Module 1 -and Statistics

7/29/2019 Module 1 -and Statistics

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PSE CERTIFIED SECURITIESPSE CERTIFIED SECURITIESSPECIALIST COURSESPECIALIST COURSE

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COURSE 1:COURSE 1:QUANTITATIVE METHODS OFQUANTITATIVE METHODS OF

FINANCE AND STATISTICSFINANCE AND STATISTICS


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WHAT IS MEANT BY TIMEWHAT IS MEANT BY TIME VALUE OF MONEY? VALUE OF MONEY?

Answer: Answer: It means that as one moves FORWARDIt means that as one moves FORWARD THROUGH TIME, the VALUE of his/her THROUGH TIME, the VALUE of his/herMONEY INCREASES OR GROWS.MONEY INCREASES OR GROWS.

the future value F of ones money is alwaysthe future value F of ones money is alwaysbigger than the present value P of the money;bigger than the present value P of the money;i.e., F > P or P < F (Illustrate on the whiteboardi.e., F > P or P < F (Illustrate on the whiteboardusing a timeline!)using a timeline!)

2


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ACCUMULATING VS. ACCUMULATING VS.DISCOUNTING:DISCOUNTING:

1.) The process of obtaining the future value F (using1.) The process of obtaining the future value F (using

either the simple interest method or the simpleeither the simple interest method or the simplediscount method or the compound interestdiscount method or the compound interestmethod) is called ACCUMULATING.method) is called ACCUMULATING.

2.) The process of getting the present value P (using 2.) The process of getting the present value P (using either the simple interest method or the simpleeither the simple interest method or the simplediscount method or the compound interestdiscount method or the compound interestmethod) is called DISCOUNTING.method) is called DISCOUNTING.

3


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HOW DO WE ACCUMULATE OR HOW DO WE ACCUMULATE OR

DISCOUNT UNDER THE COMPOUNDDISCOUNT UNDER THE COMPOUNDINTEREST METHOD?INTEREST METHOD?

( )

discounttovFF

PequationtheUseb.)

accumulatetoi1PFequationtheUsea.)

:Answers

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ILLUSTRATIVE EXERCISES ONILLUSTRATIVE EXERCISES ON ACCUMULATING AND DISCOUNTING: ACCUMULATING AND DISCOUNTING:

1.) Accumulate1.) Accumulate PhpPhp 100,000 for 5 years at an100,000 for 5 years at aninterest rate of 15%interest rate of 15% p.a.c.mp.a.c.m..

2.) Discount2.) Discount PhpPhp 500,000 over a period of 8 years500,000 over a period of 8 yearsat an interest rate of 16%at an interest rate of 16% p.a.c.q p.a.c.q..

(answer:(answer: PhpPhp 142,528.97)142,528.97)

5


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ILLUSTRATIVE PROBLEMS ONILLUSTRATIVE PROBLEMS ONDISCOUNTING:DISCOUNTING:

11..) ) The The commoncommon stocksstocks of of ABC ABC Company Company areare projectedprojected toto givegive

annualannual perper shareshare dividendsdividends of of PhpPhp 55,, PhpPhp 66,, PhpPhp 66..5050,, PhpPhp 88,,andand PhpPhp 1010 during during thethe nextnext oneone year,year, 22 years,years, 33 years,years, 44 yearsyearsandand 55 years,years, respectively respectively.. ItIt isis alsoalso projectedprojected thatthat by by thethe endend of of fivefive earsears thethe ricerice of of thesethese stocksstocks would would bebe PhPh 8080 erer shareshare..Using Using anan interestinterest raterate of of 2020%% pp..aa.. effectiveeffective (i(i..ee..,, 2020%% pp..aa..cc..aa..), ),determinedetermine thethe intrinsicintrinsic value value of of thesethese stocksstocks.. (answer(answer:: PhpPhp5252..1212) )

(N(N..BB.:.: aa..) ) DoDo thisthis exerciseexercise without without using using v v

!!bb..) ) InIn determining determining thethe intrinsicintrinsic value value of of aa commoncommon stock,stock, oneonemay may alsoalso useuse thethe investorsinvestors targettarget raterate of of returnreturn oror thethecompanyscompanys ROEROE..) )

6


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ILLUSTRATIVE PROBLEMSILLUSTRATIVE PROBLEMSON DISCOUNTING:ON DISCOUNTING:

2.) The projected annual yearend operating free cash flows (OFCF2.) The projected annual yearend operating free cash flows (OFCF

of XYZ Corporation are as follows:of XYZ Corporation are as follows: Year Year Yearend OFCF Yearend OFCF11 PhpPhp 100 M100 M

22 P pP p 11 M11 M33 PhpPhp 138 M138 M44 PhpPhp 149 M149 M

55 PhpPhp 170 M170 MDiscount (i.e., find the total of the present values of) these cashDiscount (i.e., find the total of the present values of) these cashflows using XYZs weighted average cost of capital (WACC) of flows using XYZs weighted average cost of capital (WACC) of

15% p.a. effective (i.e., 15%15% p.a. effective (i.e., 15% p.a.c.ap.a.c.a.). (answer:.). (answer: PhpPhp 434.362 M)434.362 M) 7


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WHAT IS STATISTICS?WHAT IS STATISTICS? A science which deals with the A science which deals with the

collection, organization, analysis,collection, organization, analysis,interpretation and presentation of interpretation and presentation of

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quant tat ve ata n or er to extractquant tat ve ata n or er to extractinformation and/or conclusions thatinformation and/or conclusions thathelp in the decisionhelp in the decision--making processmaking process..


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DIVISIONS OF STATISTICS:DIVISIONS OF STATISTICS:1.) DESCRIPTIVE STATISTICS:1.) DESCRIPTIVE STATISTICS:

Covers the procedures for organizing,Covers the procedures for organizing,summarizing, describing and presenting quantitativesummarizing, describing and presenting quantitativedata as well as the computation of statistical ordata as well as the computation of statistical or

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numerical measures such as:numerical measures such as:

1.1) Measures of Central Location1.1) Measures of Central Location1.2) Measures of Relative Location1.2) Measures of Relative Location1.3) Measures of Variability 1.3) Measures of Variability 1.4) Measures of Relationship Between/Among 1.4) Measures of Relationship Between/Among

Variables Variables


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DIVISIONS OF STATISTICS:DIVISIONS OF STATISTICS:2.) INFERENTIAL STATISTICS:2.) INFERENTIAL STATISTICS:

Dwells on drawing out conclusions about theDwells on drawing out conclusions about thecharacteristics of a population by applying thecharacteristics of a population by applying theappropriate statistical techniques and methods onappropriate statistical techniques and methods on

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quantitative data from samples that were randomly quantitative data from samples that were randomly taken from the population.taken from the population.


