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IEEN 5329 Homework 5 7

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IEEN 5329

Homework 5

7.3, 7.6, 7.11, 7.13, 7.25, 7.26, 7.31, 7.37

 

than one ROR possibility exists. The maximum number of is values is equal to the number of changes in the signs of the net cash flow series (Descartes' rule of signs). Also, a single positive rate can be found if the cumulative net cash flow series starts negatively and has only one sign change (Norman's criterion),                 For all cash flow, SCI ICS whew Mew is an indication M. multiple routs, a decision must be made about whether to calculate the multiple i* internal rates. or the one composite rate of return using an externally-determined reinvestment rate. This rate is commonly set at the MARR. While the internal rate is usually easier to calculate, the composite rate is the correct approach with two advantages: multiple rates of return are eliminated, and released project net cash flows are accounted for using a realistic reinvestment rate. However, the calculation of multiple rates, or the composite rate of return is often computationally involved.

If an exact ROR is not necessary, it is strongly recommended that the PR' or AW method at the MARR be used to judge economic justification.

- PROBLEMS

Understanding ROR

7.1 What does a rate of return of —100% mean?

7.2 A $10,000 loan amortized over 5 years at an interest rate of 10% per year would require payments of $2638 to completely repay the loan when interest is charged on the unrecovered balance. If interest is charged on the principal instead of the unrecovered balance, what will be the balance after 5 years if the same $2638 payments are made each year?

7.3 A-1 Mortgage makes loans with the interest paid on the loan principal rather than on the unpaid balance. For a 4-year loan of $10,000 at 10% per year, what annual payment would be required to repay the loan in 4 years if interest is charged on (a) the principal and (b) the unrecovered balance?

7.4 A small industrial contractor purchased a warehouse building for storing equipment and materials that are not immediately needed at construction job sites. The cost of the building was $100,000 and the contractor made an agreement with the seller to finance the purchase over a 5-year period. The agreement stated that monthly payments would be made based on a 30-year amortization, but the balance owed at the end of year 5 would be paid in a lump-sum balloon payment. What was the size of the balloon payment if the interest rate on the loan was 6% per year, compounded monthly?

Determination of ROR

7.5 What rate of return per month will an entrepreneur make over a 2½ year project period if he invested $150,000 to produce portable monthly costs are $27,000 with income of $33,000 per month.

7.6 The Camino Real Landfill was required to install a plastic liner to prevent leachate from migrating into the groundwater. The fill area was 50,000 square meters, and the installed liner cost was $8 per square meter. To recover the investment, the owner charged $10 for pickup loads. $25 for dump truck loads, and $70 for compactor truck loads. If the monthly distribution is 200 pickup loads. 50 dump truck loads, and 100 compactor Huck loads, what rate of return will the landfill owner make on the investment if the fill area is adequate for 4 years?

7.7 Swagelok Enterprises is a manufacturer of miniature fittings and valves. Over a 5-year period, the costs associated with one product line were as follows: first cost of $30,000 and annual costs of $18.000. Annual revenue was $27,000 and the used equipment was salvaged for $4000. What rate of return did the company make on this product?

7.8 Barron Chemical uses a thermoplastic polymer to enhance the appearance of certain RV panels. The initial cost of one process was $130.000 with annual costs of $49.000 and revenues of $78.000 in year I. increasing by $1000 per year. A salvage value of $23,000 was realized when the process was discontinued alter 8 years. What rate of return did the company make on the process?

7.9 A graduate of New Mexico State University who built a successful business wanted to start an endowment in her name that would provide scholarships to IE students. She wanted the scholarships to amount to $10,000 per year and she wanted the first one to be given on the day she made the donation (i.e., at time 0). If she planned to donate $100.000, what rate of return would the university have to make in order to be able to award the $10.000 per year scholarships forever?

7.10 PPG manufactures an epoxy amine that is used to protect the contents of polyethylene terephthalate (PET) containers from reacting with oxygen. The cash flow (in millions) associated with the process is shown below. Determine the rate of return.

Year	Cost, $	Revenue, $
0	-10	-
1	-4	2
2	-4	3
3	-4	9
4	-3	9
5	-3	9
6	-3	9

 

7.11 An entrepreneurial mechanical engineer started a tire shredding business to take advantage of a Texas state law that out-laws the disposal of whole tires in sanitary landfills. The cost of the shredder was $220.000. She spent $15.000 to get 460-volt power to the site and another $76,000 in site preparation. Through con-tracts with tire dealers, she was paid $2 per tire and handled an average of 12.000 tires per month for 3 years. The annual operating costs for labor, power, and repairs. etc.. Amounted to $1.05 per tire. She also sold some of the tire chips to septic tank In-stallers for use in drain fields. This endeavor netted $2000 per month. After 3 years she sold the equipment for $100,000. What rate of return did she make (a) per month and (b) per year (nominal and effective)?

