Homework answers / question archive / Please complete the questions 2

Please complete the questions 2

Economics

Share With

---

Please complete the questions 2.32, 2.42, 2.53, 2.57, 3.4, 3.15, 3.19, 3.26, 3.29, 3.36, 3.42, 3.49, 4.18, 4.25 in the textbook 6th edition

 

 

CHAPTER 1 Foundations of Engineering Economy

PROBLEMS Basic Concepts 1.1 What is meant by the term time value of money?

1.2 List three intangible factors.

1.3 (a) what is meant by evaluation criterion?

      (b) What is the primary evaluation criterion used in economic analysis?

1.4 List three evaluation criteria besides the economic one for selecting the best restaurant.

1.5 Discuss the importance of identifying alternatives in the engineering economic process.

1.6 What is the difference between simple and compound interest?

1.7 What is meant by minimum attractive rate NJ of return?

1.8 What is the difference between debt and equity financing? Give an example of each.

Interest Rate and Rate of Return

1.9 Trucking giant Yellow Corp agreed to purchase rival Roadway for $966 million in order to reduce so-called back-office costs (e.g., payroll and insurance) by $45 million per year. If the savings were realized as planned, what would be the rate of return on the investment?

1.10 If Ford Motor Company's profits in-creased from 22 cents per share to 29 cents per share in the April—June quarter com-pared to the previous quarter, what was the rate of increase in profits for that quarter? 1.11 A broadband service company borrowed $2 million for new equipment and repaid the principal of the loan plus $275,000

Interest after 1 year. What was the interest rate on the loan?

1.12 A design-build engineering firm completed a pipeline project wherein the company realized a profit of $2.3 million in 1 year. If the amount of money the company had invested was $6 million, what was the rate of return on the investment?

1.13 US Filter received a contract for a small water desalting plant whereby the company expected to make a 28% rate of return on its investment. If the company invested $8 million in equipment the first year, what was the amount of the profit in that year? 1.14 A publicly traded construction company reported that it just paid off a loan that it received 1 year earlier. If the total amount of money the company paid was $1.6 mil-lion and the interest rate on the loan was 10% per year, how much money did the company borrow 1 year ago? 1.15 A start-up chemical company has established a goal of making at least a 35% per year rate of return on its investment. If the company acquired $50 million in venture capital, how much did it have to earn in the first year?

Equivalence 1.16 At an interest rate of 8% per year, $10,000 today is equivalent to how much (a) 1 year from now and (b) 1 year ago? 1.17 A medium-size consulting engineering \/ firm is trying to decide whether it should replace its office furniture now or wait and do it 1 year from now. If it waits 1 year, the cost is expected to be $16,000. At an interest rate of 10% per year, what would be the equivalent cost now?

Page 2

1.18 An investment of $40,000 one year ago and $50,000 now are equivalent at what interest rate?

1.19 At what interest rate would $100,000 now be equivalent to $80,000 one year ago?

Simple and Compound Interest

 1.20 Certain certificates of deposit accumulate interest at 10% per year simple interest. If a company invests $240,000 now in these certificates for the purchase of a new machine 3 years from now, how much will the company have at the end of the 3-year period? I.21 /A local bank is offering to pay compound V interest of 7% per year on new savings accounts. An e-bank is offering 7.5% per year simple interest on a 5-year certificate of deposit. Which offer is more attractive to a company that wants to set aside $1.000, 000 now for a plant expansion 5 years from now? 1.22 Badger Pump Company invested 8500.000 five years ago in a new product line that is now worth $1.000, 000. What rate of return did the company earn (a) on a simple interest basis and (6) on a com-pound interest basis?

1.23 How long will it take for an investment to double at 5% per year (a) simple interest and (b) compound interest?

1.24 A company that manufactures regenerative thermal oxidizers made an investment 10 years ago that is now worth $1,300,000. How much was the initial investment at an interest rate of 15% per year (a) simple interest and (b) compound interest?

1.25 Companies frequently borrow money under an arrangement that requires them to make periodic payments 4 (only interest and then pay the principal of the loan all at once. A company that manufactures odor control chemicals borrowed $400.000 for 3 years at 10% per year compound interest under such an arrangement. What is the difference in the total amount paid between this arrangement (identified as plan I) and plan 2, in which the company makes no interest payments until the loan is due and then pays it off in one lump sum?

1.26 A company that manufactures in-line mix-as for bulk manufacturing is considering borrowing $1.75 million to update a production line. If it borrows the money now, it can do so at an interest rate of 7.5% per year simple interest for 5 years. If it borrows next year, the interest rate will be 8% per year compound interest, but it will be for only 4 years. (a) How much interest (total) will be paid under each scenario. and (b) should the company borrow now or I year from now? Assume the total amount due will be paid when the loan is due in either case.

Symbols and Spreadsheets

1.27 Define the symbols involved when a construction company wants to know how much money it can spend 3 years from now in lieu of spending $50,000 now to purchase a new truck, when the compound interest rate is 15% per year.

1.28 State the purpose for each of the following built-in Excel functions:

(a) FV(i%,nd 4.p)

(a) FV(i%,n,A,P)

(b) IRR(First_cell: Last_cell)

(c ) PMT(i%,n,P,F)

(d) PV)(i%n,A,F)

1.29 What are the values of the engineering economy symbols P, F, A, i. and n in the following Excel functions? Use a ? for the symbol that is to be determined.

(a) FV(7%,10,5000,9000)

(b) PMT (11%,20,14000)

(c)PV(8%,15,1000,800)

 

130 Write the engineering economy symbol that corresponds to each of the following Excel functions. (a) PV

(b) PMT

(c) NPER

(d) IRR

(e) FV

1.31 In a built-in Excel function, If a certain parameter does not apply, under what circumstances can it be left blank? When must a comma be entered in its place?

MARR and Cost of Capital

1.32 Identify each of the following as either a safe investment or a risky one.

(a) New restaurant business

(b) Savings account in a bank

(c) Certificate of deposit

(d) Government bond

(e) Relatives "get-rich-quick" idea

1.33 Identify each of the following as either equity or debt financing.

(a) Money from savings

(b) Money from a certificate of deposit

(c) Money from a relative who is a partner in the business

(d) Bank loan

(e) Credit card

1.34 Rank the following from highest to lowest rate of return or interest rate: government bond, corporate bond, credit card, bank loan to new business, interest on checking account.

