Risk Return

11

LECTURE 1LECTURE 1 Risk, Return, and Risk, Return, and

Uncertainty:Uncertainty: A Review of Principles A Review of Principles

Useful in FinanceUseful in Finance

22

The Concept of ReturnThe Concept of Return

• Measurable returnMeasurable return

• Expected returnExpected return

• Return on investmentReturn on investment

33

Measurable ReturnMeasurable Return

• DefinitionDefinition

• Holding period returnHolding period return

• Arithmetic mean returnArithmetic mean return

• Geometric mean returnGeometric mean return

• Comparison of arithmetic and Comparison of arithmetic and geometric mean returnsgeometric mean returns

44

DefinitionDefinition

• A general definition of A general definition of return return is the is the benefit associated with an benefit associated with an investmentinvestment– In most cases, return is measurableIn most cases, return is measurable– E.g., a $100 investment at 8%, E.g., a $100 investment at 8%,

compounded continuously is worth compounded continuously is worth $108.33 after one year$108.33 after one year•The return is $8.33, or 8.33%The return is $8.33, or 8.33%

55

Holding Period ReturnHolding Period Return

• The calculation of a The calculation of a holding period returnholding period return is independent of the passage of timeis independent of the passage of time

– E.g., you buy a bond for $950, receive $80 in E.g., you buy a bond for $950, receive $80 in interest, and later sell the bond for $980interest, and later sell the bond for $980• The return is ($80 + $30)/$950 = 11.58%The return is ($80 + $30)/$950 = 11.58%

• The 11.58% could have been earned over one year or The 11.58% could have been earned over one year or one weekone week

pricePurchase

GainCapitalIncomeReturn

66

Arithmetic Mean ReturnArithmetic Mean Return

• The The arithmetic mean returnarithmetic mean return is the is the arithmetic average of several holding arithmetic average of several holding period returns measured over the period returns measured over the same holding period:same holding period:

iR

n

R

i

n

i

i

periodin return of rate the~

~mean Arithmetic

1

77

Arithmetic Mean Return Arithmetic Mean Return (cont’d)(cont’d)

• Arithmetic means are a useful proxy Arithmetic means are a useful proxy for expected returnsfor expected returns

• Arithmetic means are not especially Arithmetic means are not especially useful for describing historical useful for describing historical returnsreturns– It is unclear what the number means It is unclear what the number means

once it is determinedonce it is determined

88

Geometric Mean ReturnGeometric Mean Return

• The The geometric mean returngeometric mean return is the is the nnth root of the product of th root of the product of nn values: values:

1)~

1(mean Geometric/1

1

nn

iiR

99

Arithmetic and Arithmetic and Geometric Mean ReturnsGeometric Mean Returns

ExampleExample

Assume the following sample of weekly stock Assume the following sample of weekly stock returns:returns:

WeekWeek ReturnReturn Return Return RelativeRelative

11 0.00840.0084 1.00841.0084

22 -0.0045-0.0045 0.99550.9955

33 0.00210.0021 1.00211.0021

44 0.00000.0000 1.0001.000

1010

Arithmetic and Arithmetic and Geometric Mean Returns Geometric Mean Returns (cont’d)(cont’d)

Example (cont’d)Example (cont’d)

What is the arithmetic mean return?What is the arithmetic mean return?

Solution:Solution:

0015.04

0000.00021.00045.00084.0

~mean Arithmetic

1

n

i

i

n

R

1111

Arithmetic and Arithmetic and Geometric Mean Returns Geometric Mean Returns (cont’d)(cont’d)Example (cont’d)Example (cont’d)

What is the geometric What is the geometric mean return?mean return?

Solution:Solution:

001489.0

10000.10021.19955.00084.1

1~

1(mean Geometric

4/1

/1

1

nn

iiR

1212

Comparison of Comparison of Arithmetic &Arithmetic &Geometric Mean ReturnsGeometric Mean Returns• The geometric mean reduces the The geometric mean reduces the

likelihood of nonsense answerslikelihood of nonsense answers– Assume a $100 investment falls by 50% in Assume a $100 investment falls by 50% in

period 1 and rises by 50% in period 2period 1 and rises by 50% in period 2

– The investor has $75 at the end of period 2The investor has $75 at the end of period 2•Arithmetic mean = (-50% + 50%)/2 = 0%Arithmetic mean = (-50% + 50%)/2 = 0%

