quantative techniques

7/21/2019 Quantative Techniques

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Topics

Time Value of Money Concepts

Bond Valuation Theory Concepts

Sampling Concepts Regression and Correlation Concepts

Linear Programming Concepts

Simulation Concepts


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Time Value of Money


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Objecties

!hat do "e mean by Time alue ofmoney

Present Value# $iscounted Value#%nnuity


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Time Value approach

Time alue of money is the conceptof measuring the alue of moneyoer time&

!hy do "e consider'

Because alue of money changes"ith time and it(s crucial to analyseour inestment to be able tomeasure and sole for thosechanges&


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Time Value approach

People prefer present consumption tofuture consumption ) demand more infuture to gie up present consumption

*n+ation e,ect ) -reater in+ation anderosion of alue

.ncertainty of receiing cash +o" in future) -reater the ris/# greater the erosion inalue

Process by "hich future cash +o"s areadjusted to re+ect these factors is calleddiscounting and magnitude of thesefactors is called discount rate


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$iscount Rate

Rate at "hich present and future cash+o"s are traded o,&

*t incorporates

The preference for current consumption0greater preference 1111 2igher discount rate3& 45pected in+ation 0higher in+ation 1111 higher

discount rate3&The uncertainty in the future cash +o"s 0higher

ris/ 1111 higher discount rate3& % higher discount rate "ill lead to a lo"er

present alue for future cash +o"s


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Compounding concepts

Compounding e,ect increases "ith bothrate and compounding period

%s length of holding period increases#small di,erences in rate can lead tolarge di,erences in future alues

Common rule of 67 ) $oubling the alue

n

n

iPFV )1(* +=


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Time Value of Money

!hat is Time Value of Money' 8uture Value

Present Value

8uture Value9 Compounding9

How would you

do

Compounding?


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Compounding

Compounding 8ormula

!hat if compounding is done on monthly

basis'

n

n iPFV )1(* +=

tn

nt

iPFV

*

1*

+=

Microsoft O:ce

45cel !or/s heet


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4,ectie *nterest Rate

True rate of interest ) Ta/es intoaccount compounding e,ects ofmore fre;uent interest payments

4,ectie *nterest Rate < 0=>Stated%nnual *nterest Rate?@3nA=

%s compounding becomes fre;uent#e,ectie rate increases and presentalue of future cash +o" decreases


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Charting of Cash+o"

8or any nancial proposition prepare a chart of cash+o"9e&g&

Timeline

01.01.0

8

Invested in Bonds

(1,000)

30.0.08

Inte!est "e#eived $%0

31.1&.08

Inte!est "e#eived $ %0

'ew Bond u!#*sed (1,0&0)

'et ( +0)

30.0.08

Inte!est "e#eived $ 100

-old Bond $&,0%0

Tot*l $&,1%0


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$iscount Rate

Rate at "hich present and futurecash +o"s are traded o,

2igher discount rate ) lo"er thepresent alue for future cash +o"s


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$iscounting

Present Value ou hae an option to receie Rs& =#DDD?A either today or

after one year& !hich option you "ill select' !hy'

$ecision "ill depend upon the present alue of moneyE

"hich can be calculated by a process calledDiscounting (opposite of Compounding)

*nterest Rate and Time of Receipt of money decidePresent Value

!hat is the present alue of Rs& =#DDD?A today and a year

later'

Let us find out Present Value?


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$iscounting contd 8ormula to nd Present Value of 8uture Cash Receipt

!here PV < Present Value# P < Principal# i < Rate of *nterest# n

He!e, ' 3

ene!i#*lly e;p!essed,

te 2o!mul* is>

He!e, ' 3

!rinci"a# ! 1,000 1,000 1,000

$nterest Rate i 10% 10% 10%

&' n 1 2 3

(i)es *iscountin+ in a 'ear t 2 2 2*iscount actor * 0.52 0.0/0 0.3

!resent a#ue !!4* 52.3 0/.03 3.

u) of !resent a#ue

6ssu)in+ *iscountin+ *one e)i-6nnua##y

2,/23.25

3&1

&

6101

1000

&

6101

1000

&

6101

1000&%.&&3

+

+

+

+

+

=

=

+

=N

nn

n

t

i

x

PV

11


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Types of Cash +o"s

Simple Cash +o"

