part_2_tkm0844_11e_im_ch11

©2011 Pearson Education, Inc. Publishing as Prentice Hall

Chapter 11 Investment Decision Criteria

11-1. Dowling Sportswear has the following project:

initial cash outlay $5,000,000

net cash inflow per year $1,000,000

#of years 8

===

The NPV of this project depends on the rate at which we discount the cash flows, as we can see from equation 11-1:

1 20 1 2

NPV ... .(1 ) (1 ) (1 )

n

n

CFCF CFCF

k k k= + + + +

+ + +

A. For example, if we use k = 9%, we have:

NPV = −$3,000,000 + 1 2 8

$1,000,000 $1,000,000 $1,000,000...

(1.09) (1.09) (1.09)+ + + = $534,819.

B.–D. Using the other discount rates given in the problem, plus two others, we have:

rate NPV0% $3,000,0009% $534,819

11% $146,12311.815% $0

13% ($201,230)15% ($512,678)

Solutions to End of Chapter Problems—Chapter 11 283


11.815%

($1,000,000)

($500,000)

$0

$500,000

$1,000,000

$1,500,000

$2,000,000

$2,500,000

$3,000,000

$3,500,000

0% 2% 4% 6% 8% 10% 12% 14% 16%

NP

V

Note that as the discount rate rises, NPV falls, as we can see in the graph to the left. At 0%, the NPV is simply the sum of the cash flows. From there, as k rises, the NPV falls, becoming negative once we pass 11.815%. At 11.815%, the NPV is zero—this, then, is the IRR.

11-2. Carson Trucking has the following project:

initial cash outlay $10,000,000

net cash inflow per yrs1 7 $2,500,000

net cash inflow per year, yr 8 $3,500,000

# of years 8

=− =

==

A. The NPV of this project depends on the rate at which we discount the cash flows, as we can see from equation 11-1:

1 20 1 2

NPV ...(1 ) (1 ) (1 )

n

n

CFCF CFCF

k k k= + + + +

+ + +.

For example, if we use k = 9%, we have:

1 2 7 8

$2,500,000 $2,500,000 $2,500,000 $3,500,000NPV $10,000,000

(1.09) (1.09) (1.09) (1.09)

$4,338,914.

= − + + + + +

=

K

284 Titman/Keown/Martin • Financial Management, Eleventh Edition


B.–D. Using the other discount rates given in the problem, plus two others, we have:

rate NPV0% $11,000,0009% $4,338,914

11% $3,299,23313% $2,373,08615% $1,545,206

19.429% ($0)

19.429%

($2,000,000)

$0

$2,000,000

$4,000,000

$6,000,000

$8,000,000

$10,000,000

$12,000,000

0% 5% 10% 15% 20%

NP

V

Note that as the discount rate rises, NPV falls, as we can see in the graph above. At 0%, the NPV is simply the sum of the cash flows. From there, as k rises, the NPV falls, becoming negative once we pass 19.429%. At 19.429%, the NPV is zero—this, then, is the IRR.

11-3. A. Big Steve’s is considering buying a stamping machine. This machine would cost Big Steve’s $100,000, but would generate cash flows of $18,000/year for 10 years.

If Big Steve’s discount rate is 10%, then we can find the project’s NPV using equation 11–1:

1 20 1 2

1 2 9 10

NPV(1 ) (1 ) (1 )

$18,000 $18,000 $18,000 $18,000$100,000

(1.10) (1.10) (1.10) (1.10)

$10,602.

n

n

CFCF CFCF

k k k= = + + +

+ + +

= − + + + + +

=

K

K

Since the NPV is positive in this case, Big Steve would want to accept the project.



B. If Big Steve’s discount rate is 15%, however, we have:

1 2 9 10

$18,000 $18,000 $18,000 $18,000NPV $100,000 ...

(1.15) (1.15) (1.15) (1.15)

$9662.

= − + + + + +

= −

Thus at 15%, the project is not acceptable, since its NPV is negative.

C. We know that NPV falls as the discount rate rises. We now also know, for this project, that the rate that makes the NPV equal zero (that is, the IRR) lies between 10% (where NPV is positive) and 15% (where NPV is negative). We can solve for IRR using equation 11-4:

1 2 9 10

$18,000 $18,000 $18,000 $18,0000 $100,000 ... .

(1 IRR) (1 IRR) (1 IRR) (1 IRR)= − + + + + +

+ + + +

Using Excel’s IRR function, we can solve for IRR as:

=IRR(−100000, 18000, 18000, 18000, 18000, 18000, 18000, 18000, 18000, 18000, 18000)

� IRR = 12.41%.

We can’t tell by simply looking at this IRR whether or not this project is acceptable. Instead, we must compare the IRR to Big Steve’s cost of capital. If its cost of capital is less than 12.41%, then the project is acceptable; if the discount rate is greater than 12.41%, however, the project should be rejected.

11-4. Dossman Metal Works is considering two mutually exclusive configurations for a new plasma cutter. Alternative A costs $80,000 and generates annual cost cash flows of $20,000/year for 7 years. Alternative B also costs $80,000, but generates annual cost cash flows of $6000/year for three years.

