702341 QA in Finance/ Ch 3 Probability in Finance Probability.

702341 QA in Finance/ Ch 3 Probability in Finance

Probability


Probability

Probability is a measure of the possibility of an event happening

Measure on a scale between zero and one Probability has a substantial role to play in financial

analysis as the outcomes of financial decisions are uncertain

e.g. Fluctuation in share prices


The classical approach to probability

The range of possible uncertain outcomes is known and equally likely

EXPERIMENT, SAMPLE, EVENT

consider the tossing of a fair coin:

the range is limited to two

the tossing of the coin is the experiment

the two possible outcomes refer to the sample space

the outcome whether it is head or tail is the event


The classical approach to probability

omeser of outcTotal numb

th the evenciated witcomes assoNo. of outP(A)


The empirical approach to probability

In finance, we cannot rely on the exactness of a process to determine the probabilities

Consider ‘the return of financial assets’ The range is unlimited In such situations the probability of a given outcome

Z, P(Z), is

erimentsNo. of

ccurencesNo. of Z oP(Z)

exp


The empirical approach to probability

E.g. Consider a sample of 100 daily movements in a share price. Assume that of the 100 absolute movements, five movements were 0.5 Baht each, 15 were 1 Baht each, 20 were 1.5 Baht each, 30 were 2 Baht each, 20 were 2.5 Baht and 10 were 3 Baht each


Basic rules of probability

These rules are : the addition rule concerned with A or B happening the multiplication rule concerned with A and B occurring

Which of these rules is applicable will depend on whether the combined events are INDEPENDENT or MUTUALLY EXCLUSIVE ???


Mutually exclusive

Two events cannot occur together

Sample space = {1,2,3,4,5,6}

A is the event that the face of die shows odd number:

A = {1,3,5}

B is the event that the face of die is even number:

B = {2,4,6}

A Λ B = { } = Ø

A and B is MUTUALLY EXCLUSIVE


Mutually exclusive

A B


The addition rule applied to non-mutually exclusive events

P(A or B) = P(A) + P(B) – P(A and B)

Assume that the FTSE 100 index may rise with a probability of 0.55 and fall with the probability of 0.45. Also assume that a particular time interval the S&P index may rise with a probability of 0.35 and fall with a probability of 0.65. There is also a probability of 0.3 that both indices rise together. What is the probability of wither the FTSE 100 index or the S&P 500 index rising

A B


The multiplication rule applied to non-independent events

P(A and B) = P(A) * P(B A) P(B A) is the conditional probability of B occurring

given that A has occurred

Suppose the probability of the recession is 25% and long-term bond yields have an 80% chance of declining during a recession What is the probability that a recession will occur and bond yields will decline?


Bayes’ theorem

Manipulation of the general multiplication rule The probability of the updated event An can be

updated to P(A|B) if Scenario B is known to have occurred by using the following relationships

Ni

iii

kkk

ABPAP

ABPAPBAP

1

))/().((

)/().()/(


Bayes’ theorem

Suppose the economy is in an uptrend three out of every four years (75%). Furthermore, when the economy is in an uptrend, the stock market advances 80% of the time. Conversely, the economy declines one out of every four years (25%), and the stock market declines 70% of the time when the economy is in a recession.


Random variable

Random Variable

A variable that behave in an uncertain manner

As this behavior is uncertain we can only assign probabilities to the possible values of these variables.

Thus the random variable is defined by its probability distribution and possible outcomes.

Two types of random variable: discrete and continuous


Discrete probability distribution

Variables that have only a finite number of possible outcomes

For example …a six-sided die is thrown

Possibilities r=1 1 2 3 4 5 6

Probability that Z=r 1/6 1/6 1/6 1/6 1/6 1/6

0 1 1j jP X P X


Discrete probability distribution

Probability Distribution

Values Probability

0 1/4 = .25

1 2/4 = .50

2 1/4 = .25

Event: Toss two coins Count the number of tails

T

T

T T


Continuous probability distribution

Variables that can be subdivided into an infinite number of subunits for measurement

For example …speed, asset returns Consider: a movement in an asset price from 105 to

109 will give a return of … %


Continuous probability distribution

To overcome this practical problem, we must define our continuous random variable by integrating what is know as a probability density function (pdf)

1)( dXXf


Expected value of a random discrete variable

Expected value (the mean)

Weighted average of the probability distribution

e.g.: Toss 2 coins, count the number of tails, compute

j jj

E X X P X

0 2.5 1 .5 2 .25 1

j jj

X P X


Expected value of a random discrete variable

Expected value (the mean)– Weight average squared deviation about the mean

– e.g. Toss two coins, count number of tails, compute variance

222j jE X X P X

22

2 2 2 0 1 .25 1 1 .5 2 1 .25 .5

j jX P X


Computing the Mean for Investment Returns

Return per $1,000 for two types of investments

P(XiYi) Economic condition Dow Jones fund X Growth Stock Y

.2 Recession -$100 -$200

.5 Stable Economy + 100 + 50

.3 Expanding Economy + 250 + 350

Investment

100 .2 100 .5 250 .3 $105XE X

200 .2 50 .5 350 .3 $90YE Y


Computing the Mean for Investment Returns

Return per $1,000 for two types of investments

P(XiYi) Economic condition Dow Jones fund X Growth Stock Y

.2 Recession -$100 -$200

.5 Stable Economy + 100 + 50

.3 Expanding Economy + 250 + 350

Investment

2 2 22 100 105 .2 100 105 .5 250 105 .3

14,725 121.35X

X

2 2 22 200 90 .2 50 90 .5 350 90 .3

37,900 194.68Y

Y


Probability Distribution

Important probability distributions in finance

Discrete: BINOMIAL POISSON

Continuous: NORMAL LOG NORMAL


Binomial probability distribution

Only two possible outcomes can be taken on by the variable in a given time period or a given event.

