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Centre for Management of Technology andEntrepreneurship
Centre for Management ofTechnology and
Entrepreneurship
University of Toronto
Copyright: Joseph C. Paradi199!"##$
Course: C%E&$9'ile: C%E&$9(Cash'lo)*$
Cash 'lo) *nalysis Part *
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2Course:Centre for Management of Technology and
Categories of Cash 'lo)s
First cost! e+pense to ,uild or to ,uy and install
Operations and maintenance O!M"! annuale+pense- can include electricity- la,our- repairs- etc.
#al$age $alues"! receipt at pro/ect termination fordisposal of the e0uipment can ,e a salvage cost2
%e$enues! annual receipts due to sale of products orservices
O$erhauls! ma/or capital e+penditure that occurs part)ay through the life of the asset
&repaid e'penses! annual e+penses- such as leasesand insurance payments- that must ,e paid in advance
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3Course:Centre for Management of Technology and
Economic E0uivalence
%o) do )e measure and compare economic )orth ofvarious cash 3o) pro4les5 6e need to 7no):
Magnitude of cash 3o)s
Their direction receipt or dis,ursement2
Timing )hen does transaction occur2
*pplica,le interest rates2 during time period under consideration
6e should ,e economically indi8erent to choosing,et)een t)o alternatives that are economically
e0uivalent and could therefore ,e traded for one anotherin the 4nancial mar7etplace.
*ny cash 3o) can ,e converted to an e0uivalent cash3o) at any point in time.
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4Course:Centre for Management of Technology and
*n E+ample of E0uivalence
uppose you are o8ered the alternative of receivingeither &-### at the end of ; yrs guaranteed2 or Pdollars today. interest. 6hat value of P )ould
ma7e you indi8erent to your choice ,et)een P dollarstoday and the &-### at the end of ; yrs5
?etermine the present amount that is economicallye0uivalent to &-### in ; yrs given the investment
potential of => per year.
' @ &-###- A @ ; years- i @ => per year'ind PP @ '(1Bi2A@ "-#$"
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Economic E0uivalence:eneral Principles E0uivalence calculations to compare alternatives
re0uire a common time ,asis
E0uivalence depends on the interest rate
May re0uire conversion of multiple payment cash 3o)to a single cash 3o)
E0uivalence is maintained regardless of the individualDspoint of vie) ,orro)er or lender2
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)Course:Centre for Management of Technology and
E0uivalence ! * 'actor*pproach Ao) )e can loo7 at organiing the approach to
Engineering Economy ,y de4ning its languageandnotations.
Engineering Economy 'actors apply compound interest
to calculate e0uivalent cash 3o) values. The ta,ulatedvalues at the end of the ,oo7 or you can use thefunctions in spreadsheets2 convert from one cash 3o)0uantity P- '- *- or 2 to another.
*ssumptions:1. Fnterest is compounded once per period". Cash 3o) occurs at the end of the period&. Time # is period # or the start of period 1$. *ll periods are the same length
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?e4nitions
The follo)ing are the de4nitions for the varia,les used:
i interest rate per period- e+pressed as a decimalN Aum,er of periods also called the study horion2
P Present cash 3o)- or value e0uivalent to a cash 3o)series
F 'uture cash 3o) at the end of period A- or future )orthat the end of period A e0uivalent to a cash 3o) series
A Uniform periodic cash 3o) annuity2 at the end of everyperiod from 1 to A. *lso a uniform constant amounte0uivalent to a cash 3o) series.
G radient or constant period!,y!period change in cash3o)s from period 1 to A arithmetic series2
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'actor Aotation
These have their roots in the pre!computer age )henprepared ta,les )ere used ,y engineers for manydesign needs. %o)ever- they still serve to state thepro,lem and can ,e used to solve it too.