A. At a 10% discount rate, the PV cost of these alternatives are:

PVA = (80,000) + (20,000)/ (1.1)1 + (20,000)/ (1.1)2 +….. (20,000)/ (1.1)7

PVA = ($ 177,368)

EAC is the effective annual cost and is found by dividing the PV costs of the project by the annuity factor for the discount rate and term of the project, as follows:

EACA = ($ 177,368) / 4.8684

EACA = ($ 36,432)

PVB = (80,000) + (6000)/ (1.1)1 + (6000)/ (1.1)2 + (6000)/(1.1)3

PVB = ($94,921)

EACB = ($94,921) / 2.4869

EACB = ($38,168)

B. Even though alternative B has lower annual cost cash flows, the project has a very short useful life, and thus has a higher EAC than alternative A (−$38,168 versus −$36,432). Accordingly, alternative A should be selected, since it results in a lower equivalent annual cost to operate.



11-5. A. Templeton Manufacturing is considering two mutually exclusive conveyer belt systems. The Eclipse model costs $1.4M, then provides services—at various annual costs—for 10 years. The Sabre model costs $800,000, then provides services for 5 years.

If the firm uses a 12% discount rate, we can find the NPVs of these alternatives as:

1 2 3 4 5

6 7 8 9 10

$25,000 $30,000 $30,000 $30,000 $40,000NPV $1.4M

(1.12) (1.12) (1.12) (1.12) (1.12)

$40,000 $40,000 $40,000 $40,000 $40,000

(1.12) (1.12) (1.12) (1.12) (1.12)

$1,591,171.

$50,000NPV $800,000

(1.

E

S

= − − − − − − −

− − − −

= −

= − −1 2 3 4 5

$50,000 $60,000 $60,000 $80,000

12) (1.12) (1.12) (1.12) (1.12)

$1,010,735.

− − − −

= −

If these alternatives were directly comparable, we would want to choose the one with the smaller (negative) NPV (that is, the one with the lower present value of costs), the Sabre model.

B. However, these projects have different lives, and therefore are not perfectly comparable. Since there are no IRRs for these projects (since the sign of the cash flows never changes), we will try the EAC comparison.

NPVEAC

(present value of annuity factor, 12%, 10)

$1,591,171$281,612.07

(5.6502)

NPVEAC

(present value of annuity factor, 12%, 5)

$1,010,735$280,387.73.

(3.6048)

EE

SS

i n

i n

== =

−= = −

== =

−= = −

Thus choosing the Sabre model means an equivalent annual cost of −$280,388, while the Eclipse’s EAC is −$281,612. Since the Sabre model provides the conveyor services at a lower EAC, Templeton should choose it over the Eclipse model.

11-6. A. The IRR is a single interest rate that sets the present value of future cash flows equal to the initial cost, i.e. sets the NPV of the project equal to zero. It is very easy to find this rate when there is only one future cash flow. For example, using part (a)—an initial outflow of $10,000, then $17,182 in 8 years:

8

8

1/8

FVPV

(1 IRR)

$17,182$10,000

(1 IRR)

$17,182(1 IRR) 1.7182

$10,000

(1.7182) 1 IRR 7%.

n=

+

=+

+ = =

− = =



Using the same approach for the rest of the parts of the problem, we find:

1/10$48,077

(b) IRR 1 17%$10,000 = − =

1/ 20$115,231

(c) IRR 1 13%$10,000

= − =

1/3$13,680

(d) IRR 1 11.01%.$10,000 = − =

11-7. The IRR is a single interest rate that sets the present value of future cash flows equal to the initial cost, i.e., sets the NPV equal to zero. When there are multiple future cash flows (i.e., more than two), the only way to find the IRR is through a search procedure (as described in Checkpoint 11-4). As we showed in Problems 11-1 and 11-2, one method is to find the cash flow stream’s NPVs at various discount rates; the rate at which the NPV profile (the graph of NPV [y] against rate [x]) crosses the x axis is the stream’s IRR. We can also simply use a financial calculator or Excel’s IRR function. We will use Excel.

A. Conceptually, we’re finding the discount rate that sets NPV equal to zero. Using the stream in part (a) as an example:

1 2 9 10

$1,993 $1,993 $1,993 $1,9930 $10,000 ...

(1 IRR) (1 IRR) (1 IRR) (1 IRR)= − + + + + +

+ + + +.

Using Excel, we can solve for IRR as:

=IRR(−10000, 1993, 1993, 1993, 1993, 1993, 1993, 1993, 1993, 1993, 1993) � IRR = 15.01%.

15.01%

($3,000)

($1,000)

$1,000

$3,000

$5,000

$7,000

$9,000

0% 5% 10% 15% 20% 25% 30%

NP

V

rate

We can see that the NPV profile for this cash flow stream crosses the x axis at 15.01%.



Repeating this process for the rest of the streams we were given, we have:

B. = IRR(−10000, 2054, 2054, 2054, 2054, 2054, 2054, 2054, 2054, 2054, 2054, 2054, 2054, 2054, 2054, 2054, 2054, 2054, 2054, 2054, 2054) � IRR = 20%.

C. = IRR(−10000, 1193, 1193, 1193, 1193, 1193, 1193, 1193, 1193, 1193, 1193, 1193, 1193) � IRR = 6%.

D. = IRR(−10000, 2843, 2843, 2843, 2843, 2843) � IRR = 13%.

11-8. East Coast Television’s project has an IRR of 14%. Thus at a 14% discount rate, the project has a zero NPV. It must therefore be that at 14%, the present value of the 15, $50,000 cash flows equals the initial outlay. We can therefore solve for this outlay as follows:

Since:

1 2 14 15

1 2 14 15

$50,000 $50,000 $50,000 $50,0000 $ ...

(1.14) (1.14) (1.14) (1.14)

$50,000 $50,000 $50,000 $50,000$ ...