7.12 An Internet B to C company projected the cash flows (in millions) What annual rate of return will be realized if the cash flows occur as projected?

Year	Expenses, $	Revenue, $
0	-40	-
1	-40	12
2	-43	15
3	-45	17
4	-46	51
5	-48	63
6-10	-50	80

 

7.13 The University of California at San Diego is considering a plan to build an 8-megawatt cogeneration plant to provide for part of its power needs. The cost of the plant is expected to be $41 million. The university consumes 55,000 megawatt-hours per year at a cost of $120 per megawatt-hour (a) If the university will be able to produce power at one-half the cost that it now pays. What rate of return will it make on its investment if the power plant lasts 30 years? (b) if the university can sell an average of 12,000 megawatt-hours per year back to the utility at $90 per megawatt-hour. What rate of re-turn will it make?

7.14 A new razor from Gillette called the M3Power emits pulses that cause the skin to prop up hair so that it can be cut off more easily. This might make the blades last longer because there would be less need to repeatedly shave over the same surface. The M3Power system (including batteries) sells for $14.99 at some stores. The blades cost $10.99 for a package of four. The more conventional M3Turbo blades cost $7.99 for a package of four. If the blades for the M3Power system last 2 months while the blades for the M3Turbo last only 1 month. What rate of return (a) per month and (b) per year (nominal and effective) will be made if a person purchases the M3Power system? Assume the person already has an M3Turbo razor but needs to purchase blades at time 0. Use a 1-year project period.

7.15 Techstreet.com is a small Web design business that provides services for two main types of websites: brochure sites and e-commerce sites. One package involves an up-front payment of $90,000 and monthly payments of 1.4c per "hit." A new CAD software company is considering the package. The company expects to have at least 6000 hits per month, and it hopes that 1.5% of the hits will result in a sale. If the average income from sales (after fees and expenses) is $150. What rate of return per month will the CAD software company realize if it uses the website for 2 years?

7.16 A plaintiff in a successful lawsuit was awarded a judgment of $4800 per month for 5 years. The plaintiff needs a fairly large sum of money now for an investment and has offered the defendant the opportunity to pay off the award in a lump-sum amount of $110,000. If the defendant accepts the offer and pays the $110.000 now, what rate of return will the defendant have made on the "investment"? Assume the next $4800 payment is due 1 month from now.

7.17 Army Research Laboratory scientists developed a diffusion-enhanced adhesion process that is expected to significantly improve the performance of multifunction hybrid composites. NASA engineers estimate that composites made using the new process will result in savings in many space exploration projects. The cash flows for one project are shown below. Deter-mine the rate of return per year.

Year t	Cost ($1000)	Saving ($1000)
0	-210	-
1	-150	-
2-5	-	100+60(t-2)

 

7.18 ASM International, an Australian steel company, claims that a savings of 40% of the cost of stainless steel threaded bar can he achieved by replacing machined threads with precision weld depositions. A U.S. manufacturer of rock bolts and grout-in fittings plans to purchase the equipment. A mechanical engineer with the company has prepared the following cash flow estimates. Determine the expected rate of return per quarter and per year (nominal).

Quarter	Cost, $	Saving, $
0	-450,000	-
1	-50,000	10,000
2	-40,000	20,000
3	-30,000	30,000
4	-20,000	40,000
5	-10,000	50,000
6-12	-	80,000

 

7.19 An indium-gallium-arsenide-nitrogen alloy developed at Sandia National Laboratory is said to have potential uses in electricity-generating solar cells. The new material is expected to have a longer life, and it is believed to have a 40% efficiency rate. Which is nearly twice that of standard silicon solar cells. The useful life of a telecommunications satellite could be ex-tended from 10 to 15 years by using the new solar cells. What rate of return could be realized if an extra Investment now of $950,000 would result in extra revenues of $450,000 in year 11. $500.000 in year 12 and amounts increasing by $50,000 per year through year 15?

7.20 A permanent endowment at the University of Alabama is to award scholarships to engineering students. The awards are to be made beginning 5 years after the $10 million lump-sum donation is made. If the interest from the endowment is to fund 100 students each year in the amount of $10,000 each. What annual rate of return must the endowment fund earn?

