6.11 Exercises

A. Problem:  Project Evaluation Methods at Topley

1. Payback period

[latex]\frac{{220,000}}{{50,000}}[/latex] = 4.4 years

2. Discounted payback period

Discounted payback period = 8 +  [latex]\frac{{2,820.46}}{{13,147.65}}[/latex] = 8.21 years

3. NPV

50,000 [latex](\frac{{1 - {{\left( {1 + .16} \right)}^{ - 10}}}}{{.16}}[/latex]) = 241,661.37

241,661.37 – 220,000 = 21,661.37

NPV is what remains after compensating investors for the RRR of 16%. This is the excess profit of the project in dollar terms and is equivalent to 2.60% in Part 4.

4. IRR

220,000 = 50,000 [latex](\frac{{1 - {{\left( {1 + i} \right)}^{ - 10}}}}{i}[/latex])

i = .1860 or 18.60%

The RRR of the project is 16.00%, so the project is earning 2.60% more than required.  This is the excess profit of the project in per cent terms.

Note:  The IRR function in Excel can be used to solve for i.

5. Modified IRR

[latex]\begin{array}{l}{}^1\;220,000 = \frac{{1,066,073.50}}{{{{\left( {1 + i} \right)}^{10}}}}\quad \quad i = .1709\quad \\{}^2\;220,000 = \frac{{1,211,326.80}}{{{{\left( {1 + i} \right)}^{10}}}}\quad \quad i = .1860\quad \end{array}[/latex]

MIRR is 17.09%

6. PI

[latex]{\rm{\;}}\frac{{241,661.37}}{{220,000}}[/latex] = 1.10

B. Problem:  Project Evaluation Methods at Cott Beverages

1. Payback period
   

   Year	Cumulative Cash Flows
   1	10,000
   2	18,000
   3	24,000
   4	29,000
   5	33,000
   6	36,000
   7	39,000

   3 + [latex](\frac{{28,000 - 24,000}}{{\begin{array}{*{20}{c}}{29,000 - 24,000}\\{}\end{array}}}[/latex]) (1) = 3.8 years

2. Discounted payback period
   

   Year	Cash Flow	Discount Factor	Present Value	Cumulative Present Value
   0	(28,000)	(1 .16)^0	-28,000.00	-28,000.00
   1	10,000	(1 .16)^1	8,620.69	-19,379.31
   2	8,000	(1 .16)^2	5,945.30	-13,434.01
   3	6,000	(1 .16)^3	3,843.95	-9,590.06
   4	5,000	(1 .16)^4	2,761.46	-6,828.60
   5	4,000	(1 .16)^5	1,904.45	-4,924.15
   6	3,000	(1 .16)^6	1,231.33	-3,692.82
   7	3,000	(1 .16)^7	1,061.49	-2,631.33

   Project does not break even on a present value basis.

3. NPV
   

   Year	Cash Flow	Discount Factor	Present Value
   1	10,000	(1 .16)^1	8,620.69
   2	8,000	(1 .16)^2	5,945.30
   3	6,000	(1 .16)^3	3,843.95
   4	5,000	(1 .16)^4	2,761.46
   5	4,000	(1 .16)^5	1,904.45
   6	3,000	(1 .16)^6	1,231.33
   7	3,000	(1 .16)^7	1,061.49
   Total			25,368.67

   25,368.67 – 28,000.00 = <2,631.33>

   The project is not earning its RRR of 16% as the NPV is negative.

4. IRR

28,000 = [latex](\frac{{10,000}}{{{{\left( {1 + i} \right)}^1}}}[/latex]) + [latex](\frac{{8,000}}{{{{\left( {1 + i} \right)}^2}}}[/latex]) + [latex](\frac{{6,000}}{{{{\left( {1 + i} \right)}^3}}}[/latex]) + [latex](\frac{{5,000}}{{{{\left( {1 + i} \right)}^4}}}[/latex]) + [latex](\frac{{4,000}}{{{{\left( {1 + i} \right)}^5}}}[/latex]) + [latex](\frac{{3,000}}{{{{\left( {1 + i} \right)}^6}}}[/latex]) + [latex](\frac{{3,000}}{{{{\left( {1 + i} \right)}^7}}}[/latex])

i = .1186 or 11.86%

Project is not earning the RRR of 16%.

