1 managerial finance professor andrew hall statistics in finance probability

1

Managerial Finance Professor Andrew Hall

Statistics In Finance

Probability

2Managerial Finance I


Probability

A Simple Event• is an outcome of an experiment which cannot be

decomposed into a simpler outcome.

An Event• is a collection of one or more simple events.

Random Sample• is taken in such a way that any possible sample of specific

size has the same probability as any other of being selected.



Probability

An Experiment• Results in ...

Outcomes• which are made up of Simple Events and Events…

Classical Probability attempts to assess the whole population assigning probabilities by calculating relative likely frequencies.

Number of Event E(Event E)

Number of All EventsP



Probability Notation

The probability of event A is written…

• P(A)• with P being “the probability” and • the parentheses being “of event”

All the probabilities of events in a sample space add up to 1, and

No event can occur less than zero times so…

• 0 >= P(Event) >= 1



Probability Notation

No event can occur less than zero times so…

0 >= P(Event) >= 1

Don’t ever, ever, ever answer a probability question with a number less than 0 or greater than 1



Probability

If an Experiment• Can results in only two outcomes...

P(A) + P(B) = 1

P(B) = 1 - P(A)

and

P(A) = 1 - P(B)

7



Samples and Populations



Basics

Population is the whole group of interest

Sample is a subset of the population that you may have data about.

Elements are the individual members of the population or sample studied.

Variable is used to refer to a particular characteristic of an element which can take on different values for each element of the population.

9



Mean, Average or Expected Value of a Sample



Mean, Average or Expected Value Ten students

Ages

If you threw a stone, what age would you expect the person it hit, to be?

Simple answer, 21

No variability in the values

21

21

21

21

21

21

21

21

21

21

1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

Age

Age



Mean, Average or Expected Value

A formula

1

n

iiX ExpectedValue E Value

n

Value

1 2 3 4 5 6 7 8 9 102

121

21

21

21

21

21

21

21

21

Index, iValue

Number of values, n

1

n

iiValue

+ + + + + + + + +

210 210

10X 21




Ten more students

Ages

If you threw a stone!

Less simple answer, 21 or 22

Some variability in the values

22

21

22

21

22

21

22

21

22

21

Say, 21½

1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

30

Age

Age




A formula

1

n


n

Value

1 2 3 4 5 6 7 8 9 102

221

22

21

22

21

22

21

22

21

Index, iValue

Number of values, n

1

n

iiValue

+ + + + + + + + +

215 215

10X 21 ½




Ten more students

Ages

If you threw a stone!

How to answer? Guess?

A good deal of variability in the values

22

12

30

3 27

6 29

8 28

21

Say, 14½

1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

30

Age

Age




A formula

1

n


n

Value

1 2 3 4 5 6 7 8 9 102

212

30

3 27

6 29

8 28

21

Index, iValue

Number of values, n

1

n

iiValue

+ + + + + + + + +

186 186

10X 18.6

16



Variability, Volatility and Variance of a Sample



Variability, Volatility and Variance

How far is each value from the mean value?

22

12

30

3 27

6 29

8 28

21

Value

1 2 3 4 5 6 7 8 9 10

Index, i

18.6X

Student 1 2 3 4 5 6 7 8 9 10Age 22.0 12.0 30.0 3.0 27.0 6.0 29.0 8.0 28.0 21.0Less Mean -18.6 -18.6 -18.6 -18.6 -18.6 -18.6 -18.6 -18.6 -18.6 -18.6Deviation 3.4 -6.6 11.4 -15.6 8.4 -12.6 10.4 -10.6 9.4 2.4

Sum of Deviations 0.0

iDeviation XValue



1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

30

Age

Age


Use deviation²

22

12

30

3 27

6 29

8 28

21

Value

1 2 3 4 5 6 7 8 9 10

Index, i

22

i XDeviation Value Student 1 2 3 4 5 6 7 8 9 10Age 22.0 12.0 30.0 3.0 27.0 6.0 29.0 8.0 28.0 21.0Less Mean -18.6 -18.6 -18.6 -18.6 -18.6 -18.6 -18.6 -18.6 -18.6 -18.6Deviation 3.4 -6.6 11.4 -15.6 8.4 -12.6 10.4 -10.6 9.4 2.4

Deviation^2 11.56 43.56 130 243.4 70.56 158.8 108.2 112.4 88.36 5.76

Sum of Deviations Squared 972.4




22

2 1 1

n n

i ii

n n

XDeviation Values

2 972.497.24

10s

Student 1 2 3 4 5 6 7 8 9 10Age 22.0 12.0 30.0 3.0 27.0 6.0 29.0 8.0 28.0 21.0Less Mean -18.6 -18.6 -18.6 -18.6 -18.6 -18.6 -18.6 -18.6 -18.6 -18.6Deviation 3.4 -6.6 11.4 -15.6 8.4 -12.6 10.4 -10.6 9.4 2.4