7.21 A charitable foundation received a donation from a wealthy building contractor in the amount of $5 million. It specifies that $200.000 is to be awarded each year for 5 years starting now (i.e., 6 awards) to a university engaged in research pertaining to the development of layered composite materials. Thereafter grants equal to the amount of interest earned each year are to be made. If the size of the grants from year 6 into the indefinite future is expected to be $1,000,000 per year, what annual rate of return is the foundation earning?

Multiple ROR Values

7.22 What is the difference between a conventional and a nonconventional cash flow series?

7.23 What cash flows are associated with Descartes' rule of signs and Norstrom's criterion?

7.24 According to Descartes' rule of signs, how many possible i* values are there for net cash flows that have the following signs?

 (a) ---+++-+ 

(b)  --------------+++++ 

(c) ++---------+-+---- 

7.25 The cash flow (in 1000s) associated with a new method of manufacturing box cutters is shown on page 268 for a 2-year period. (a) Use Descartes' rule to determine the maximum number of possible rate of re-turn values. (b) Use Norstrom’s criterion to determine if there is only one positive rate of return value.

Quarter	Expense, $	Revenue, $
1	-20	0
1	-20	5
2	-10	10
3	-10	25
4	-10	26
5	-10	20
6	-15	17
7	-12	15
8	-15	2

 

7.26 RKI Instruments manufactures a ventilation controller designed for monitoring and controlling carbon monoxide in parking garages, boiler rooms, and tunnels. etc. The net cash flow associated with one phase of the operation is shown below. (a) How many possible rate of return values are there for this cash flow series? (b) Find all the rate of return values between 0% and 100%,

Year	Net Cash Flow, $
0	-30,000
1	20,000
2	15,000
3	-2,000

 

7.27 A manufacturer of heavy-tow carbon fibers (used for sporting goods. thermo-plastic compounds. windmill blades. etc.) reported the net cash flows below. (a) Determine the number of possible rate of return values and (b) find all rate of return values between -50% and 120%.

Year	Net Cash Flow, $
0	-17000
1	20,000
2	-5,000
3	8,000

 

7.28 Are-hot Technologies. manufacturers of six-axis, electric servo-driven robots, has experienced the cash flows below in a shipping department. (a) Determine the number of possible rate of return values. (b) Find all i* values between 0% and 100%

Year	Expense, $	Saving, $
0	-33,000	0
1	-15,000	18,000
2	-40,000	38,000
3	-20,000	55,000
4	-13,000	12,000

 

 

7.29 Five years ago, a company made a $5 million investment in a new high-temperature material. The product was not well accepted after the first year on the market. However, when it was reintroduced 4 years later, it did sell well during the year. Major research funding to broaden the applications has cost $15 million in year 5. Deter-mine the rate of return for these cash flows (shown below in $1000s).

Year	Net Cash Flow, $
0	-5,000
1	4,000
2	0
3	0
4	20,000
5	-15,000

 

 

Composite Rate of Return

 7.30 What is meant by the term reinvestment rate?

7.31 An engineer working for GE invested his bonus money each year in company stock. His bonus has been $5000 each year for the past 6 years (i.e.. at the end of years I through 6) At the end of year 7. he sold $9000 worth of his stock to remodel his kitchen (he didn't purchase any stock that year). In years 8 through 10, he again invested his $5000 bonus. The engineer sold all his remaining stock for $50,000 immediately after the last investment at the end of year 10. (a) Determine the number of possible rate of return values in the net cash flow series. (b) Find the internal rate of return (s). (c) Determine the composite rate of return. Use a reinvestment of 20% per year.

 7.32 A company that makes clutch disks for race cars had die cash flows shown below for one department. Calculate (a) the internal rate of return and (b) the composite rate of return, using a reinvestment rate of 15% per year.

Year	Cash Flow, $1000
0	-65
1	30
2	84
3	-10
4	-12

 

7.33 For the cash flow series below. Calculate the composite rate of return, using a reinvestment rate of 14% per Year.

Year	Cash Flow, $
0	3000
1	-2000
2	1000
3	-6000
4	3800

 

7.34 For the high-temperature material project in Problem 7.29, determine the composite rate or return if the reinvestment rate is 15% per year. The cash flows (repeated below) are in $1000 units.

