quantative techniques
TRANSCRIPT
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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&
%& .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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& 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&
%& .sing the most basic formula for 8V9
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