(1.14) (1.14) (1.14) (1.14)

$307,108.

X

X

= − + + + + +

= + + + +

=

Now that we know the initial cash flow, we can find the NPV of the stream at 10%:

1 2 14 15

$50,000 $50,000 $50,000 $50,000NPV $307,108 ...

(1.10) (1.10) (1.10) (1.10)

$73,196.

= − + + + + +

=

The NPV is positive because the 10% discount rate we used is less than the stream’s IRR.

11-9. A.–C. We are asked to find the IRRs for three series of cash flows, as shown below:

t (a) (b) (c)0 ($10,000) ($10,000) ($10,000)1 $2,000 $8,000 $2,0002 $5,000 $5,000 $2,0003 $8,000 $2,000 $2,0004 $2,0005 $2,0006 $5,000

IRR 18.79% 30.20% 11.20%sum of CFs $5,000 $5,000 $5,000

Note that we used Excel’s IRR function to find the IRRs. For example, with the series in (a), we have: = IRR(−10000, 2000, 5000, 8000) � IRR = 18.79%.



Note the interesting thing here: All three series offer the same number of dollars—the sums of their cash flows are all $5000. However, since the timing of those cash flows is different, each series offers a different IRR. If you’re going to pay $10,000 today to get $15,000 spread over future years, based on the concept that money has a time value, you’re going to be better off if those future dollars come sooner. Thus, series (b) offers the highest IRR: With (b), you get $8000 in one year, compared to only $2000 for (a) and (c). In fact, it takes over 2 years to receive $8000 from (a), and 4 years from (c).

18.79% 30.20%11.20%

($5,000)

($3,000)

($1,000)

$1,000

$3,000

$5,000

0% 5% 10% 15% 20% 25% 30% 35%

NP

V

series (a)

series (b)

series (c)

We can see how important the timing issue is by looking at the three series’ NPV profiles: All start at the same point on the y axis (this is simply the sum of the cash flows, since it represents the NPV with the discount rate of 0%), but they decline at different rates. The x intercepts are the IRRs; stream (b)’s flatter slope translates into the highest IRR.

t CFt

0 ($800,000)1 $175,0002 $175,0003 $175,0004 $175,0005 $175,0006 $175,0007 $175,0008 $175,0009 $175,000

10 $175,000



11-10. Jella Cosmetics’ project has the following cash flows:

To find its IRR, we set up equation 11-4:

1 2 9 10

$175,000 $175,000 $175,000 $175,0000 $800,00 ...

(1 IRR) (1 IRR) (1 IRR) (1 IRR)= − + + + + +

+ + + +.

Using Excel, we can solve for IRR as:

= IRR(−800000, 175000, 175000, 175000, 175000, 175000, 175000, 175000, 175000, 175000, 175000) � IRR = 17.52%.

Note that we do not need the “appropriate discount rate” of 12% for calculation of the IRR. The whole purpose of IRR is to abstract from an outside interest rate—it’s an “internal” rate of return, because it considers only the cash flows from the project. The analyst still must be able to discern whether the “appropriate discount rate,” whether calculated or not, is higher than the IRR or lower (since the project will be rejected in the former case and accepted in the latter). Also, IRR has problems in certain cases: There may be no IRR for a series of cash flows; there may be multiple IRRs for a series of cash flows (see Checkpoint 11.5); the IRR may not give the correct ranking of mutually exclusive projects. (The last problem can occur if, for example, the timing of the alternatives’ cash flows are different, as demonstrated in Figure 11.2, or if the scale of the projects is very different.)

If we were to discount the cash flows at 12%, we would find an NPV of $188,789:

= NPV(12%, 175000, 175000, 175000, 175000, 175000, 175000, 175000, 175000, 175000, 175000) − 800000

NPV = $188,789.

This is, of course, positive, since 12% < 17.52% = IRR.



8.10%

t CFt PV(CFt)

0 ($200) ($200)1 ($200) ($185)2 ($200) ($171)3 ($200) ($158)4 ($200) ($146)5 ($200) ($136)6 ($200) ($125)7 ($200) ($116)8 ($200) ($107)9 ($200) ($99)10 ($200) ($92)11 ($200) ($85)12 ($200) ($79)13 ($200) ($73)14 ($200) ($67)15 ($200) ($62)16 ($200) ($58)17 ($200) ($53)18 ($200) ($49)19 ($200) ($46)20 $10,000 $2,107

IRR 8.10%sum of cash flows ($0)

11-11. Your investment advisor offers you the opportunity shown in the first column to the right: You pay $200/year from t = 0 through t = 19, then receive $10,000 at t = 20. We can find the IRR with Excel’s IRR function:

= IRR(−200, −200, −200, −200, −200, −200, −200, −200, −200, −200, −200, −200, −200, −200, −200, −200, −200, −200, −200, −200, 10000) � IRR = 8.10%.

The righthand column shows the present value of the cash flows, discounted at 8.10%. Since their sum equals $0, we know we’ve found the IRR.

Note that it doesn’t matter that we have so many negative cash flows in a row: We’re simply trying to find the rate at which the PV of the series is zero. And since we only have one change of sign, we don’t need to worry about multiple IRRs in this case.