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KEY STATISTICAL CONCEPTS:KEY STATISTICAL CONCEPTS:1.) POPULATION or UNIVERSE1.) POPULATION or UNIVERSE

2.) SAMPLE2.) SAMPLE

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3.) PARAMETER 3.) PARAMETER

4.) STATISTIC4.) STATISTIC

5.) RANDOM VARIABLE5.) RANDOM VARIABLE

)( theand the

)( stheand xthe


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TYPES OF DATA:TYPES OF DATA:1.) CATEGORICAL DATA 1.) CATEGORICAL DATA

1.1) Yes or No or Dont Know 1.1) Yes or No or Dont Know 1.2) Equity or Fixed1.2) Equity or Fixed--Income or BalancedIncome or Balanced1.3) Banks and Financial Services, Telecom,1.3) Banks and Financial Services, Telecom,

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, , , ,, , , ,1.4) Freshman, Sophomore, Junior, Senior1.4) Freshman, Sophomore, Junior, Senior1.5) S, M , L, XL, XXL, XXXL, XXXXL1.5) S, M , L, XL, XXL, XXXL, XXXXL1.6) Rural, Thrift, Commercial, Universal1.6) Rural, Thrift, Commercial, Universal1.7) AAA, AA, A, BBB, BB, B, CCC, CC, C1.7) AAA, AA, A, BBB, BB, B, CCC, CC, C


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TYPES OF DATA:TYPES OF DATA:2.) NUMERICAL DATA:2.) NUMERICAL DATA:

2.1)2.1)DISCRETE DATA:DISCRETE DATA: age, income, expenses,age, income, expenses,sales volume, productionsales volume, productionoutput, no. of telephone callsoutput, no. of telephone calls

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per monthper month

2.2)2.2)CONTINUOUS DATA:CONTINUOUS DATA: height, IQ, Rate of height, IQ, Rate of Return, waiting time fromReturn, waiting time fromarrival up to time of being arrival up to time of being attended to, lifetime of aattended to, lifetime of a

fluorescent bulbfluorescent bulb


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CHARACTERISTICS OF THECHARACTERISTICS OF THEMEANMEAN

1.) Most commonly used1.) Most commonly used2.) Easy to compute2.) Easy to compute3.) Its value is AFFECTED by extremely big or3.) Its value is AFFECTED by extremely big or

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N.B.: These extreme values are referred to asN.B.: These extreme values are referred to asoutliersoutliers

4.) Appropriate to use when values or observations4.) Appropriate to use when values or observationsare taken from a HOMOGENEOUS populationare taken from a HOMOGENEOUS population


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HOW TO COMPUTE THEHOW TO COMPUTE THEMEANMEAN

X N i

i======== 1:MEANPOPULATION.1

16

n X

x MEAN SAMPLE

n

ii======== 1:.2


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ILLUSTRATION : COMPUTING THE MEANILLUSTRATION : COMPUTING THE MEAN

OnOn August August 2828,, 20072007,, thethe closing closing shareshare pricesprices of of banksbanks inin thethe locallocal stock stock marketmarket were were asas followsfollows:: Asiatrust Asiatrust Dev. Bank Dev. Bank : P9.5: P9.5 Security Bank : P73Security Bank : P73BDO / EPCIBDO / EPCI : 57.5 PB: 57.5 PB CommComm : 52: 52BPIBPI : 58.5 PNB : 46.5: 58.5 PNB : 46.5China Bank China Bank : 820.0* PSB : 63.5: 820.0* PSB : 63.5

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na rusna rus : . rus o. : .: . rus o. : .CitystateCitystate Savings : 23 RCBC : 25Savings : 23 RCBC : 25EIBEIB : 0.4* Union Bank : 55: 0.4* Union Bank : 55Metrobank Metrobank : 55: 55

1.) What is the mean of these share prices?1.) What is the mean of these share prices?2.) Calculate the mean excluding the outliers China Bank and EIB2.) Calculate the mean excluding the outliers China Bank and EIB

Answers: 1.) P94.89 Answers: 1.) P94.892.) P46.382.) P46.38


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THE WEIGHTED MEAN:THE WEIGHTED MEAN:

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X W X

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1:MEANWEIGHTED

18

. .. . ii ..

ILLUSTRATION: A student obtained the following grades (in %) inILLUSTRATION: A student obtained the following grades (in %) insubjects A, B, C, D, and Esubjects A, B, C, D, and E

A (5 units) A (5 units) : 75: 75

B (3 units)B (3 units) : 78: 78C (4 units)C (4 units) : 56: 56D (3 units)D (3 units) : 80: 80E (1 unit)E (1 unit) : 95: 95

Calculate the students GPA.( Answer : 73% )Calculate the students GPA.( Answer : 73% )


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MEAN OR EXPECTED VALUEMEAN OR EXPECTED VALUEOF A RANDOM VARIABLEOF A RANDOM VARIABLE

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ILLUSTRATION: EXPECTED VALUEILLUSTRATION: EXPECTED VALUEOF A RANDOM VARIABLE:OF A RANDOM VARIABLE:

Within the next 3 months, the probable rate of return on twoWithin the next 3 months, the probable rate of return on twoassets A and B (stocks, for example) are as follows:assets A and B (stocks, for example) are as follows:

RATE OF RETURNRATE OF RETURNProbabilityProbability ASSET A ASSET A ASSET B ASSET B

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.. .. ..0.30.3 0.120.12 0.230.230.10.1 0.050.05 0.100.10

Determine: 1.) The expected rate of return on Asset A.Determine: 1.) The expected rate of return on Asset A.(answer: 13.1%)(answer: 13.1%)

2.) The expected rate return on Asset B2.) The expected rate return on Asset B(answer: 11.5%)(answer: 11.5%)


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TIMETIME--WEIGHTED AVERAGE RATE OF RETURNWEIGHTED AVERAGE RATE OF RETURN

OR GEOMETRIC MEAN RATE OF RETURN:OR GEOMETRIC MEAN RATE OF RETURN:

ILLUSTRATION:ILLUSTRATION:During a certain 5 month period, a stock or issue hadDuring a certain 5 month period, a stock or issue had

n...3,2,1,ii,periodtheduringreturnof ratethe:

1)1...()1)(1(21

========

++++++++++++====

i

n

p

n p p pT

Rwhere

R R R R

21

e o ow ng mon y ra es o re urn:e o ow ng mon y ra es o re urn:Month 1Month 1 : 15%: 15%Month 2Month 2 : 12%: 12%Month 3Month 3 :: --5%5%Month 4Month 4 :: --10%10%Month 5Month 5 : 8%: 8%

What was the timeWhat was the time- -weighted average monthly rate of return weighted average monthly rate of returnduring the entire 5during the entire 5- -month period? (answer: 3.53%)month period? (answer: 3.53%)


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CHARACTERISTICS OF THECHARACTERISTICS OF THEMEDIANMEDIAN

1.) The value or observation compared to which 50%1.) The value or observation compared to which 50%

of the total number of observations are SMALLER of the total number of observations are SMALLER and 50% are BIGGER.and 50% are BIGGER.

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2.) Not affected by extremely big or extremely small2.) Not affected by extremely big or extremely small values/ observations values/ observations

3.) Finds application when the population is NOT3.) Finds application when the population is NOTHOMOGENEOUS.HOMOGENEOUS.


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HOW TO DETERMINE THEHOW TO DETERMINE THEMEDIANMEDIAN

STEPS:STEPS:1.) Arrange the values / observations in ASCENDING order.1.) Arrange the values / observations in ASCENDING order.2.) Apply the following RULES:2.) Apply the following RULES:

RULE 1: If there is an ODD number of observationsRULE 1: If there is an ODD number of observations

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(i.e.(i.e. nn = no. of observations = an odd number)= no. of observations = an odd number)

nobservation

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number EVEN anisn If RULE

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,:2


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ILLUSTRATION: OBTAININGILLUSTRATION: OBTAININGTHE MEDIANTHE MEDIAN

Obtain the median of each of the followingObtain the median of each of the following

sets of observations:sets of observations:

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. , , , , , , , ,. , , , , , , , ,

2.) 1.9%, 1.8%, 1.6%, 2.1%, 1.5%, 1.7%, 2.0%,2.) 1.9%, 1.8%, 1.6%, 2.1%, 1.5%, 1.7%, 2.0%,1.8%1.8%


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CHARACTERISTICS OF THE MODE:CHARACTERISTICS OF THE MODE:1.) It is the observation that appears1.) It is the observation that appears

MOSTMOST frequently frequently

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..extremely small valuesextremely small values

