Lecture Engineering economics - Chapter 4: Return on investment and single ‐ Payment calculations

Upon completion of this chapter you should understand: Applying return on investment analysis to decision problems, management goals, efficiency and productivity; time value of money and the application application of single‐payment interest calculations to single-and multiple-payment problems; time value of money or cash flow diagrams; application of compound, effective, nominal and continuous interest calculations; inflation and the time value of money.

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Chapter 4 – Unit 1

Return on Investment and Single‐

Payment Calculations

IET 350 Engineering Economics

Learning Objectives – Chapter 4

Upon completion of this chapter you should understand:

Applying return on investment analysis to decision

problems, management goals, efficiency and productivity

Time value of money and the application of single

Time value of money and the application of single‐

payment interest calculations to single‐ and multiple‐

payment problems

Time value of money or cash flow diagrams

Application of compound, effective, nominal and

continuous interest calculations

Inflation and the time value of money

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Learning Objectives – Unit 1

Upon completion of this unit you should understand:

payment problems

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The concept of interest being charged is not new having been

traced as far back as 1900 B.C. in Babylon

Except when made illegal because of religious objections

(1200‐1500 A.D.) interest has been a normal part of business

in the Western economies

Return or return on investment (ROI) may include all types

returns: interest on a savings account at the bank; interest on

investments such as certificates of deposit; dividends from

the ownership of stock; profit from selling stock; etc

4

Return

Return is measured by a percentage determined over a

specified time period:

⎟

⎜

×

= Investment

Earnings 100

Return

Return can also be used to measure gain from purchase and

sale of an investment:

5

⎠

⎝ Investment

⎟

⎜

×

=

Investment

Price Purchase ‐ Price Selling 100

Return

Costs associated with an investment are typically deducted:

⎟

⎜

×

=

Investment Expenses ‐ Revenue 100

Return

Shareholders use various return ratios as key measures of a

corporation’s financial performance:

6

⎟

⎜

×

=

Assets Total Costs ‐ Revenue 100

Assets

on

Return

⎠

⎝

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Return on Investment (ROI) can be used as an analysis and

evaluation tool for a multitude of applications.

Refer to the list on page 130 of the Bowman text

ROI is applied to decision making by using estimated returns

and the time value of money

Estimated returns may use historical data

Time value of money places all financial events at a

common point or points in time using appropriate

interest rates.

7

Return on Investment

ROI is an important benchmark used by management to

determine if changes (investments) are producing the desired

results.

ROI determination and how it is used for analysis is important y p

to all areas of an organization that employs team‐based

decision making. Examples include:

Making make vs. buy decisions must consider return

implications of investing in equipment to make components

Making investment decision on individual products by

comparing ROI rather than profit

8

Return on Investment

ROI can be thought of as the financial efficiency of the

organization. Efficiency is the measure of output over input:

Input

Output Efficiency =

ROI is the measure of profit (output) over investment (input):

9

Investment

Profit Investment

on

Input

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ROI can be determined using income before or after taxes

Since taxes may represent a significant cost to the

organization, after‐tax income is frequently used

The affect of inflation should be included in ROI calculations

since the income associated with the investment will occur

over time. Economic periods of high inflation can significantly

affect ROI

Economic conditions also affect the prevailing interest rate.

Assuming the organization is borrowing funds to finance an

investment, interest rates will also significantly affect ROI.

10

Microsoft Excel® Hints

Excel® has several built‐in ROI functions:

IRR(values, guess) → returns the internal rate of return for

a series of cash flows

XIRR(values dates guess) → returns the internal rate of

11

XIRR(values, dates, guess) → returns the internal rate of

return for a schedule of cash flows

MIRR(values, finance_rate, reinvest_rate) → returns the

internal rate of return for a series of periodic cash flows

including both the cost of investment and interest on

reinvestment of cash

End Unit 1 Material

Go to Unit 2 Time Value of Money

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Time Value of Money

payment problems

14

Time Value of Money

The value of a dollar today is different from the value of a

dollar in the future. Value of the dollar over time is affect by:

Interest paid on borrowed money

Interest received on money loaned

Interest received on money loaned.

