## Problems

For additional practice, please see the problems (with selected solutions) provided on the Student CD-ROM that accompanies this book.

2.1 Using 12% simple interest per year, how much interest will be owed on a loan of S500 at the end of two years?

2.2 If a sum of £3000 is borrowed for six months at 9% simple interest per year, what is the total amount due (principal and interest) at the end of six months?

2.3 \\Tiat principal amount will yield S150 in interest at the end of three months when the interest rate is 1% simple interest per month?

2.4 If 2400 rupees in interest is paid on a two-year simple-interest loan of 12 000 rupees, what is the interest rate per year?

2.5 Simple interest of $190.67 is owed on a loan of $550 after four years and four months. What is the annual interest rate?

2.6 How much will be in a bank account at the end of five years if €2000 is invested today at 12% interest per annum, compounded yearly?

2.7 How much is accumulated in each of these savings plans over two vears?

(a) Deposit 1000 yuan today at 10% compounded annually.

(b) Deposit 900 yuan today at 12% compounded monthly.

2.8 Greg wants to have $50 000 in five vears. The bank is offering five-year investment certificates that pay 8% nominal interest, compounded quarterly. How much money should he invest in the certificates to reach his goal?

2.9 Greg wants to have $50 000 in five years. He has S20 000 todav to invest. The bank is offering five-year investment certificates that pay interest compounded quarterly. ~\\ nat is the minimum nominal interest rate he would have to receive to reach his goal?

2.10 Greg wants to have $50 000. He will invest S20 000 today in investment certificates that pay 8% nominal interest, compounded quarterly. How long will it take him to reach his goal?

2.11 Greg will invest $20 000 today in five-year investment certificates that pay 8% nominal interest, compounded quarterly. How much money will this be in five years?

2.12 You bought an antique car three years ago for 500 000 yuan. Today it is worth 650 000 yuan.

(a) What annual interest rate did you earn if interest is compounded yearly?

(b) What monthly interest rate did you earn if interest is compounded monthly?

2.13 You have a bank deposit now worth S5000. How long will it take for your deposit to be worth more than S8000 if

(a) the account pays 5% actual interest every half-year, and is compounded even,7 half-year?

(b) the account pays 5% nominal interest, compounded semiannually?

2.14 Some time ago, you put £500 into a bank account for a "rainy day." Since then, the bank has been paying you 1% per month, compounded monthly. Today, you checked the balance, and found it to be £708.31. How long ago did you deposit the £500?

2.15 (a) If you put S1000 in a bank account today that pays 10% interest per year, how much monev could be withdrawn 20 years from now?

(b) If you put SI 000 in a bank account today that pays 10% simple interest per year, how much monev could be withdrawn 20 vears from now?

2.16 How long will it take any sum to double itself,

(a) with an 11 % simple interest rate?

(b) with an 11 % interest rate, compounded annually?

(c) with an 11 % interest rate, compounded continuously?

2.17 Compute the effective annual interest rate on each of these investments.

(a) 25% nominal interest, compounded semiannually

(b) 25% nominal interest, compounded quarterly

(c) 25% nominal interest, compounded continuously

2.18 For a 15% effective annual interest rate, what is the nominal interest rate if

(a) interest is compounded monthly?

(b) interest is compounded daily (assume 365 days per year)?

(c) interest is compounded continuously ?

2.19 A Studebaker automobile that cost S665 in 1934 was sold as an antique car at S14 800 in 1998. What was the rate of return on this "investment"?

2.20 Clifford has X euros right now. In 5 years, X will be €3500 if it is invested at 7.5%, compounded annually. Determine the present value ofX. If Clifford invested X euros at 7.5%, compounded daily, how much would the value of Xbe in 10 years?

2.21 You have just won a lottery prize of Si 000 000 collectable in 10 yearly installments of 100 000 rupees, starting today. Why is this prize not really Si 000 000? What is it really worth today if money can be invested at 10% annual interest, compounded monthly? Use a spreadsheet to construct a table showing the present worth of each installment, and the total present worth of the prize.