Note:  IRR function in Excel can be used to solve for i.

5. PI

[latex]\frac{{25,368.67}}{{28,000.00}}[/latex]  = .91

C. Problem:  Standalone Decision at Rogers

1. No
   

   Initial investment	-120,000.00
   Tax Shield (120,000) (.45) [latex](\frac{{.25}}{{.25 + .12}}[/latex]) [latex](\frac{{2 + .12}}{{2\left( {1 + .12} \right)}}[/latex])	34,531.85
   Increase in working capital	-5,000.00
   1Annual savings	71,332.15
   Salvage value (10,000) / (1 .12)4	6,355.18
   Lost tax shield (10,000) (.45) [latex](\frac{{.25}}{{.25 + .12}}[/latex]) [latex](\frac{{2 + .12}}{{2\left( {1 + .12} \right)}}[/latex]) / (1+.12)4	-1,828.80
   Decrease in working capital (5,000) / (1 .12)4	3,177.59
   NPV	-11,432.03

   ¹90,000 – 22,000 – 6,000 – 3,300 – 7,000 – 9,000 = 42,700

   (42,700) (1 – .45) [latex](\frac{{1 - {{\left( {1 + .12} \right)}^{ - 4}}}}{{.12}}[/latex])  = 71,332.15

2. Yes

(200) (90,000/450) – (22,000) – (200) (6,000/450) – (200) (7,000/450) – (200) (9,000/450) = 8,222.22

(8,222.22) (1 – .45) [latex](\frac{{1 - {{\left( {1 + .12} \right)}^{ - 4}}}}{{.12}}[/latex])  = 13,735.57

-11,432.03 + 13,735.57 = 2,303.54

D. Problem:  Replacement Decision at Ruby

1. Yes
   

   Net investment (500,000 – 50,000)	-450,000.00
   Tax shield (450,000) (.35) [latex](\frac{{.2}}{{.2 + .10}}[/latex]) [latex](\frac{{2 + .10}}{{2\left( {1 + .10} \right)}}[/latex])	100,277.27
   Increase in net working capital	-10,000.00
   Annual saving (200,000) (2) (1-.35) [latex](\frac{{1 - {{\left( {1 + .10} \right)}^{ - 8}}}}{{.10}}[/latex]) + (20,000) (4 + 2) (1-.35)) [latex](\frac{{1 - {{\left( {1 + .10} \right)}^{ - 8}}}}{{.10}}[/latex])	1,803,205.05
   Salvage value (80,000-10,000) / (1 .10)8	32,655.52
   Lost tax shield (80,000-10,000) (.35) [latex](\frac{{.2}}{{.2 + .10}}[/latex]) [latex](\frac{{2 + .10}}{{2\left( {1 + .10} \right)}}[/latex]) / (1+.10)8	-7,273.27
   Decrease in net working capital (10,000) / (1 .10)8	4,655.07
   NPV	1,473,469.64

E. Problem:  Replacement Decision at Zebra

1. Yes
   

   Net investment (141,000 – 10,000)	-131,000.00
   Tax shield (131,000) (.31) [latex](\frac{{.2}}{{.2 + .115}}[/latex]) [latex](\frac{{2 + .115}}{{2\left( {1 + .115} \right)}}[/latex])	24,454.45
   Decrease in net working capital	30,000.00
   1 Annual savings (191,250) (1-.31) [latex](\frac{{1 - {{\left( {1 + .115} \right)}^{ - 6}}}}{{.115}}[/latex])	550,322.43
   Salvage value (18,000) / (1 .115)6	9,367.49
   Lost tax shield (18,000) (.31) [latex](\frac{{.2}}{{.2 + .115}}[/latex]) [latex](\frac{{2 + .115}}{{2\left( {1 + .115} \right)}}[/latex]) / (1+.115)6	-1,748.68
   Increase in net working capital (30,000) / (1+.115)6	-15,612.49
   NPV	465,783.20