11-12. We are given the cash flows for three independent projects, as shown in the chart below.

t A B C0 ($50,000) ($100,000) ($450,000)1 $10,000 $25,000 $200,0002 $15,000 $25,000 $200,0003 $20,000 $25,000 $200,0004 $25,000 $25,0005 $30,000 $25,000

IRR 23.29% 7.93% 15.89%

NPV at 10% $22,217 ($5,230) $47,370

NPV at 20% $4,437 ($25,235) ($28,704)

PROJECT

We can find the projects’ IRRs and NPVs as follows (using project A as an example):

1 2 3 4 5

$10,000 $15,000 $20,000 $25,000 $30,000IRR :0 $50,000

(1 IRR) (1 IRR) (1 IRR) (1 IRR) (1 IRR)A = − + + + + ++ + + + +

.

Using Excel’s IRR function, we find:

= IRR(−50000, 10000, 15000, 20000, 250000, 30000) � IRR = 23.29%.

NPVA at 10% is found as:

1 2 3 4 5

$10,000 $15,000 $20,000 $25,000 $30,000NPV $50,000 .

(1.10) (1.10) (1.10) (1.10) (1.10)= − + + + + +

Using Excel’s NPV function, we find:

=NPV(10%, 10000, 15000, 20000, 250000, 30000) – 50000 � NPV = $22,217.

Since A’s NPV is positive at 10%, we would accept it if 10% were our appropriate discount rate.

As shown in the table above, A’s NPV is positive at both 10% and 20%, so that it is an acceptable project in either case. B’s NPV is negative at both rates, so is rejected in both cases. However C’s decision is split: Its NPV is positive at 10%, so it’s acceptable at this rate; however, at 20%, its NPV is negative, so it is then unacceptable.

Thus at 10%, choose A and C; at 20%, choose only A.



11-13. Mode Publishing’s project has the following cash flows:

t PROJECT0 ?1 $800,000,0002 $400,000,0003 $300,000,0004 $500,000,000

If this project’s payback period is 2.5 years, then its initial cost equals the total (undiscounted) of the first two years’ cash flows, plus half of the third year’s: ($800M + $400M + 0.5 ∗ $300M) = $1350M. We can now find the project’s IRR as:

1 2 3 4

$800 $400 $300 $500IRR: 0

(1 IRR) (1 IRR) (1 IRR) (1 IRR)

M M M M= − + + ++ + + +

= IRR(−13500000, 800000000, 400000000, 300000000, 500000000) � IRR = 20.41%.

t PROJECT0 ($10,000,000)1 $3,000,0002 $3,000,0003 $3,000,0004 $3,000,0005 ($5,000,000)6 $5,000,0007 $5,000,0008 $5,000,0009 $5,000,000

10 $5,000,000

11-14. Emily’s Soccer Mania has the following project opportunity:

The project has a second cash outflow at t = 5. We will discount that cash flow back to t = 0, add it to the initial outflow of $10M, then find the project’s modified IRR (MIRR). Using 10%, 12%, and 14% to perform the discounting of the t = 5 cash flow, we find the results below. The value of the discounted 5th year investment is incorrect in the 12% and 14% calculations. At 12% the revised t0 investment s/b ($12,837,134) and at 14% the investment s/b ($12,596,843). The corrected MIRR’s for the 12% case and 14% case are 21.88% and 22.39%, respectively.



11-15. Carraway Trucking has the following project opportunity:

t CFt

0 ($20,000,000)1 $4,000,0002 $4,000,0003 $4,000,0004 ($5,000,000)5 $2,000,0006 $2,000,0007 $2,000,0008 $2,000,0009 $2,000,000

10 $2,000,000

The project has a second cash outflow at t = 4. We will discount that cash flow back to t = 0, add it to the initial outflow of $20M, then find the project’s modified IRR (MIRR). Using 12% to perform the discounting of the t = 4 cash flow, we find the result shown at the bottom of the “12%” column below. The MIRR is 0.74%.



t CFt 12% PV(CFt)

0 ($20,000,000) ($23,177,590) ($20,000,000)1 $4,000,000 $4,000,000 $3,571,4292 $4,000,000 $4,000,000 $3,188,7763 $4,000,000 $4,000,000 $2,847,1214 ($5,000,000) $0 ($3,177,590)5 $2,000,000 $2,000,000 $1,134,8546 $2,000,000 $2,000,000 $1,013,2627 $2,000,000 $2,000,000 $904,6988 $2,000,000 $2,000,000 $807,7669 $2,000,000 $2,000,000 $721,220

10 $2,000,000 $2,000,000 $643,946

MIRR = 0.74% ($8,344,518) = NPV

IRR = -1.02%

The last column above shows the present value of the cash flows, discounted at 12%. The NPV is very negative: −$8,344,518. This is to be expected, given that the IRR is 0.74% (IRR < WACC � NPV < 0).

The IRR for this project is −1.02%. (We found this using Excel’s IRR function, as illustrated in earlier problems.) While projects with three sign changes, like this one, can possibly have three IRRs, between −100% and 219%.

11-16. Microwave Oven Programming, Inc. has a project whose cash flows are shown in the second column of the table below.

28.53%

t CFt PV(CFt)

0 ($7,000,000) ($7,000,000)1 $3,000,000 $2,334,0352 $4,000,000 $2,421,2083 $2,000,000 $941,8644 $2,000,000 $732,7815 $2,000,000 $570,112

IRR = 28.53% $0 = NPV



We can find the IRR for this project as:

1 2 3 4 5

$3 $4 $2 $2 $2IRR: 0 $7M

(1 ) (1 ) (1 ) (1 ) (1 )

M M M M M

IRR IRR IRR IRR IRR= − + + + + +

+ + + + +.

= IRR(−7000000, 3000000, 4000000, 2000000, 2000000, 2000000) � IRR = 28.53%.