•Geometric mean = (0.50 x 1.50)Geometric mean = (0.50 x 1.50)1/21/2 –1 = -13.40% –1 = -13.40%

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Comparison of Comparison of Arithmetic &Arithmetic &Geometric Mean ReturnsGeometric Mean Returns• The geometric mean must be used to The geometric mean must be used to

determine the rate of return that determine the rate of return that equates a present value with a series equates a present value with a series of future valuesof future values

• The greater the dispersion in a series The greater the dispersion in a series of numbers, the wider the gap between of numbers, the wider the gap between the arithmetic and geometric meanthe arithmetic and geometric mean

1414

Expected ReturnExpected Return

• Expected returnExpected return refers to the refers to the futurefuture– In finance, what happened in the past is In finance, what happened in the past is

not as important as what happens in the not as important as what happens in the futurefuture

– We can use past information to make We can use past information to make estimates about the futureestimates about the future

1515

Standard Deviation and Standard Deviation and VarianceVariance

• Standard deviation and variance are Standard deviation and variance are the most common measures of total the most common measures of total riskrisk

• They measure the dispersion of a set They measure the dispersion of a set of observations around the mean of observations around the mean observationobservation

1616

Standard Deviation and Standard Deviation and Variance (cont’d)Variance (cont’d)

• General equation for variance:General equation for variance:

• If all outcomes are equally likely:If all outcomes are equally likely:

2

2

1

Variance prob( )n

i ii

x x x

2

2

1

1 n

ii

x xn

1717

Standard Deviation and Standard Deviation and Variance (cont’d)Variance (cont’d)

• Equation for standard deviation:Equation for standard deviation:

2

2

1

Standard deviation prob( )n

i ii

x x x

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Semi-VarianceSemi-Variance

• Semi-variance considers the dispersion Semi-variance considers the dispersion only on the adverse sideonly on the adverse side– Ignores all observations greater than the Ignores all observations greater than the

meanmean– Calculates variance using only “bad” returns Calculates variance using only “bad” returns

that are less than averagethat are less than average– Since risk means “chance of loss” positive Since risk means “chance of loss” positive

dispersion can distort the variance or dispersion can distort the variance or standard deviation statistic as a measure of standard deviation statistic as a measure of riskrisk

1919

Some Statistical Facts of Some Statistical Facts of LifeLife

• DefinitionsDefinitions

• Properties of random variablesProperties of random variables

• Linear regressionLinear regression

• R squared and standard errorsR squared and standard errors

2020

DefinitionsDefinitions

• ConstantsConstants

• VariablesVariables

• PopulationsPopulations

• SamplesSamples

• Sample statisticsSample statistics

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ConstantsConstants

• A constant is a value that does not A constant is a value that does not changechange– E.g., the number of sides of a cubeE.g., the number of sides of a cube– E.g., the sum of the interior angles of a E.g., the sum of the interior angles of a

triangletriangle

• A constant can be represented by a A constant can be represented by a numeral or by a symbolnumeral or by a symbol

2222

VariablesVariables

• A variable has no fixed valueA variable has no fixed value– It is useful only when it is considered in It is useful only when it is considered in

the context of other possible values it the context of other possible values it might assumemight assume

• In finance, variables are called In finance, variables are called random variablesrandom variables

2323

Variables (cont’d)Variables (cont’d)

• Discrete random variablesDiscrete random variables are are countablecountable– E.g., the number of trout you catchE.g., the number of trout you catch

• Continuous random variablesContinuous random variables are are measurablemeasurable– E.g., the length of a troutE.g., the length of a trout

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• Quantitative variablesQuantitative variables are are measured by real numbersmeasured by real numbers– E.g., numerical measurementE.g., numerical measurement

• Qualitative variablesQualitative variables are are categoricalcategorical– E.g., hair colorE.g., hair color

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• Independent variablesIndependent variables are are measured directlymeasured directly– E.g., the height of a boxE.g., the height of a box

• Dependent variablesDependent variables can only be can only be measured once other independent measured once other independent variables are measuredvariables are measured– E.g., the volume of a box (requires E.g., the volume of a box (requires

length, width, and height)length, width, and height)

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PopulationsPopulations

• A A populationpopulation is the entire collection of is the entire collection of a particular set of random variablesa particular set of random variables

• The nature of a population is described The nature of a population is described by its by its distributiondistribution– The The medianmedian of a distribution is the point of a distribution is the point

where half the observations lie on either where half the observations lie on either sideside