1.35 Rank the following from highest to lowest interest rate: cost of capital, acceptable rate of return on a risky investment, mini-mum attractive rate of return, rate of return on a safe investment, interest on checking account, interest on savings account.

1.36 Five separate projects have calculated rates of return of 8. 11, 12.4, 14, and 19% per year. An engineer wants to know which projects to accept on the basis of rate of return. She learns from the finance department that company funds, which have a cost of capital of 18% per year are commonly used to fund 25% of all capital projects. Later, she is told that borrowed money is currently costing 10% per year if the MARR is established at exactly the weighted average cost of capital, which projects should she accept?

Cash Flows

 1.37 What is meant by the end of period convention?

1.38 Identify the following as cash inflows or outflows to Daimler-Chrysler income taxes, loan interest, salvage value, rebates to dealers, sales revenues, accounting services, coat reductions.

1.39 Construct a cash flow diagram for the following cash flows: $10,000 outflow at time zero, $33030 per year outflow in years 1 through 3 and $9000 inflow in years 4 through 8 at an interest rate of 10% per year, and an unknown future amount in year 8.

1.40 Construct a cash flow diagram to find the present worth of a future outflow 01 $40,000 in year 5 at an interest rate of 15% per year.

Doubling the Value

 1.41 Use the rule of 72 to estimate the time i would take for an initial investment $10.000 to accumulate to $20.000 at: compound rate of 8% per year.

1.42 Estimate the time it would take (according to the rule of 72) for money to quadruple in value at a compound interest rate 9% per year.

 (b) Since than is no A = $550 in year 5, the F/ A factor cannot be used_ The relation F = 550(F/A,I,4) would furnish the future worth in year 4, out year 5.

(c) The first gradient amount G - $100 occurs in year 3. Use of the relation P_g - 100(P/G,i%,4) will compute P_G in year 1, not year 0. (The present wash of the base amount of $1000 is not included here) (d) The receipt values are unequal: thus the relation F= A (F/A,I,3) cannot be used to compute F.

CHAPTER SUMMARY

Formulas and factors derived and applied in this chapter perform equivalence calculations for present, future, annual, and gradient cash flows. Capability in using these formulas and their standard notation manually and with spreadsheets is critical to complete an engineering economy study. Using these formulas and spreadsheet functions, you can convert single cash flows into uniform cash flows,gradients into present worth’s, and much more. You can solve for rate of return i or time it. A thorough understanding of how to manipulate cash flows using the material in this chapter will help you address financial questions in professional practice as well as in everyday living.

PROBLEMS

Use of interest Tables

2.1 Find the correct numerical value for the following factors from the interest tables.

I. (F/P, 8%.25)

2. (P/A, 4, 3%,8)

3. (P/G, 9%,20)

4. (F/A.15%, 18)

5. (A/P,30% ,15)

Determination of F, P, and A

2.2 The U.S. Border Patrol is considering the purchase of a new helicopter for aerial surveillance of the New Mexico-Texas border with Mexico. A similar helicopter was purchased 4 years ago at a cost of $140,000. At an interest rate of 7% per year, what would be the equivalent value today of that $140,000 expenditure?

2.3 Pressure Systems, Inc., manufactures high-accuracy liquid-level transducers. It is investigating whether it should update certain equipment now or wait to do it later. If the cost now is $200,000, what will the equivalent amount be 3 years from now at an interest rate of 10% per year?

2.4 Petroleum Products. Inc., is a pipeline company that provides petroleum products to wholesalers in the northern United States and Canada. The company is considering purchasing insertion turbine flow meters to allow for better monitoring of pipeline integrity. If these meters would prevent one major disruption (through early detection of product loss) valued at $600,000 four years from now, how much could the company afford to spend now at an interest rate of 12% per year?

2.5 Sensotech Inc., a maker of micro electro mechanical systems, believes it can reduce product recalls by 10% if it purchases new software for detecting faulty parts. The cost of the new software is $225,000 (a) How much would the company have to save each year 4 years to recover its in-vestment if it uses a minimum attractive rate of return of 15% per year? (b) What was the cost of recalls per year before the software was purchased if the company did exactly recover its investment in 4 years from the 10% reduction?

2.6 Thompson Mechanical Products is planning to set aside $150,000 now for possibly replacing its large synchronous refiner motors whenever it becomes necessary. If the replacement isn't needed for 7 years, how much will the company have in its in-vestment set-aside account if it achieves a rate of return of 18% per year?

2.7 French car maker Renault signed a $75 million contract with ABB of Zurich, Switzerland, for automated underbody assembly lines, body assembly workshops, and fine control systems. If ABB will be paid in 2 years (when the systems am ready), what is the present worth of the contract at 18% per year interest?

2.8 Atlas Long-Haul Transportation is considering installing Valutemp temperature log-gets in all of its refrigerated trucks for monitoring temperatures during transit. If the systems will reduce insurance claims by $100,000 two years from now, how much should the company be willing to spend now if it uses an interest rate of 12% per year?

2.9 GE Marine Systems is planning to supply a Japanese shipbuilder with aero-derivative gas turbines to power II DD-class destroyers for the Japanese Self-Defense Force. The buyer can pay the total contract price of $1,700,000 now or an equivalent amount 1 year from now (when the turbines will be needed). At an interest rate of 18% per year. What is the equivalent future amount?

2.10 What is the present worth of a future cost of $162,000 to Corning, Inc., 6 years from now at an an interest rate of 12% per year?

2.11 How much could Cryogenics Inc., a maker of superconducting magnetic energy storage systems, afford to spend now on new equipment in lieu of spending $125,000 five years from now if the company's rate of return is 14% per year?

2.12 V-Tek Systems is a manufacturer of vertical compactors, and it is examining its cash flow requirements for the next 5 years. The company expects to replace office machines and computer equipment at various times over the 5-year planning period. Specifically, the company expects to spend $9000 two years from now, $8000 three years from now, and $5000 five years from now. What is the present worth of the planned expenditures at an interest rate of 10% per year?