%nnuities

-ro"ing %nnuities Perpetuities

-ro"ing Perpetuities


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Simple Cash +o"

Single Cash +o" in a specied future timeperiod

$iscounting9 process by "hich a cash +o"

is e5pected to occur in the future isbrought to its present alue

Compounding9 *s the process by "hich acash +o" today is conerted to itse5pected future alue


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%nnuities

Constant cash +o"occurring atregular interals of

time %n annuity can

occur at the end ofeach period# as in

this time line# or atthe beginning ofeach period&

+

=

r

rCAPV

t)1(

11


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e5ample

Outright Buy V?s $eferred Payment

Choice of Rs& J#DD#DDD upfront or payKDDDD for e years

PV for KD#DDD using earlier formula )G#7J#JGD

Therefore choice&

!hen the present alues of yourinstalment payments e5ceed the cashdo"n price it is better to pay cash do"nand ac;uire the asset&


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Perpetuity and %nnuity

Perpetuity Present Value

t => PVIF(r, ) = 1/r=> APV = C/r

Annuity Future Value

+=r

rCAFV

t 1)1(


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Annuity present value interest factors

Number

ofperiods

Interest rate

5% 10% 15% 20%

1 0.95! 0.9091 0."#9# 0."$$$

2 1."59! 1.%$55 1.#5% 1.5%"

3 .%$ .!"#9 ."$ .10#5

4 $.5!#0 $.1#99 ."550 .5""%

5 !.$95 $.%90" $.$5 .990#

+

=r

rtrPVIFA

t)1(

11

),(


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45amples9 %nnuity PresentValue

Annuity Present Value Suppose you need 2! eac" year

for t"e next t"ree years to make

your fees payments#Assume you need t"e $rst 2!in exactly one year# Suppose youcan place your money in a sa%in&s

account yieldin& '( compoundedannually# )o* muc" do you need to"a%e in t"e account today


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45amples9 %nnuity Present Value0continued3

Annuity Present Value , Solution)ere *e kno* t"e periodic cas" o*s are

2! eac"# -sin& t"e most .asicapproac"/

PV = 2!01#' + 2!01#'2+2!01#'

= 1'!31'#32 + 14!156#44+ 13!'46#63

= 31!351#75

)ere8s a s"ortcut met"od for sol%in& t"e pro.lemusin& t"e annuity present value factor/

PV = 2! 9::::::::::::;0::::::::::

= 2! x 2#34474

= ::::::::::::::::


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45amples9 %nnuity Present Value 0continued3

Annuity Present Value , Solution)ere *e kno* t"e periodic cas" o*s are2! eac"# -sin& t"e most .asicapproac"/

PV = 2!01#' + 2!01#'2

+ 2!01#' ;0#'

= 2! 2#34474

= 31!351#75


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%nother problem

Suppose *e expect to recei%e 1 at t"e end ofeac" of t"e next 3 years# ?ur opportunity rate is 6(#@"at is t"e %alue today of t"is set of cas" o*s

PV = 1 1 , 101#6>3B0#6 = 1 1 , #45426B0#6

= 1 5#21265 = 5212#6

No* suppose t"e cas" o* is 1 per year forever#C"is is called aperpetuity# And t"e PV is easy tocalculate/

PV = C0r= 10#6 = 16!666#66D So! payments in years 6 t"ru "a%e a total PV of

12!535#


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8inding C

Example: Finding C F# Gou *ant to .uy a motorcycle# It costs 23!#

@it" a 1( do*n payment! t"e .ank *ill loan yout"e rest at 12( per year 1( per mont"> for 6mont"s# @"at *ill your mont"ly payment .e

A# Gou *ill .orro* #7 23! = 22!3# C"is ist"e amount today! so it8s t"e present %alue# C"e rateis 1(! and t"ere are 6periods/

22!3= C 1 , 101#1> B0#1 = C1 , #3353B0#1

= C 55#733C = 22!3055#733

C = 3#3 per mont"