3.) May not be unique3.) May not be unique4.) May not exist4.) May not exist


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ILLUSTRATION: OBTAINING THE MODEILLUSTRATION: OBTAINING THE MODE

Find the mode of the following sets of values:Find the mode of the following sets of values:

1.) 20%, 18%, 22%, 11%, 10%, 12%, 5%, 11%,1.) 20%, 18%, 22%, 11%, 10%, 12%, 5%, 11%,

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14%14%

2.) 1.9, 1.8, 1.5, 2.1, 1.5, 1.7, 2.0, 1.82.) 1.9, 1.8, 1.5, 2.1, 1.5, 1.7, 2.0, 1.8


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MEASURES OF RELATIVEMEASURES OF RELATIVEPOSITION / LOCATION:POSITION / LOCATION:

1.)1.)QUARTILES:QUARTILES: measures that divide themeasures that divide thedistribution of the ordered data into four (4)distribution of the ordered data into four (4)

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2.2. :: measures t at v e t emeasures t at v e t edistribution of ordered data into ten (10)distribution of ordered data into ten (10)

3.)3.)PERCENTILES:PERCENTILES: measures that divide themeasures that divide thedistribution of ordered data into 100distribution of ordered data into 100


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THE QUARTILES:THE QUARTILES:FIRST QUARTILEFIRST QUARTILE == QQ 11 = the observation compared to which= the observation compared to which

25% of the total no. of observations are SMALLER and 75% a25% of the total no. of observations are SMALLER and 75% aBIGGER.BIGGER.==

SECOND QUARTILESECOND QUARTILE = Q= Q22 = the observation compared to which= the observation compared to whicht arrangemenordered theinnobservatio

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+

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50% of the total no of observations are SMALLER and 50% ar50% of the total no of observations are SMALLER and 50% arBIGGER.BIGGER.==

THIRD QUARTILETHIRD QUARTILE == QQ33 = the observation compared to which= the observation compared to which75% of the total no. of observations are SMALLER and 25% a75% of the total no. of observations are SMALLER and 25% aBIGGER .BIGGER .== t arrangemenordered theinnobservation

th

+ )1(43

Medianthe


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HOW TO OBTAINHOW TO OBTAINQUARTILESQUARTILES

STEPS:STEPS:1.) Arrange the observations in ASCENDING order1.) Arrange the observations in ASCENDING order

2.) Follow the following RULES:2.) Follow the following RULES:

RULE 1: if the resulting positioning point (i.e., the value of or of ) isRULE 1: if the resulting positioning point (i.e., the value of or of ) isan INTEGER, thean INTEGER, the observation corresponding to that positioning pointobservation corresponding to that positioning point

41+n

4)1(3 +n

29

s c osen as e quar e.s c osen as e quar e.

RULE 2: If or is HALFWAY BETWEEN TWO INTEGERS,RULE 2: If or is HALFWAY BETWEEN TWO INTEGERS,the quartile is the AVERAGE of their corresponding observations.the quartile is the AVERAGE of their corresponding observations.

RULE 3: If or is NEITHER ANRULE 3: If or is NEITHER AN INTEGER NOR INTEGER NOR HALFWAY BETWEEN TWO INTEGERS, round off to the NEARESTHALFWAY BETWEEN TWO INTEGERS, round off to the NEARESTINTEGER and select the value of the corresponding observation as theINTEGER and select the value of the corresponding observation as thequartile.quartile.

4)1(3 +n

41+n

4)1(3 +n

41+n


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ILLUSTRATION 1: OBTAININGILLUSTRATION 1: OBTAININGTHE QUARTILESTHE QUARTILES

Obtain QObtain Q 11 , Q, Q 33 , and the median for the following, and the median for the following

observations:observations:1.4, 1.5, 1.6, 1.7, 1.8, 1.8, 1.9, 2.0, 2.0, 2.0, 2.1, 2.1, 2.2, 2.2, 2.31.4, 1.5, 1.6, 1.7, 1.8, 1.8, 1.9, 2.0, 2.0, 2.0, 2.1, 2.1, 2.2, 2.2, 2.3:Solution

300.28

1.212

124

)115(34

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nobservatiotheQ Median

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ILLUSTRATION 2: OBTAININGILLUSTRATION 2: OBTAININGTHE QUARTILESTHE QUARTILES

Obtain QObtain Q11 , Q, Q33 , and the median for the following , and the median for the following

observations:observations:1.4, 1.5, 1.6, 1.7, 1.8, 1.8, 1.9, 2.0, 2.0, 2.1, 2.1, 2.2, 2.2,1.4, 1.5, 1.6, 1.7, 1.8, 1.8, 1.9, 2.0, 2.0, 2.1, 2.1, 2.2, 2.2,:Solution

3195.187

1.211

)25.11(4

)114(34

)1(3

7.14

)75.3(4

1144

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THE 5THE 5--NUMBER SUMMARY NUMBER SUMMARY A summary consisting of the following 5 A summary consisting of the following 5

numbers or values:numbers or values:XXsmallestsmallest , Q, Q11 , Median, Q, Median, Q 33 ,, XXbiggestbiggest

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This summary helps to determine if theThis summary helps to determine if thedistribution of a given set of observations isdistribution of a given set of observations is

symmetric or rightsymmetric or right- - skewed or leftskewed or left--skewed.skewed.


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USING THE 5USING THE 5- -NUMBER SUMARY TONUMBER SUMARY TORECOGNIZE DATA SYMMETRY:RECOGNIZE DATA SYMMETRY:

1.) The distance from1.) The distance from xxsmallestsmallest to the Median =to the Median =the distance from the Median tothe distance from the Median to x xbiggestbiggest

33

2.) The distance from2.) The distance from x xsmallestsmallest to Qto Q 1==the distance from Qthe distance from Q 33 toto xxbiggestbiggest


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USING THE 5USING THE 5- -NUMBER SUMMARY NUMBER SUMMARY TO RECOGNIZE DATA SKEWNESS:TO RECOGNIZE DATA SKEWNESS:

1.) RIGHT1.) RIGHT--SKEWED DISTRIBUTION OF DATA:SKEWED DISTRIBUTION OF DATA:

1.1) Distance from median to x1.1) Distance from median to xbiggestbiggest> distance from> distance fromxxsmallestsmallestto medianto median1.2) Distance from Q1.2) Distance from Q33 to xto xbiggestbiggest> distance from x> distance from xsmallestsmallest

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2.)2.) LEFTLEFT--SKEWED DISTRIBUTION OF DATA:SKEWED DISTRIBUTION OF DATA:2.1) Distance from x2.1) Distance from xsmallestsmallestto Median > distance fromto Median > distance from

Median to xMedian to xbiggestbiggest2.2) Distance from x2.2) Distance from xsmallestsmallestto Qto Q11 > distance from> distance fromQQ33 to xto xbiggestbiggest


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THE BOXTHE BOX- -AND AND--WHISKER PLOTWHISKER PLOT(or simply, BOX PLOT)(or simply, BOX PLOT)

A graphical representation of the A graphical representation of thedistribution of a given set of data based ondistribution of a given set of data based on

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-- ..illustration on the white board)illustration on the white board)


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MEASURES OF VARIABILITY OR MEASURES OF VARIABILITY OR VARIATION VARIATION

1.) The RANGE: Range =1.) The RANGE: Range = XX biggestbiggest X X smallestsmallest

2.) The VARIANCE and the STANDARD2.) The VARIANCE and the STANDARD

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3.) The COEFFICIENT OF VARIATION3.) The COEFFICIENT OF VARIATION

THE VARIANCE AND THETHE VARIANCE AND THE


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THE VARIANCE AND THETHE VARIANCE AND THE

STANDARD DEVIATIONSTANDARD DEVIATION)(

:

2

1

2

2

variance DeviationStandard

N

X Variance

POPULATION FOR N

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37etc.liters,squaredyears,squared pesos,squaredi.e.,units,

SQUAREDinmeasuredareandthatnotePlease:N.B.