Deviation^2 11.56 43.56 130 243.4 70.56 158.8 108.2 112.4 88.36 5.76




1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

30

Age

Age


Use deviation²

22

21

22

21

22

21

22

21

22

21

Value

1 2 3 4 5 6 7 8 9 10

Index, i

22

i XDeviation Value

Student 1 2 3 4 5 6 7 8 9 10Age 22.0 21.0 22.0 21.0 22.0 21.0 22.0 21.0 22.0 21.0Less Mean -21.5 -21.5 -21.5 -21.5 -21.5 -21.5 -21.5 -21.5 -21.5 -21.5Deviation 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5

Deviation^2 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25




1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

30

Age

Age


22

2 1 1

n n

i ii

n n

XDeviation Values

Student 1 2 3 4 5 6 7 8 9 10Age 22.0 21.0 22.0 21.0 22.0 21.0 22.0 21.0 22.0 21.0Less Mean -21.5 -21.5 -21.5 -21.5 -21.5 -21.5 -21.5 -21.5 -21.5 -21.5Deviation 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5

Deviation^2 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25


2 2.50.25

10s

22



Standard Deviation of a Sample



Sample Standard Deviation

The Standard Deviation is the square root of the Variance

The Sample Standard Deviation is denoted by S

2

2 1

n

ii

sn

XValues

24



Population Statistics



Population Mean

The Population Mean, μ, is:

Where n is the size of the population

1

n

ii

n

Value



Population Variance

The Population Variance is:


2

2 1

n

ii

n

Value



Population Standard Deviation

The Population Standard Deviation is the square root of the Population Variance

The Population Standard Deviation is denoted by σ


2

2 1

n

ii

n

Value

28



Covariance and Correlation



Covariance – Start with a time series

1 2 3 4 5 6 7 8 9 10

-30

-20

-10

0

10

20

30

A



Covariance – Varying Apart

1 2 3 4 5 6 7 8 9 10

-30

-20

-10

0

10

20

30

AC



Covariance – Varying Together

1 2 3 4 5 6 7 8 9 10

-30

-20

-10

0

10

20

30

AB



Covariance – No Clear Pattern

1 2 3 4 5 6 7 8 9 10

-30

-20

-10

0

10

20

30

AD



Covariance – A Formula

1

,

*B

A

A

n

iB n

DeviationDeviation

1

,

*iforif B Bor

Bi

A

A A

n

n

ValuValu ee



Covariance – Numeric Results

Covariance Between A and B

Covariance Between A and C

Covariance Between A and D

1 2 3 4 5 6 7 8 9 10

-30

-20

-10

0

10

20

30

1 2 3 4 5 6 7 8 9 10

-30

-20

-10

0

10

20

30

1 2 3 4 5 6 7 8 9 10

-30

-20

-10

0

10

20

30

,386

A B

,386

A C

,10.1

A D

Positive

Negative

Positive

Relatively Small



Covariance – Has No Scale

Covariance has no scale

What did 386, minus 386 and 10.1 mean?

Comparing two covariances is therefore no very meaningful.

Correlation is a “normalized” covariance

Correlation is covariance on a scale from minus one to plus one.



Correlation – A Formula

,

, *Covariance

Correlation StdDev StdDevA B

A BA B

,

, *A B

A BA B



Correlation– Numeric Results

Correlation Between A and B

Correlation Between A and C

Correlation Between A and D

1 2 3 4 5 6 7 8 9 10

-30

-20

-10

0

10

20

30

1 2 3 4 5 6 7 8 9 10

-30

-20

-10

0

10

20

30

1 2 3 4 5 6 7 8 9 10

-30

-20

-10

0

10

20

30

,1

A B

,1

A C

,0.02722

A D

Plus One

Minus One

Positive

Nearly Zero

38



Using Probabilities in Calculations




1 2 3 4 5 6 7 8 9 102

221

22

21

22

21

22

21

22

21

Index, iValue

Number of values, n

1

n

iiValue

+ + + + + + + + +

215 215

10X 21 ½

1

n

iiX

n

Value

Number of 21's in List("Value is 21")

Number of Items in ListP

Number of 22's in List("Value is 22")

Number of Items in ListP

50.50

10

50.50

10




22

21

22

21

22

21

22

21

22

21

Value

21X ½

("Value is 21") 0.50P ("Value is 22") 0.50P

1

*m

i ii

X PValue Value

2 Values, so m=2

1 1 2 2* *X P PValue Value Value Value

21*0.50 22*0.50X 10.50 11.00X

21.50X

41



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1 managerial finance professor andrew hall statistics in finance probability

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