2.13 A proximity sensor attached to the tip of an endoscope could reduce risks during eye surgery by alerting surgeons to the location of critical retinal tissue. If a certain eye surgeon expects that by using this technology, he will avoid lawsuits of $1.25 and $0.5 million 2 and 5 years from now, respectively, how much could he afford to spend now if his out-of-pocket costs for the lawsuits would be only 10% of the mat amount of each suit? Use an interest rate of 8% per year.

2.14 The current cost of liability insurance for a certain consulting firm is $65,000. If the insurance cost is expected to increase by 4% each year, what will be the cost 5 years from now?

2.15 American Gas Products manufactures a device called a Can-Emitor that empties the contents of old aerosol cans in 2 to 3 seconds. This eliminates having to dispose of the cans as hazardous wastes. If a certain paint company can save $75,000 per year in waste disposal costs, how much could the company afford to spend now on the Can-Emitor if it wants to recover its investment in 3 years at an interest rate of 20% per year?

2.16 Atlantic Metals and Plastic uses austenitic nickel-chromium alloys to manufacture resistance heating wire. The company is considering a new annealing-drawing process to reduce costs. If the new process will cost 51.8 million now, how much must be saved each year to recover the in-vestment in 6 years at an interest rate of 12%per year?

2.17 A green algae, Chlamydomonas reinhardtii, can produce hydrogen when temporarily deprived of sulfur for up to 2 days at a time a small company needs to purchase equipment costing $3.4 million to commercialize the process. If the company wants to earn a rate of return of 20% per year and recover it investments in 8 years, what must be the net value of the hydrogen produced each year?

2.18How much money could RTT Environmental services borrow to finance a site reclamation project if it expects revenues of $280,000 per year over a 5-year cleanup period? Expenses associated with the project art expected to be $90,000 per year. Assume the interest rate is 10% per year.

2 19 Western Playland and Aquatics Park spends $75,000 each year in consulting services for ride inspection. New actuator element technology enables engineers to simulate complex computer-controlled movements in any direction. How much could the amusement park afford to spend now on the new technology if the annual consulting services will no longer be needed? Assume the park uses an interest rate of 15% per year and it wants to re-cover its investment in 5 years.

2.20 Under an agreement with the Internet Service Providers (ISPs) Association, SBC Communications reduced the price it charges ISPs to resell its high-speed digital subscriber line (DSL) service from $458 to $360 per year per customer line. A particular ISR which has 20,000 customers, plans to pass 90% of the savings along to its customers. What is the total future worth of these savings over a 5-year horizon at an interest rate of 8% per year?

2.21 To improve crack detection in aircraft, the U.S. Air Force combined ultrasonic inspection procedures with laser heating to identify fatigue cracks. Early detection of cracks may reduce repair costs by as much as $200,000 per year. What is the present worth of these savings over a 5-year period at an interest rate of 10% per year?

2.22 A recent engineering graduate passed the FE exam and was given a raise (beginning in year 1) of 8% per year. What is the present value of the $2000 per year over the expected 35 year career?

2.23 Southwestern Moving and Storage wants to have enough money to purchase a new tractor-trailer in 3 years. If the unit will cost $250,000, how much should the company set aside each year if the account earns 9% per year?

2.24 Vision Technologies, Inc., is a small company that uses ultra-wideband technology to develop devices that can detect objects (including people) inside buildings, be-hind walls, or below ground. The company expects to spend $100,000 per year for labor and $125,000 per year for sup-plies before a product can be marketed. At an interest rate of 15% per year, what is the total equivalent future amount of the company's expenses at the end of 3 years?

Factor Values

2.25 Find the numerical value of the following factors by (a) interpolation and (b) using the appropriate formula.

1. (P/F, 18%, 33)

2. (A/G), 12%, 54)

2.26 Find the numerical value of the following factors by (a) interpolation and (b) using the appropriate formula. I. (F/A, 19%, 20) 2. (P/A, 26%, 15)

 

Arithmetic Gradient

2.27 A cash flow sequence starts in year 1 at $3000 and decreases by $200 each year through year 10. (a) Determine the value of the gradient G; (b) determine the amount of cash flow in year 8; and (c) determine the value of n for the gradient.

2.28 Cisco Systems expects sales to be de-scribed by the cash flow sequence (6000 + 5k), where k is in years and cash flow is in millions. Determine (a) the value of the gradient G; (b) the amount of cash flow in year 6; and   (c) the value of n for the gradient if the cash flow ends in year 12.

 

2.29 For the cash flow sequence that starts in year 1 and is described by 900 — 100k, where k represents years 1 through 5, (a) determine the value of the gradient G and (b) determine the cash flow in year 5.

2.30 Omega Instruments has budgeted $300,000 per year to pay for certain ceramic parts over the next 5 years. If the company expects the cost of the parts to increase uniformly according to an arithmetic gradient of $10,000 per year, what is it expecting the cost to be in year 1, if the interest rate is 10% per year?

2.31 Chevron-Texaco expects receipts from a group of stripper wells (wells that produce less than 10 barrels per day) to decline according to an arithmetic gradient of $50,000 per yea: This year's receipts are expected to be $280,000 (i.e., end of year 1), and the company expects the useful life of the wells to be 5 years. (a) What is the amount of the cash flow in year 3, and (b) what is the equivalent uniform annual worth in years 1 through 5 of the in-come from the wells at an interest rate of 12% per year?

2.32 Income from cardboard recycling at Fort Bliss has been increasing at a constant rate of $1000 in each of the last 3 years. If this year's income (i.e., end of year 1) is expected to be $4000 and the increased income trend continues through year 5, (a) what will the income be 3 years from now (i.e., end of year 3) and (b) what is the present worth of the income over that 5-year period at an interest rate of 10% per year?

2.42 Hughes Cable Systems plans to offer its employees a salary enhancement package that has revenue sharing as its main component. Specifically, the company will set aside 1% of total sales for year-end bonuses for all its employees. The sales are expected to be $5 million the first year, $6 million the second year, and amounts increasing by 20% cash year for the next 5 years. At an interest rate of 10% per year. What is the equivalent annual worth in years 1 through 5 of the bonus package?

2.43 Determine how much money would be in a savings account that started with a deposit of $2000 in year 1 with each succeeding amount increasing by 10% per year. Use an interest rate of 15% per year and a 7-year period.