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8inding t

F# Suppose you o*e 2 on aVisa card! and t"e interest rateis 2( per mont"# If you make t"e

minimum mont"ly paymentsof 3! "o* lon& *ill it take you to

pay oH t"e de.t Assume youuit c"ar&in& immediately>


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6. 6 lonti)e:

2000 50 71 - 1/81.029t/.02

.0 1 - 1/1.02

t

1.02t 5.0t #n81.029 5.0t #n85.09/#n81.029

t 1.3 )onths, or about.years


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Annuity future value interest factorsNumber ofperiods

Interest rate

5% 10% 15% 20%

1 1.0000 1.0000 1.0000 1.0000

2 .0500 .1000 .1500 .000

3 $.155 $.$100 $.!%5 $.#!00

4 !.$101 !.#!10 !.99$! 5.$#"0

5 5.55# #.1051 #.%!! %.!!1#

+=

r

rtrFVIFA

t1)1(

),(


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45amples for future alue ofannuities & Suppose you deposit 7DDD each year for the ne5t

three years into an account that pays NF& 2o" much"ill you hae in G years' *mportant9 ou ma/e the rstdeposit in e5actly one year&

%& .sing the most basic formula for 8V9

8V < 7DDD =&DN11 > 7DDD =&DN11 > 7DDD < 7GG7#ND > 7=D > 7DDD < #JK7#ND

.sing the shortcut formula at the top of the page9

8V< 7DDD Q11111111111 ? D&DN < 7DDD G&7JJ < JK7#ND

+=

r

rtrFVIFA

t 1)1(),(


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45ample contd

& Suppose you deposit 7DDD each year for the ne5t threeyears into an account that pays NF& 2o" much "ill youhae in G years' *mportant9 ou ma/e the rst deposit ine5actly one year&


8V < 7DDD =&DN11 > 7DDD =&DN11 > 7DDD < 7GG7#ND > 7=D > 7DDD < #JK7#ND

.sing the shortcut formula at the top of the page9

8V< 7DDD Q11111111111 ? D&DN < 7DDD G&7JJ < JK7#ND

+=

r

rtrFVIFA

t 1)1(),(


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& Suppose you deposit 7DDD each year for the ne5t threeyears into an account that pays NF& 2o" much "ill youhae in G years' *mportant9 ou ma/e the rst deposit ine5actly one year&


8V < 7DDD =&DN7 > 7DDD =&DN= > 7DDD < 7GG7#ND > 7=D > 7DDD < #JK7#ND

.sing the shortcut formula at the top of the page9 8V< 7DDD Q0= > D&DN3GA = ? D&DN < 7DDD G&7JJ < JK7#ND

+=

r

rtrFVIFA

t 1)1(),(


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Perpetuity

% perpetuity is a constant cash +o" paid0or receied3 at regular time interalsforeer&

Thus a lifetime pension can be consideredas a perpetuity or rentals receied frome5ploitation of land "hich is passed onfrom generation to generation&

The present alue of a perpetuity can be"ritten as C?r


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Console Bond

% is a bond that has no maturity and paysa 5ed coupon 0rate of interest3&

%ssume that you hae a per cent couponconsole bond& The original face alue < Rs

=DDD& The current alue of this bond if theinterest rate is K per cent is as follo"s&

Current alue of Console Bond < RsD?D&DK < Rs 6

The alue of a Console bond "ill be e;ualto its face alue only if the coupon rate ise;ual to the interest rate& *n this case Rs=DDD# i&e& D?D&D


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-ro"ing %nnuity

% gro"ing %nnuity is a cash +o" that ise5pected to gro" at a constant rateforeer

PV < C=?0r-g3 A 0=?0r-g3300=>g3?0=>r33t # %lthough a gro"ing annuity and a gro"ing

perpetuity share seeral features# the factthat a gro"ing perpetuity lasts foreer

puts constraints on the gro"th rate& *t hasto be less than the discount rate for theformula to "or/&


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Suppose you hae just "on the rst priUein a lottery& The lottery o,ers you t"opossibilities for receiing your priUe& The

rst possibility is to receie a payment of=D#DDD at the end of the year# and then#for the ne5t =H years this payment "ill berepeated# but it "ill gro" at a rate of HF&