e.g. getting head is success while getting tail is failure

For each of a succession of trials the probability of two outcome is constant

e.g. Probability of getting a tail is the same each time we toss the coin



Each binomial trial is identical

e.g. 15 tosses of a coin; ten light bulbs taken from a warehouse

Each trial is independent

the outcome of one trial does not affect the outcome of the other



Sd

Su

S

Su2

Sud = Sdu

Sd2

j=2

j=1

j=0

J = number of success



The probability of achieving each outcome depends on:

1. the probability of achieving a success

2. the total number of ways of achieving that outcome

e.g. consider the case of j = 1 (Sdu = Sud)

1. each way has a probability of 0.25

2. there are two ways to achieving an outcome



!1

! !

: probability of successes given and

: number of "successes" in sample 0,1, ,

: the probability of each "success"

: sample size

n XXnP X p p

X n X

P X X n p

X X n

p

n

Combination rule

or the number of binomial trials


A binomial tree of asset prices

The most common application of the binomial distribution in finance is ‘security price change’

It is assumed that over the next small interval of time security price will wither rise (‘a success’) or fall (‘a failure’) by a given amount

The binomial distribution is an assumption in some option pricing models



3 stages in developing the expected value of asset price: create a binomial lattice determine the probabilities of each outcome multiply each possible outcome by the appropriate

probability and sum the products to arrive at the expected value



Sd

Su

S=50

Su2

Sud = Sdu

Sd2

T0 T1 T2

In each of the time period the asset may rise with probability of 0.5, or it may fall with a probability of 0.5

suppose: u= 1.10, d = 1/1.10



Su (45.45)

Su (55)

S=50

Su2

(60.50)

Sud = Sdu (50)

Sd2

(41.32)T0 T1 T2



The expected value is calculated as:

(60.50 x 0.25) + (50.0 x 0.50) + (41.32 x 0.25)

= 50.46

The variance is

(60.50-50.46)2 x 0.25 + (50.0-50.46)2 x 0.50 + (41.32-50.46)2 x 0.25

= 46.18


The Poisson distribution

Discrete events in an interval The probability of One Success in an interval is

stable The probability of More than One Success in this

interval is 0 e.g. number of customers arriving in 15 minutes e.g. information which causes market price to move

arrive at a rate of 10 pieces per minute



!

: probability of "successes" given

: number of "successes" per unit

: expected (average) number of "successes"

: 2.71828 (base of natural logs)

XeP X

XP X X

X

e



ex. Find the probability of 4 customers arriving in 3 minutes when the mean is 3.6.

ex. information which causes market price to move arrive at a rate of 10 pieces per minute. Find the

probability of only eight pieces of information arriving in the next minute ???

3.6 43.6

.19124!

eP X


The normal distribution

Most important continuous probability distribution Bell shaped Symmetrical Mean, median and

mode are equal

Mean Median Mode

X

f(X)

1)( dXXf



21

2

2

1

2

: density of random variable

3.14159; 2.71828

: population mean

: population standard deviation

: value of random variable

X

f X e

f X X

e

X X



Probability is the area under the curve!

c dX

f(X) ?P c X d



There are an infinite number of normal distributions By varying the parameters and µ, we obtain

different normal distributions

An infinite number of normal distributions means an infinite number of tables to look up !!!


Standardizing example

6.2 50.12

10

XZ

Normal Distribution

Standardized Normal

Distribution10 1Z

5 6.2 X Z0Z

0.12


Normal Distribution

Standardized Normal

Distribution10 1Z

5 7.1 X Z0Z

0.21

2.9 5 7.1 5.21 .21

10 10

X XZ Z

2.9 0.21

.0832

2.9 7.1 .1664P X

.0832


Z .00 .01

0.0 .5000 .5040 .5080

.5398 .5438

0.2 .5793 .5832 .5871

0.3 .6179 .6217 .6255

.5832.02

0.1 .5478

Cumulative Standardized Normal Distribution Table (Portion) 0 1Z Z

Z = 0.21

2.9 7.1 .1664P X (continued)

0



Example:

We wish to know the probability of a given asset, which is assumed to have normally distributed returns, providing a return of between 4.9% and 5%. The mean of the return on that asset to date is 4%, and the standard deviation is 1%



Example:

The earnings of a company are expected to be $4.00 per share, with a standard deviation of $40. Assuming earnings per share are a continuous random variable that is normally distributed, calculate the probability of actual EPS will be $3.90 or higher.



Example:

The earnings of a company are expected to be $4.00 per share, with a standard deviation of $40. Assuming earnings per share are a continuous random variable that is normally distributed, calculate the probability of actual EPS will be between $3.60 and $4.40.


The lognormal distribution

Modern portfolio theory assumes that investment return are normally distributed random variable.

Is that true ?



However, this is not true, investment return can only take on values between -100% and % which are not symmetrically distributed, but skewed.

%100

)(rE



Mathematical trick !!!

0 0/VVt

0V

VP t

)/ln( 0VVt

0

lnV

VP t

Distribution of Vt/Vo is Lognormal Distribution of ln(Vt/Vo) is Normal

702341 QA in Finance/ Ch 3 Probability in Finance Probability.

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Transcript of 702341 QA in Finance/ Ch 3 Probability in Finance Probability.