The third column in the table above shows what happens when we discount the project’s cash flows at 28.53%: the NPV = $0.

11-17. Cash flows for this project are as follows:

T0 ($400,000)

1–4 $150,000

5 $150,000 plus outflow of ($200,000)

6–7 $150,000

NPV @ 12% = $334,048

To find the MIRR, we discount the ($50,000) cash flow in year 5 to t0 at the 12% discount rate. The revised t0 investment becomes ($428,371) and the revised cash flows become $150,000 in years 1–4, $0 in year 5, and $150,000 in years 6–10. The MIRR = 29.55%

t CFt 12% PV(CFt)

0 ($400,000) ($490,470) ($400,000)1 $150,000 $150,000 $133,9292 $150,000 $150,000 $119,5793 $150,000 $150,000 $106,7674 $150,000 $150,000 $95,3285 $150,000 $150,000 $85,1146 $0 $0 $07 ($200,000) $0 ($90,470)

MIRR = 16.06% $50,247 = NPV

11-18. Star Industries builds and operates landfills. Every 5 years it must pay $10M to build a new landfill; each new landfill can then be used for 5 years, generating $3M/year. Columns A, B, and C in the spreadsheet below illustrate how this works: column A gives the outflows for the new landfills, which occur at t = 0, t = 5, t = 10, and t = 15; column B shows the $3M/year inflows from the landfills in use; column C shows the total cash flow to Star (which is the sum of columns A and B). (The color-coding in the spreadsheet simply links the construction costs for a landfill in column A with the cash flows that landfill generates over the next 5 years.)



Column D finds the MIRR. First, all of the 4, $10M outflows from column A are discounted back to t = 0 at Star’s WACC of 10%. The total PV of these outflows is shown as a single t = 0 outflow of $22,458,567. The rest of column D is simply the PV of the $3M inflows each year, at 10%. The MIRR is then the IRR of this revised cash flow series. The IRR for the cash flows in column D is 11.96%.

Column E simply finds the NPV of the actual cash flows from column C. The NPV of this project is $3,082,125, so the project should be accepted.

The IRR for the project is 15.24%. This unadjusted IRR would have given Star the correct decision: accept, since IRR > WACC.

Although the cash flows from this project change signs many times, between −100% and 289% there is only one IRR, 15.24%.

11-19. Fijisawa, Inc. is considering an expansion project whose cash flows are shown in the second column of the table below:

9%

t CFt PV(CFt)

0 ($1,950,000) ($1,950,000)1 $450,000 $412,8442 $450,000 $378,7563 $450,000 $347,4834 $450,000 $318,7915 $450,000 $292,4696 $450,000 $268,320

$68,663 = NPV

IRR = 10.17%

PI = 1.04



The firm’s discount rate is 9%. Discounting the project’s cash flows at 9% gives the NPV shown in the third column above. The NPV is $68,663, so the project should be accepted.

Since the NPV is positive, the profitability index (PI) will be greater than 1. Fijisawa’s PI is found as follows, using equation 11-3:

PV(future CFs) $2,018,663PI 1.04.

initial outflow 1,950,000= = =

(The numerator of this ratio is the sum of the PVs in column 3 above, from t = 1 through t = 6.)

Finally, we can find the IRR using equation 11-4:

1 2 5 6

$450,000 $450,000 $450,000 $450,0000 $1,950,000 ...

(1 IRR) (1 IRR) (1 IRR) (1 IRR)= − + + + + +

+ + + +.

Using Excel’s IRR function, we can solve for IRR as:

=IRR(−950000, 450000, 450000, 450000, 450000, 450000, 450000) � IRR = 10.17%.

The IRR is greater than Fijisawa’s discount rate of 9%, which it would be, given the positive NPV.

All of the three measures are saying the same thing: Fijisawa should accept the expansion opportunity.

11-20. Gio’s Restaurant is considering a project whose cash flows are shown in the second column of the table below:

12% cumulative

discounted

t CFt PV(CFt) cash flows

0 ($150,000,000) ($150,000,000) ($150,000,000)1 $90,000,000 $80,357,143 ($69,642,857)2 $70,000,000 $55,803,571 ($13,839,286)3 $90,000,000 $64,060,222 $50,220,9374 $100,000,000 $63,551,808

A payback method adds cash flows from different periods together, looking for the number of periods it takes to recoup the initial cash outflow. In Gio’s case, we need to find out how many years’ worth of cash flows equal −$150M. In addition, we will be discounting the future years’ cash flows before we add them.

Years’ 1 and 2 have cash flows that total $136,160,714 in discounted value, so that we have only ($150,000,000 − $136,160,714) = $13,839,286 left to recoup in year 3. Year 3’s total discounted cash flow is $64,060,222, so we won’t need the full year’s worth of t = 3 cash flows. Instead, we will need only $13,839,286

$64,060,222( ) = 22% of year 3’s cash flows. Thus, Gio’s discounted payback period is

2.22 years.



11-21. The Callaway Cattle Company is considering a project whose cash flows are shown in the second column of the table below:

10% cumulative

discounted

t CFt PV(CFt) cash flows

0 ($500,000) ($500,000) ($500,000)1 $200,000 $181,818 ($318,182)2 $200,000 $165,289 ($152,893)3 $200,000 $150,263 ($2,630)4 $200,000 $136,603

A payback method adds cash flows from different periods together, looking for the number of periods it takes to recoup the initial cash outflow. In Callaway’s case, we need to find out how many years’ worth of cash flows equal −$500,000. In addition, we will be discounting the future years’ cash flows before we add them.