2.44 The future worth in year 10 of a geometric gradient series of cash flows was found to be $80,000. If the interest rate was 15% per year and the annual rate of increase was 9% per year. What was the cash flow amount in year 1?

2.45 Thomasville Furniture Industries offers several types of high-performance fabrics that are capable of withstanding chemicals as harsh as chlorine. A certain mid-western manufacturing company that uses fabric in several products has a report showing that the present worth of fabric purchases over a certain 5-year period was $900,000. If the costs were known to geometrically increase by 5% per year during that time and the company used an interest rate of 15% per year for investments, what was the cost of the fabric in year 2?

2.46 Find the present worth of a series of investments that starts at $1000 in year I and increases by 10% per year for 20 years. Assume the interest rate is 10% per year.

2.47 A northern California consulting firm wants to start saving money for replacement of network servers. If the company invests $3000 at the end of year 1 and increases the amount invested by 5% each year, How much will be in the account 4 years from now if it cams interest at a rate of 8% per year?

2.48 A company that manufactures purgable hydrogen sulfide monitors is planning to make deposits such that each one is 5% larger than the preceding one. How large must the first deposit be (at the end of year 1) if the deposits extend through year 10 and the fourth deposit is $1250? Use an interest rate of 10% per year.

Interest Rate and Rate of Return

2.49 What compound interest rate per year is equivalent to a 12% per year simple interest rate over a 15-year period?

2.50 A publicly traded consulting engineering firm pays a bonus to each engineer at the end of the year based on the company's profit for that year. If the company's initial investment was $1.2 million, what rate of return has it made on its investment if each engineer's bonus has been $3000 per year for the past 10 years? Assume the company has six engineers and that the bonus money represents 5% of the company's profit.

2.51 Damon Iron Works, Inc., manufactures angular contact ball bearings for pumps that operate in harsh environments. If the company invested $2.4 million in a process that resulted in profits of $760,000 per year for 5 years, what rate of return did the company make on its investment?

2.52 An investment of $600,000 increased to $1,000,000 over a 5-year period. What was the rate of return on the investment?

233 A small company that specializes in powder coating expanded its building and purchased a new oven that is large enough to handle automobile frames. The building and oven cost $125.000, but new business from hot-rodder has increased annual in-come by $520,000. If operating expenses for gas, materials, labor, etc., amount to $470,000 per years, what rate of return will be made on the investment if only the cash flows that occur over the next 4 years are included in the calculation?

2.54 The business plan for a start-up company that manufactures multigas portable detectors showed equivalent annual cash flows of $400,000 for the first 5 years. If the cash flow in year I was $320,000 and the increase thereafter was $50,000 per year, what interest rate was used in the calculation?

2.55 A new company that makes medium-voltage soft starters spent $85,000 to build a new website. Net income was $60,000 the first year, increasing by $15,000 each year. What rate of return did the company make in its first 5 years?

Number of Years

2.56 A company that manufactures plastic control valves has a fund for equipment replacement that contains $500,000 if the company spends $75.000 per year on new equipment, how many years will it take to reduce the fund to less than $75,000 at an interest rate of 10% per year?

2.57 An A&E firm is considering purchasing the building it currently occupies under a long-term lease because the owner of the building suddenly put it up for sale. The building is being offered at a price of $170,000. Since the lease is already paid for this year, the next annual lease payment of $30,000 isn't due until the end of this year. Because the A&E firm has been a good tenant, the owner has offered to sell to them for $160,000. If the firm purchases the building with no down payment, how long will it be before the company recovers its investment at an interest rate of 12% per year?

2.58 An engineer who invested very well plans to retire now because she has $2,000,000 in her ORP account. How long will she be able to withdraw $100,000 per year (beginning I year from now) if her account earns interest at a mate of 4% per year?

2.59 A company that manufactures ultrasonic wind sensors invested $1.5 million 2 years ago to acquire pan ownership in an innovative chip-making company. How long would it take (from the date of the initial investment) for its share of the chip company to be worth $3 million if that company is growing at a rate of 20% per year?

2.60 A certain mechanical engineer plans 10 retire when he has $1.6 million in his brokerage account. If he started with $100,000 in the account, how long will it be (from the time he started) before he can retire if the account makes a rate of return of 18% per year?

 

From one year (or interest period) to the next, interest rates will very. To accurately perform equivalence calculations for P and A when rates vary signify-cantly, the applicable interest rate should be used, not an average or constant rate. Whether performed by hand or by computer, the procedures and factors are the same as those for constant interest rates; however, the number of calculations increases.

PROBLEMS

Nominal and Effective Rates

4.1 Identify the compounding period for the following interest statements: (a) 1% per month; (b) 2.5% per quarter, and (c) 9.3% per year compounded semiannually.

4.2 Identify the compounding period for the following interest statements: (a) Nominal 7% per year compounded quarterly: (b) effective 6.8% per year compounded monthly; and (c) effective 3.4% per quarter compounded weekly.

4.3 Determine the number of times interest would be compounded in 1 year for the following interest statements: (a) 1% per month; (6)2% per quarter, and (c) 8% per year compounded semiannually.

4.4 For an interest rate of 10% per year com-pounded quarterly, determine the number of times interest would be compounded (a) per quarter. (b) Per year, and (c) per 3 years.

4.5 For an interest rate of 0.50% per quarto, determine the nominal interest rate per (a) semiannual period. (b) year, and (c) 2 years.

4.6 For an interest rate of 12% pa year com-pounded every 2 months, determine the nominal interest rate per (a) 4 months, (6) 6 months, and (c) 2 years.

4.7 For an interest rate of 10% per year, com-pounded quarterly, determine the nominal rate per (a) 6 months and (6) 2 years.

4.8 Identify the following interest rate statements as either nominal or effective: (0)1.3% per month; (b) 1% per week, compounded weekly; (c)nominal 15% per year, compounded monthly; (c) effective 1.5% per month, compounded daily; and (r)15% per year, compounded semiannually.

4.9 What effective interest rate per 6 months is equivalent to 14% per year, compounded semiannually? 4.10 An interest rate of 16% per year, com-pounded quarterly, is equivalent to what effective interest rate per year?