   ¹

   Additional units (12.00-5.75) (15,000)	93,750
   Savings – VC (7.50-5.75) (50,000)	87,500
   Savings – FC	10,000
   Total	191,250

F. Problem:  Standalone Decision with Inflation at Weatherly

1. Nominal approach
   

   Investment	-3,500,000
   Tax shield (3,500,000) (.21) [latex](\frac{{.3}}{{.3 + .09}}[/latex]) [latex](\frac{{2 + .09}}{{2\left( {1 + .09} \right)}}[/latex])	542,043
   1Annual savings	2,776,826
   Salvage value (450,000) (1 .02)5 / (1 .09)5	322,910
   Lost tax shield (496,836) (.21) [latex](\frac{{.3}}{{.3 + .09}}[/latex]) [latex](\frac{{2 + .09}}{{2\left( {1 + .09} \right)}}[/latex]) / (1 + .09)5	-50,009
   NPV	91,770

   ¹ (6,000) (16) (17.21 – 5.24) – (2) (115,000) – 65,000 = 854,120

   (854,120) (1 – .21) = 674,755

    

   Year 1 (674,755) (1 + .02)1 / (1 + .09)¹ = 631,422

   Year 2 (674,755) (1 + .02)2 / (1 + .09)² = 590,872

   Year 3 (674,755) (1 + .02)3 / (1 + .09)³ = 552,926

   Year 4 (674,755) (1 + .02)4 / (1 + .09)⁴ = 517,417

   Year 5 (674,755) (1 + .02)5 / (1 + .09)⁵ = 484,189

    

   Total PV = 2,776,826

2. Real approach

(1 + .09) = (1 + .02) (1 + Real rate)

Real rate = .0686

¹ (6,000) (16) (17.21 – 5.24) – (2) (115,000) – 65,000 = 854,120

(854,120) (1 – .21) = 674,755

Year 1 (674,755) / (1 + .0686)¹ = 631,438

Year 2 (674,755) / (1 + .0686)² = 590,902

Year 3 (674,755) / (1 + .0686)³ = 552,969

Year 4 (674,755) / (1 + .0686)⁴ = 517,470

Year 5 (674,755) / (1 + .0686)⁵ = 484,251

Total PV = 2,777,030

Why are the NPVs for the nominal and real approach not equal?  It is due to more than simply rounding error.  Under the nominal approach, it is assumed that future cash flows will increase by 2.0% each year and that the nominal discount rate of 9.0% has an allowance for this inflation plus the real rate.  Under the real approach, the increase in future cash flows due to inflation is ignored and inflation is left out of the discount rate as well and all discounting is done at the real rate of 6.86% – inflation essentially cancels out in the numerator and denominator.

When the CAD 3,000,000 is put in a CCA pool, the government does not allow these amounts to be increased each year to compensate for inflation.  The government has considered indexing the contents of CCA pools but has decided against it because of the magnitude of lost tax revenues.  As a result, the RRR used in the present value of CCA tax shield calculation must be 9.0% and not 6.86% since the value of the pool does not rise by the inflation rate each year.  There is no inflation in the numerator to cancel out with the denominator.

The correct calculation of the NPV under the real approach is:

¹ (6,000) (16) (17.21 – 5.24) – (2) (115,000) – 65,000 = 854,120

(854,120) (1 – .21) = 674,755

Year 1 (674,755) / (1 + .0686)¹ = 631,438

Year 2 (674,755) / (1 + .0686)² = 590,902

Year 3 (674,755) / (1 + .0686)³ = 552,969

Year 4 (674,755) / (1 + .0686)⁴ = 517,470

Year 5 (674,755) / (1 + .0686)⁵ = 484,251

Total PV = 2,776,826

3. The general inflation estimates of 2.0% is reasonable for operating costs, but not for metal prices. The price of most metals is unstable.  A more thorough analysis of the metal price over the next five years is needed before a decision can be made.  Even with this, the price of most metals is difficult to forecast meaning project risk is high.