2.22 Suppose in Problem 2.21 that you have a large mortgage you want to pay off now. You propose an alternative, but equivalent, payment scheme. You would like $300 000 today, and the balance of the prize in five years when you intend to purchase a large piece of waterfront property. How much will the payment be in five years? Assume that annual interest is 10%, compounded monthly.

2.23 You are looking at purchasing a new computer for your four-year undergraduate program. Brand 1 costs S4000 now, and you expect it will last throughout your program without any upgrades. Brand 2 costs S2500 now and will need an upgrade at the end of two years, which you expect to be Si700. With 8% annual interest, compounded monthly, which is the less expensive alternative, if they provide the same level of sendee and will both be worthless at the end of the four years?

2.24 The Kovalam Bank advertises savings account interest as 6% compounded daily. Mdiat is the effective interest rate?

2.25 The Bank of Brisbane is offering a new savings account that pays a nominal 7.99% interest, compounded continuously. Will your money earn more in this account than in a daily interest account that pays 8%?

2.26 You are comparing two investments. The first pays 1% interest per month, compounded monthly, and the second pays 6% interest per six months, compounded every six months.

(a) What is the effective semiannual interest rate for each investment?

(b) "What is the effective annual interest rate for each investment?

(c) On the basis of interest rate, which investment do you prefer? Does your decision depend on whether you make the comparison based on an effective six-month rate or an effective one-year rate?

2.27 The Crete Credit Union advertises savings account interest as 5.5% compounded weekly and chequing account interest at 7% compounded monthly. What are the effective interest rates for the two types of accounts?

2.28 Victory Visa, Magnificent Master Card, and Amazing Express are credit card companies that charge different interest on overdue accounts. Victor}' Visa charges 26% compounded daily, Magnificent Master Card charges 28% compounded weekly, and Amazing Express charges 30% compounded monthly. On the basis of interest rate, which credit card has the best deal?

2.29 April has a bank deposit now worth S796.25. A year ago, it was $750. Wnat was the nominal monthly interest rate on her account?

2.30 You have S50 000 to invest in the stock market and have sought the advice of Adam, an experienced colleague who is willing to advise you, for a fee. Adam has told you that he has found a one-year investment for you that provides 15% interest, compounded monthly.

(a) Wnat is the effective annual interest rate, based on a 15% nominal annual rate and monthly compounding?

(b) Adam says that he will make the investment for you for a modest fee of 2% of the investment's value one year from now. If you invest the S50 000 today, how much will you have at the end of one year (before Adam's fee)?

(c) W nat is the effective annual interest rate of this investment including Adam's fee?

2.31 May has 2000 yuan in her bank account right now. She wanted to know how much it would be in one year, so she calculated and came up with 2140.73 yuan. Then she realized she had made a mistake. She had wanted to use the formula for monthly compounding, but instead, she had used the continuous compounding formula. Redo the calculation for May and find out how much will actually be in her account a year from now.

2.32 Hans now has $6000. In three months, he will receive a cheque for $2000. He must pay $900 at the end of each month (starting exactly one month from now). Draw a single cash flow diagram illustrating all of these payments for a total of six monthly periods. Include his cash on hand as a payment at time 0.

2.33 Margaret is considering an investment that will cost her $500 today. It will pay her $100 at the end of each of the next 12 months, and cost her another $300 one year from todav. Illustrate these cash flows in two cash flow diagrams. The first should show each cash flow element separately, and the second should show only the net cash flows in each period.

2.34 Heddy is considering working on a project that will cost her $20 000 today. It will pay her $10 000 at the end of each of the next 12 months, and cost her another $15 000 at the end of each quarter. An extra $10 000 will be received at the end of the project, one year from now. Illustrate these cash flows in two cash flow diagrams. The first should show each cash flow element separately, and the second should show only the net cash flow in each period.

2.35 Illustrate the following cash flows over 12 months in a cash flow diagram. $how only the net cash flow in each period.