4.11 What nominal interest rate per year is equivalent to an effective 16% per year, compounded semiannually?

4.12 What effective interest rate per year is equivalent to an effective 18% per year, compounded semiannually?

4.13 What compounding period is associated with nominal and effective rates of 18% and 18.81% per year, respectively?

4.14 An interest rate of 1% per month is equivalent to what effective rate per 2 months?

4.15 An interest rate of 12% per year, com-pounded monthly, is equivalent to what nominal and effective interest rates per 6 months?

4.16 (a) an interest rate of 6.8% per semi-annual period compounded weekly, is equivalent to what weekly interest rate?

       (b) Is the weekly rate a nominal or effective rate? Assume 26 weeks per 6 months.

Payment and Compounding Periods

4.17 Deposits of $100 per week are made into a savings account that pays interest of 6% per year, compounded quarterly. Identify the payment and compounding periods.

4.18 A certain national bank advertises quarterly compounding for business checking accounts. What payment and compounding periods are associated with deposits of daily receipts?

4.19 Determine the F/P factor for 3 years at an interest rate of 8% per year, compounded quarterly.

4.20 Determine the P/G factor for 5 years at an effective interest rate of 6% per year, compounded semiannually.

Equivalence for Single Amounts and Series

4.21 A company that specializes in online security software development wants to have $85 million available in 3 years to pay stock dividends. How much money must the company set aside now in an account that earns interest at a rate of 8% per year, compounded quarterly?

4.22 Because testing of nuclear bombs was halted in 1992, the U.S. Department of Energy has been developing a laser project that will allow engineers to simulate (in a laboratory) conditions in a thermonuclear reaction. Due to soaring cost overruns, a congressional committee undertook an investigation and discovered that the estimated development coat of the project increased at an average rate of 1% per month over a 5-year period. If the original cost was estimated to be $2.7 billion 5 years ago, what is the expected cost today?

4.23 A present sum of $5000 at an interest ram of 8% per year, compounded semiannually, is equivalent to how much money 8 years ago?

4.24 In an effort to ensure the safety of all phone users, the Federal Communications Commission (FCC) requires cell phases to have a specific absorbed radiation (SAR) number of 1.6 watts per kilogram (W/Kg) of tissue or less. A new cell phone company estimates that by advertising its favorable 1.2 SAR number. It will increase sales by $1.2 million 3 months from now when its phones go on sale. At an interest rate of 20% per year, compounded quarterly, what is the maximum amount the company can afford to spend now for advertising in order to break even?

4.25 Radio Frequency Identification (RFID) is technology that is used by drivers with speed passes at toll booths and ranchers who track livestock from farm to fork. Wal-Mart expects to begin using the technology to track products within its stores. If RFID-tagged products will result in better inventory control that will save the company $1.3 million per month beginning 3 months from now, how much could the company afford to spend now to implement the technology at an interest rate of 12% per year, compounded monthly, if it wants to recover its investment in 2½ years?

4.35 Magnetek Instrument and Controls, a manufacturer of liquid-level sensors, expects sales for one of its models to increase by 20% every 6 months into the foreseeable future. If the sales 6 months from now are expected to be $I50.000, determine the equivalent semiannual worth of sales for a 5-year period at an interest rate of 14% per year, compounded semiannually.

4.36 Metal fab Pump and Filter projects that the cost of steel bodies for certain valves will increase by $2 every 3 months. If the coat for the first quarter is expected to be $80, what is the present worth of the costs for a 3-year period at an interest rate of 3% per quarter?

4.37 Fieldsaver Technologies, a manufacturer of precision laboratory equipment, borrowed $2 million to renovate one of its testing labs. The loan was repaid in 2 years through quarterly payments that increased by $50,000 each time. At an interest rate of 3% per quarter, what was the size of the first quarterly payment?

4.38 For the cash flows shown below, deter-mine the present worth (time 0), using an interest rate of 18% pa year, compounded monthly.

 

 

Month	Cash Flow, S/Month
0 1-12 13-28	1000 2000 3000

 

4.39 The cash flows (in thousands) associated with Fisher-Price's Touch learning system are shown below. Determine the uniform quarterly series in quarters 0 through 8 that would be equivalent to the cash flows shown at an interest rate of 16% per year, compounded quarterly.

Quarter	Cash Flow, S/ Quarter
1 2-3 5-8	1000 2000 3000

 

Equivalence When PP < CP

4.40 An engineer deposits $300 per month into a savings account that pays interest at a rate of 6% per year, compounded semiannually. How much will be in the account at the end of 15 years? Assume no interperiod compounding.

4.41 At time t= 0, an engineer deposited %10,000 into an account that pays interest at 8% per year compounded semiannually. If she withdrew $1000 in months 2, 11, and 23, what was the total value of the account at the end of 3 years? Assume no interperiod compounding.

4.42 For the transactions shown below, determine the amount of money in the account at the end of year 3 if the interest rate is 8% per year. Compounded semiannually. Assume no interperiod compounding.

End of Quarter	Amount of Deposit, s/Quarter	Amount of Withdrawal. S/Quarter
1 2-4 7 11	900 700 1000 -	2600 1000

 

4.43 The New Mexico State Police and Public Safety Department owns a helicopter that it uses to provide transportation and logistical support for high-level state officials. The $495 hourly rate coven operating expenses and the pilot's salary. If the governor uses the helicopter an average of 2 days per month for 6 hours each day, what is the equivalent future worth of the costs for 1 year at an interest rate of 6% per year, compounded quarterly (treat the costs as deposits)?

Continuous Compounding

4.44 What effective interest rate per year, compounded continuously, is equivalent to a nominal rate of 13% per year?

4.45 What effective interest rate per 6 months is equal to a nominal 2% per month, compounded continuously?

4.46 What nominal rate per stunner is equivalent to an effective rate of 12.7% per year, compounded continuously?

4.47 Corrosion problems and manufacturing defects rendered a gasoline pipeline be-tween El Paso and Phoenix subject to longitudinal weld seam failures. Therefore, the pressure was reduced to 80% of the de-sign value. If the reduced pressure results in the delivery of $100, 000 per month less product, what will be the value of the lost revenue after a 2-year period at an interest rate of 15% per year, compounded continuously?