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svariances DeviationStandard

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x xsVariance

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ILLUSTRATIVE EXERCISEILLUSTRATIVE EXERCISEDetermine the variance and the standardDetermine the variance and the standard

deviation of the following SAMPLE data:deviation of the following SAMPLE data:

38

. , . , . , . , . , . , . , . , . , . , . ,. , . , . , . , . , . , . , . , . , . , . ,2.2, 2.2, 2.32.2, 2.2, 2.3(answers: s(answers: s 22 = 0.076923; s = 0.277350)= 0.076923; s = 0.277350)


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THE COEFFICIENT OFTHE COEFFICIENT OF


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THE COEFFICIENT OFTHE COEFFICIENT OF

VARIATION OR CV: VARIATION OR CV:

:SAMPLEAFOR

100

:POPULATIONAFOR

xCV =

40 units.differentinmeasured

dataof setsmoreortwocomparinginusefulisCVThe2.

mean.theof unit valueperdataof essscatteredn ordispersionthemeasuresCVThe1.

:N.B.

100 x xs

CV =


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MEASURES OF LINEAR MEASURES OF LINEAR RELATIONSHIP:RELATIONSHIP:

sample).a(forrbyor

)populationa(forbydenotedisitY;andXvariablesrandomtwobetweeniprelationshlineartheof strengththeof measure

:NCORRELATIOLINEAROFTCOEFFICIEN1.

41

X.R.V.TINDEPENDEN

theof valueswith theiprelationshlinearby theexplainedorforaccountedbecanthatYR.V.DEPENDENTtheof valuesin the

tyvariabilitheof percentageorproportionthemeasures

:)ror(IONDETERMINATOFTCOEFFICIEN2. 22

COMMENTS ON THE VALUES OF rCOMMENTS ON THE VALUES OF r


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COMMENTS ON THE VALUES OF rCOMMENTS ON THE VALUES OF r

AND r : AND r :1.)1.) --1 r 11 r 12.) r = o there is no linear relationship between2.) r = o there is no linear relationship between

the values of the R.V.s X and Y the values of the R.V.s X and Y r = 1 the values of the R.V.s X and Y arer = 1 the values of the R.V.s X and Y arePERFECTLY POSITIVELY PERFECTLY POSITIVELY

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CORRELATEDCORRELATEDr =r = --1 the values of the R.V.s X and Y are1 the values of the R.V.s X and Y arePERFECTLY NEGATIVELY PERFECTLY NEGATIVELY

CORRELATEDCORRELATED ..3.) The value of r3.) The value of r22 is always positive and is expressed as ais always positive and is expressed as a

percentage.percentage.


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WORKSHEET FOR COMPUTING r:WORKSHEET FOR COMPUTING r:XX Y Y XY XY XX Y Y

XX11 Y Y 11 XX11 Y Y 11 XX11 Y Y 11

XX22 Y Y 22 XX22 Y Y 22 XX22 Y Y 22

44

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

XXnn Y Y nn XXnn Y Y nn XXnn Y Y nn

XX Y Y (XY)(XY) (X)(X) (Y)(Y)

INDIVIDUAL HOMEWORKINDIVIDUAL HOMEWORK


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INDIVIDUAL HOMEWORK INDIVIDUAL HOMEWORK

(To be submitted next meeting):(To be submitted next meeting):DETERMINING r AND r:DETERMINING r AND r:

Compute and interpret the coefficient of linearCompute and interpret the coefficient of linearcorrelation and the coefficient of determinationcorrelation and the coefficient of determination

45

Give your answer up to 3 significant digits.Give your answer up to 3 significant digits.

XX 44 55 99 1414 1818 2222 2424 Y Y 1616 2222 1111 1616 77 33 1717

STATISTICAL EXPERIMENTSTATISTICAL EXPERIMENT


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STATISTICAL EXPERIMENT,STATISTICAL EXPERIMENT,SAMPLE SPACE AND EVENTS:SAMPLE SPACE AND EVENTS:

STATISTICAL EXPERIMENTSTATISTICAL EXPERIMENT : process of generating random: process of generating randomoutcomes or dataoutcomes or data

Examples:Examples: drawing a card from a welldrawing a card from a well--shuffled deck;shuffled deck;tossing a pair of dicetossing a pair of dice

SAMPLE SPACE:SAMPLE SPACE: set of all ossible outcomes of a statisticalset of all ossible outcomes of a statistical

46

experimentexperimentSAMPLE POINTS:SAMPLE POINTS: the elements of a sample spacethe elements of a sample spaceEVENT:EVENT: a subset of a sample spacea subset of a sample spaceNULL SPACE OR EMPTY SPACE:NULL SPACE OR EMPTY SPACE: subset of the sample space;subset of the sample space;

has no elements; denoted by has no elements; denoted by oo VENN DIAGRAM: VENN DIAGRAM: pictorial representation of a sample space andpictorial representation of a sample space andthe events in this sample spacethe events in this sample space

ASSIGNING PROBABILITYASSIGNING PROBABILITY


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ASSIGNING PROBABILITY ASSIGNING PROBABILITY

WEIGHTS TO THE ELEMENTS OF A WEIGHTS TO THE ELEMENTS OF A SAMPLE SPACE:SAMPLE SPACE:

1.) Drawing one card from a well1.) Drawing one card from a well- -shuffled deck shuffled deck

47

3.) Tossing a LOADED DIE once (the die is3.) Tossing a LOADED DIE once (the die isloaded such that the even numbers areloaded such that the even numbers areTHRICE as likely to turn up as the oddTHRICE as likely to turn up as the oddnumbers)numbers)

PROBABILITY OF AN EVENT APROBABILITY OF AN EVENT A


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PROBABILITY OF AN EVENT A PROBABILITY OF AN EVENT A (MARGINAL PROBABILITY OF A):(MARGINAL PROBABILITY OF A):

P(A) = sum of the probability weights of thoseP(A) = sum of the probability weights of thosesample points belonging to event Asample points belonging to event A

N.B.: 0 P(an Event) 1;N.B.: 0 P(an Event) 1;If P(an Event) = 1, the event is certain toIf P(an Event) = 1, the event is certain to

48

uuIf P(an Event) = 0, it is an impossibleIf P(an Event) = 0, it is an impossibleeventevent

Example: A die is loaded such that the number 1 isExample: A die is loaded such that the number 1 is

thrice as likely to turn up as the otherthrice as likely to turn up as the othernumbers. If the die is tossed, find:numbers. If the die is tossed, find:1.) P(getting a 1)1.) P(getting a 1)2.) P(getting a number less than 4)2.) P(getting a number less than 4)


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PROBABILITY RULES:PROBABILITY RULES:1.)1.) ADDITIVE RULE: ADDITIVE RULE: For any two events A and B,For any two events A and B,

P(A U B) = P(A) + P(B)P(A U B) = P(A) + P(B) P(A B)P(A B)

N.B.: a.) Illustrate using theN.B.: a.) Illustrate using the VENN DIAGRAM VENN DIAGRAMb.)b.)P( A B)P( A B) = joint probability of A and B= joint probability of A and B

50

= the probability that events A and B will= the probability that events A and B willboth occurboth occur

2.) If A and B are2.) If A and B areMUTUALLY EXCLUSIVEMUTUALLY EXCLUSIVE events (i.e., events whichevents (i.e., events whichhave no sample point in common, or events that can not occurhave no sample point in common, or events that can not occurjointly), thenjointly), then

P (A U B) = P(A) + P(B)P (A U B) = P(A) + P(B)


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COMPLEMENT OF AN EVENT A:COMPLEMENT OF AN EVENT A:

The complement of an event A with respect The complement of an event A with respect

to some sample space S is the set of all theto some sample space S is the set of all theelements or sample points in S that are not in Aelements or sample points in S that are not in A

51

..