The interest rate is =7F during the entireperiod& The second possibility is to receie=#DD#DDD right no"& !hich of the t"opossibilities "ould you ta/e'


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C < =D#DDDr < D&=7g < D&DH

t < = PV < =D#DDD 0=?D&D63 A

0=?D&D630=&DH?=&=73= < WK=#KNK&J=

X W=DD#DDD# therefore# you "ouldprefer to be paid out right no"&


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%ssume the same situation as in45ample *# but "ith the di,erencethat you can no" ma/e a choice

bet"een receiing a payment of=D#DDD at the end of year =# "hich"ill then gro" at HF per year# and be

paid out to you for the ne5t =H years&Or# you can receie NH#DDD right no"&!hat "ould you do'


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!e /no" from 45ample * that the presentalue of the gro"ing annuity is e;ual toK=#KNK&J=& 2o"eer# the annuity starts

only at the end of year =# and hence# "eneed to bring this alue bac/ oneadditional period before "e can compare itto the NH#DDD to receied right no"& Thus9

PV < K=#KNK&J= ? 0=&=73 < N7#=GG&JD XNH#DDD# so "e still prefer to be paid outimmediately&


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Jro*in& Perpetuity

% gro"ing perpetuity is the same asa regular perpetuity 0C?r3#but thecash +o" is gro"ing 0or declining3

each year& % perpetuity has no limit to the

number of cash +o"s# it "ill go

indenitely& The gro"ing perpetuityis in that "ay just the same as agro"ing annuity "ith an e5tremely

large t.


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PV < 7H ? 0D&DJ6H A D&D=3 < &6


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Capital Budgeting

4ery business has four basic decisions toma/e9

@"ic" proKects to take In%estment

decisions> 2o" to nance these projects' Linancin&decisions>

2o" much to return to inestors'

Mi%idend decisions> 2o" to manage "or/ing capital and itscomponents' iuidity decisions>


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@et Present Value Net Present Value means the dierence !et"een the PV of

Cash #n$o"s % Cash &ut$o"s

'o" do you compute NPV Prepare Cash$o" Chart

Net o #n$o" % &ut$o" for each period separately #f #n$o" &ut$o"* positive cash

#f #n$o" + &ut$o"* negative cash

Find present values of #n$o"s % &ut$o"s !y applying

Discount Factor (or Present Value Factor) NPV , (PV of #n$o"s) -E.. (PV of &ut$o"s)/ 0esult can !e 1ve

&0 2ve

Continuing "ith our example of 3ond #nvestment:


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@PV contd

If Oas"o*s are discounted at say 1(! t"e sum of PV is23#3! a positi%e num.er

!hat is I""?

*escri"tion *ate 6)ount $n / Out ! Outf#ow ! $nf#ow

Invested in 106 Bonds 01@*n08 (1,000) ut2low (1,000.00)

Inte!est !e#eived 30@un08 %0 In2low A.&

Inte!est !e#eived 31e#08 %0 In2low A%.3%'ew Bond u!#*sed 2!om

pen *!=et31e#08 (1,0&0) ut2low (+&%.1)

Inte!est !e#eived 30@un08 100 In2low 8.38

-old Bond in pen *!=et 30@un08 &,0%0 In2low 1,0.8

-um (1,+&%.1) 1,+%0.&&

'et !esent D*lue &%.0%#ow these values are arrived at?


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@PV

If... It means... Ten...

&PV > 0

t'e inestent *+ula alue t+ t'e-ir t'e pr+et ay e aepte

&PV 0

t'e inestent *+ulsutrat alue-r+ t'e -ir t'e pr+et s'+ul e reete

&PV = 0

t'e inestent *+ulneit'er 2ain n+rl+se alue -+r t'e-ir

3e s'+ul e ini--erent int'e eisi+n *'et'er t+aept +r reet t'epr+et. 4'is pr+et asn+ +netary alue.eisi+n s'+ul e ase+n +t'er riteria, e.2.I66et.