The format of engineering economy factors is: G(- A2 )here G and < are chosen from the cash
3o) sym,ols P-'-* and . o- if you have < multiplied ,y a factor- you get the
e0uivalent value of G: P @ *P(*- i -A2 e.g. convert froma cash 3o) '2 in year 1# to an e0uivalent presentvalue P2- the factor is:
P('- i A2 ! see te+t pages after ;;9 for the ta,les
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Aames of the EE 'actors !see pp =& in te+t o no) )e 7no) ho) all this )or7s together- ,ut these factors
have names as follo)s:P('-i-A2 Present )orth factor'(P-i-A2 Compound amount factorP(*-i-A2 eries present )orth factor
*(P-i-A2 Capital recovery factor- ho) much an investment has toreturn to recover its cost ! no salvage value*('-i-A2 in7ing fund factor ! )here a savings account is used toaccumulate funds for future investment'(*-i-A2 eries compound amount factor
Aote that the 4rst letter is the varia,le you are see7ing and thesecond the one you have P(* )ant 'irst Cost investment2- haveannuity payment
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Compound Fnterest 'actorsfor ?iscrete Compounding The four discrete cash 3o) patterns are:
a single dis,ursement or receipt a set of e0ual amounts in(out over a se0uence of periods !
annuity a set of e0ual amounts in(out that change ,y a constant
amountfrom one period to the ne+t in a se0uence of periods !arithmetic gradient series
a set of e0ual amounts in(out that change ,y a constantproportionfrom one period to the ne+t in a se0uence of periods! geometric gradient series
The follo)ing assumptions apply: compounding periods are e0ual cash 3o)s at the end of the period consider payment at # to
,e at period !1 annuities and gradients occur at period ends
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The Hasic ingle Payment'actors P and ' are the ,asic single payment 0uantities and
they are related as follo)s:' @ P1 B i2A can ,e developed as:'n@ P1B in2 ! )here: iis the annual interest rate !
simple interest ! ,ut this is not very useful- so )edevelop the compound interest model:'n@ 'n!11B i2
o )e can see this in the EE notation as:
'(P-i-A2 @ 1 B i2A
and for the reverseP('-i-A2 @ 1 B i2!A
The limits of P(' and '(P factors are: 1 )hen i and A approach # P(' approaches # '(P in4nity )hen i and A approach in4nity.
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The Ielationship ,et)een Pand '
P
'
# 1 " AA!1
' occurs A periods after P
Ff you had "-###no) and investedit at 1#>- ho)much )ould it ,e
)orth in = years5
P @ "-###- i @ 1#> per year- A @ = years' @ P1Bi2A@"-###1B#.1#2=@ $-"=.1=' @ P '(P-1#>- =2@"-###K".1$& @$-"=."#
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*nother P and ' E+ample
P
'
# 1 " AA!1
' occurs A periods after P
Ff 1-### is to ,e received in ; years. *t an annual
interest rate of 1">- )hat is the P of this amount5
'@1###- i @ 1">- A @ ; yearsP @ '(1Bi2A@ 1###(1B.1"2;@ ;.$#P @ 'P('- 1">-;2 @ 1####.;$2 @ ;.$#
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?iscrete Compounding-?iscrete Cash 'lo)s P. ;1
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Com,ining 'actors
6e can com,ine these factors in various )ays to createa model for the real pro,lem at hand: delayed income stream ,ecause of startup time
another need may ,e prepaid e+penses ! also a )ay to deal
)ith startup delays or construction delays. 6e can lin7 formulas together to model the actual
proposition ! in fact deriving one formulaLs factor froman otherLs
*lso- many times the pro,lem has to ,e de4ned inmore than one part
Then- there are many approaches to a speci4c pro,lem! one may ,e longer ,ut the ans)er must ,e the same
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* imple E+ample
6e invest 1"#-### P@!1"#-###N 6e pay for ; years -###(year P@!