G. Problem:  Standalone Decision with Inflation at Quaker

1. No

.05 .03 = .08

(1 + .08) = (1 .0250) (1 + X)

X = .0537

¹Years 1–6

(2,900,000 – 1,900,000 – 800,000) = 200,000

(200,000) (1 – .3) [latex](\frac{{1 - {{\left( {1 + .0537} \right)}^{ - 6}}}}{{.0537}}[/latex]) = 702,265.44

Years 7–12

(3,500,000 – 2,100,000 – 800,000) = 600,000

(600,000) (1 – .3) [latex](\frac{{1 - {{\left( {1 + .0537} \right)}^{ - 6}}}}{{.0537}}[/latex]) / (1 + .0537)⁶ = 1,539,290.79

H. Problem:  Capital Rationing at Bosie

1. D and F
   

   Project	Investment	Profitability Index	NPV	Selection	Total Investment	Total NPV
   A	4,000,000	1.18	720,000	0
   B	3,000,000	1.08	240,000	0
   C	5,000,000	1.33	1,650,000	0
   D	6,000,000	1.31	1,860,000	1	6,000,000	1,860,000
   E	4,000,000	1.19	760,000	0
   F	6,000,000	1.20	1,200,000	1	6,000,000	1,200,000
   G	4,000,000	1.18	720,000	0
   				Total	12,000,000	3,060,000

   See Excel spreadsheet

I. Problem:  Projects of Varying Lives at Wilson

1. Project 1 (Best Project)

[latex]\begin{array}{l}23,000\left( {\frac{{1 - {{\left( {1 + .07} \right)}^{ - 5}}}}{{.07}}} \right) - 55,000 = 39,304.54\\39,304.54 + \frac{{39,304.54}}{{{{\left( {1 + .07} \right)}^5}}} = 67,328.13\end{array}[/latex]

Project 2

[latex]15,000\left( {\frac{{1 - {{\left( {1 + .07} \right)}^{ - 10}}}}{{.07}}} \right) - 60,000 = 45,353.72[/latex]

2. Project 1 (Best Project)

[latex]39,304.54 = {\rm{P}}\left( {\frac{{1 - {{\left( {1 + .07} \right)}^{ - 5}}}}{{.07}}} \right)\quad \quad {\rm{P}} = 9,586.01[/latex]

Project 2

[latex]45,353.72 = {\rm{P}}\left( {\frac{{1 - {{\left( {1 + .07} \right)}^{ - 10}}}}{{.07}}} \right)\quad \quad {\rm{P}} = 6,457.35[/latex]

3. The assumption is that once the project is complete it can be repeated with the same return. For routine projects like machinery replacement this is reasonable, but for one-time projects like new products this is not reasonable.

J. Problem:  Projects of Varying Lives at Jensen

1. Project 1 (Best Project)

[latex]\frac{{40,000}}{{{{\left( {1 + .08} \right)}^1}}} + \frac{{38,000}}{{{{\left( {1 + .08} \right)}^2}}} + \frac{{42,000}}{{{{\left( {1 + .08} \right)}^3}}} - 65,000 = 37,956.87[/latex]

[latex]37,956.87 + \frac{{37,956.87}}{{{{\left( {1 + .08} \right)}^3}}} = 68,088.25[/latex]

Project 2

[latex]27,000\left( {\frac{{1 - {{\left( {1 + .08} \right)}^{ - 6}}}}{{.08}}} \right) - 79,000 = 45,817.75[/latex]

2. Project 1 (Best Project)

[latex]37,956.87 = P\left( {\frac{{1 - {{\left( {1 + .08} \right)}^{ - 3}}}}{{.08}}} \right)\quad \quad P = 14,728.54[/latex]

Project 2

[latex]45,817.75 = P\left( {\frac{{1 - {{\left( {1 + .08} \right)}^{ - 6}}}}{{.08}}} \right)\quad \quad P = 9,911.08[/latex]

4. The assumption is that once the project is complete it can be repeated with the same return. For routine projects like machinery replacement this is reasonable, but for one-time projects like new products this is not reasonable.