Years’ 1, 2, and 3 have cash flows that total $497,370 in discounted value, so that we have only ($500,000 − $497,370) = $2630 left to recoup in year 4. Year 4’s total discounted cash flow is $136,603, so we won’t need the full year’s worth of t = 4 cash flows. Instead, we will need only

$2,630$136,603( ) = 2% of year 4’s cash flows. Thus Callaway’s discounted payback period is 3.02 years.

11-22. Bar-None Manufacturing has three investment proposals for fence panels. It does not want to accept any project that has longer than 3 years to pay off. It will consider both the payback criterion and the discounted payback criterion to see which of its project opportunities are acceptable.

We show the results below:

10% cumulative

cumulative discounted

t CFt cash flows PV(CFt) cash flows

0 ($1,000) ($1,000) ($1,000) ($1,000)1 $600 ($400) $545 ($455)2 $300 ($100) $248 ($207)3 $200 $150 ($56)4 $100 $685 $500 $310

PROJECT A



Project A must recoup its $1000 investment. After years 1 and 2, it has generated undiscounted cash flows of $900, and must therefore use $1000 $900

$200( )− = 50% of year 3’s cash flows. Thus its undiscounted payback period is 2.5 years, which meets the company’s cutoff. Its discounted payback is longer, of course. It takes 3 years’ worth of cash flows to get $944 back;1 it therefore needs $1000 $944

$68( )− = 83% of year 4’s cash flows. Thus the discounted payback is 3.83, which would make the project unacceptable under this criterion.

For project B, we have:

10% cumulative



0 ($10,000) ($10,000) ($10,000) ($10,000)1 $5,000 ($5,000) $4,545 ($5,455)2 $3,000 ($2,000) $2,479 ($2,975)3 $3,000 $2,254 ($721)4 $3,000 $2,0495 $3,000 $1,863

PROJECT B

Undiscounted, project B needs years 1 and 2 (for a total of $5,000 + $3,000 = $8,000), plus $10,000 $8,000

$3,000( )− = 67% of year 3’s; the payback period is therefore 2.67. For the discounted cash flows, the project needs cash flows from years 1 through 3 (for a total of $9,279), plus

$10,000 $9,279$2,049( )− = 35% of year 4’s. The discounted payback is 3.35 years, making the project

unacceptable on a discounted basis, but acceptable if the criterion is undiscounted.

For project C:

10% cumulative



0 ($5,000) ($5,000) ($5,000) ($5,000)1 $1,000 ($4,000) $909 ($4,091)2 $1,000 ($3,000) $826 ($3,264)3 $2,000 ($1,000) $1,503 ($1,762)4 $2,000 $1,366 ($396)5 $2,000 $1,242

PROJECT C

Project C needs years 1 through 3 (total = $4000), plus $5000 $4000$2000( )− = 50% of year 4’s; the payback

period is 3.50, which is unacceptable. On a discounted basis, the project needs 4 years’ worth of cash flows (total = $4604), plus $5000 $4604

$1242( )− = 32% of year 5’s. The discounted payback period is 4.32 years, which again is unacceptable.

1 There is a rounding issue here; the numbers displayed in the table are different by $1.



We can summarize the results as follows:

project: A B C

payback 2.50 2.67 3.50

discounted payback 3.83 3.35 4.32

NPV $1,279 $3,191 $846

None of these projects is acceptable using discounted payback; only C is also unacceptable using regular payback. However, we have also included the NPVs in the table above. (These can be found simply by summing the PV(CFt) columns in the tables for each project.) All of the three projects’ NPVs are positive. Thus if these were independent projects, we’d advise Bar-None to accept them all. If they were mutually exclusive, the firm should choose project B—even though its payback is a bit longer than A’s.

This discrepancy points out one of the less obvious problems with payback—even the discounted kind. Why is a firm so anxious to get its money back? If a project is producing strong returns, why is a firm so anxious to see it end? (For example, if you had an old CD that’s still earning 8%, would you want it to mature, so that you could reinvest the money at the current rates of around 1%?) Firms with good future projects can finance them by accessing the capital markets (assuming no hard credit rationing); they don’t need to rely on getting their money back fast from old projects to fund current ones. Payback is inherently flawed because of this myopic perspective. Payback (discounted especially) does become relevant and insightful when considering the implications of when competition might enter the market and negatively impact the project, or in the case of expected obsolescence. In either case, the firm may want to understand how long it takes to “get their money back” before extreme negative consequences occur in the market.

11-23. Plato Energy has two drilling opportunities, one in the Barnett Shale region and the other in the Gulf Coast. The cash flows, discounted cash flows, and NPVs for these two projects are shown below:

t CFt cumulative CF PV(CFt) CFt cumulative CF PV(CFt)

0 ($5,000,000) ($5,000,000) ($5,000,000) ($1,500,000) ($1,500,000) ($1,500,000)1 $2,000,000 ($3,000,000) $1,666,667 $800,000 ($700,000) $666,6672 $2,000,000 ($1,000,000) $1,388,889 $800,000 $100,000 $555,5563 ($1,000,000) ($2,000,000) ($578,704) $400,000 $231,4814 $2,000,000 $0 $964,506 $100,000 $48,2255 $1,500,000 $602,8166 $1,500,000 $502,3477 $1,500,000 $418,6228 $800,000 $186,0549 $500,000 $96,903

10 $100,000 $16,151

NPV at 20% = $264,252 NPV at 20% = $1,929

BARNETT

SHALE

PROJECT

COAST

GULF



A. The Barnett Shale project pays off in exactly 4 years. The Gulf Coast project pays off in less than 2 years: 1 year + $1,500,000 $800,000

$800,000( )− = 1.875 years.