– The The mode mode is the value in a distribution is the value in a distribution that occurs most frequentlythat occurs most frequently

2727

Populations (cont’d)Populations (cont’d)

• A distribution can have A distribution can have skewnessskewness– There is more dispersion on one side of There is more dispersion on one side of

the distributionthe distribution– Positive skewnessPositive skewness means the mean is means the mean is

greater than the mediangreater than the median•Stock returns are positively skewedStock returns are positively skewed

– Negative skewnessNegative skewness means the mean means the mean is less than the medianis less than the median

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Positive Skewness Negative Skewness

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• A A binomial distributionbinomial distribution contains contains only two random variablesonly two random variables– E.g., the toss of a coinE.g., the toss of a coin

• A A finite populationfinite population is one in which is one in which each possible outcome is knowneach possible outcome is known– E.g., a card drawn from a deck of cardsE.g., a card drawn from a deck of cards

3030


• An An infinite populationinfinite population is one where is one where not all observations can be countednot all observations can be counted– E.g., the microorganisms in a cubic mile E.g., the microorganisms in a cubic mile

of ocean waterof ocean water

• A A univariate populationunivariate population has one has one variable of interestvariable of interest

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• A A bivariate populationbivariate population has two has two variables of interestvariables of interest– E.g., weight and sizeE.g., weight and size

• A A multivariate populationmultivariate population has has more than two variables of interestmore than two variables of interest– E.g., weight, size, and colorE.g., weight, size, and color

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SamplesSamples

• A A samplesample is any subset of a is any subset of a populationpopulation– E.g., a sample of past monthly stock E.g., a sample of past monthly stock

returns of a particular stockreturns of a particular stock

3333

Sample StatisticsSample Statistics

• Sample statisticsSample statistics are are characteristics of samplescharacteristics of samples– A true population statistic is usually A true population statistic is usually

unobservable and must be estimated unobservable and must be estimated with a sample statisticwith a sample statistic•ExpensiveExpensive

•Statistically unnecessaryStatistically unnecessary

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Properties of Properties of Random VariablesRandom Variables

• ExampleExample

• Central tendencyCentral tendency

• DispersionDispersion

• LogarithmsLogarithms

• ExpectationsExpectations

• Correlation and covarianceCorrelation and covariance

3535

ExampleExample

Assume the following monthly stock returns Assume the following monthly stock returns for Stocks A and B:for Stocks A and B:

MonthMonth Stock AStock A Stock BStock B

11 2%2% 3%3%

22 -1%-1% 0%0%

33 4%4% 5%5%

44 1%1% 4%4%

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Central TendencyCentral Tendency

• Central tendency is what a random Central tendency is what a random variable looks like, on averagevariable looks like, on average

• The usual measure of central The usual measure of central tendency is the population’s tendency is the population’s expected valueexpected value (the mean) (the mean)– The average value of all elements of the The average value of all elements of the

populationpopulation

1

1( )

n

i ii

E R Rn

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The The expected returnsexpected returns for Stocks A and B are: for Stocks A and B are:

1

1 1( ) (2% 1% 4% 1%) 1.50%

4

n

A ii

E R Rn

1

1 1( ) (3% 0% 5% 4%) 3.00%

4

n

B ii

E R Rn

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DispersionDispersion

• Investors are interest in the best and Investors are interest in the best and the worst in addition to the averagethe worst in addition to the average

• A common measure of dispersion is A common measure of dispersion is the variance or standard deviationthe variance or standard deviation

22

22

i

i

E x x

E x x

3939


The variance ad standard deviationThe variance ad standard deviation for Stock for Stock A are:A are:

22

2 2 2 2

2

1(2% 1.5%) ( 1% 1.5%) (4% 1.5%) (1% 1.5%)

41

(0.0013) 0.0003254

0.000325 0.018 1.8%

iE x x

4040


The variance ad standard deviationThe variance ad standard deviation for Stock for Stock B are:B are:

22

2 2 2 2

2

1(3% 3.0%) (0% 3.0%) (5% 3.0%) (4% 3.0%)

41

(0.0014) 0.000354

0.00035 0.0187 1.87%

iE x x

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LogarithmsLogarithms

• Logarithms reduce the impact of Logarithms reduce the impact of extreme valuesextreme values– E.g., takeover rumors may cause huge E.g., takeover rumors may cause huge

price swingsprice swings– A A logreturnlogreturn is the logarithm of a return is the logarithm of a return