4.48 Because of a chronic water shortage in Santa Fe, new athletic fields must use artificial turf or xeriscape landscaping. If the value of the water saved each month is $6000, how much can a private developer afford to spend on artificial turf it he wants to recover his investment in 5 years at an interest rate of 18% per year, compounded continuously?

4.49 A Taiwan-based chemical company had to file for bankruptcy because of a nation- wide phase-out of methyl tertiary butyl ether (MTBE). If the company reorganizes and invests $50 million in a new ethanol production facility, how much money must it make each month if it wants to re-cover its investment in 3 years at an interest rate of 2% per month, compounded continuously?

4.50 In order to have $85,000 four years from now for equipment replacement, a construction company plans to set aside money today in government-insured bonds. If the bonds earn interest at a rate of 6% per year, compounded continuously, how much money must the company invest?

4.51 How long would it take for a lump-sum investment to double in value at an interest rate of 1.5% per month, compounded con-tenuously?

4.52 What effective interest rate per month, compounded continuously, would be required for a single deposit to triple in value in 5 years?

Varying Interest Rates

4.53 How much money could the maker of fluidized-bed scrubbers afford to spend now instead of spending $150,000 in year 5 if the interest rate is 10% in years 1 through 3 and 12% in years 4 and 5?

4.54 What is the future worth in year 8 of a present sum of $50, 000 if the interest rate is 10% per year in years 1 through 4 and 1% per month in years 5 through 8?

CHAPTER SUMMARY

In Chapter 2, we derived the equations to calculate the present, future, or annual worth of specific cash flow series. In this chapter, we have shown that these equations apply to cash flow series different from those for which the basic relations are derived. For example, when a uniform series does not begin in period 1, we still use the P/A factor to find the "present worth" of the series, except the P-value is located one period ahead of the first A value, not at time 0. For arithmetic and geometric gradients, the P-value is two periods ahead of where the gradient starts. With this information, it is possible to solve for any symbol —P, A, or F—for any conceivable cash flow series.

 We have experienced some of the power of spreadsheet functions in determining P, F, and A values once the cash flow estimates are entered into spreadsheet cells.

PROBLEMS

Present Worth Calculations

3.1 Because unintended lane changes by distracted drivers are responsible for 43% of all highway fatalities, Ford Motor Co. and Volvo launched a program to develop technologies to prevent accidents by sleepy drivers. A device costing $260 tracks lane markings and sounds an alert during lane changes. If these devices are included in 100,000 new cars per year beginning 3 years from now, what would be the present worth of their cost over a 10-year period at an interest rate of 10% per year?

3.2 One plan to raise money for Texas schools involves an "enrichment tax" that could collect $56 for every student in certain school district. If there are 50,000 students in the district and the cash flow begins 2 years from now, what is the present worth of the enrichment plan over a 5-year planning period at an interest rate of 8% per year?

3.3 Amalgamated Iron and Steel purchased a new machine for ram cambering large I-beams. The company expects to bend 80 beams at $2000 per beam in each of the first 3 years, after which the company expects to bend 100 beams per year at $2500 per beam through year 8. If the company's minimum attractive rate of return is 18% per year, what is the present worth of the expected income?

3.4 Rubbermaid Plastics plans to purchase a rectilinear robot for pulling parts from an injection molding machine. Because of the robot's speed, the company expects production costs to decrease by $100,000 per year in each of the first 3 years and by $200,000 per year in the next 2 years. What is the present worth of the cost savings if the company uses an interest rate of 15% per year on such investments?

3.5 Toyco Watercraft has a contract with a pans supplier that involves purchases amounting to $150,000 per year, with the first purchase to be made now, followed by similar purchases over the next 5 years. Determine the present worth of the con-tract at an interest rate of 10% per year.

3.6 Calculate the present worth in year 0 of the following series of disbursements. Assume that i = 10% per year.

Year       Disbursement, $                                                            Year                                   Disbursement, $

0                 0                                                                                     6                                     5000                                  1                 3500                                                                               7                                      5000 2                 3500                                                                               8                                      5000 3             3500                                                                                    9                                     5000 4             5000                                                                                   10                                    5000 5             5000

 

 

Annual Worth Calculations

3.7 Cisco's gross revenue (the percentage of revenue left after subtracting the cost of goods sold) was 70.1% of total revenue over a certain 4-year period. If the tom/ revenue was $5.4 billion for the first 2 years and $6.1 billion for the last 2 years. What was the equivalent annual worth of the gross revenue over that 4-year period at an interest rate of 20% per year?

3.8 BKM Systems sales revenues are shown below. Calculate the equivalent annual worth (years 1 through 7), using an interest rate of 10% per year.

Year	Disbursement, $	Year	Disbursement, $
0		4	5000
1	4000	5	5000
2	4000	6	5000
3	4000	7	5000

 

3.9 A metallurgical engineer decides to set aside money for his newborn daughter's college education. He estimates that her needs will be $20,000 on her 17^th, 18^th, 19^th, and 20^th, birthdays. If he plans to make uniform deposits starting 3 years from now and continue through year 16, what should be the size of each deposit, if the account earns interest at a rate of 8% per year?

3.10 Calculate the annual worth in years 1 through 10 of the following series of in-comes and expenses if the interest rate is 10% per year.

Year	Income, $/Year	Expense, $Year
0	10,000	2000
1-6	800	200
7-10	900	300

 

3.11 How much money would you have to pay each year in 8 equal payments, starting 2 years from today, to repay a $20,000 loan received from a relative today, if the interest rate is 8% per year?

3.12 An industrial engineer is planning for his early retirement 25 years from now. He believes he can comfortably set aside $10,000 each year starting now. If he plans to start withdrawing money 1 year after he makes his last deposit (i.e., year 26), what uniform amount could he withdraw each year for 30 years, if the account earns interest at a rate of 8% per year?

3.13 A rural utility company provides standby power to pumping stations using diesel-powered generators. An alternative has arisen whereby the utility could use natural gas to power the generators, but it will be a few years before the gas is available at remote sites. The utility estimates that by switching fo gas, it will save $15,000 per year, starting 2 years from now. At an interest rate of 8% per year, determine the equivalent annual worth (years 1 through 10) of the projected savings.