N.B.: note thatN.B.: note thatP(A) + P(A) = 1P(A) + P(A) = 1

P(A) = 1P(A) = 1 P(A)P(A)

ILLUSTRATION: DETERMININGILLUSTRATION: DETERMINING


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PROBABILITIES:PROBABILITIES:From past experiences, a stockbroker believesFrom past experiences, a stockbroker believes

that under present economic conditions a customerthat under present economic conditions a customer will invest in bonds with a probability of 0.45, wil will invest in bonds with a probability of 0.45, wilinvest in mutual funds with a robabilit of 0.5 wilinvest in mutual funds with a robabilit of 0.5 wil

52

invest in both bonds and mutual funds with ainvest in both bonds and mutual funds with aprobability of 0.15. At this time, find the probabilitprobability of 0.15. At this time, find the probabilitthat a customer will investthat a customer will invest

a.) in either bonds or mutual funds;a.) in either bonds or mutual funds;b.) in neither bonds nor mutual funds.b.) in neither bonds nor mutual funds.


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CONDITIONAL PROBABILITY:CONDITIONAL PROBABILITY: The TheCONDITIONAL PROBABILITY CONDITIONAL PROBABILITY of an event A, given thatof an event A, given that

another event B has occurred, denoted by P(A|B), is defined as:another event B has occurred, denoted by P(A|B), is defined as:P(A|B) = P(A B),P(A|B) = P(A B), provided thatprovided thatP(B) 0P(B) 0

P(B)P(B) the the JOINT PROBABILITY JOINT PROBABILITY that both events A and Bthat both events A and B

53

P(A B) = P(B) P(A|B)P(A B) = P(B) P(A|B)N.B.:N.B.: 1) Since it does not matter which one of the two events is call1) Since it does not matter which one of the two events is call

A and which one is called B, we can interchange A and B in the A and which one is called B, we can interchange A and B in theabove equation;above equation; i,.ei,.e.,.,

P(B|A) = P(B A) P(B A) = P(A) P(B|A)P(B|A) = P(B A) P(B A) = P(A) P(B|A)P(A)P(A)

2)2)P(A B) = P(B A) P(B) P(A|B) = P(A) P(B|A)P(A B) = P(B A) P(B) P(A|B) = P(A) P(B|A)



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CONDITIONAL PROBABILITIES:CONDITIONAL PROBABILITIES: A random sample of 100 call center employees, all college A random sample of 100 call center employees, all college

graduates, are classified below according to sex and college degreegraduates, are classified below according to sex and college degree

MALEMALE FEMALEFEMALE

PSYCHOLOGY PSYCHOLOGY 77 1212

54

If a person is chosen at random from this group of employees, findIf a person is chosen at random from this group of employees, find

probability thatprobability thata.) The person is a male, given that the person has a degree ina.) The person is a male, given that the person has a degree inEducationEducation

b.) The person is a nonb.) The person is a non--psychology degree holder, given that thepsychology degree holder, given that the

person is a female.person is a female.

EDUCATIONEDUCATION 1818 2020

COMMERCECOMMERCE 1515 2828


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INDEPENDENT EVENTS:INDEPENDENT EVENTS:

Events A and B are INDEPENDENT if eitherEvents A and B are INDEPENDENT if either

P(A|B) = P(A) or P(B|A) = P(B)P(A|B) = P(A) or P(B|A) = P(B)

55

Otherwise, A and B are dependent.Otherwise, A and B are dependent.

JOINT PROBABILITY OFJOINT PROBABILITY OF


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JOINT PROBABILITY OF JOINT PROBABILITY OF

INDEPENDENT EVENTS:INDEPENDENT EVENTS:1.) If A and B are INDEPENDENT events:1.) If A and B are INDEPENDENT events:

P(A B) = P(B A) = P(A) P(B)P(A B) = P(B A) = P(A) P(B)2.) In general, if there are2.) In general, if there arekk INDEPENDENTINDEPENDENT

56

EVENEVEN11,,

22,,

33,, kk , t e r J IN, t e r J IN

PROBABILITY isPROBABILITY isP(AP(A11 A A22 A A33 A Akk ) = ) =

P(AP(A11 ) P(A ) P(A22 ) P(A ) P(A33 ) P(A ) P(Akk ) )

TOTAL PROBABILITYTOTAL PROBABILITY


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TOTAL PROBABILITY TOTAL PROBABILITY

THEOREM:THEOREM:If the events BIf the events B11, B, B22,, BB33,, BBkk constitute a PARTITIONconstitute a PARTITION

of the sample space S such that P(Bof the sample space S such that P(Bii ) ) 0 for 0 for ii = 1, 2,= 1, 2,kk , then for any event A in S, then for any event A in SP(A) = P(BP(A) = P(B11 A) A) + P(B+ P(B22 A) + A) + PP (B(B33 A) + + P A) + + P ( (BBkk A A ) )

57

= P(B= P(B11 ) P(A|B ) P(A|B11 ) + P(B ) + P(B22 ) P(A|B ) P(A|B22 ) + + P( ) + + P(BBkk ) P(A| ) P(A| BBkk ) )

N.B.: Illustrate the theorem using the VENNN.B.: Illustrate the theorem using the VENNDIAGRAM, first withDIAGRAM, first withkk = 2 and then with= 2 and then withkk = 3.= 3.

ILLUSTRATION: APPLICATION OFILLUSTRATION: APPLICATION OF


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ILLUSTRATION: APPLICATION OFILLUSTRATION: APPLICATION OFTHE TOTAL PROBABILITY THE TOTAL PROBABILITY

THEOREM:THEOREM:

ABC Corporation is aiming to close a P500M sales ABC Corporation is aiming to close a P500M salestransaction with XYZ Company. ABC has three topnotchtransaction with XYZ Company. ABC has three topnotchsales executives who could negotiate with XYZ: Peter, Pasales executives who could negotiate with XYZ: Peter, Pa

58

and Mary. Their probabilities of being selected to brokerand Mary. Their probabilities of being selected to brokerthe deal are 0.5, 0.2 and 0.3, respectively. If ABC choosesthe deal are 0.5, 0.2 and 0.3, respectively. If ABC choosesPeter, the probability of closing the deal is 0.8; if Paul werPeter, the probability of closing the deal is 0.8; if Paul werchosen, ABCs probability of closing the deal is 0.6 whilechosen, ABCs probability of closing the deal is 0.6 whileMary were selected, the probability that ABC will get theMary were selected, the probability that ABC will get thedeal is 0.7. What is the probability that ABC will close thedeal is 0.7. What is the probability that ABC will close thedeal? (answer: 0.73)deal? (answer: 0.73)

BAYES RULEBAYES RULE


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BAYES RULE:BAYES RULE:

If the events BIf the events B11, B, B22,, BB33,, BBkk constitute aconstitute a

PARTITION of the sample space S where P(BPARTITION of the sample space S where P(Bii ) 0 ) 0for i = 1, 2, ,for i = 1, 2, ,kk , then for any event A in S such, then for any event A in S such

59

,,P( P(BBrr|A|A) = P(B ) = P(Brr ) P( ) P(A|B A|Brr ) )

P(BP(B11 ) P(A|B ) P(A|B11 ) + P(B ) + P(B22 ) P(A|B ) P(A|B22 ) ++ P( ) ++ P(BBkk ) P( ) P(A|B A|Bkk ) )for r = 1, 2, 3, for r = 1, 2, 3, kk ..N.B.: Note that the denominator is equal to P(A) according to theN.B.: Note that the denominator is equal to P(A) according to the

Total Probability Theorem (please go back to slide no. 56) Total Probability Theorem (please go back to slide no. 56)

ILLUSTRATION: APPLICATIONILLUSTRATION: APPLICATION


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OF BAYES RULE:OF BAYES RULE:

Let us go back to the illustration of slide no. 58.Let us go back to the illustration of slide no. 58.