• Logarithms make other statistical Logarithms make other statistical tools more appropriatetools more appropriate– E.g., linear regressionE.g., linear regression

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Logarithms (cont’d)Logarithms (cont’d)

• Using logreturns on stock return Using logreturns on stock return distributions:distributions:– Take the raw returnsTake the raw returns

– Convert the raw returns to return relativesConvert the raw returns to return relatives

– Take the natural logarithm of the return Take the natural logarithm of the return relativesrelatives

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ExpectationsExpectations

• The expected value of a constant is a The expected value of a constant is a constant:constant:

• The expected value of a constant The expected value of a constant times a random variable is the times a random variable is the constant times the expected value of constant times the expected value of the random variable:the random variable:

( )E a a

( ) ( )E ax aE x

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Expectations (cont’d)Expectations (cont’d)

• The expected value of a combination The expected value of a combination of random variables is equal to the of random variables is equal to the sum of the expected value of each sum of the expected value of each element of the combination:element of the combination:

( ) ( ) ( )E x y E x E y

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Correlations and Correlations and CovarianceCovariance• CorrelationCorrelation is the degree of association is the degree of association

between two variablesbetween two variables

• CovarianceCovariance is the product moment of is the product moment of two random variables about their meanstwo random variables about their means

• Correlation and covariance are related Correlation and covariance are related and generally measure the same and generally measure the same phenomenonphenomenon

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Correlations and Correlations and Covariance (cont’d)Covariance (cont’d)

( , ) ( )( )ABCOV A B E A A B B

( , )AB

A B

COV A B

4747


The covariance and correlation for Stocks A The covariance and correlation for Stocks A and B are:and B are:

1(0.5% 0.0%) ( 2.5% 3.0%) (2.5% 2.0%) ( 0.5% 1.0%)

41

(0.001225)40.000306

AB

( , ) 0.0003060.909

(0.018)(0.0187)ABA B

COV A B

4848

Correlations and Correlations and CovarianceCovariance• Correlation ranges from –1.0 to +1.0. Correlation ranges from –1.0 to +1.0.

– Two random variables that are Two random variables that are perfectly perfectly positivelypositively correlated have a correlation correlated have a correlation coefficient of +1.0coefficient of +1.0

– Two random variables that are Two random variables that are perfectly perfectly negativelynegatively correlated have a correlation correlated have a correlation coefficient of –1.0coefficient of –1.0

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1

23456789101112131415

A B C D E F G H I J K

Year

AdamsFarm stock

return

MorganSausage

stockreturn

1990 30.73% 21.44% <-- =3%+0.6*B31991 55.21% 36.13%1992 15.82% 12.49%1993 33.54% 23.12%1994 14.93% 11.96%1995 35.84% 24.50%1996 48.39% 32.03%1997 37.71% 25.63%1998 67.85% 43.71%1999 44.85% 29.91%

Correlation 1.00 <-- =CORREL(B3:B12,C3:C12)

CORRELATION +1Adams Farm and Morgan Sausage Stocks

rMorgan Sausage,t = 3% + 0.6*rAdams Farm,t Annual Stock Returns, Adams Farm and Morgan Sausage

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

10% 20% 30% 40% 50% 60% 70%Adams Farm

Mor

gan

Saus

age

5050

181920212223242526272829303132333435363738394041

A B C D E F G H I

Stock A Stock BMonth Price Return Price Return

0 25.00 45.001 24.12 -3.58% 44.85 -0.33%2 23.37 -3.16% 46.88 4.43% <-- =LN(E23/E22)3 24.75 5.74% 45.25 -3.54%4 26.62 7.28% 50.87 11.71%5 26.50 -0.45% 53.25 4.57%6 28.00 5.51% 53.25 0.00%7 28.88 3.09% 62.75 16.42%8 29.75 2.97% 65.50 4.29%9 31.38 5.33% 66.87 2.07%

10 36.25 14.43% 78.50 16.03%11 37.13 2.40% 78.00 -0.64%12 36.88 -0.68% 68.23 -13.38%

Monthly mean 3.24% 3.47% <-- =AVERAGE(F22:F33)Monthly variance 0.23% 0.65% <-- =VARP(F22:F33)Monthly stand. dev. 4.78% 8.03% <-- =STDEVP(F22:F33)