3.14 The operating cost of a pulverized coal cyclone furnace is expected to be $80,000 per year. If the steam produced will be needed only for 5 years beginning now (i.e., years 0 through 5), what is the equivalent annual worth in years 1 through 5 of the operating cost at an interest rate of 10% per year?

3.15 An entrepreneurial electrical engineer has approached a large water utility with a proposal that promises to reduce the utility's power bill by at least 15% per year for the next 5 years through installation of patented surge protectors. The proposal states that the engineer will get $5000 now and annual payments that are equivalent to 75% of the power savings achieved from the devices. Assuming the savings are the same every year (i.e., 15%) and that the utility's power bill is $1 million per year, what would be the equivalent uniform amount (years 1 through 5) of the payments to the engineer? Assume the utility uses an interest rate of 6% per year.

3.16 A large water utility is planning to upgrade its SCADA system for controlling well pumps, booster pumps, and disinfection equipment so that everything can be controlled from one site. The first phase will reduce labor and travel costs by $28,000 per year. The second phase will reduce costs by an additional $20,000 per year. If phase I savings occur in years 0, 1, 2, and 3 and phase II occurs in years 4 through 10, what is the equivalent annual worth of the upgraded system in years 1 through 10 at an interest rate of 8% per year?

3.17 A mechanical engineer who recently graduated with a master's degree is contemplating starting his own commercial heating and cooling company. He can purchase a Web page design package aimed at delivering information only for $600 per year. If his business is successful, he will purchase a more elaborate e-commerce package costing $4000 per year. If the engineer purchases the less expensive page now (beginning-of-year payments) and he purchases the e-commerce package 1 year from now (also beginning-of-year payments), what is the equivalent annual worth of costs for the website for a 5-year period (years 1 through 5) at an interest rate of 12% per year?

Future Worth Calculations

3.18 Lifetime savings accounts, known as LSA_s, would allow people to invest after tax money without being taxed on any of the gains. If an engineer invests $10,000 now and $10,000 each year for the next 20 years, how much will be in the account immediately after the last deposit if the account grows by 15% per year?

3.19 How much money was deposited each year for 5 years if the account is now worth $100,000 and the last deposit was made 10 years ago? Assume the account earned interest at 7% per year.

3.20 Calculate the future worth (in year 11) of the following income and expenses if the interest rate is 8% per year.

Year	Income, $	Expense, $
0	12,000	3000
1-6	800	200
7-11	900	200

 

Random Placement and Uniform Series

3.21 What is the equivalent worth in year 5 of the following series of income and disbursements, if the interest rate is 12% per year?

Year	Income, $	Expense, $
0	0	9000
1-5	6000	6000
6-8	6000	3000
9-14	8000	5000

 

3.22 Use the cash flow diagram below to calculate the amount of money in year 5 that is equivalent to all the cash flows shown if the interest rate is 12% per year.

3.23 By spending $10,000 now and $25,000 three years from now, a plating company can increase its income in years 4 through 10. At an interest rate of 12% per year, how much extra income per year would be needed in years 4 through 10 to recover the investment?

3.24 Sierra Electric Company is considering the purchase of a hillside ranch for possible use as a windmill farm sometime in the future. The owner of the 500-acre ranch will sell for $3000per acre ranch company will pay her in two payments—one payment now and an-other that is twice as large 3 years from now. If the transaction interest rate is 8% per year, what is the amount of the first payment?

3.25 Two equal deposits made 20 and 21 years ago, respectively, will allow a retiree to withdraw $10,000 now and $10,000 per year for 14 more years. If the account earned interest at 10% per year, how large was each deposit?

3.26 A concrete and building materials company is trying to bring the company-funded portion of its employee retirement fund into compliance with 1413-301. The company has already deposited $20,000 in each of the last 5 years. How much must be deposited now in order for the fund to have $350,000 three years from now, if the fund grows at a rate of 15% per year? 3.27 Find the value of x below such that the positive cash flows will be exactly equivalent to the negative cash flows if the interest rate is 14% per year.

 $700 $700

0 1 2 3 4 51 6 7 8 9 • 10 11 12 Year

 

$800       $800       $800

3.28 In attempting to obtain a swing loan from a local bank, a general contractor was asked to provide an estimate of annual expenses. One component of the expenses is shown in the cash flow diagram below. Convert the amounts shown into an equivalent uniform annual amount in years 1 through 8, using an interest rate of 12% per year.

3.29 Determine the value in year 8 that is equivalent to the cash flows below. Use an interest rate of 12% per year.

3.30 Find the value of x in the diagram below that will make the equivalent present worth of the cash flow equal to $15,000 if the interest rate is 15% per year.

 

3.31 Calculate the amount of money in year 3 that is equivalent to the following cash flows, if the interest rate is 16% per year.

Year	Amount,$	Year	Amount, $
0	900	5	3000
1	900	6	-1500
2	900	7	500
3	900	8	500
4	3000

 

3.32 Calculate the annual worth (years 1 through 7) of the following series of disbursements. Assume that i = 12% per year.

Year	Disbursement,$	Year	Disbursement, $
0	5000	4	5000
1	3500	5	5000
2	3500	6	5000
3	3500	7	5000

 

3.33 Calculate the value of x for the cash flows below such that the equivalent total value in year 8 is $20,000, using an interest rate of 15% per year.

Year	Cash Flow, $	Year	Cash Flow $
0	2000	6	X
1	2000	7	X
2	X	8	X
3	X	9	1000
4	X	10	1000
5	X	11	1000

 

 Shifted Arithmetic Gradients

3.34 San Antonio is considering various options for providing water in its 50-year plan, including desalting. One brackish aquifer is expected to yield desalted water that will generate revenue of $4.1 million per year for the first 4 years, after which less production will decrease revenue each year by $50,000 per year. If the aquifer will be totally depleted in 25 years, what is the present worth of the desalting option at an interest rate of 6% per year?

3.35 Exxon-Mobil is planning to sell a number of producing oil wells. The wells are expected to produce 100,000 barrels of oil per year for 8 more years at a selling price of $28 per barrel for the next 2 years,in creasing by $1 per barrel through year 8. How much should an independent refiner be willing to pay for the wells now, if the interest rate is 12% per year?