Cash Payments |
$20 every three months, starting now |

Cash Receipts |
Receive S30 at the end of the first month, and from that point on, receive 10% more than the previous month at the end of each month |

2.36 There are two possible investments, A and B. Their cash flows are shown in the table below. Illustrate these cash flows over 12 months in two cash flow diagrams. Show only the net cash flow in each period. Just looking at the diagrams, would you prefer one investment to the other? Comment on this.

S2400 now and a closing fee of $200 at the end of month 12 |
Investment B S500 every two months, starting two months from now | |

Receipts |
$250 monthly payment at the end of each month |
Receive $50 at the end of the first month, and from that point on, receive S50 more than the previous month at the end of each month |

2.37 You are indifferent between receiving 100 rand today and 110 rand one year from now. The bank pays you 6% interest on deposits and charges you 8% for loans. Name the three types of equivalence and comment (with one sentence for each) on whether each exists for this situation and why.

2.38 June has a small house on a small street in a small town. If she sells the house now, she will likely get €110 000 for it. If she waits for one year, she will likely get more, say, €120 000. If she sells the house now, she can invest the money in a one-year guaranteed growth bond that pays 8% interest, compounded monthly. If she keeps the house, then the interest on the mortgage payments is 8% compounded daily. June is indifferent between the two options: selling the house now and keeping the house for another year. Discuss whether each of the three types of equivalence exists in this case.

2.39 Using a spreadsheet, construct graphs for the loan described in part (a) below.

(a) Plot the amount owed (principal plus interest) on a simple interest loan of Si00 for .Vvears for .V = 1, 2, ... 10. On the same graph, plot the amount owed on a compound interest loan of SI00 for A"years for N = 1, 2, ... 10. The interest rate is 6% per year for each loan.

(b) Repeat part (a), but use an interest rate of 18%. Observe the dramatic effect compounding has on the amount owed at the higher interest rate.

2.40 (a) At 12% interest per annum, how long will it take for a penny to become a million dollars? How long will it take at 18%?

(b) Show the growth in values on a spreadsheet using 10-year time intervals.

2.41 Use a spreadsheet to determine how long it will take for a £100 deposit to double in value for each of the following interest rates and compounding periods. For each, plot the size of the deposit over time, for as many periods as necessary for the original sum to double.

(a) 8% per year, compounded monthly

(b) 11 % per year, compounded semiannually

(c) 12% per year, compounded continuously

2.42 Construct a graph showing how the effective interest rate for the following nominal rates increases as the compounding period becomes shorter and shorter. Consider a range of compounding periods of your choice from daily compounding to annual compounding.

2.43 Today, an investment you made three years ago has matured and is now worth 3000 rupees. It was a three-year deposit that bore an interest rate of 10% per year, compounded monthly. You knew at the time that you were taking a risk in making such an investment because interest rates vary over time and you "locked in" at 10% for three years.

(a) How much was your initial deposit? Plot the value of your investment over the three-year period.

(b) Looking back over the past three years, interest rates for similar one-year investments did indeed van,-. The interest rates were 8% the first year, 10% the second, and 14% the third. Plot the value of your initial deposit over time as if you had invested at this set of rates, rather than for a constant 10% rate. Did you lose out by having- locked into the 10% investment? If so, bv how much?

### More Challenging Problem

2.44 Alarlee has a choice between X pounds today or Y pounds one year from now. X is a fixed value, but Yvaries depending on the interest rate. At interest rate /, X and Fare mathematically equivalent for Marlee. At interest ratey, A and Fhave decisional equivalence for Marlee. At interest rate k, A and Fhave market equivalence for Marlee. W nat can be said about the origins, nature, and comparative values of i,j, and k?

MINI-CASE 2.1

### Student Credit Cards

Most major banks offer a credit card sendee for students. Common features of the student credit cards include no annual fee, a S500 credit limit, and an annual interest rate of 19.7% (in Canada as of 2007). Also, the student cards often come with many of the perks available for the general public: purchase security or travel-related insurance, extended warranty protection, access to cash advances, etc. The approval process for getting a card is relatively simple for university and college students so that thev can start building a credit histon" and enjoy the convenience of having a credit card while still in school.