B. Thus under this criterion, Plato would want to choose the Gulf Coast opportunity (assuming that the payback cutoff for acceptable projects is more than 1.875 years).

One of the deficiencies of the payback method is its lack of consideration of any cash flows after the payback period. Thus the fact that the Barnett Shale project continues to offer positive cash flows for many years after the Gulf Coast project has terminated is ignored. By choosing based on payback, Plato would accept the Gulf Coast opportunity (assuming these are mutually exclusive), thereby choosing a project whose NPV is less than 1% of that available with the alternative.

Of course, there are other deficiencies with payback: It doesn’t consider time value (a decision criterion killer by itself—any method used should consider time value), and it assumes that faster payback is necessarily preferred (see answer to Problem 11-22). NPV, of course, has none of these problems.

C. The NPVs are also shown in the table above: Barnett Shale’s is $264,252, while Gulf Coast’s is only $1929. Of course, this is an odd comparison: The timing of the projects is different, so we should adjust for that. The EAC (or equivalent annual benefit in this case) is $63,030 for the Barnett Shale, but only $745 for the Gulf Coast. Thus, adjusting for timing will not make a difference: Barnett Shale still wins.

D. NPV is a measure of the value that is created by the acceptance of a project. Thus, Plato’s current shareholders would be $264,252 better off if the company chooses the Barnett Shale project, but only $1929 better off with the Gulf Coast project. If these projects are independent, then the firm should choose both. However, if they are mutually exclusive, it absolutely should choose the Barnett Shale.

11-24. The project you are considering has the cash flows identified in the second column below:

10% cumulative



0 ($80,000) ($80,000) ($80,000) ($80,000)1 $20,000 ($60,000) $18,182 ($61,818)2 $20,000 ($40,000) $16,529 ($45,289)3 $20,000 ($20,000) $15,026 ($30,263)4 $20,000 $0 $13,660 ($16,603)5 $20,000 $12,418 ($4,184)6 $20,000 $11,289

NPV = $7,105

IRR = 13.0%

PI = 1.09

A. As shown in the third column above, the undiscounted payback is 4 years [($20,000/year) ∗ (4 years) = $80,000 = initial cash flow]. The fourth column shows the discounted version: After 5 years, there is still $4184 to recover; this will occur after $4,184

$11,289( ) = 37% of year 6’s cash flows are received. The discounted payback is therefore 5.37 years. These payback values would indicate acceptance if they are lower than your set payback cutoff.



B. The NPV is the sum of all of the project’s cash flows. As shown in the fourth column, this project’s NPV is $7105. Since this is positive, this project is acceptable.

D. The project’s IRR is the rate that sets the PV of the future cash flows equal to the initial cost. Using Excel’s IRR function, we find:

= IRR(−80000, 20000, 20000, 20000, 20000, 20000, 20000) � IRR = 13%.

Since this is greater than your discount rate of 10%, this project is acceptable (as we already learned from the positive NPV, which tells exactly the same story).

C. Finally, the profitability index is found as:

$87,105PI 1.09.

$80,000= =

Since this is greater than 1, the project is acceptable. (Again, NPV already told us that.)

11-25. You are evaluating projects A and B, whose cash flows are shown below (in the two “CFt” columns):

10%

t CFt PV(CFt) CFt PV(CFt)

0 ($100,000) ($100,000) ($100,000) ($100,000)1 $33,000 $30,000 $0 $02 $33,000 $27,273 $0 $03 $33,000 $24,793 $0 $04 $33,000 $22,539 $0 $05 $33,000 $20,490 $220,000 $136,603

NPV = $25,096 NPV = $36,603

project Bproject A

To find the NPVs of these two projects, we would use equation 11-1:

1 20 1 2

NPV ...(1 ) (1 ) (1 )

n

n

CFCF CFCF

k k k= + + + +

+ + +.

For example, for project A, we have:

1 2 5

$33,000 $33,000 $33,000NPV= $100,000 ... $25,096.

(1.10) (1.10) (1.10)− + + + + =


5

$220,000NPV= $100,000 $36,603.

(1.10)− + =

Since these projects are mutually exclusive, we would choose the one that has the higher (positive) NPV: project B.



11-26. You are evaluating two independent projects, A and B, whose cash flows are shown in the chart below in the two “CFt” columns:

12%

t CFt PV(CFt) CFt PV(CFt)

0 ($50,000) ($50,000) ($70,000) ($70,000)1 $12,000 $10,714 $13,000 $11,6072 $12,000 $9,566 $13,000 $10,3643 $12,000 $8,541 $13,000 $9,2534 $12,000 $7,626 $13,000 $8,2625 $12,000 $6,809 $13,000 $7,3776 $12,000 $6,080 $13,000 $6,586

NPV = ($663) NPV = ($16,552)

PI = 0.99 PI = 0.76

IRR = 11.53% IRR = 3.18%

project A project B

To find the projects’ NPVs, we use equation 11-1:

1 20 1 2

NPV ... .(1 ) (1 ) (1 )

n

n

CFCF CFCF

k k k= + + + +

+ + +

For example, for project A, we have:

1 2 6

$12,000 $12,000 $12,000NPV $50,000 ... $663.

(1.12) (1.12) (1.12)= − + + + + = −


1 2 6

$13,000 $13,000 $13,000NPV $70,000 ... $16,552.