3.36 Burlington Northern is considering the elimination of a railroad grade crossing by constructing a dual-track overpass. The railroad subcontracts for maintenance of its crossing gates at $11,500 per year. Be-ginning 4 years from now, however, the costs are expected to increase by $1000 per year into the foreseeable future (that is, $12,500 4 years from now, $13,500 five years from now, etc.). The overpass will cost $1.4 million (now) to build, but it will eliminate 100% of the auto-train collisions that have cost an average of $250,000 per year. If the railroad uses a 10-year study period and an interest rate of 10% per year, determine whether the railroad should build the overpass. 3.37 Levi Strauss has some of its jeans stone-washed under a contract with independent U.S. Garment Corp. If U.S. Garment's operating cost per machine is $22,000 per year for years 1 and 2 and then it increases by $1000 per year through year 5, what is the equivalent uniform annual cost per machine (years 1 through 5) at an interest rate of 12% per year?

3.38 Herman Trucking Company's receipts and disbursements (in $1000) are shown below. Calculate the future worth in year 7 at an interest rate of 10% per year.

Year	Cash Flow,$	Year	Cash Flow,$
0	-10,000	4	5000
1	4000	5	-1000
2	3000	6	7000
3	4000	7	8000

 

3.39 Peyton Packing has a ham cooker that has the cost stream below. If the interest rate is 15% per year, determine the annual worth (in years 1 through 7) of the costs.

Year	Cost,$	Year	Cost,$
0	4,000	4	6,000
1	4,000	5	8,000
2	3,000	6	10,000
3	2,000	7	12,000

 

3.40 A start-up company selling color-keyed carnuba car wax borrows $40,000 at an interest rate of 10% per year and wishes to repay the loan over a 5-year period with annual payments such that the third through fifth payments are $2000 greater than the first two. Determine the size of the first two payments.

3.41 For the cash flows below, find the value of x that makes the present worth in year 0 equal to $11,000 at an interest rate of 12% per year.

 

 

 

Year	Cost, Flow$	Year	Cost, Flow$
0	200	5	700
1	300	6	800
2	400	7	900
3	X	8	1000
4	600	9	1100

 

Shifted Geometric Gradients

3.42 In an effort to compensate for shrinking land-line customers, SBC and Bell South (owners of angular Wireless LLC) got into a bidding war with Vodaphone to acquire AT&T Wireless. The initial $11 per share offer escalated to $13 for the 2.73 billion shares of AT&T Wireless. If the buyout took exactly 1 year to close (i.e., end of year 1), what would be the present worth today (time 0) of the acquisition for profits of $5.3 billion in year 2, increasing by 9% per year through year 11? Assume SBC and well south us a rate of return of 15% per year.

3.43 A successful alumnus is planning to make a contribution to the community college from which he graduated. The donation is to be made over a 5-year period beginning now, a total of six payments. It will support five pre-engineering students per year for 20 years, with the first scholarship to be awarded immediately (a total of 21 scholarships). The cost of tuition at the school is $4000 per year and is expected to stay at that amount for 3 more years. After that time (i.e., year 4), the tuition is expected to increase by 8% per year. If the college can invest the money and earn interest at a rate of 10% per year, what size must the donations be?

3.44 Calculate the present worth (year 0) of a lease that requires a payment of $20,000 now and amounts increasing by 5% per year through year 10. Use an interest rate of 14% per year.

3.45 Calculate the present worth for a machine that has an initial cost of $29,000, a life of 10 years, and an annual operating cost of $13,000 for the first 4 years, increasing by 10% per year thereafter. Use an interest rate of 10% per year.

3.46 A-1 Box Company is planning to lease a computer system that will cost (with service) $15,000 in year 1, $16,500 in year 2, and amounts increasing by 10% each year thereafter. Assume the lease payments must be made at the beginning of the year and that a 5-year lease is planned. What is the present worth (year 0) if the company uses a minimum attractive rate of return of 16% per year?

3.47 Dakota Hi-C Steel signed a contract that will generate revenue of $210,000 now, $222,600 in year 1, and amounts increasing by 8% per year through year 5. Calculate the future worth of the contract at an interest rate of 8% per year.

Shifted Decreasing Gradients

3.48 Find the present worth (at time 0) of the chrome plating costs in the cash flow diagram.

Assume i = 12% per year.

 P=?

 

3.49 Compute the present worth (year 0) of the following cash flows at i = 12% per year.

Year	Amount, $	Year	Amount, $
0	5000	8	700
1-5	1000	9	600
6	900	10	500
7	800	11	400

 

 3.50 For the cash flow tabulation, calculate the equivalent uniform annual worth in periods 1 through 10, if the interest rate is 10% per year.

Year	Amount, $	Year	Amount,$
0	2000	6	2400
1	2000	7	2300
2	2000	8	2200
3	2000	9	2100
4	2000	10	2000
5	2500

 

 

3.51 Prudential Realty has an escrow account for one of its property management clients that currently contains $20,000. How long will it take to deplete the account if the client withdraws $5000 now, $4500 one year from now, and amounts decreasing by $500 each year thereafter, if the account earns interest at a rate of 8% per year?

3.52 The cost of spacers used around fuel rods in liquid-metal fast breeder reactors has been decreasing because of the availability of improved temperature-resistant ceramic materials. Determine the present worth (in year 0) of the costs shown in the diagram below, using an interest rate of 15% per year.

0 1 2 3 4 5 Year

 

3.53 Compute the future worth in year 10 at i = 10% per year for the cash flow shown below.

PS?

0 I 2 3 4 5 6 7 8 9 10 Year

 

FE REVIEW PROBLEMS

3.54 The annual worth in years 4 through 8 of an amount of money x that will be received 2 years from now is $4000. At an interest rate of 10% per year, the value of x is closest to

(a) Less than $12,000 (b) $12,531 (c) $12,885 (d) More than $13,000

 

 

3.55 The multistate Powerball Lottery, worth $182 million, was won by a single individual who had purchased five tickets at $1 each. The individual was given two choices: Receive 26 payments of $7 mil-lion each, with the first payment to be made now and the rest to be made at the end of each of the next 25 years; or receive