eeconomics_chapter_4.ppt

7/18/2019 eeconomics_chapter_4.ppt

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Copyright 2009 by Pearson Education, Inc.

Upper Saddle Rier, !e" #ersey 0$%&'(ll rights resered.

Engineering Economy, )ourteenth Edition*y +illia -. Sullian, Elin . +ic/s, and C. Patric/ oelling

Engineering Economy

Chapter 4: The Time Value of Money


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The objective of Chapter 4 is to

explain time value of money

calculations and to illustrateeconomic equivalence


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Money has a time value

! Capitalrefers to "ealth in the form of

money or property that can be used to

produce more "ealth

! Engineering economy studies involve the

commitment of capital for extended periods

of time

! # dollar today is "orth more than a dollar

one or more years from no" $for several

reasons%


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&eturn to capital in the form of interest and

profit is an essential ingredient of

engineering economy studies

! 'nterest and profit pay the providers of capital forforgoing its use during the time the capital is being

used! 'nterest and profit are payments for the riskthe

investor ta(es in letting another use his or hercapital

! #ny project or venture must provide a sufficientreturn to be financially attractive to the suppliersof money or property


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)imple 'nterest: infrequently

used*hen the total interest earned or charged is linearly

proportional to the initial amount of the loan

$principal%+ the interest rate+ and the number of

interest periods+ the interest and interest rate are said

to besimple


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Computation of simple interest

The total interest+ '+ earned or paid may be computed

using the formula belo"

P, principal amount lent or borro"ed

N, number of interest periods $eg+ years%

i, interest rate per interest period

The total amount repaid at the end ofNinterest

periods isP-I


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'f ./+000 "ere loaned for five years at a

simple interest rate of 12 per year+ theinterest earned "ould be

)o+ the total amount repaid at the end

of five years "ould be the originalamount $./+000% plus the interest

$.3+1/0%+ or .+1/0


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Compound interest reflects both the remaining principal

and any accumulated interest 5or .3+000 at 3026

7eriod

$3%

#mount o"edat beginning of

period

$8%,$3%x302

'nterestamount for

period

$9%,$3%-$8%

#mounto"ed at end

of period3 .3+000 .300 .3+300

8 .3+300 .330 .3+830

9 .3+830 .383 .3+993

Compound interest is commonly used in personal and

professional financial transactions


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Economic equivalenceallo"s us to

compare alternatives on a common basis! Each alternative can be reduced to an

equivalent basisdependent on

interest rate+amount of money involved+ and

timing of monetary receipts or expenses

! ;sing these elements "e can


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*e need some tools to find economic

equivalence

! >otation used in formulas for compound interestcalculations i , effective interest rate per interest period

> , number of compounding $interest% periods

7 , present sum of money? equivalentvalue of one ormore cash flo"s at a reference point in time? the present

5 , future sum of money? equivalentvalue of one ormore cash flo"s at a reference point in time? the future

# , end@of@period cash flo"s in a uniform seriescontinuing for a certain number of periods+ starting atthe end of the first period and continuing through thelast


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# cash flo" diagram is an indispensable

tool for clarifying and visualiAing aseries of cash flo"s


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Cash flo" tables are essential to

modeling engineering economyproblems in a spreadsheet


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*e can apply compound interest

formulas to


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't is common to use standard notation for

interest factors

This is also (no"n as thesingle payment

compound amountfactor The term on theright is read


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*e can use these to find economically

equivalent values at different points in time.8+/00 at time Aero is equivalent to ho" much after six

years if the interest rate is B2 per year

.9+000 at the end of year seven is equivalent to ho"

much today $time Aero% if the interest rate is 2 peryear


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Engineering Economy, )ourteenth Edition

*y +illia -. Sullian, Elin . +ic/s, and C. Patric/ oelling

There are interest factors for a series of

end@of@period cash flo"s

Do" much "ill you have in 40 years if you

save .9+000 each year and your account

earns B2 interest each year


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5inding the present amount from a series

of end@of@period cash flo"s

Do" much "ould is needed today to provide

an annual amount of ./0+000 each year for 80

years+ at 2 interest each year


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7roblem F3

! The manager of a canned food processing plant

must decide bet"een t"o different labeling

machines Machine # "ill have a first cost of.48+000+ an annual operating cost of .8B+000+ and

a service life of 4 years Machine G "ill cost

./3+000 to buy and "ill have an annual operating

cost of .31+000 during its 4@year life #t aninterest rate of 302 per year+ "hich should be

selected on the basis of a present "orth analysis


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5inding # "hen given 5

Do" much "ould you need to set aside each

year for 8/ years+ at 302 interest+ to have

accumulated .3+000+000 at the end of the 8/

years


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5inding # "hen given 7

'f you had ./00+000 today in an account

earning 302 each year+ ho" much could you

"ithdra" each year for 8/ years


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't can be challenging to solve forNor i