(1.12) (1.12) (1.12)= − + + + + = −

Since both of these NPVs are negative, neither project should be accepted.

We also can see this using both the IRRs and profitability index measures, using equations 11-4 and 11-3, respectively:

1 2 5 6

$12,000 $12,000 $12,000 $12,0000 $50,000 ...

(1 IRR) (1 IRR) (1 IRR) (1 IRR)= − + + + + +

+ + + + � IRRA = 11.53%.

1 2 5 6

$13,000 $13,000 $13,000 $13,0000 $70,000 ...

(1 IRR) (1 IRR) (1 IRR) (1 IRR)= − + + + + +

+ + + + � IRRB = 3.18%.

As we knew from the negative NPVs, these IRRs are both less than the firm’s cost of capital, 12 (if IRR < WACC, NPV < 0).



Finally, the PI:

PV(future CFs) $49,337PI 0.99.

initial outflow $50,000

PV(future CFs) $53,448PI 0.76.

initial outflow $70,000

A

B

= = =

= = =

Both of these are less than 1, meaning that they should be rejected.

All three of these measures tell the same story: The sums of the present values of the future cash flows from these projects are less than their costs, so accepting these projects would reduce the wealth of current shareholders. Neither project should be accepted.

11-27. Garmen Technologies is considering a 10-year project. This project costs $250,000 at t = 0, and has cash flows spanning 10 more years (through t = 10). At t = 5, it has a second negative cash flow, for $45,000. All of the project’s cash flows are shown in the “CFt” column in the table below:

12%

cumulative

t CFt cash flows PV(CFt)

0 ($250,000) ($250,000) ($250,000)1 $60,000 ($190,000) $53,5712 $60,000 ($130,000) $47,8323 $60,000 ($70,000) $42,7074 $60,000 ($10,000) $38,1315 ($45,000) ($55,000) ($25,534)6 $65,000 $32,9317 $65,000 $29,4038 $65,000 $26,2529 $65,000 $23,440

10 $90,000 $28,978

NPV = $47,710

IRR = 16.4%

payback period = 5.85

A. The payback period for this project is 5.85 years. We can see this by summing the first 5 years’ cash flows: ($60,000 + $60,000 + $60,000 + $60,000 − $45,000) = $195,000. To fully recoup the initial investment of $250,000, the firm needs another ($250,000 − $195,000) = $55,000; it will take $55,000

$65,000( ) = 85% of year 6’s $65,000 cash flow. Thus the payback is 5 years + 0.85 years = 5.85 years.



If Mr. Garmen asked me what useful information payback period provides, I’d say none. The accept/reject criterion associated with payback requires an arbitrary cutoff. Payback ignores time value, which is sufficient to reject it wholeheartedly. Payback is myopic, ignoring all cash flows that occur after the payback period is reached. And payback is based on the flawed assumption that a firm wants its money back fast. I would advise Mr. Garmen to focus on NPV to make his accept/reject decision. On the other hand, if Mr Garmen was concerned about the long-term need for this facility and was wondering if he could break even and exit the warehouse early, the discounted payback would be a good indicative measure in tandem with the NPV.

B. Mr. Garmen is also interested in IRR. This is the interest rate that sets the present value of the future cash flows equal to the initial investment. For this project, IRR will solve the following equation:

1 2 3 4 5

6 7 8 9 10

$60,000 $60,000 $60,000 $60,000 $45,0000 $250,000

(1 IRR) (1 IRR) (1 IRR) (1 IRR) (1 IRR)

$65,000 $65,000 $65,000 $65,000 $90,000

(1 IRR) (1 IRR) (1 IRR) (1 IRR) (1 IRR)

= − + + + + − ++ + + + +

+ + + ++ + + + +

However, since there are three sign changes for these cash flows, the project might have up to three IRRs. However, between −100% and 597%, there is only 1: 16.4%. The values for the rates given in the problem are shown below:

0%, $295,000

20%, ($30,602)

50%, ($144,331)

100%, ($193,164)

($300,000)

($200,000)

($100,000)

$0

$100,000

$200,000

$300,000

0% 20% 40% 60% 80% 100% 120%

NP

V

rate

rate NPV0% $295,00020% ($30,602)50% ($144,331)

100% ($193,164)



C. The NPV of this project is $47,710. This figure was calculated as:

1 2 3 4 5

6 7 8 9 10

$60,000 $60,000 $60,000 $60,000 $45,000NPV $250,000

(1.12) (1.12) (1.12) (1.12) (1.12)

$65,000 $65,000 $65,000 $65,000 $90,000

(1.12) (1.12) (1.12) (1.12) (1.12)

= − + + + + − +

+ + + +

Since the NPV is positive, the project should be accepted.

The NPV discounts the expected future cash flows from a project by the cost of funds for that project. That is, it expresses the future cash flows in t = 0 dollars by bringing them back (discounting them) by enough each year to ensure that they repay the required return for the new investors who provided the funds to finance the project in the first place. If there is any money left over after removing all of these required payments (that is, if the sum of the cash flows remaining after this discounting is positive—if NPV > 0), that “extra” money belongs to the original shareholders—the ones for whom the manager was working when he went ahead with the project. The NPV measures the value created for old shareholders by the acceptance of a project. Garmen’s current project can pay back the new investors who provide the funds to go forward with the project, and still have $47,710 left over. That $47,710 belongs to the original shareholders. If Garmen does not go forward with this project, it will force its shareholders to forego this $47,710 economic profit.

part_2_tkm0844_11e_im_ch11

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