-###P(*-1">-;2N Then pay &;-### at year & P@!&;-###P('-1">-&2N
Ieceive $#-### at year $ P@$#-###P('-1">-$2N
$#-###
1"#-###
-###
&;-###
-###
http://d/Engineering%20Economics/Library1/HiddenFiles.ppt#A%20Simple%20Examplehttp://d/Engineering%20Economics/Library1/HiddenFiles.ppt#A%20Simple%20Example7/25/2019 E02 CashFlow
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Compound Fnterest 'actorsfor *nnuities: Uniform eries *ll cash 3o)s in series are e0ual annuity2
, 2 ../2 ./,
0 0 0 0 0
&P 1*1 i2/1*1 i2/"O. *1 i2/A/12*1 i2/A
1 *1 i2/
P 1*
eries Present 6orth 'actor P 1*P(* i- A2 p." Te+t
K
1
n
+()i
+
n
N
)i1i
11(
Aote: P occurs 1 periodHefore 1st*- ' )ould occur
*t same time as last *.
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Compound Fnterest 'actorsfor *nnuities: Uniform eries 'rom ,efore- P @ * 1Bi2A!1N(i1Bi2AN
Then to get '(*- i- A2 )e can convert this to a futuresingle payment ,y multiplying ,oth sides ,y 1Bi2A
1Bi2AK P @ *1Bi2A!1N(i resulting in
' @ *1Bi2A!1N(i p.# te+t
Thus- )e get the P(*-i-A2 and the '(*-i-A2 factors
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Honds
'inancial instrument used ,y large 4rms andgovernment to raise funds to 4nance pro/ects- de,to,ligations-
* special form of loan- usually )ith a long term
The creditor 4rm or government2 promises to pay astated amount of interest at speci4ed intervals for ade4ned period and then to repay the principal at aspeci4c date maturity date2
Canada savings ,onds usually represent the lo)estinterest ,ecause they are the safest set tone forinterest rates2- then are provincial ,onds- ne+t areQHlue!ChipR corporates ,an7s- large 4rms2
QJun7R ,onds are very high interest and ris7
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Hond Terminology
P @ Purchase price of a ,ond' @ ales price or redemption value2 of a ,ondS @ Par face2 value the stated value on the ,ondr @ annual interest sho)n or coupon rate2
n @ compounding period 0uarterly- si+ months- etc.2i @ the yield rate per annum usually the mar7et rate2A @ num,er of remaining years to maturityIedemption )hen the issuer pays for it in cashMaturity ?ate date on )hich the par value is to ,e
repaid date ,ond e+pires2Mar7et Salue the price one has to pay to ,uy a ,ond* @ Sr(n2 @ the value of a single interest payment acoupon2P @ Sr(n2 P(*- i(n- nA2 B ' P('- i- A2
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Hond Saluation
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22Course:Centre for Management of Technology and
Canadian Honds
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23Course:Centre for Management of Technology and
The
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24Course:Centre for Management of Technology and
E+ample of *rithmeticradients 6e have t)o propositions for investment:
Ae) Multi Processor Computer Aatural as Pipeline
Cash 'lo) Pro4les
End of
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2(Course:Centre for Management of Technology and
C' ?iagrams for Computer Aatural as Vptions
0 1 2 3 4
$50,000
$20,000$15,000
$10,000
$5,000
(+)
(-)
0 1 2 3 4
$50,000
$20,000$15,000
$10,000
$5,000
(+)
(-)ComputerVption
Aatural asVption
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2)Course:Centre for Management of Technology and
*rithmetic radients
*n easy de4nition of an arithmetic gradient is: revenuegro)s ,y 1#-### 2 each year.
Hut the rate is not al)ays constant
There are four possi,ilities ,esides @#
1. *W# and W# ! means positive and increasing
". *W# and X# ! means positive ,ut decreasing
&. *X# and W# ! means negative ,ut ,ecoming less so
$. *X# and X# ! means negative and ,ecoming more so
# 1 " & A
"
&
$
A!12
Aote: Ao cash 3o) at end of period 1Each successive cash 3o) increases,y a 4+ed amount e0ual to
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2*Course:Centre for Management of Technology and
*rithmetic radientscontDd... * series of receipts or dis,ursements that start at # at
the end of the 1st period and then increase ,y aconstant amount from period to period. e.g. increasing operating costs for an aging machine
The sum of an annuity plus an arithmetic gradientseries is a common pattern.