Annual mean 38.88% 41.62% <-- =12*F35Annual variance 2.75% 7.75% <-- =12*F36Annual stand. dev. 16.57% 27.83% <-- =SQRT(F40)

CALCULATING THE RETURNS

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444546474849505152535455565758596061626364

A B C D E F G H I JCOVARIANCE AND VARIANCE CALCULATION

Stock A Stock BReturn Return-mean Return Return-mean Product

-0.0358 -0.0682 -0.0033 -0.0380 0.00259 <-- =E48*B48-0.0316 -0.0640 0.0443 0.0096 -0.000610.0574 0.0250 -0.0354 -0.0701 -0.001750.0728 0.0404 0.1171 0.0824 0.00333

-0.0045 -0.0369 0.0457 0.0110 -0.000410.0551 0.0227 0.0000 -0.0347 -0.000790.0309 -0.0015 0.1642 0.1295 -0.000190.0297 -0.0027 0.0429 0.0082 -0.000020.0533 0.0209 0.0207 -0.0140 -0.000290.1443 0.1119 0.1603 0.1257 0.014060.0240 -0.0084 -0.0064 -0.0411 0.00035

-0.0068 -0.0392 -0.1338 -0.1685 0.00660

Covariance 0.00191 <-- =AVERAGE(G48:G59)0.00191 <-- =COVAR(A48:A59,D48:D59)

Correlation 0.49589 <-- =G62/(F37*C37)0.49589 <-- =CORREL(A48:A59,D48:D59)

=D48-$F$35

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Linear RegressionLinear Regression

• Linear regression is a mathematical Linear regression is a mathematical technique used to predict the value technique used to predict the value of one variable from a series of of one variable from a series of values of other variablesvalues of other variables– E.g., predict the return of an individual E.g., predict the return of an individual

stock using a stock market indexstock using a stock market index

• Regression finds the equation of a Regression finds the equation of a line through the points that gives the line through the points that gives the best possible fitbest possible fit

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Linear Regression Linear Regression (cont’d)(cont’d)

ExampleExample

Assume the following sample of weekly stock Assume the following sample of weekly stock and stock index returns:and stock index returns:

WeekWeek Stock ReturnStock Return Index ReturnIndex Return

11 0.00840.0084 0.00880.0088

22 -0.0045-0.0045 -0.0048-0.0048

33 0.00210.0021 0.00190.0019

44 0.00000.0000 0.00050.0005

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Linear Regression Linear Regression (cont’d)(cont’d)


-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

-0.01 -0.005 0 0.005 0.01

Return (Market)

Re

turn

(S

tock

)

Intercept = 0Slope = 0.96R squared = 0.99

5555

R Squared and R Squared and Standard ErrorsStandard Errors

• ApplicationApplication

• R squaredR squared

• Standard ErrorsStandard Errors

5656

ApplicationApplication

• R-squaredR-squared and the and the standard errorstandard error are used to assess the accuracy of are used to assess the accuracy of calculated statisticscalculated statistics

5757

R SquaredR Squared

• R squared is a measure of how good a fit R squared is a measure of how good a fit we get with the regression linewe get with the regression line– If every data point lies exactly on the line, R If every data point lies exactly on the line, R

squared is 100%squared is 100%

• R squared is the square of the correlation R squared is the square of the correlation coefficient between the security returns coefficient between the security returns and the market returnsand the market returns– It measures the portion of a security’s It measures the portion of a security’s

variability that is due to the market variabilityvariability that is due to the market variability

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1

23456789101112131415161718192021222324252627282930

A B C D E F G H I J K L

Date

S&P 500IndexSPX

MirageResorts

MIRJan-97 6.13% 16.18%Feb-97 0.59% 0.00%Mar-97 -4.26% -15.42%Apr-97 5.84% -5.29%

May-97 5.86% 18.63%Jun-97 4.35% 5.76%Jul-97 7.81% 5.94%

Aug-97 -5.75% 0.23%Sep-97 5.32% 12.35%Oct-97 -3.45% -17.01%Nov-97 4.46% -5.00%Dec-97 1.57% -4.21%Jan-98 1.02% 1.37%Feb-98 7.04% -0.54%Mar-98 4.99% 5.99%Apr-98 0.91% -9.25%

