Why Net Present Value Leads to Better Investment Decisions Than Other
Criteria
1.
a. _{}
_{}
_{}
b. Payback_{A} = 1 year
Payback_{B} = 2 years
Payback_{C} = 4 years
c. A and B.
2. The discounted payback period is the number of periods a project must last in order to achieve a zero net present value. It is marginally preferable to the regular payback rule because it uses discounted cash flows, thereby overcoming the criticism that all cash flows prior to the cutoff date have equal weight. However, the discounted payback period still does not account for cash flows occurring after the cutoff date.
3. Book rate of return uses the accounting definition of income and investment (i.e., book value of assets). Both of these accounting concepts differ from cash flow measures. In addition, book rate of return does not recognize the time value of money. Hence, decisions based on book rate of return can, and often do, lead to choices that are unacceptable when analyzed on a net present value basis.
4. a. When using the IRR rule, the firm must still compare the IRR with the opportunity cost of capital. Thus, even with the IRR method, one must think about the appropriate discount rate.
b. Risky cash flows should be discounted at a higher rate than the rate used to discount less risky cash flows. Using the payback rule is equivalent to using the NPV rule with a zero discount rate for cash flows before the payback period and an infinite discount rate for cash flows thereafter.
5. In general, the discounted payback rule is slightly better than the regular payback rule. But, in this case, it might actually be worse: with the same cutoff period, fewer long‑lived investment projects will make the grade.
6.

r = 
17.44% 
0.00% 
10.00% 
15.00% 
20.00% 
25.00% 
45.27% 
Year 0 
3,000.00 
3,000.00 
3,000.00 
3,000.00 
3,000.00 
3,000.00 
3,000.00 
3,000.00 
Year 1 
3,500.00 
4,239.34 
3,500.00 
3,181.82 
3,043.48 
2,916.67 
2,800.00 
2,409.31 
Year 2 
4,000.00 
5,868.41 
4,000.00 
3,305.79 
3,024.57 
2,777.78 
2,560.00 
1,895.43 
Year 3 
4,000.00 
7,108.06 
4,000.00 
3,005.26 
2,630.06 
2,314.81 
2,048.00 
1,304.76 

PV = 
0.31 
500.00 
482.35 
437.99 
379.64 
312.00 
0.02 
The two IRRs for this project are (approximately): –17.44% and 45.27%. The NPV is positive between these two discount rates.
7. a. The figure on the next page was drawn from the following points:

Discount Rate 


0% 
10% 
20% 
NPV_{A} 
+20.00 
+4.13 
8.33 
NPV_{B} 
+40.00 
+5.18 
18.98 




b. From the graph, we can estimate the IRR of each project from the point where its line crosses the horizontal axis:
IRR_{A} = 13.1% and IRR_{B} = 11.9%
c. The company should accept Project A if its NPV is positive and higher than that of Project B; that is, the company should accept Project A if the discount rate is greater than 10.7 percent and less than 13.1 percent.
d. The cash flows for (B – A) are:
Therefore:

Discount Rate 


0% 
10% 
20% 
NPV_{BA} 
+20.00 
+1.05 
10.65 
IRR_{BA} = 10.7%
The company should accept Project A if the discount rate is greater than 10.7% and less than 13.1%. As shown in the graph, for these discount rates, the IRR for the incremental investment is less than the opportunity of cost of capital.
8. a. Because Project A requires a larger capital outlay, it is possible that Project A has both a lower IRR and a higher NPV than Project B. (In fact, NPV_{A} is greater than NPV_{B} for all discount rates less than 10 percent.) Because the goal is to maximize shareholder wealth, NPV is the correct criterion.
b. To use the IRR criterion for mutually exclusive projects, calculate the IRR for the incremental cash flows:

C_{0} 
C_{1} 
C_{2} 
IRR 
A  B 
200 
+110 
+121 
10% 
Because the IRR for the incremental cash flows exceeds the cost of capital, the additional investment in A is worthwhile.
c. _{}
_{}
9. Use incremental analysis:

C_{1} 
C_{2} 
C_{3} 

Current arrangement 
250,000 
250,000 
+650,000 

Extra shift 
550,000 
+650,000 
0 

Incremental flows 
300,000 
+900,000 
650,000 
The IRRs for the incremental flows are approximately 21.13 and 78.87 percent. If the cost of capital is between these rates, Titanic should work the extra shift.
10. The statement is true because more immediate cash flows will be discounted less than cash flows that are further into the future. Hence, projects with quick paybacks and low investments will be preferred on an IRR basis, even though longerterm projects might have larger NPVs.
11. a. _{}
_{}
b. Both projects have a Profitability Index greater than zero, and so both are acceptable projects. In order to choose between these projects, we must use incremental analysis. For the incremental cash flows:
_{}
The increment is thus an acceptable project, and so the larger project should be accepted, i.e., accept Project F. (Note that, in this case, the better project has the lower profitability index.)
12. Because there are three sign changes in the sequence of cash flows, we know that there can be as many as three internal rates of return. Using trial and error, graphical analysis, or solving analytically (the easiest way to solve for the IRR is with a spreadsheet program such as Excel), we can show that there is only one IRR, 5.24 percent.
A project with an IRR equal to 5.24 percent is not attractive when the opportunity cost of capital is 14 percent. (Alternatively, we can say that, with a discount rate of 14 percent, the project’s NPV is $2,443 so that the project is not attractive.)
13. Using the fact that Profitability Index = (Net Present Value/Investment), we find that:
Project 

Profitability Index 
1 

0.22 
2 

0.02 
3 

0.17 
4 

0.14 
5 

0.07 
6 

0.18 
7 

0.12 
Thus, given the budget of $1 million, the best the company can do is to accept Projects 1, 3, 4, and 6.
If the company accepted all positive NPV projects, the market value (compared to the market value under the budget limitation) would increase by the NPV of Project 5 and the NPV of Project 7: ($7,000 + $48,000) = $55,000. Thus, the budget limit costs the company $55,000 in terms of its market value.
14. Maximize: NPV = 6,700x_{W} + 9,000x_{X} + 0X_{Y}  1,500x_{Z}
subject to: 10,000x_{W} + 0x_{X} + 10,000x_{Y} + 15,000x_{Z} £ 20,000
10,000x_{W} + 20,000x_{X}  5,000x_{Y}  5,000x_{Z} £ 20,000
0x_{W}  5,000x_{X}  5,000x_{Y}  4,000x_{Z} £ 20,000
0 £ x_{W} £ 1
0 £ x_{X} £ 1
0 £ x_{Z} £ 1
Challenge Questions
1. The IRR is the discount rate which, when applied to a project’s cash flows, yields NPV = 0. Thus, it does not represent an opportunity cost. However, if each project’s cash flows could be invested at that project’s IRR, then the NPV of each project would be zero because the IRR would then be the opportunity cost of capital for each project. The discount rate used in an NPV calculation is the opportunity cost of capital. Therefore, it is true that the NPV rule does assume that cash flows are reinvested at the opportunity cost of capital.
2. a.
C_{0} 
= 3,000 
C_{0} 
= 3,000 
C_{1} 
= +3,500 
C_{1} 
= +3,500 
C_{2} 
= +4,000 
C_{2} + PV(C_{3} 
) = +4,000 – 3,571.43 = 428.57 
C_{3} 
= 4,000 
MIRR = 27.84% 
b. _{}
(1.12^{2})(x C_{1}) + (1.12)(x C_{2}) = C_{3}
(x)[(1.12^{2})(C_{1}) + (1.12)(C_{2})] = C_{3}
_{}
_{}
_{}
_{}
Now, find MIRR using either trial and error or the IRR function (on a financial calculator or Excel). We find that MIRR = 23.53%.
It is not clear that either of these modified IRRs is at all meaningful. Rather, these calculations seem to highlight the fact that MIRR really has no economic meaning.
3. A project with all positive cash flows has no IRR. For example:
C_{0} = 100
C_{1} = 100
C_{2} = 100
C_{3} = 100
4. Using Excel Spreadsheet Addin Linear Programming Module:
Optimized NPV = $13,450
with x_{W} = 1; x_{X} = 0.75; x_{Y} = 1 and x_{Z} = 0
If the financing available at t = 0 is $21,000:
Optimized NPV = $13,500
with x_{W} = 1; x_{X} = (23/30); x_{Y} = 1 and x_{Z} = (2/30)
Here, the shadow price for the constraint at t = 0 is $50, the increase in NPV for a $1,000 increase in financing available at t = 0.
In this case, the program viewed x_{Z} as a viable choice, even though the NPV of Project Z is negative. The reason for this result is that project Z provides a positive cash flow in periods 1 and 2.
If the financing available at t = 1 is $21,000:
Optimized NPV = $13,900
with x_{W} = 1; x_{X} = 0.8; x_{Y} = 1 and x_{Z} = 0
Hence, the shadow price of an additional $1,000 in t = 1 financing is $450.
5. a. The constraint in the second period would become:
30x_{A}  5x_{B}  5x_{C} + 40x_{D}  (10 10x_{A}  5x_{B}  5x_{C})(1 + r) £ 10
b. The constraint in the first period would become:
10x_{A} + 5x_{B} + 5x_{C} + 0x_{D} + cost of hiring & training £ 10