The gradient formulas can ,e derived /ust as theannuity formulas )ere. ee Te+t pages =!#.
To convert gradient to uniform series:
* @ *(- i- A2
Then can convert * to P or '
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2+Course:Centre for Management of Technology and
*rithmetic radient )ith*W#
# 1 " & A
*LB
*LB"
*LB&
$
*LBA!12
*L
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29Course:Centre for Management of Technology and
eometric radient eries
Sery useful to model in3ation- change in productivity-shrin7age of mar7et sie
Each successive cash 3o) increases or decreases2 ,y
a constant > each period.
The ,ase value of the series is *. The Qgro)thR raterate of increase or decrease2 is referred to as g.
Ff g is positive- the terms are increasing in value.
Ff g is negative- terms are decreasing in value.
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*rithmetic radient )ith*W# and W# @ eometric
eries ee 'actor development in the te+t on pp 1.
# 1 " & A!1
*1Bg2*1Bg2"
$
*
A!"
*1Bg2&
*1Bg2A!1Aote: There is a cash 3o)at end of period 1
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eometric radients
6e can de4ne a gro)th ad/usted interest rate i-:
There are $ cases
i g # ro)th is B ,ut less than interest rate i-is g W i W # ro)th is B and greater than interest rate i-is g 1i W # ro)th is B and 1to interest rate
g X # ro)th is negative series is decreasing i-is
0 01
Ai P N
g
= = +
0
1 1
1 1
g
i i
+=
+ +
0 1 11
ii
g
+=
+ o that
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eometric radients'ormulas
g1
1
)i1(i
1)i1()N,i,g,A/P(
or
)g1(
)N,i,A/P()N,i,g,A/P(
N
N
+
+
+=
+
=
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?eferred *nnuity
Ff the cash 3o) does not ,egin until some later date-the annuity is 7no)n as deferred annuity.
*nnuity is deferred for Jperiods. P#@ * P(*- i>- A!J2 P('- i>- J2
# 1 "
J
JB1
JB"
JB& A!1 A
P @ 5
*
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Aon!standard *nnuities andradients Treat each cash 3o) individually Convert the non!standard annuity or gradient to
standard form ,y changing the compounding period Convert the non!standard annuity to standard ,y
4nding an e0ual standard annuity for the compoundingperiod
%o) much is accumulated over "# years in a fund thatpays $> interest- compounded yearly- if 1-### isdeposited at the end of every fourth year5
# $ = 1" 1 "#
1###
' @ 5
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?i8erent olutions
Method 1: consider each cash 3o)s separately' @ 1### '(P-$>-12 B 1### '(P-$>-1"2 B 1###'(P-$>-=2 B 1### '(P-$>-$2 B 1### @ #1&
Method ": convert the compounding period from annualto every four years
ie@ 1B#.#$2$!1 @ 1.99>
' @ 1### '(*- 1.99>- ;2 @ #1&
Method &: convert the annuity to an e0uivalent yearlyannuity
* @ 1###*('-$>-$2 @ "&;.$9 ' @ "&;.$9 '(*-$>-"#2 @ #1"
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P6 Computations )hen .
These cases are also called YperpetuitiesY ! e.g.scholarships- endo)ments- etc.
There are many engineering pro/ects )here the life ofthe pro/ect is so long as to ,e considered ZforeverL.
Usually ;# years as a rule of thum,- ,ut assume cash3o)s continue inde4nitely.
There are other pro/ects )here a sum of money isprovided at the ,eginning of the pro/ect and it is then
invested to yield the amount used for the pro/ectannuities for ;# years2
'or ,oth- the P6 of the in4nitely long uniform series ofcash 3o)s ,ecomes the Capitalied 5alue
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P6 )hen .
Until no)- )e assumed that the horion A is a 4+ednum,er of years.
The present )orth of a very long and uniform series ofcash 3o)s is calculated as:
= +
= +
+=
=
lim ( - - 2
1 2 1lim
1 2
11
1 2lim
N
N
NN
N
N
P A P A i N
iP A
i i
iP Ai
AP
i