The printed information does not use the term nominal or effective, nor does it define the compounding period. However, it is common in the credit card business for the annual interest rate to be divided into daily rates for billing purposes. Hence, the quoted annual rate of 19.7% is a nominal rate and the compounding period is daily. The actual effective interest rate is then (1 + 0.197/365)3" - 1 = 0.2 1 77 or21.77%.

### Discussion

Interest information must be disclosed by law. but lenders and borrowers have some latitude as to how and where they disclose it. Moreover, there is a natural desire to make the interest rate look lower than it really is for borrowers, and higher than it really is for lenders.

In the example of student credit cards, the effective interest rate is 21.77%, roughly 2% higher than the stated interest rate. The actual effective interest rate could even end up being higher if fees such as late fees, over-the-limit fees, and transaction fees are charged.

### Questions

1. Go to your local bank branch and find out the interest rate paid for various kinds of savings accounts, chequing accounts, and loans. For each interest rate quoted, determine if it is a nominal or effective rate. If it is nominal, determine the compounding period and calculate the effective interest rate.

2. Have a contest with your classmates to see who can find the organization that will lend money to a student like you at the cheapest effective interest rate, or that will take investments which provide a guaranteed return at the highest effective interest rate. The valid rates must be generally available, not tied to particular behaviour by the client, and not secured to an asset (like a mortgage).

3. If you borrowed S1000 at the best rate you could find and invested it at the best rate you could find, how much money would you make or lose in a year? Explain why the result of your calculation could not have the opposite sign.

Engineering Economics in Action, Part 3A: Apples and Oranges

3.1 Introduction

3.2 Timing of Cash Flows and Modelling

3.3 Compound Interest Factors tor Discrete Compounding

3.4 Compound Interest Factors for Single Disbursements or Receipts

3.5 Compound Interest Factors for Annuities

3.6 Conversion Factor for Arithmetic Gradient Series

3.7 Conversion Factor for Geometric Gradient Series

3.8 Non-Standard Annuities and Gradients

3.9 Present Worth Computations When /V-> <>

Review Problems Summary

Engineering Economics in Action, Part 3B: No Free Lunch

Problems Mini-Case 3.1

Appendix 3A: Continuous Compounding and Continuous Cash Flows Appendix 3B: Derivation of Discrete Compound Interest Factors

Engineering Economics in Action, Part 3A:

### Apples and Oranges

The flyer was slick, all right. The information was laid out so anybody could see that leasing palletizing equipment through the Provincial Finance Company (PFC) made much more sense than buying it. It was something Naomi could copy right into her report to Clem.

Naomi had been asked to check out options for automating part of the shipping department. Parts were to be stacked and bound on plastic pallets, then loaded onto trucks and sent to one of Global Widgets' sister companies. The saleswoman tor the company whose equipment seemed most suitable for Global Widgets' needs included the leasing flyer with her quote.

Naomi looked at the figures again. They seemed to make sense, but there was something that didn't seem right to her. For one thing, if it was cheaper to lease, why didn't everybody lease everything? She knew that some things, like automobiles and airplanes, are often leased instead of bought, but generally companies buy assets. Second, where was the money coming from to give the finance company a profit? If the seller was getting the same amount and the buyer was paying less, how could PFC make money?

"Got a recommendation on that palletizer yet, Naomi?" Clem's voice was cheery as he suddenly appeared at her doorway. Naomi knew that the shipping department was the focus of Clem's attention right now and he wanted to get improvements in place as soon as possible.

"Yes, I do. There's really only one that will do the job. and it does it well at a good price. There is something I'm trying to figure out, though. Christine sent me some information about leasing it instead of buying it, and I'm trying to figure out where the catch is. There has got to be one, but I can't see it right now."

"OK, let me give you a hint: apples and oranges. You can't add them. Now, let's get the paperwork started for that palletizer. The shipping department is just too much of a bottleneck." Clem disappeared from her door as quickly as he had arrived, leaving Naomi musing to herself.

"Apples and oranges? Apples and oranges? Ahh... apples and oranges, of course!"

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