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*nternal Rate of Return 0*RR3

$enition9 The Rate at which the NPV is Zero. It can also betermed as Eective 0ate

*f "e "ant to nd out *RR of the bond inestment cash+o"9


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*RR Contd

To proe that at *RR of ==&GNF the @PV of *nestmentCash+o" is Uero# see the formula Y table9

3&10

&

638.111

&1%0

&

638.111

+0

&

638.111

%0

&

638.111

10000

+

+

+

+

+

+

+

=


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*RR contd

%s an inestment decision tool# thecalculated *RR should notbe used torate mutually e5clusie projects# but

only to decide "hether a singleproject is "orth inesting in&

Since *RR does not consider cost of

capital# it should not be used tocompare projects of di,erentduration


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BO@$ V%L.%T*O@


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Objecties

$istinguish bonds coupon rate#current yield# yield to maturity

8ind the mar/et price of a bond gien

its yield to maturity# nd a bond(syield gien its price# anddemonstrate "hy prices and yieldsmay ary inersely

!hy bonds *nterest rate ris/ Bond ratings and inestors demand

for appropriate interest rates


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Bond characteristics

BondA eidence of debt issued by a bodycorporate or -ot& *n *ndia# -otpredominantly

% bond represents a loanmade by inestors tothe issuer.*n return for his?her money# theinestor receies a lega* claim on future cash+o"s of the borro"er&

The issuer promises to9 Ma/e regular couponpayments eery period until the

bond matures# and

Pay the face?par?maturity alueof the bond "hen itmatures


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4lements of Bondonds reuire coupon or interest payments

determined as part of t"e contract

Ooupon payments represent interest on t"e

.ond

Linal interest payment and principal are paid

at speci$c date of maturity

face par> %alue/ amount paid to bondholder at

maturity

coupon payments/ interest paid

maturity or term>/ the end of life time of a

bond


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Bond Concepts

*ssuer9 company# state or country

Coupon9 5ed interest rate that issuerpays to lender 0inestor3

Maturity date9 date "hen borro"er "illpay the lenders 0inestor3 principal bac/

Bid price9 price that someone is "illing topay the lenders

ield9 indicates annual returnuntil thebond matures


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2o" do bonds "or/'

*f a bond has e years to maturity# an Rs&ND annualcoupon# and a Rs&=DDD face alue# its cash +o"s "ould loo/li/e this9

Time D = 7 G J H Coupons Rs&ND Rs&ND Rs&ND Rs&ND Rs&ND

8ace Value=DDD Mar/et Price Rs&1111 2o" much is this bond "orth' *t depends on the leel of

current mar/et interest rates& *f the going rate on bonds li/ethis one is =DF# then this bond has a mar/et alue ofRs&K7J&=N& !hy'


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!oupon payments "ace value#aturity

Annuity component$ump sumcomponent

nrFI

rI

rIPV

ondaforformula!eneral

)1()1(1

>

& ++++

++

+=

%%A3& )10.01(

1000

)10.01(

80

)10.01(

80

)10.01(

80

)10.01(

80

10.01

80)(

++

++

++

++

++

+= ondof"ri#ePV

Bond prices and *nterest


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Bond prices and *nterestRates *nterest rate same as coupon rate

Bond sells for face alue

*nterest rate higher than coupon rate Bond sells at a discount

*nterest rate lo"er than coupon rate Bond sells at a premium


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Bond terminology

ield to Maturity $iscount rate that ma/es present alue

of bond(s payments e;ual to its price

Current ield%nnual coupon diided by the

current mar/et price of the bond

Current yield < ND ? K7J&=N price change

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

*nestment

e&g& you buy F bond at =D=D&66 and sellne5t year at =D7D

Rate of return < D>K&GG?=D=D&66

O l ti


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Oorrelation

Oorrelation exists .et*eent*o %aria.les *"en one of

t"em is related to t"e ot"erin some *ay

Assumptions


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Assumptions

1# C"e sample of paired datax*y> is a random sample#

2# C"e pairs of x*y> data "a%ea .i%ariate normaldistri.ution#

S tt di


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Scatter dia&ram

Scatterplot or scatterdia&ram>

is a &rap" in *"ic" t"e

paired x*y> sample data

are plotted *it" a"oriontalxaxis and a%ertical yaxis# Eac"

