Lecture

2

Engineering Economics ENGR 3300

Department of Mechanical EngineeringInter American University of Puerto Rico

Bayamon CampusDr. Omar E. Meza Castillo

omeza@bayamon.inter.eduhttp://www.facultad/bayamon.inter.edu/omeza

Lecture

2 Time Value of Money

Interest and Equivalence: Time Value of Money, Simple Interest,

Equivalence, Single Payment Compounded Interest

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Time Value of Money

The economic value of a sum of money depends of WHEN it is received.

A dollar in our hands today is worth more than a dollar promised to us in the future.o The dollar we have today is ours for

certain.o The dollar can be “consumed”

immediately if desired.o The dollar has EARNING POWER – we

can invest or lend the dollar and receive INTEREST.

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Joe Cool’s parents are offering him a college loan of $5K per year for 4 years to get a UG degree in Engineering.

Beginning one year after graduation, Joe has to repay the loan at $5K per year. Is this a good deal for Joe?

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Question 2 - Power-Ball Lottery

A suburban Chicago couple won the Power-ball.

They had to choose between a single lump sum $104 million, or $198 million paid out over 25 years (or $7.92 million per year).

The winning couple opted for the lump sum.

Did they make the right choice?

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To make such comparisons (the lottery decision problem), we must be able to compare the value of money at different point in time.

To do this, we need to develop a method for reducing a sequence of benefits and costs to a single point in time. Then, we will make our comparisons on that basis.

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Time Value of Money

Money has a time value because it can earn more money over time (earning power).

Time value of money is measured in term of interest rate.

Interest is the cost of money – a cost to the borrower and an earning to the lender.

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Elements of Transactions Involving Interest

The initial amount of money invested or borrowed in a transaction is called the principal (P).

The interest rate (i) measures the cost or price of money and is expressed as a percentage per period of time.

A period of time called the interest period (n) determines how frequently interest is calculated.

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….

A specified length of time marks the duration of the transaction and thereby establishes a certain number of interest periods (N).

A plan for receipts or disbursements (An) yields a particular cash flow pattern over a specified length of time.

A future amount of money (F) results from the cumulative effects of the interest rate over a number of interest periods.

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An = A discrete payment or receipt occurring at the end of some interest period.

i = The interest rate per interest period. N = The total number of interest periods. P = a sum of money at a time chosen for

purposes of analysis as time zero, sometimes referred to as the present value or present worth.

F = A future sum of money at the end of the analysis period. This sum may be specified as Fn.

A = An end-of-period payment or receipt in a uniform series that continues for N periods. This is a special situation where A1 = A2 = …= AN.

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Example of an Interest Transaction

An electronics manufacturing company buys a machine for $25,000 and borrows $20,000 from a bank at a 9% annual interest rate. In addition, the company pays a $200 loan origination fee when the loan commences.

Lecture

2Cash Flow Diagrams

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Cash Flow Diagrams

Represent time by a horizontal line marked off with the number of interest period specified.

The cash flows over time are represented by arrows at relevant periods: Upward arrows denote positive flows (receipts), downward arrows negative flows (disbursements).

Cash flow diagrams function in a manner similar to free-body diagrams or circuit diagrams, which most engineers frequently use.

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Cash Flow Diagrams

Lecture

2Simple and

Compound Interest

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Methods of Calculating Interest

Simple interest is interest earned on only the principal amount during each interest period.

In general, for a deposit of P dollars at a simple interest rate of i for N periods, the total earned interest would be

The total amount available at the end of N periods thus would be

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LET’S PRACTICE

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Question #1

Suppose a bank is offering its customers 3% interest on savings accounts. If a customer deposits $1500 in the account, how much interest does the customer earn in 5 years?

In this problem, we are given the interest rate (i), the amount put into the account (P), and the amount of time (N). However, before we can put these values into our formula, we must change the 3% to a decimal and make it 0.03. Now we are ready to go to the formula.

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5*)500,1*03.0(I225I

So after 5 years, the account has earned $225 in interest.

If we want to find out the total amount in the account, we would need to add the interest to the original amount.

In this case, there would be $1725 in the account.

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Question #2

Jamie wants to earn $500 in interest so she’ll have enough to buy a used car. She puts $2000 into an account that earns 2.5% interest. How long will she need to leave her money in the account to earn $500 in interest?

years10

)2000*025.0(

500

)*(

*)*(

N

N

Pi

IN

NPiI

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Question #3

A local bank is advertising that you can double your money in eight years if you invest with them. Suppose you have $1000 to invest. What interest rate is the bank offering?

%5.12

)1000*8(

1000

)*(

*)*(

i

i

PN

Ii

NPiI

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TRY THESE

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Question #1

Kelly plans to put her graduation money into an account and leave it there for 4 years while she goes to college. She receives $750 in graduation money that she puts it into an account that earns 4.25% interest. How much will be in Kelly’s account at the end of four years?

A.$127.50B.$754.0425C. $877.50D.$1275

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Question #2

Randy wants to move his savings account to a new bank that pays a better interest rate of 3.5% so that he can earn $100 in interest faster than at his old bank. If he moves $800 to the new bank, how long will it take for him to earn the $100 in interest?