May-98 -1.88% -5.67%Jun-98 3.94% 2.40%Jul-98 -1.16% 0.88%

Aug-98 -14.58% -30.81%Sep-98 6.24% 12.61% Slope 1.469256 <-- =SLOPE(C3:C26,B3:B26)Oct-98 8.03% 1.12% 1.469256 <-- =COVAR(C3:C26,B3:B26)/VARP(B3:B26)Nov-98 5.91% -12.18%Dec-98 5.64% 0.42% Intercept -0.042365 <-- =INTERCEPT(C3:C26,B3:B26)

-0.042365 <-- =AVERAGE(C3:C26)-B28*AVERAGE(B3:B26)

R-squared 0.500072 <-- =RSQ(C3:C26,B3:B26)0.500072 <-- =CORREL(C3:C26,B3:B26)^2

SIMPLE REGRESSION EXAMPLE IN EXCEL

MIR Returns vs S&P500 ReturnsMonthly Returns, 1997-1998

y = 1.4693x - 0.0424

R2 = 0.5001-40%

-30%

-20%

-10%

0%

10%

20%

30%

-20% -15% -10% -5% 0% 5% 10%S&P500

MIR

5959

Standard ErrorsStandard Errors

• The standard error is the standard The standard error is the standard deviation divided by the square root deviation divided by the square root of the number of observations:of the number of observations:

Standard errorn

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Standard Errors (cont’d)Standard Errors (cont’d)

• The standard error enables us to The standard error enables us to determine the likelihood that the determine the likelihood that the coefficient is statistically different from coefficient is statistically different from zerozero– About 68% of the elements of the About 68% of the elements of the

distribution lie within one standard error of distribution lie within one standard error of the meanthe mean

– About 95% lie within 1.96 standard errorsAbout 95% lie within 1.96 standard errors– About 99% lie within 3.00 standard errorsAbout 99% lie within 3.00 standard errors

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Runs TestRuns Test

• A runs test allows the statistical testing A runs test allows the statistical testing of whether a series of price movements of whether a series of price movements occurred by chance.occurred by chance.

• A run is defined as an uninterrupted A run is defined as an uninterrupted sequence of the same observation. sequence of the same observation. ExEx: if : if the stock price increases 10 times in a the stock price increases 10 times in a row, then decreases 3 times, and then row, then decreases 3 times, and then increases 4 times, we then say that we increases 4 times, we then say that we have three runs.have three runs.

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NotationNotation

• R = number of runs (3 in this example)R = number of runs (3 in this example)

• nn11 = number of observations in the first = number of observations in the first category. For instance, here we have a total category. For instance, here we have a total of 14 “ups”, so nof 14 “ups”, so n11=14.=14.

• nn22 = number of observations in the second = number of observations in the second category. For instance, here we have a total category. For instance, here we have a total of 3 “downs”, so nof 3 “downs”, so n22=3.=3.

• Note that nNote that n11 and n and n22 could be the number of could be the number of “Heads” and “Tails” in the case of a coin “Heads” and “Tails” in the case of a coin toss.toss.

6363

Statistical TestStatistical Test

1 2

1 2

2 1 2 1 2 1 22

1 2 1 2

The z statistic computed is:

(thus z is a standard normal variable)

where

2 1

2 (2 )

( ) ( 1)

R xz

n nx

n n

n n n n n n

n n n n

6464

ExampleExample

• Let the number of runs R=23Let the number of runs R=23

• Let the number of ups nLet the number of ups n11=20=20

• Let the number of downs nLet the number of downs n22=30=30

Then the mean number of runs 25

The standard deviation 3.36

Yielding a z statistic of: 0.595

x

z

6565

About 2.5% of the area under About 2.5% of the area under the normal curve is below a z the normal curve is below a z score of -1.96.score of -1.96.

6666

InterpretationInterpretation

• Since our z-score is not in the lower tail Since our z-score is not in the lower tail (nor is it in the upper tail), the runs we (nor is it in the upper tail), the runs we have witnessed are purely the product have witnessed are purely the product of chance.of chance.

• If, on the other hand, we had obtained If, on the other hand, we had obtained a z-score in the upper (2.5%) or lower a z-score in the upper (2.5%) or lower (2.5%) tail, we would then be 95% (2.5%) tail, we would then be 95% certain that this specific occurrence of certain that this specific occurrence of runs didn’t happen by chance. (Or that runs didn’t happen by chance. (Or that we just witnessed an extremely rare we just witnessed an extremely rare event)event)

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Risk Return