60

to close the deal, what is the probability that Mato close the deal, what is the probability that Ma was chosen to broker the deal? was chosen to broker the deal?

(answer: 0.2877)(answer: 0.2877)INDIVIDUAL HOMEWORK: Please refer to handout.INDIVIDUAL HOMEWORK: Please refer to handout.

APPLICATION: DETERMINING APPLICATION: DETERMINING


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ASSET RISK AND PORTFOLIO RISK: ASSET RISK AND PORTFOLIO RISK:1.) MEASURE OF RISK =1.) MEASURE OF RISK = = variance = = variance =

2.)2.) where: p where: pii = the probability that outcome= the probability that outcome ii will will

E(R)][Ripin

1i

22 ==

61

occuroccurn = the no. of possible outcomesn = the no. of possible outcomesR R ii = the rate of return on the asset if outcome= the rate of return on the asset if outcome ii occursoccursE(R) = the expected rate of return on the assetE(R) = the expected rate of return on the asset

==

i

n

ii R p .

1

====

ILLUSTRATION:ILLUSTRATION:


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DETERMINING ASSET RISK:DETERMINING ASSET RISK: The probable rates of return on Assets A, B, C are as follows: The probable rates of return on Assets A, B, C are as follows:

ProbabilityProbabilityRATE OF RETURNRATE OF RETURN

on Asset A on Asset A (R (R A A ))

on Asset Bon Asset B(R (R BB))

on Asset Con Asset C(R (R CC))

0.200.20 0.110.11 0.140.14 0.080.08

62

Find: a.) the expected rate of return on each assetFind: a.) the expected rate of return on each assetb.) the risk on each assetb.) the risk on each asset

N.B.: 1.) show the solution on the whiteboard for Asset AN.B.: 1.) show the solution on the whiteboard for Asset A2.) Individual homework for Asset B and Asset C2.) Individual homework for Asset B and Asset C

0.300.30 0.150.15 0.090.09 0.160.16

0.500.50 0.120.12 0.110.11 0.140.14

028355.0017321.0

134.0)E(R11.0)E(R:answers CB==

==

C B R R



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PORTFOLIO RISK:PORTFOLIO RISK: The probable rates of return on Assets A, B, and C are as follows: The probable rates of return on Assets A, B, and C are as follows:

Probability Probability RATE OF RETURNRATE OF RETURNon Asset Aon Asset A

(R (R A A ) )on Asset Bon Asset B

(R (R BB ) )on Asset Con Asset C

(R (R CC ) )0.200.20 0.110.11 0.140.14 0.080.08

63

Determine: a.) the expected rate of return and the risk on PORTFOLIO 1 composedDetermine: a.) the expected rate of return and the risk on PORTFOLIO 1 composedof 30% A, 30% B and 40% C.of 30% A, 30% B and 40% C.

b.) the expected rate of return and the risk on PORTFOLIO 2 composedb.) the expected rate of return and the risk on PORTFOLIO 2 composedof 30% A, 10% B, and 60% C.of 30% A, 10% B, and 60% C.N.B.: 1.) Show the solutions for PORTFOLIO 1 on the whiteboardN.B.: 1.) Show the solutions for PORTFOLIO 1 on the whiteboard

2.) Individual homework for PORTFOLIO 22.) Individual homework for PORTFOLIO 2

0.300.30 0.150.15 0.090.09 0.160.160.500.50 0.120.12 0.110.11 0.140.14

019112.0;1295.0)E(R:answers22P

==P R

COVARIANCE BETWEEN ASSETS:COVARIANCE BETWEEN ASSETS:


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COVARIANCE BETWEEN ASSETS:COVARIANCE BETWEEN ASSETS:COVARIANCE BETWEEN TWO ASSETS =COVARIANCE BETWEEN TWO ASSETS = Cov Cov(R (R A A,R ,R BB ): a ): a

measure of the movements of the rates of return of two (2) assetsmeasure of the movements of the rates of return of two (2) assetssecurities A and B.securities A and B.

Positive Covariance R Positive Covariance R A A and R and R BBtend to move in the sametend to move in the samedirectiondirection

64

A A BBdirectionsdirections

where: p where: pii = probability that outcome= probability that outcome ii will occur will occurR R A Aii = rate of return on asset A if outcome= rate of return on asset A if outcome ii occursoccursR R BBii = rate of return on asset B if outcome= rate of return on asset B if outcome ii occursoccursE(R E(R A A ) = expected rate of return on asset A ) = expected rate of return on asset A

E(R E(R BB ) = expected rate of return on asset B ) = expected rate of return on asset B

=

=n

i B B A Ai B A )]}i E(R) ][Ri E(R{[R p) ,RCov(R

1

ILLUSTRATION: DETERMINING COVARIANCES:ILLUSTRATION: DETERMINING COVARIANCES:


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The probable rates of return on assets A, B, and C, are as follows: The probable rates of return on assets A, B, and C, are as follows:Probability Probability RATE OF RETURNRATE OF RETURN

on Asset Aon Asset A(R (R A A ) )

on Asset Bon Asset B(R (R BB ) )

on Asset Con Asset C(R (R CC ) )

0.200.20 0.110.11 0.140.14 0.080.080.300.30 0.150.15 0.090.09 0.160.160.500.50 0.120.12 0.110.11 0.140.14

65

Determine: a.)Determine: a.) Cov Cov ( ( R R A A,, R R BB ) )b.)b.) Cov Cov ( ( R R A A,, R R CC ) )c.)c.) Cov Cov ( ( R R BB,, R R CC ) )

N.B.: 1.) Show on the whiteboard the solution forN.B.: 1.) Show on the whiteboard the solution for Cov Cov ( ( R R A A,, R R BB ) )2.)2.)INDIVIDUAL HOMEWORK forINDIVIDUAL HOMEWORK for Cov Cov ( ( R R A A,, R R CC ) and ) and Cov Cov ( ( R R BB,, R R CC ) )answers:answers: Cov Cov (R (R A A, R , R BB ) = 0.000342 ) = 0.000342

Cov Cov (R (R BB, R , R CC ) = ) = -- 0.000480.00048

PORTFOLIO RISK ON A 2PORTFOLIO RISK ON A 2- -ASSET ASSET


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PORTFOLIO USING COVARIANCE:PORTFOLIO USING COVARIANCE:1.)1.) pp = W = W A AR R A A + W + W BBR R BB + 2W + 2W A A W W BB Cov Cov (R (R A A, R , R BB ) )

where: where: 22BB = variance of the portfolio= variance of the portfolio W W A A = weight or proportion of asset A= weight or proportion of asset A

66

BB = we g or propor on o asse= we g or propor on o asse22R R A A = variance of the rates of return on Asset A= variance of the rates of return on Asset A22 R R BB = variance of the rates of return on Asset B= variance of the rates of return on Asset BCov Cov (R (R A A, R , R BB ) = covariance between R ) = covariance between R A A and R and R BB

2.) PORTFOLIO RISK =2.) PORTFOLIO RISK = PP= = PP

ILLUSTRATION: DETERMININGILLUSTRATION: DETERMININGPORTFOLIO RISK ON A 2PORTFOLIO RISK ON A 2 ASSETASSET


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PORTFOLIO RISK ON A 2PORTFOLIO RISK ON A 2- -ASSET ASSETPORTFOLIO (USING COVARIANCE):PORTFOLIO (USING COVARIANCE):

The probable rates of return on assets A, B, and C are as follows: The probable rates of return on assets A, B, and C are as follows:

Probability Probability RATE OF RETURNRATE OF RETURN




0.200.20 0.110.11 0.140.14 0.080.08

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Determine: a.) the risk on the portfolio composed of 50% B and 50% CDetermine: a.) the risk on the portfolio composed of 50% B and 50% Cusing covarianceusing covariance

b.) the risk on the portfolio composed of 30% A and 70% Cb.) the risk on the portfolio composed of 30% A and 70% Cusing covarianceusing covarianceN.B.: 1.) show on the whiteboard the solution to a.)N.B.: 1.) show on the whiteboard the solution to a.)