A.3.57 yearsB.0.357 yearsC. 0.28 yearsD.2.8 years

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Methods of Calculating Interest

Compound interest, the interest earned in each period is calculated on the basis of the total amount at the end of the previous period. This total amount includes the original principal plus the accumulated interest that has been left in the account.

In general, if you deposited (invested) P dollars at the interest rate i, you would have P + iP = P(1 + i) dollars at the end of one period.

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If the entire amount (principal and interest) is reinvested at the same rate i for another period, at the end of the second period you would have

P(1 + i) + i[P(1 + i) ] = P(1 + i) (1 + i) = P(1 + i)2

Continuing, we see that the balance after the third period is

P(1 + i)2 + i[P(1 + i)2 ] = P(1 + i)3

This interest-earning process repeats, and after N periods the total accumulated value (balance) F will grow to F = P(1 + i)N

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NiPF )1(

If the interest is compounded annually

If the interest is compounded a number of interest periods different to 1 (annually). Where n is the number of interest periods per year.

nN

n

iPF )1(

If the interest rate is compounded monthly, then n=12If the interest rate is compounded quarterly, then n=4If the interest rate is compounded semi-annually, then n=2If the interest rate is compounded annually, then n=1

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LET’S PRACTICE

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Question #1

Suppose you deposit $1000 in a bank savings account that pays interest at a rate of 10% compounded annually. Assume that you don’t withdraw the interest earned at the end period (one year), but let it accumulate. How much would you have at the end of year 3?

331,1$

)1.01(*1000

)1(3

F

F

iPF N

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Question #1

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Question #2

If you deposit $4,000 into an account paying 6% annual interest compounded quarterly, how much money will be in the account after 5 years?

42.5387

)346855007.1(*4000

)015.1(*4000

)4

06.01(*4000

)1(

20

5*4

F

F

F

F

n

iPF nN

After 5 years there will be $5,387.42 in the account

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Question #3

How much money would you need to deposit today at 9% annual interest compounded monthly to have $12,000 in the account after 6 years?

08.7007

)712552707.1(*000,12

)0075.1(*000,12

)12

09.01(*000,12

)1(

72

6*12

P

P

P

P

n

iPF nN

You would need to deposit $7007.08 in the account

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Question #1

If you deposit $5,000 into an account paying 6% annual interest compounded monthly, how long until there is $8,000 in the account?

A.6 yearsB.7.9 yearsC. 8.1 yearsD.10 years

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Question #2

If you deposit $6,500 into an account paying 8% annual interest compounded monthly, how much money will be in the account after 7 years?

A.$8,534.12B.$10,358.24C. $11,358.24D.$12,000

Lecture

2Economic

Equivalence

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Economic Equivalence

What do we mean by economic equivalence?

Why do we need to establish an “economic equivalence?”

How do we establish an “economic equivalence?”

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Economic Equivalence

Various dollar amounts that will be economically equivalent to $3,000 in 5 years, given an interest rate of 8%.

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Principles of Economic Equivalence

Equivalence calculations made to compare alternatives require a common time basis.

Equivalence depends on interest rate. Equivalence calculations may require

the conversion of multiple payment cash flows to a single cash flow.

Equivalence in maintained regardless of point-of-view as long we use the same interest rate.

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Types of Cash Flows

Lecture

2

Interest Formulas for Single Cash

Flows

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Single Amounts: Find F, given P, i, N

Single-Payment Compound-Amount Factor (growth factor)

Given:

FFind

P

yearsN

i

000,2$

8

%10

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Continued …

Solution:

18.287,4$)1436.2(*000,2$

)8%,10,/(000,2$

)10.01(*000,2$ 8

PF

FAppendix B

Calculator

Microsoft Excel

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Present-Worth FactorSingle Amounts: Find P, given

F, i, N Single-Payment Present-Worth

Factor (discounting factor) Given:

PFind

F

yearsN

i

000,1$

5

%12

F

P

NiFPFP

iFP N

,,/

1

0

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Continued …

Solution:

40.567$)5674.0(*000,1$

)5%,12,/(000,1$

)12.01(*000,1$ 5

FP

PAppendix B

Calculator

Microsoft Excel

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Solving for Time and Interest Rates

Solving for i Given:

iFind

yearsN

F

P

5

000,4$

000,3$

F

P

?i

0

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Continued …

Solution:

%6)5%,,/(3333.1

)5%,,/(3

4

)1(*000,3$000,4$ 5

iiPF

iPF

iAppendix B

Calculator

Microsoft Excel

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Solving for Time and Interest Rates

Single Amounts: Find N, given P, F, i

Given:

NFind

APRi

F

P

%12

000,5$

000,3$

F

P

i

0

N

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Continued …

Solution:

yearsNorN

N

5.412.1log

3/5log)12.1(

3

5

)12.01(*000,3$000,5$

Calculator

Microsoft Excel

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Homework 1 www.bc.inter.edu/facultad/omeza