Interest can be classified as:

Simple – applied to the principle amount only

Compound – applied to principle and interest

Continuous Compounding – similar to compound interest

except the number of time periods = infinity

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Simple Interest is paid on the amount invested or charged on

the amount loaned only.

Interest is not applied to accumulated interest. Rarely used

16

( 1 nisimple)

P

Where: F = Future Value ($)

P = Principle ($)

n = Time (#time periods)

isimple= Simple Interest (% per year)

Simple Interest ‐ Example

You loan your brother‐in‐law $2,000 at 5% simple interest per

year to be repaid in 6 years

How much will you receive if and when payment is made?

17

$2,600 F

0.30) (1

$2,000

0.05) 6 (1

$2,000 ni 1 P

=

+

×

=

× +

×

= +

×

=

F = Future Value = ?

P = Principle = $2,000

n = Time = 6 years

isimple= Simple Interest = 5% per year

Remember mathematical priorities require that you multiple

before adding. For (1+6x0.05), first multiply 6 by 0.05, then add 1.

Compound Interest

Compound interest is paid on the amount invested and the

accumulated interest Ö interest on interest.

Compound interest is an exponential function of interest

over time.

Compounding period can be daily, monthly, quarterly,

annually or any other period of time.

Compound interest is the basis for engineering economic

analysis and will be used through‐out IET 350 unless

otherwise indicated

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Compound interest is determined by the following equation:

( )n

i 1 P

19

Where: F = Future Value ($)

P = Principle ($)

n = Time (years)

i = Interest (% per time period)

Compound Interest ‐ Example

You loan your brother‐in‐law $2,000 at 5% compound

interest per year to be repaid in 6 years

20

$2,680.19 F

(1.340096)

$2,000

0.05) (1

$2,000

i 1 P F

6 n

=

×

=

+

×

= +

×

=

F = Future Value = ?

n = Time = 6 years

i = Interest = 5% per year

Interest has increased $80.19 compared to simple interest. This amount

is the interest paid on the accumulated interest Ö interest on interest.

Single Payments

Single payment calculations cover situations where a single

amount of money is borrowed or invested for some period of

time with interest compounded

Using the compound interest formula, if three of the four g p ,

factors are known, the fourth factor can be determined. For

example:

The amount to be invested today (P) can be found for a

desired future amount (F) for a given interest and time

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( ) 1 i P

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Single Payment ‐ Example

You have $5,000 to invest for 6 years at which time you need

$7,500 for graduate school tuition.

What interest rate compounded annually is required?

( )n

22

6.99%

0.069913

i 1 1.069913

i 1 1.50 )i (1 1.50

)i (1

$5,000

$7,500

i) (1

$5,000

$7,500

i 1 P F

6 n

=

= +

=

+

= +

×

= +

×

=

6 6 6

F = Future Value = $7,500

n = Time = 6 years

i = Interest = ? per year

Single Payment ‐ Example

You have $5,000 to invest in a fund that pays 4.5%

compounded annually.

When will your investment grow to $7,500?

F F t V l $7 500 F P ( )1 + i n

23

F = Future Value = $7,500

n = Time = ? years

i = Interest = 4.5% per year

( )

years 9.2 n

) n(0.044017 0.405465

ln(1.045) n ln(1.50) (1.045) 1.50

(1.045)

$5,000

$7,500

0.045) (1

$5,000

$7,500

i 1 P F

n n n n

=

×

=

= +

×

= +

×

=

You can use logarithms to simply a

factor to a power:

ln(x Y )= Yln(x)

Solution Methods

Several methods are available to determine solutions to

compound interest problems including:

Solving the compound interest equation with any calculator

with exponential function capability.p p y F=P×( )1+in

Using interest tables

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( )1 i P

F= × +

Bowman page 139

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Several methods are available to determine solutions to

compound interest problems including:

Using built‐in compound interest functions

on a calculator (example from an HP‐39gs)

25

on a calculator (example from an HP 39gs)

Using built‐in spreadsheet functions in Excel such as the

future value function FV( )

Microsoft Excel® Hints

Excel® has several built‐in functions useful for single‐payment:

FV(rate, nper, pmt, pv, type) → returns the future value of

an investment for a specified time period and interest rate

PV(rate nper pmt fv type) → returns the present value of

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PV(rate, nper, pmt, fv, type) → returns the present value of

an investment for a specified time period and interest rate

RATE(nper, pmt, pv, fv, type, guess) → returns the interest

rate per period for a cash flow

NPER(rate, pmt, pv, fv, type) → returns the number of

periods for a cash flow with a constant interest rate

Note that the PV value must be entered as a negative number.