Year	Cash Flow, $
0	-5000
1	4000
2	0
3	0
4	20,000
5	-15,000

 

 Bonds

7.35 A municipal bond that was issued by the city of Phoenix 3 years ago has a face value of $25,000 and a bond interest rate of 6% per year payable semiannually. If the bond is due 25 years after it was issued. (a) What are the amount and frequency of the bond interest payments and (b) what value of n must be used in the P/A formula to find the present worth of the remaining bond interest payments? Assume the market interest rate is 8% per year. Compounded semiannually.

7.36 A $10,000 mortgage bond with a bond interest rate of 8% per year, payable quarterly, was purchased for $9200. The bond was kept until it was due, a total of 7 years. What rate of return was made by the purchaser per 3 months and per year (nominal)?

7.37 A plan for remodeling the downtown area of the city of Steubenville. Ohio, required the city to issue $5 million worth of general obligation bonds for infrastructure re-placement. The bond interest rate was set at 6% per year, payable quarterly with the principal repayment date 30 years into the future. The brokerage fees for the transactions amounted to $100.000. If the city received $4.6 million (Wore paying the brokerage fees) from the bond issue, (a) what interest rate (per quarter) did the investors require to purchase the bonds and (b) what are the nominal and effective rates of return per year to the investors?

7.38 A collateral bond with a face value of $5000 was purchased by an investor for $4100. The bond was due in 11 years, and it had a bond interest rate of 4% per year, payable semiannually. If the investor kept the bond to maturity, what rate of return per semiannual period did she make?

7.39 An engineer planning for his child's college education purchased a rem coupon corporate bond (i.e., a bond that has no interest payments) for $9250. The bond has a face value of $50.000 and is due in 18 years. If the bond is held to maturity, what rate of return will the engineer make on the investment?

7.40 Four years ago. Texaco issued $5 million worth of debenture bonds having a bond interest rate of 10% per year, Payable semiannually. Market interest rates dropped, and the company called the bonds (i.e.. paid them off in advance) at a 10% premium on the face value (i.e.. paid $5.5 million to retire the bonds). What semiannual rate of return did an investor make if he purchased one $5000 bond at face value 4 years ago and held it until it was called 4 years later?

7.41 Five years ago. GSI. an oil services company, issued $10 million worth of 12%. 30-year bonds with interest payable quarterly. The interest rate in the marketplace decreased enough that the company is considering calling the bonds. If the company buys the bonds back now for $11 million, (a) what rate of return per quarter will the company make on the $11 million expenditure and (b) what nominal rate of return per year will it make on the $11 million investment? Hint: By spending $11 million now the company will not have to make the quarterly bond interest payments or pay the face value of the bonds when they come due 25 years from now.

FE REVIEW PROBLEMS

7.42 When the net cash flow for an alternative changes signs more than once, the cash flow is said to be (a) Conventional

(b) Simple

(c) Extraordinary

(d) Nonconventional

7.43 According to Descartes' rule of signs, how many possible rate of return values are there for net cash flow that has the following signs?

++++------+-+---++

(a) 3

(b) 5

(c) 6

(d) Less than 3

 

7.44 A small manufacturing company borrowed $1 million and repaid the loan through monthly payments of $20.000 for 2 years plus a single lump-sum payment of $1 million at the end of 2 years. Thc interest rate on the loan was closest to

(a) 0.5% per month

(b) 2% per month

(c) 2% per year

(4) 84% per year

7.45 According to Norstrom's criterion, there is only one positive rate of return value in a cash flow series when

(a) The cumulative cash flow starts out positive and changes sign only once

(b) The cumulative cash flow starts out negative and changes sign only once.

(c) The cumulative cash flow total is greater than zero.

(d) The cumulative cash flow total is less than zero.

7.46 An investment of $60,000 resulted in uniform income of $10,000 per year for 10 years. The rate of return on the investment was closest to

(a) 10.6% per year (6) 14.2% per year (c) 16.4% per year (d) 18.6% per year

7.47 For the net cash flows shown below, the maximum number of possible rate of return solutions is (a) 0 (b) 1 (c) 2 (d) 3

Year	Net Cash Flow,$
0	-60,000
1	20,000
2	22,000
3	15,000
4	35,000
5	13,000
6	-2,000

 

7.48 A bulk materials hauler purchased a used dump truck for $50,000. The operating cost was $5000 per month with average revenues of $7500 per month. After 2 years, the truck was sold for $11,000. The rate of return was closest to

(a) 2.6% per month (b) 2.6% per year (e) 3.6% per month (d) 3.6% per year

7.49 Assume you are told that by investing $100,000 now, you will receive $10,000 per year starting in year 5 and continuing forever. If you accept the offer, the rate of return on the investment is