Positi%e inear Oorrelation


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Positi%e inear Oorrelation

x x

yy y

x

a &ositive b tron' positive

c &erfect positive

Ne&ati%e inear Oorrelation


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Ne&ati%e inear Oorrelation

x x

yy y

x

() &e2atie (e) 7tr+n2 ne2atie (-) Per-et ne2atie

No inear Oorrelation


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No inear Oorrelation

x x

yy

' No !orrelation Nonlinear !orrelation

Correlation %nalysis


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Correlation %nalysis

Statistical tool to describe the degree to"hich one ariable is linearly related toanother

Often used in conjunction "ith regression

analysis Three measures

Coe:cient of determination 8or measuring e5tent or strength of association

Coariance 8or direction and strength of the relationship

Coe:cient of correlation $imensionless alue sho"ing e5tent and direction of

relationship


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T*M4 S4R*4S

Objecties


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Objecties

.nderstanding four components oftime series

Compute seasonal indices

Regression based techni;ues

Time series


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Time series

-roup of data or statisticalinformation accumulated at regularinterals

Variations in Time series


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Variations in Time series

Secular trend % persistent trend in a single direction& % mar/et

moement oer the long term "hich does not re+ectcyclicalseasonal or technical factors&

Cyclical +uctuation

The term .usiness cycleor economic cyclerefers tothe +uctuations of economic actiity 0.usinessuctuations3 around its longAterm gro"th trend& Thecycleinoles shifts oer time bet"een periods ofrelatiely rapid gro"th of output 0recoery andprosperity3# and periods of relatie stagnation or decline0contraction or recession3&

Seasonal ariation Pattern of change "ithin a year

*rregular ariation .npredictable# changing in a random manner

Secular Trend
http://glossary.reuters.com/?title=Cyclicalhttp://en.wiktionary.org/wiki/cyclehttp://en.wikipedia.org/wiki/Recessionhttp://en.wikipedia.org/wiki/Recessionhttp://en.wiktionary.org/wiki/cyclehttp://glossary.reuters.com/?title=Cyclical


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Secular Trend

Cyclical Trend


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Cyclical Trend

Seasonal


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Seasonal

Trend analysis


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Trend analysis

To describe historical patterns Past trends "ill help us project future


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L*@4%R PRO-R%MM*@-

Objecties


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Objecties

.nderstanding Linear programmingbasics

-raphic and Simple5 methods

Linear Programming


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Linear Programming

Mathematical techni;ue used toallocate limited resources amongcompeting demands in an optimal

"ay 4&g& resource and mar/eting

constraints

Certain !or/ing capital re;uirements Capacity constraints

Labour aailability

Ra" materials aailability

Linear Programming


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Linear Programming

Problem formulation if %ll e;uations are linear ) if J persons

produce = unit# for G# =7 persons are

needed Constraints are /no"n and deterministic

) probability of occurrence is ta/en as=&D

Variables should hae non negatiealues

$ecision alues are also diisible

Types of LP problems


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Types of LP problems

Ma5imisation A Prot Minimisation A Costs

TransportationA to minimise cost of

shipping products and at the same timema5imise shipping m units to ndestinations

$ecision ma/ing

8or Sensitiity of results -oal programming ) Objectie function

8inancial Budgeting


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Simulation

Simulation


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Simulation

Studying e,ects of changes in real "orldthrough models

%dantages9

45periments can be conducted before realsystem is operational# reduces costssubstantially

%ppropriate to situations "here siUe andcomple5ity of problem ma/e use of techni;uesdi:cult

Training needs

Sensitiity analysis

Simulation


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Simulation

$isadantages9Time consuming

Re;uires substantial computer

e5perience and e5pertise Chances of oerloo/ing seemingly

di:cult scenarios

More art than science

Simulation %pplications


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Simulation %pplications

%ir tra:c control ;ueuing %ircraft maintenance scheduling

%ssembly line scheduling

Rail freight carriers

8acility layout

8light simulators?$riing simulator

Simulation Methodology


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Simulation Methodology

Start $ene Problem

Construct simulation model

Specify alues of parameters and ariables Run simulation

4aluation of results

Propose ne" e5periment Stop

Simulation A 8eatures


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Simulation A 8eatures

Model ) representatie of system' Time incrementing procedure ) 5ed

time or ariable

quantative techniques

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