2.) INDIVIDUAL HOMEWORK for b.)2.) INDIVIDUAL HOMEWORK for b.)

answer for b.) :answer for b.) :pp= 0.023649= 0.023649

0.300.30 0.150.15 0.090.09 0.160.160.500.50 0.120.12 0.110.11 0.140.14

PORTFOLIO RISK ON A 3PORTFOLIO RISK ON A 3- -ASSET ASSET


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PORTFOLIO USING COVARIANCE:PORTFOLIO USING COVARIANCE:

1.)1.) pp = W = W A AR R A A + W + W BBR R BB + W + W CCR R CC + 2W + 2W A A W W BB Cov Cov (R (R A A, R , R BB ) )+ 2W + 2W A A W W CC Cov Cov (R (R A A, R , R CC ) + 2W ) + 2W BB W W CC Cov Cov (R (R BB, R , R CC ) )

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2.) PORTFOLIO RISK =2.) PORTFOLIO RISK =PP== PP

ILLUSTRATION: DETERMINING PORTFOLIO RISK ILLUSTRATION: DETERMINING PORTFOLIO RISK


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ON A 3ON A 3--ASSET PORTFOLIO (USING COVARIANCES): ASSET PORTFOLIO (USING COVARIANCES):

The probable rates of return on assets A, B and C are as follows: The probable rates of return on assets A, B and C are as follows:Probability Probability RATE OF RETURNRATE OF RETURN




0.200.20 0.110.11 0.140.14 0.080.080.300.30 0.150.15 0.090.09 0.160.16

69

Using Using covariancescovariances, determine:, determine:a.)a.)the risk on PORTFOLIO 1 composed of 30% A, 30% B, 40% C.the risk on PORTFOLIO 1 composed of 30% A, 30% B, 40% C.

b.)b.) the risk on PORTFOLIO 2 composed of 30% A, 10% B, 60% C.the risk on PORTFOLIO 2 composed of 30% A, 10% B, 60% C.N.B.: 1.) show solution on the whiteboard for PORTFOLIO 1N.B.: 1.) show solution on the whiteboard for PORTFOLIO 1

2.) INDIVIDUAL HOMEWORK: risk on PORTFOLIO 22.) INDIVIDUAL HOMEWORK: risk on PORTFOLIO 2

answer p for Portfolio 2 = 0.019112

0.500.50 0.120.12 0.110.11 0.140.14

CORRELATION COEFFICIENT BETWEEN THE RATESCORRELATION COEFFICIENT BETWEEN THE RATESOF RETURN ON TWO ASSETS A AND B:OF RETURN ON TWO ASSETS A AND B:


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OF RETURN ON TWO ASSETS A AND B:OF RETURN ON TWO ASSETS A AND B:

1.) Measures the strength or degree of dependency of the direction1.) Measures the strength or degree of dependency of the directionmovements of the rates of return on Assets A and Bmovements of the rates of return on Assets A and B

2.)2.)3)3) => there is a perfect positive linear correlation between R => there is a perfect positive linear correlation between R A Aand R and R BB1, = B A R R

11 , B A R R

70

=> R => R A A and R and R BB move in the SAME DIRECTION all of themove in the SAME DIRECTION all of thetimetime4)4) =>=> there is a perfect negative linear correlation between R there is a perfect negative linear correlation between R A A

and R and R BB=> R => R A A and R and R BB move in OPPOSITE DIRECTIONS all of themove in OPPOSITE DIRECTIONS all of the

timetime5)5) =>=> there is no dependency between the movements of thethere is no dependency between the movements of the values of R values of R A A and R and R BB

6)6) => R => R A A and R and R BB move in the SAME DIRECTIONmove in the SAME DIRECTION

80% of the time80% of the time

1, = B A R R

0, = B A R R

%8080.0, == B A R R

COMPUTING THE CORRELATIONCOMPUTING THE CORRELATIONCOEFFICIENT BETWEEN RCOEFFICIENT BETWEEN R AND RAND R ::


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Where : Cov (R A, R B ) = the covariance between R A and R B R A = the risk on Asset A

= the standard deviation of R A

COEFFICIENT BETWEEN R COEFFICIENT BETWEEN R A A

AND R AND R BB

::( )

B A R R

B A B A

R RCov R R

= ,,

71

R B = the risk on Asset B

= the standard deviation of R BN.B: Please note that the sign of follows the sign of B A R R , B A R RCov ,

( )( )

977328..)

89259..):

,.

,.

:determineand 65No.slideback toreferPlease:SEATWORK INDIVIDUAL

b

aanswers

R Rb

R Ra

C B

B A

COMPUTING THE CORRELATIONCOMPUTING THE CORRELATIONCOEFFICIENT BETWEEN RCOEFFICIENT BETWEEN R AA AND RAND R BB


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COEFFICIENT BETWEEN R COEFFICIENT BETWEEN R A A AND R AND R BBUSING HISTORICAL DATA:USING HISTORICAL DATA:

Let the values of R Let the values of R A A and R and R BB be given by:be given by:

R R A A = X = X 11, X , X 22, X , X 33, , X X nnR R BB = Y = Y 11, Y , Y 22, Y , Y 33, , Y Y nn

72

en, as s own a so n s e o. , e corre a on coe c enen, as s own a so n s e o. , e corre a on coe c en

between X and Y or between R between X and Y or between R A A and R and R BB is:is:

N.B. : In obtaining , use the worksheet suggested inN.B. : In obtaining , use the worksheet suggested inslide No. 44 for the computation of r.slide No. 44 for the computation of r.

B A R R ,

( ) ( ) ( ) ( )( )( )[ ] ( )( )[ ]2222

,

=

Y Y n X X n

Y X XY n R R B A

INDIVIDUAL HOMEWORK: COMPUTING THE CORRELATIONINDIVIDUAL HOMEWORK: COMPUTING THE CORRELATIONCOEFFICIENT BETWEEN R COEFFICIENT BETWEEN R AA AND R AND R BB USING HISTORICAL DATA USING HISTORICAL DATA


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A A BB

During a certain year, the monthly rates of return (in %) on Stocks A and B are tabulated below:During a certain year, the monthly rates of return (in %) on Stocks A and B are tabulated below:

Month R A or X R B or Y

1 7.2 7.3

2 - 6.1 - 8.6

3 - 10.2 1.4

73

. .

5 - 2.8 8.6

6 2.1 1.8

7 2.8 6.8

8 2.9 - 5.0

9 - 6.7 - 0.5

10 - 8.2 - 7.2

11 - 1.4 3.4

12 - 3.6 5.1

Give your answer up to5 decimal places.

