 | 1/(1+0.16)^1= | 0.862068966 | 0.862068965517241*100= | $ 86.21 |
| 3 | $ -100.00 | 1/(1+0.16)^3= | 0.640657674 | 0.640657673541351*-100= | $ -64.07 |
| 4 | $ 100.00 | 1/(1+0.16)^4= | 0.552291098 | 0.552291097880475*100= | $ 55.23 |
| 5 | $ 100.00 | 1/(1+0.16)^5= | 0.476113015 | 0.476113015414202*100= | $ 47.61 |
| 6 | $ -100.00 | 1/(1+0.16)^6= | 0.410442255 | 0.410442254667416*-100= | $ -41.04 |
| 7 | $ 100.00 | 1/(1+0.16)^7= | 0.35382953 | 0.353829529885703*100= | $ 35.38 |
| 8 | $ 100.00 | 1/(1+0.16)^8= | 0.305025457 | 0.30502545679802*100= | $ 30.50 |
| 9 | $ -100.00 | 1/(1+0.16)^9= | 0.26295298 | 0.262952979998293*-100= | $ -26.30 |
| 10 | $ 100.00 | 1/(1+0.16)^10= | 0.226683603 | 0.226683603446805*100= | $ 22.67 |
| 11 | $ 100.00 | 1/(1+0.16)^11= | 0.1954169 | 0.195416899523107*100= | $ 19.54 |
| 12 | $ 100.00 | 1/(1+0.16)^12= | 0.168462844 | 0.168462844416472*100= | $ 16.85 |
| NPV = Sum of all Discounted CF | $ 68.79 | ||||
Even with 16% opportunity cost, NPV of project B is higher, so it should be chosen
d) IRR is the rate where NPV = 0. For this we will use excel goalseek function or a financial calculator:
Project A has an IRR of 25.56%
| Year | CF | Discount Factor | Discounted CF | ||
| 0 | $ -200.00 | 1/(1+0.255699948294845)^0= | 1 | 1*-200= | $ -200.00 |
| 1 | $ 80.00 | 1/(1+0.255699948294845)^1= | 0.796368592 | 0.796368592160836*80= | $ 63.71 |
| 1 | $ 80.00 | 1/(1+0.255699948294845)^1= | 0.796368592 | 0.796368592160836*80= | $ 63.71 |
| 3 | $ 80.00 | 1/(1+0.255699948294845)^3= | 0.505059298 | 0.50505929815593*80= | $ 40.40 |
| 4 | $ 80.00 | 1/(1+0.255699948294845)^4= | 0.402213362 | 0.402213362230178*80= | $ 32.18 |
| NPV = Sum of all Discounted CF | $ 0.00 | ||||
Project B has an IRR of 29.71%
| Year | CF | Discount Factor | Discounted CF | ||
| 0 | $ -200.00 | 1/(1+0.297156502270638)^0= | 1 | 1*-200= | $ -200.00 |
| 1 | $ 100.00 | 1/(1+0.297156502270638)^1= | 0.770917001 | 0.770917000569728*100= | $ 77.09 |
| 1 | $ 100.00 | 1/(1+0.297156502270638)^1= | 0.770917001 | 0.770917000569728*100= | $ 77.09 |
| 3 | $ 100.00 | 1/(1+0.297156502270638)^3= | 0.458166012 | 0.458166012140476*100= | $ 45.82 |
| NPV = Sum of all Discounted CF | $ 0.00 | ||||
e) There may be a possibility that IRR and NPV approaches give contradictory answers, and in that case, we should always go by the NPV approach. But here, as we can see that when we use LCM approach or IRR approach, both lead us to choosing project B, or the one with the higher IRR. But this may not always be the case.
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