K. Problem:  Changes in Net Working Capital at Amsterdam

1.

NWC will rise over the life of the project (2013 – CAD 155,655, 2014 – CAD 50,633, 2015 – CAD 12,658) before returning to its original level (CAD 274,725) at the end of 2015.

L. Problem:  Taxation Effects of Terminal Cash Flows

1.

Case 1

Terminal Cash Flows

Sale of land                                  4,000,000

¹Capital gains tax                          -525,000

¹(4,000,000 – 1,000,000) (.50) (.35)

What if the land is sold for CAD 500,000?

Terminal Cash Flows

Sale of land                                 500,000

¹Capital loss benefit                    87,500

¹(500,000 – 1,000,000) (.50) (.35)

Case 2

Terminal Cash Flows

Sales of building                         600,000

¹Capital gain                                   -17,500

²Recapture                                    -53,804

¹(600,000 – 500,000) (.5) (.35)

²(346,275 – 500,000) (.35)

What if the building is sold for CAD 100,000?

Terminal Cash Flows

Sale of building                           100,000

¹Terminal loss                                 86,196

¹(346,275 – 100,000) (.35)

Case 3

Terminal Cash Flows

Sale of equipment                      50,000

¹Present value of future CCA    13,040

¹(102,042 – 50,000) (.35) [latex](\frac{{.30}}{{.30 + .10}}[/latex]) [latex](\frac{{2 + .10}}{{2\left( {1 + .10} \right)}}[/latex])

What if the equipment is sold for CAD 120,000?

Terminal Cash Flows

Sales of equipment                     120,000

¹Present value of future CCA      -4,500

¹(102,042 – 120,000) (.35) [latex](\frac{{.30}}{{.30 + .10}}[/latex]) [latex](\frac{{2 + .10}}{{2\left( {1 + .10} \right)}}[/latex])

What if the equipment is sold for CAD 600,000?

Terminal Cash Flows

Sales of equipment                     600,000

¹Capital gains tax                            -17,500

²Present value of future CCA      -97,716

¹(600,000 – 500,000) (.5) (.35)

²(102,042 – 500,000) (.35) [latex](\frac{{.30}}{{.30 + .10}}[/latex]) [latex](\frac{{2 + .10}}{{2\left( {1 + .10} \right)}}[/latex])

M. Problem:  Managing Risk by Adjusting the Discount Rate at Rexall

1. Yes

(1 .09) = (1 Real rate) (1 .025)

Real rate = .0634

¹ Years 1 – 5

(55,000) (1.50 – (1.29 – .35)) – 10,000 = 20,800

(580,000) (.35) = 203,000

(20,800 203,000) (1 – .32) [latex](\frac{{1 - {{\left( {1 + .0634} \right)}^{ - 5}}}}{{.0634}}[/latex]) = 635,168.23

Years 6 – 10

(580,000) (.35) = 203,000

203,000 – 10,000 = 193,000

(193,000) (1 – .32) [latex](\frac{{1 - {{\left( {1 + .0634} \right)}^{ - 5}}}}{{.0634}}[/latex]) = 547,754.55

547,654.55 / (1 .0634)5 = 402,812.29

Total PV = 1,037,981

N. Problem:  Managing Risk by Adjusting the Discount Rate at Dodson

1.