(a) Less than 10% per year (b) 0% per year (c) 10% per year (d) Over 10% per year

7.50 Five years ago, an alumnus of a small university donated $50,000 to establish a permanent endowment for scholarships. The first scholarships were awarded 1 year after the money was donated. If the amount awarded each year (i.e.. the interest) is $4500, the rate of return carnal on the fund is closest to

(a) 7.5% per year (b) 8.5% per year (c) 9% per year (d) 10% per year

7.51 When positive net cash flows are generated before the end of a project, and when these cash flows are reinvested at an interest rate that is greater than the internal rate of return,

(a) The resulting rate of return is equal to the internal rate of return.

(b) The resulting rate of return is less than the internal rate of return.

(c) The resulting rate of return is equal to the reinvestment rate of return.

(d) The resulting rate of return is greater than the internal rate of return.

7.52 A $10,000 mortgage bond that is due in 20 years pays interest of $250 every 6 months. The bond interest rate is closest to

(a) 2.5% per year, payable quarterly

(b) 5.0% per year, payable quarterly

(c) 5% per year, payable semiannually

(d) 10% per year, payable quarterly

7.53 A $10,000 bond that matures in 20 years with interest at 8% per year payable quarterly was issued 4 years ago. 1i the bond is purchased now for $10.000 and held to maturity, what will be the effective rate of return per quarter to the purchaser?

(a) 2% (b) 2.02% (c) 4% (d) 8%

7.54 A person purchases a $5000. 5% per year bond, with interest payable semiannually, for the amount of $4000. The bond has a maturity date of 14 years from now. The equation for calculating how much the person must sell the bond for 6 years from now in order to make a rate of return of 12% per year, compounded semiannually, is

(a) 0 = -4000 + 125(P/A,6%,12) + x(P/F,6%12)

(b) 0 = -4000+ 100(P/A,6%.12)+ x(P/ F.6%.12)

(c) 0 = -5000 + 125(P/A.6%.12) + x(P/F.6%.12)

(d) ) 0 = -4000 + 125(P/A,12%.6) x(P/F.I2%.6)

7.55 A $50.000 corporate bond due in 20 years with an interest rate of 10% per year, payable quarterly, is for sale for $50.000. If an investor purchases the bond and holds it to maturity, the rate of return will be closest to

(a) Nominal 10% per year, compounded quarterly (b) 2.5% per quarter (c) Both (a) and (b) are correct (d) Effective 10% per year

EXTENDED EXERCISES

 EXTENDED EXERCISE I—THE COST OF A POOR CREDIT RATING

Two people each borrow $5000 at a 10% per year interest rate for 3 years. A portion of Charles's loan agreement states that interest "... is paid at the rate of 10% compounded each year on the declining balance." Charles is told his annual payment will be $2010.57, due at the end of each year of the loan. Jeremy currently has a slightly degraded credit rating which the bank loan officer discovered. Jeremy has a habit of paying his hills late. The bank approved the loan, but a part of his loan agreement states that interest "... is paid at a rate of 10% compounded each year on the original loan amount." Jeremy is told his annual payment will be $2166.67 due at the end of each year.

Questions

Answer the following by hand, by computer, or both.

1. Develop a table and a plot for Charles and for Jeremy of the unrecovered balances (total amount owed) just before each payment is due.

2. How much more total money and interest will Jeremy pay than Charles over the 3 years?

EXTENDED EXERCISE 2—WHEN IS IT BEST TO SELL A BUSINESS?

After Jeff finished medical school and Imelda completed a degree in engineering, the couple decided to put a substantial part of their savings into rental property.

With a hefty bank loan and a cash down payment of $120,000 of their own funds, they were able to purchase six duplexes from a person exiting the residential rental business. Net cash flow on rental income after all expenses and taxes for the first 4 years was good: $25,000 at the end of the first year. Increasing by $3000 each year thereafter. A business friend of Jeff's introduced him to a potential buyer for all properties with an estimated $225,000 net cash out after the 4 years of ownership. But they did not sell. They wanted to stay in the business for a while longer, given the increasing net cash flows they had experienced thus far.

During year 5, an economic downturn reduced net cash flow to $35,000. In response, an extra $20,000 was spent in improvements and advertising in each of years 6 and 7. But the net cash flow continued to decrease by $10,000 per year through year 7. Jeff had another offer to sell in year 7 for only $60.000. This was considered too much of a loss, so they did not take advantage of the opportunity. In the last 3 years, they have expended $20,000, $20,000, and $30,000 each year in improvements and advertising costs but the net cash flow from the business has been only $15,000. $10,000, and $10,000 each year.                             Imelda and Jeff want out, but they have no offer to buy at any price, and they have most of their savings committed to the rental property.