Example Problem 4.1

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Example Problem 4.1 Solution

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Additional Reading Ö Financial Functions:

http://www.functionx.com/excel/Lesson12.htm

Go to Unit 3 Cash Flow Diagrams

28

Cash Flow Diagrams

payment problems

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Visual representations of complex problems are excellent

tools to conceptualize the various components and

parameters of the problem

The time value of money diagram visually shows all cash flow f y g y

and noncash financial transactions

The time value of money diagram is also known as a cash

flow diagram.

31

Time Value of Money Diagrams

Diagram conventions:

Horizontal line represents the project or investment

under analysis

Horizontal line displays time with present time t0at left

end of line

32

Arrows coming from below the line with arrow head

pointing toward the line represent cash flowing into the

project

p j

The initial investment, principle, present value or present

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The future value (F) is shown at the right end of the

horizontal line.

Arrows coming from above the line with arrow head

pointing away from the line represent cash from or out of

into the project

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The length of the lines can be drawn to scale to show the

relative amount of dollar flow.

Multiple cash or noncash inflows and outflows are shown

at the appropriate time on the horizontal line

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End Unit 3 Material

Go to Unit 4 Compounding Interest

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Compounding Interest

payment problems

38

Interest

As previously discussed, interest can be classified as:

Simple – applied to the principle amount only

Compound – applied to principle and interest

Continuous Compounding – similar to compound interest

except the number of time periods = infinity

Interest can also be classified as:

Nominal interest – interest rate without the effect of

interest compounding

Effective interest – interest rate including the effect of

interest compounding.

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Nominal interest is the stated annual interest rate.

Effective or actual interest is adjusted by the number of

compounding periods to reflect the actual interest rate for

the duration of the investment or project.p j

40

1 ‐ m

r 1

i Interest

Effective

m

⎠

⎞

⎜

⎝

⎛ +

=

Where: r = nominal interest rate/year

m = #compounding periods/year

Effective Interest ‐ Example

What is the actual interest rate given a 4% annual interest

rate compounded monthly?

1 r 1 i

m

⎟

⎜

⎛ +

=

41

4.074%

0.04074 i

1 ‐ 1.00333

1 ‐ 12

0.04 1

1 ‐ m 1 i

eff

12 12 eff

=

⎟

⎜

⎛ +

=

⎟

⎠

⎜

⎝ +

=

Effective Interest

The effective interest rate can be combined with the

compound interest equation to determine the future value of

a principle amount:

nm

⎞

⎛

42

nm

m

r 1 P

⎠

⎞

⎜

⎝

⎛ +

×

=

m = #compounding periods/year

n = time (years)

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Effective Interest ‐ Example

You loan your brother‐in‐law $2,000 at 5% annual interest

compounded weekly to be repaid in 6 years

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2699.33

$ F

1.349664

$2,000 1.000962

$2,000

52

0.05 1

$2,000 m

r 1 P F

312 5 nm

=

×

=

×

=

⎟

⎜

⎛ +

×

=

⎟

⎜

⎛ +

×

=

× 2

Interest has increased

$19.14 compared to

annually compounded

interest and $99.33

compared to simple

interest.

Continuous Compounding

Continuous compounding is not typically used for

engineering economic analysis. It is used for savings accounts

and other investment areas

The number of continuous compounding periods is infinity.p g p y

The equation uses the mathematical constant e (e=2.71828).

Scientific calculators have the function as a key (eY)

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n = time (years)

1 ‐ e

eff= ( )r n

e P

Continuous Compounding ‐ Example

You loan your brother‐in‐law $2,000 at 5% annual interest

compounded continuously to be repaid in 6 years

45

( )

2699.72

$ F

1.349859

$2,000 1.0512716

$2,000

2.71828

$2,000

e P F

6

6 0.05

n r

=

×

=

×

=

×

=

×

=

Total interest received

has increased an

additional $0.39

compared to weekly

compounded interest.

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