FURTHER APPLICATIONS OFFURTHER APPLICATIONS OF


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STATISTICS IN PORTFOLIOSTATISTICS IN PORTFOLIO ANALYSIS ANALYSIS1.1. Determination of theDetermination of the and theand the of a stockof a stock2.2. = measure of the UNSYSTEMATIC= measure of the UNSYSTEMATIC

74

a stocka stock3.3. = measure of the SYSTEMATIC RISK= measure of the SYSTEMATIC RISK

(or UNDIVERSIFIABLE RISK) on a(or UNDIVERSIFIABLE RISK) on astockstock

DETERMINATION OF THEDETERMINATION OF THE OF A OF A


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STOCK USING COVARIANCE:STOCK USING COVARIANCE:

stock on thereturnof ratesR :where

),(2

=

=

=

M R

M R RCov

75

1!isindexmarkettheof the

1),(

becomesforequationabovethe,RbyreplacedisRWhenN.B.

indexmarketon thereturnof ratestheof variancethe

2

2

2

M

2

=>

===

=

M

M

M

M

R

R

R

M M

R

R RCov

ILLUSTRATION: DETERMINATION OFILLUSTRATION: DETERMINATION OF USING COVARIANCEUSING COVARIANCE


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The probable rates of return on stocks A, B and C andThe probable rates of return on stocks A, B and C and

on the market index are as follows:on the market index are as follows:

ProbabilityProbability R R A A R R BB R R CC R R MM0.200.20 0.110.11 0.140.14 0.080.08 0.180.18

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.. .. .. .. ..0.500.50 0.120.12 0.110.11 0.200.20 0.100.10

Determine the of : a.) Stock Bof : a.) Stock B

b.) Stock Cb.) Stock CN.B.: 1.) Show solution for a.) (N.B.: 1.) Show solution for a.) ( of stock B) on the whiteboardof stock B) on the whiteboard

2.)2.) INDIVIDUAL HOMEWORK INDIVIDUAL HOMEWORK for b.) (for b.) ( of stock C)of stock C)

answer:answer: of stock C =of stock C = -- 1.493451.49345

DETERMINATION OFDETERMINATION OF ANDAND


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FROM HISTORICAL DATA:FROM HISTORICAL DATA:PROCEDURE:PROCEDURE: Linear Regression based on the equationLinear Regression based on the equation Y = Y = ++ XX

Where: Y = RWhere: Y = Rtt = the stocks rate of return during the period t= the stocks rate of return during the period t

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X = RX = RMMtt = the markets rate of return during the period t= the markets rate of return during the period t = diversifiable risk on the stock or asset= diversifiable risk on the stock or asset

= undiversifiable risk on the stock or asset= undiversifiable risk on the stock or asset

FORMULA FOR ALPHA:FORMULA FOR ALPHA:2


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( ) ( )( )( ) ( )222

=

X X n XY X X Y

Where: n = the no. of observation periods in days,or weeks, or months, or years

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N.B.: 1.) is also referred to as the company is also referred to as the company--specific risk on the stock.specific risk on the stock.2.)2.) == R R actualactual R R expectedexpected == R R actualactual E(R)E(R)2.1)2.1) > 0 => the stock performed better than expected =>> 0 => the stock performed better than expected =>

stock isstock isUNDERVALUEDUNDERVALUED2.2)2.2) < 0< 0 => the stock did not perform as expected =>

stock isOVERVALUED2.3) = 0= 0=> the stock is efficiently or properly valued

FORMULA FOR BETA:FORMULA FOR BETA:


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( ) ( )( )22 = Y X XY n

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SUGGESTED WORKSHEET FOR SUGGESTED WORKSHEET FOR DETERMININGDETERMINING ANDAND ::


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t Y = Rt

X = RMt

X 2 XY

1

2

3

:

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:

:

n

Y X X 2 XY

N.B. : R M= the rate of return on the market index (i.e., the PSEi)

ANOTHER INTERPRETATION ANOTHER INTERPRETATION


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OF THE VALUE OFOF THE VALUE OF measures themeasures the SENSITIVITYSENSITIVITY oror RESPONSIVENESSRESPONSIVENESS of the market priceof the market price(or the rate of return) of a stock to the directional movement of the market(or the rate of return) of a stock to the directional movement of the marketindex. It can be positive or negative.index. It can be positive or negative.

1.)1.) PositivePositive = the market price of the stock moves in the SAME direction as= the market price of the stock moves in the SAME direction asthe market indexthe market index

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2.)2.) NegativeNegative = the market price of the stock moves in the OPPOSITE= the market price of the stock moves in the OPPOSITEdirection as the market indexdirection as the market index

3.)3.) = 0= 0 =>=> the market price of the stock is NOT RESPONSIVE to thethe market price of the stock is NOT RESPONSIVE to themovement of the market indexmovement of the market index

4.)4.) == +1+1 =>=> if the market index GOES UP (or DOWN) by say 10%, theif the market index GOES UP (or DOWN) by say 10%, themarket price of the stock GOES UP (or DOWN) by also 10%market price of the stock GOES UP (or DOWN) by also 10%

INTERPRETATION OF THEINTERPRETATION OF THE


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VALUE OF VALUE OF : (continuation): (continuation)5.)5.) == --11 =>=> if the market index GOES UP (or DOWN) by say 10%, theif the market index GOES UP (or DOWN) by say 10%, themarket price of the stock GOES DOWN (or UP) by also 10%market price of the stock GOES DOWN (or UP) by also 10%

6.)6.) = + 1.5= + 1.5 == if the market index GOES UP (or DOWN) by say 10%,if the market index GOES UP (or DOWN) by say 10%,the market price of the stock GOES UP (or DOWN) BY 15%the market price of the stock GOES UP (or DOWN) BY 15% --

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.. .. , ,the market price of the stock GOES DOWN (or UP) BY 15%the market price of the stock GOES DOWN (or UP) BY 15%

QUERIES:QUERIES:1.) If the1.) If the of a stock isof a stock is --2.0 and the index goes UP by 5%, what2.0 and the index goes UP by 5%, whathappens to the market price of the stock?happens to the market price of the stock?

2.) If the2.) If the of a stock is + 0.8 and the index goes DOWN by 3%, whatof a stock is + 0.8 and the index goes DOWN by 3%, whathappens to the market price of the stock?happens to the market price of the stock?

INDIVIDUAL HOMEWORK: DETERMINING THEINDIVIDUAL HOMEWORK: DETERMINING THE AND THE AND THE OF AOF ASTOCK FROM HISTORICAL DATA (To be submitted on March 26, 2011):STOCK FROM HISTORICAL DATA (To be submitted on March 26, 2011):

During a certain year, the monthly rates of return on stock A (R During a certain year, the monthly rates of return on stock A (R ) and the monthly rates) and the monthly rates


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g y , y (g y , y ( A A

) y) yof return on the stock market index (R of return on the stock market index (R MM), all expressed in %, are tabulated below:), all expressed in %, are tabulated below:

MonthMonth(t)(t)

Y = R Y = R A A X = R X = R MM

11 7.27.2 6.86.822 --6.16.1 7.57.5

33 --10.210.2 9.29.2

1.) Determine the and theand the of of stock A.stock A.

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44 4.54.5 6.56.5

55 --2.82.8 --5.65.666 2.12.1 --1.81.8

77 2.82.8 8.88.8

88 2.92.9 8.38.3

99 --6.76.7 8.58.51010 --8.28.2 7.37.3

1111 --1.41.4 --2.52.5

1212 --3.63.6 --5.55.5

2.) Interpret the values of the2.) Interpret the values of the and theand the that you obtained.that you obtained.

Give your answer to

1.) up to 5 decimal places


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Module 1 -and Statistics