! *e may (no"P+A+ and iand "ant to find

N

! *e may (no"P+A+ andNand "ant to findi

! These problems present special challenges

that are best handled on a spreadsheet


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5indingN#cme borro"ed .300+000 from a local ban(+ "hich

charges them an interest rate of 12 per year 'f #cme

pays the ban( .B+000 per year+ no" many years "ill it

ta(e to pay off the loan

)o+

This can be solved by using the interest tables and

interpolation+ but "e generally resort to a computer

solution


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5inding i

Hill invested .3+000 each year for five years in a local

company and sold her interest after five years for

.B+000 *hat annual rate of return did Hill earn

)o+

#gain+ this can be solved using the interest tables

and interpolation+ but "e generally resort to a

computer solution


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There are specific spreadsheet functions

to findNand i

The Excel function used to solve forNis

>7E&$rate+pmt+pv%+ "hich "ill compute the

number of payments of magnitudepmtrequired to

pay off a present amount $pv)at a fixed interestrate $rate%

Ine Excel function used to solve for iis

TE$nper+pmt+pv+v%+ "hich returns a fixedinterest rate for an annuity ofpmtthat lasts for nper

periods to either its present value $pv% or future value

$v%


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*e need to be able to handle

cash flo"s that do not occur until

some time in the future

! Jeferred annuities are uniform series that

do not begin until some time in the future

! 'f the annuity is deferred!periods then the

first payment $cash flo"% begins at the end

of period!-3


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5inding the value at time "of a

deferred annuity is a t"o@step

process

3 ;se $P#A+ i2+N$!% find the value of the

deferred annuity at the end of period!$"here there areN$!cash flo"s in theannuity%

8 ;se $P#F+ i2+!% to find the value of thedeferred annuity at time Aero


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)ometimes cash flo"s change by a

constant amount each period

*e can model these situations as a uniorm

gradientof cash flo"s The table belo"

sho"s such a gradient

End of 7eriod Cash 5lo"s

3 0

8 %

9 8%

: :

> &N$')%


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't is easy to find the present value

of a uniform gradient series)imilar to the other types of cash flo"s+ there is a

formula $albeit quite complicated% "e can use to find

the present value+ and a set of factors developed forinterest tables


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*e can also findAorF

equivalent to a uniform gradientseries


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End of Kear Cash 5lo"s $.%

3 8+000

8 9+0009 4+000

4 /+000

The annual equivalent of

this series of cash flo"s can

be found by considering an

annuity portion of the cash

flo"s and a gradient

portion

End of Kear #nnuity $.% Lradient $.%

3 8+000 0

8 8+000 3+000

9 8+000 8+000

4 8+000 9+000


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)ometimes cash flo"s change by

a constant rate+ +each period@@thisis ageometric gradient series

End of Kear Cash 5lo"s $.%

3 3+000

8 3+800

9 3+440

4 3+18B

This table presents a

geometric gradient series 't

begins at the end of year 3

and has a rate of gro"th+ +of 802


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*e can find the present value of a

geometric series by using the appropriate

formula belo"

*here is the initial cash flo" in the series


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*hen interest rates vary "ith

time different procedures arenecessary

!'nterest rates often change "ith time $eg+ avariable rate mortgage%

! *e often must resort to moving cash flo"s

one period at a time+ reflecting the interest

rate for that single period


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The present equivalent of a cash flo" occurring at

the end of periodNcan be computed "ith the

equation belo"+ "here ikis the interest rate for the

kthperiod

'fF4, .8+/00 and i3,B2+ i8,302+ and i9,332+ then


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>ominal and effective interest rates

! More often than not+ the time bet"een successivecompounding+ or the interest period+ is less than

one year $eg+ daily+ monthly+ quarterly%

! The annual rate is (no"n as a nominalrate

! # nominalrate of 382+ compounded monthly+

means an interest of 32 $38238% "ould accrue

each month+ and the annual rate "ould be

eectivelysome"hat greater than 382! The more frequent the compounding the greater

the eectiveinterest


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The effect of more frequent

compounding can be easilydetermined

Net rbe the nominal+ annual interest rate and(the

number of compounding periods per year *e canfind+ ithe effective interest by using the formula

belo"


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5inding effective interest rates

5or an 3B2 nominal rate+ compounded quarterly+ the

effective interest is

5or a 12 nominal rate+ compounded monthly+ the

effective interest is


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'nterest can be compounded

continuously! 'nterest is typically compounded at the end

of discrete periods

!'n most companies cash is al"ays flo"ing+and should be immediately put to use

! *e can allo" compounding to occur

continuously throughout the period

! The effect of this compared to discrete

compounding is small in most cases


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*e can use the effective interest

formula to derive the interestfactors

#s the number of compounding periods gets

larger $(gets larger%+ "e find that


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U S ddl Ri ! # 0$%&'


* +illi - S lli Eli +i / d C P t i / lli

Continuous compounding

interest factors

The other factors can be found from these

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eeconomics_chapter_4.ppt