Project A (Best Project)

-185,000 + [latex]\frac{{85,000}}{{{{\left( {1 + \;.19} \right)}^1}}}[/latex] + [latex]\frac{{75,000}}{{{{\left( {1 + \;.19} \right)}^2}}} +[/latex]  [latex]\frac{{75,000}}{{{{\left( {1 + \;.19} \right)}^3}}}[/latex] + [latex]\frac{{65,000}}{{{{\left( {1 + \;.19} \right)}^4}}}[/latex] + [latex]\frac{{65,000}}{{{{\left( {1 + \;.19} \right)}^5}}}[/latex] = 43,549

Project B

-240,000 + [latex]\frac{{55,000}}{{{{\left( {1 + \;.10} \right)}^1}}}[/latex] + [latex]\frac{{65,000}}{{{{\left( {1 + \;.10} \right)}^2}}} +[/latex] [latex]\frac{{75,000}}{{{{\left( {1 + \;.10} \right)}^3}}}[/latex] + [latex]\frac{{85,000}}{{{{\left( {1 + \;.1} \right)}^4}}}[/latex] + [latex]\frac{{95,000}}{{{{\left( {1 + \;.10} \right)}^5}}}[/latex] = 37,111

Project C

-315,000 + 95,000 [latex](\frac{{1 - {{\left( {1 + .12} \right)}^{ - 5}}}}{{.12}}[/latex]) = 27,454

O. Problem:  Managing Risk through Management Options at Hansen

¹The joint probability is the probability multiplied along the decision tree, so .7*.6*.5=.210 in the top row, and .7*.60*.3=.126 in the second row, .7*.6*.2=.084 in the third row, .7*.4=.28 in the fourth row, and .3 in the fifth row.

²This is the NPV of each leg of the decision tree. Starting at the bottom, if the project is abandoned after phase 1, they spent 620,000 so that is the NPV, ie -620. If they continue to phase 2, they spend an additional 1,000,000, so the NPV in the second to the bottom row is -620 – 1,000/(1+.095) = -1,533.24. If they continue to phase 3, they spend another 9,000 and then the path splits 3 ways. Starting at the bottom, they spend an additional 3,000 in year 3, so NPV is -620 – 1,000/(1+.095) – 9,000/(1+.095)^2 – 3,000/(1+.095)^3 = -11,324.30.

³From the decision tree above.

⁴NPV – E(NPV) is the deviation from the mean and is each NPV minus the expected NPV. For the top row it is 34,232.65-6,460.72=27,771.93.

⁵This is the 4th column squared (the deviation from the mean squared) times the probability in column 2.

⁶The expected NPV or E(NPV) is the weighted average of the possible NPVs where the weights are the probabilities of each NPV occurring and is the sum of the NPV X P column.

⁷This is the sum of the probabilities times the squared deviations from the mean or expected value and is the variance of the Net Present value. See the formula below:

[latex]{\sigma ^2} = \mathop \sum \limits_{i = 1}^n \left[ {{\rho _i}{{\left( {NP{V_i} - E\left( {NPV} \right)} \right)}^2}} \right][/latex]

2. Abandonment:  The project could be abandoned at three different points over its life depending on the potential outcomes.

Flexibility:  The price could be increased in Years 4 and 5 if demand for the product was high.

Growth:  Production could be expanded in Years 4 and 5 if demand for the product was high.

P. Problem:  Managing Risk through Management Options at Acme

2. Abandonment:  The project could be abandoned at four different points over its life depending on the potential outcomes.

Growth:  Production could be expanded to take advantage of high product demand in Years 4, 5, and 6.

Q. Problem:  Complex Capital Budgeting with Spreadsheets at Magnum

1. See Excel spreadsheet
2. See Excel spreadsheet
3. Electric cars are the preferred project with a positive NPV and an IRR that exceeds its RRR.
   

   Electric Cars
   NPV	CAD 67,198,203
   IRR	24.73% (RRR = 10.00%)
   Payback Period	6 Years, 3 Months (15 Year Project)
   Solar Generators
   NPV	-CAD 5,325,555
   IRR	13.18% (RRR=16.00%)
   Payback Period	None

   NPV of the solar generators project was affected by its high-risk level (more cyclical business in a new industry) and a narrow profit margin.  Increasing prices or lowering operating costs should be explored before abandoning the project in order to generate a positive NPV.  Also, a larger factory may help the company better meet demand (plant capacity was reached) in the later years of the project which will increase its NPV.