Questions

Determine the rate of return for the following:

1. At the end of year 4, first. if the $225,000 purchase offer had been accepted: second, without selling. 2 After 7 years, first, if the $60,000 "sacrifice" offer had been accepted and second, without selling.

3. Now, after 10 years, with no prospect of sale.

4. If the houses are sold and given to a charity, assume a net cash infusion to Jeff and Imelda of $25,000 after taxes at the end of this year. What is the rate of return over the 10 years of ownership?

CASE STUDY

BOB LEARNS ABOUT MULTIPLE RATES OF RETURN'

Background

'Contributed by Dr. Tep Sastri (former Amax: hue Professor. Industrial Engineering. Texas A&M University).

 

 When Bob began a 'summer internship with VAC. an electricity distribution company in an Atlantic coast city of about 275,000, he was given a project on the first day by his boss, Kathy. Home worth one of the major corporate customers, just placed a request for a lower rate per kwh. Once its minimum required usage is exceeded each month. Kathy has an internal report from the Customer Relations Department that itemizes the net cash flows below for the Home worth account during the last 10 years.

Year	Cash Flow ($1000)
1993	$200
1994	100
1995	50
1996	-1800
1997	600
1998	500
1999	400
2000	300
2001	200
2002	100

 

The report also states that the annual rate of return is between 25 and 50%, but no further information is provided. This information is not detailed enough for Kathy to evaluate the company's request.

                  Over the next few hours. Bob and Kathy had a series of discussions as Bob worked to answer Kathy's increasingly more specific questions. The following is an abbreviated version of these conversations. Luckily, both Bob and Kathy took an engineering economy course during their undergraduate work, and their professors coveted the method 10 find a unique rate of re-turn for any cash flow series.

Development of the Situation

1. Kathy asked Bob to do a preliminary study to find the correct rate of return. She wanted only one number, not a range, and not two or three possible values. She did, however, have a passing interest in initially knowing the values of multiple rates. If they do exist, in order to determine if the report from customer relations was correct or just a "shot in the dark."

               Kathy told Bob that the MARK for the company is 15% per year for these major clients. She also explained that the 1996 negative cash flow was caused by an on-site equipment upgrade when Homeworth expanded its manufacturing capacity and increased power usage about 5-fold.

2. Once Bob had finished his initial analysis, Kathy told him that she had forgotten to tell him that the rate of return earned externally on the positive cash flows from these major clients is placed into a venture capital pool headquartered in Chicago. It has been making 35% per year for the last decade. She wanted to know if a unique return still existed and if the Homeworth account was financially viable at a MARR of 35%.

In response to this request, Bob developed the four-step procedure outlined below to closely estimate the composite rate of return i' for any reinvestment rate c and two multiple rates i₁* and i₂*. He plans to apply this procedure to answer this latest question and show the result to Kathy.

Step 1. Determine the i* roots of the PW relation for the cash flow series.

Step 2. For a given reinvestment rate c and the two i* values from step 1, determine which of the following conditions applies:

(a) If c <i*₁ ,then I’ <i₁.

(b) If c> <i*₂ then I’ <i₂

(c) If  i*₁ <c< i*_1,  then I’ can be less than c or greater than c₁  and i_{1 <I’<i}2

Step 3. Guess a starting value for I’ according to the result from step 2. Apply the net-investment method from periods 1 to n. Repeat this step until F, is close to 0. If this F_n  is a small positive value, guess another i that will result in a small negative value, and vice

Step 4. Using the two F. results from step 3, linearly interpolate i' such that the corresponding F,, is approximately zero. Off course, the final I’ value can also be obtained directly in step 3, Without Interpolation.

3. Finally. Kathy asked Bob to reevaluate the cash flows for Homeworth at the MARR of 35%, but using a reinvestment rate of 45% to determine if the series is still justified.

4. If the I’ approximating procedure Bob developed is not available, use the original cash flow data to apply the basic net-investment procedure, and answer Exercises 2 and 3, where c is 35 and 45% respectively.

5. Kathy concluded from this exercise that any cash flow series is economically justified for any rein-vestment rate that is larger than the MARR. Is this a correct conclusion? Explain why or why not.

 

Case Study Exercises

1, 2, and 3. Answer the questions for Bob using spreadsheets.