QA in Finance/ Ch 3 Probability in Finance Probability

QA in Finance/ Ch 3 Probability in Finance 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

QA in Finance/ Ch 3 Probability in Finance 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

QA in Finance/ Ch 3 Probability in Finance The classical approach to probability

QA in Finance/ Ch 3 Probability in Finance 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

QA in Finance/ Ch 3 Probability in Finance 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

QA in Finance/ Ch 3 Probability in Finance 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 ???

QA in Finance/ Ch 3 Probability in Finance 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

QA in Finance/ Ch 3 Probability in Finance Mutually exclusive A B 

QA in Finance/ Ch 3 Probability in Finance 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 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 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 AB 

QA in Finance/ Ch 3 Probability in Finance 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?

QA in Finance/ Ch 3 Probability in Finance 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

QA in Finance/ Ch 3 Probability in Finance 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.

QA in Finance/ Ch 3 Probability in Finance 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

QA in Finance/ Ch 3 Probability in Finance Discrete probability distribution  Variables that have only a finite number of possible outcomes  For example …a six-sided die is thrown Possibilities r= Probability that Z=r 1/61/6 1/6 1/6 1/6 1/6 

QA in Finance/ Ch 3 Probability in Finance Discrete probability distribution Probability Distribution Values Probability 01/4 =.25 12/4 =.50 21/4 =.25 Event: Toss two coins Count the number of tails T T TT

QA in Finance/ Ch 3 Probability in Finance 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 … %

QA in Finance/ Ch 3 Probability in Finance 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)

QA in Finance/ Ch 3 Probability in Finance 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

QA in Finance/ Ch 3 Probability in Finance 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

QA in Finance/ Ch 3 Probability in Finance Computing the Mean for Investment Returns Return per $1,000 for two types of investments P(X i Y i ) Economic condition Dow Jones fund X Growth Stock Y.2 Recession-$100 -$200.5 Stable Economy Expanding Economy Investment

QA in Finance/ Ch 3 Probability in Finance Computing the Mean for Investment Returns Return per $1,000 for two types of investments P(X i Y i ) Economic condition Dow Jones fund X Growth Stock Y.2 Recession-$100 -$200.5 Stable Economy Expanding Economy Investment

QA in Finance/ Ch 3 Probability in Finance Probability Distribution Important probability distributions in finance Discrete:  BINOMIAL  POISSON Continuous:  NORMAL  LOG NORMAL

QA in Finance/ Ch 3 Probability in Finance 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

QA in Finance/ Ch 3 Probability in Finance Binomial probability distribution  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

QA in Finance/ Ch 3 Probability in Finance Binomial probability distribution Sd Su S Su 2 Sud = Sdu Sd 2 j=2 j=1 j=0 J = number of success

QA in Finance/ Ch 3 Probability in Finance Binomial probability distribution 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 there are two ways to achieving an outcome

QA in Finance/ Ch 3 Probability in Finance Binomial probability distribution Combination rule or the number of binomial trials

QA in Finance/ Ch 3 Probability in Finance 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

QA in Finance/ Ch 3 Probability in Finance A binomial tree of asset prices 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

QA in Finance/ Ch 3 Probability in Finance A binomial tree of asset prices Sd Su S=50 Su 2 Sud = Sdu Sd 2 T0T0 T1T1 T2T2 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

QA in Finance/ Ch 3 Probability in Finance A binomial tree of asset prices Su (45.45) Su (55) S=50 Su 2 (60.50) Sud = Sdu (50) Sd 2 (41.32) T0T0 T1T1 T2T2

QA in Finance/ Ch 3 Probability in Finance A binomial tree of asset prices The expected value is calculated as: (60.50 x 0.25) + (50.0 x 0.50) + (41.32 x 0.25) = The variance is ( ) 2 x ( ) 2 x ( ) 2 x 0.25 = 46.18

QA in Finance/ Ch 3 Probability in Finance 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

QA in Finance/ Ch 3 Probability in Finance The Poisson distribution

QA in Finance/ Ch 3 Probability in Finance The Poisson distribution 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 ???

QA in Finance/ Ch 3 Probability in Finance The normal distribution  Most important continuous probability distribution  Bell shaped  Symmetrical  Mean, median and mode are equal  Mean Median Mode X f(X) 

QA in Finance/ Ch 3 Probability in Finance The normal distribution

QA in Finance/ Ch 3 Probability in Finance The normal distribution  Probability is the area under the curve! c d X f(X)f(X)

QA in Finance/ Ch 3 Probability in Finance The normal distribution  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 !!!

QA in Finance/ Ch 3 Probability in Finance Standardizing example Normal Distribution Standardized Normal Distribution

QA in Finance/ Ch 3 Probability in Finance Normal Distribution Standardized Normal Distribution

QA in Finance/ Ch 3 Probability in Finance Z Cumulative Standardized Normal Distribution Table (Portion) Z = 0.21 (continued)

QA in Finance/ Ch 3 Probability in Finance The normal distribution 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%

QA in Finance/ Ch 3 Probability in Finance The normal distribution 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.

QA in Finance/ Ch 3 Probability in Finance The normal distribution 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.

QA in Finance/ Ch 3 Probability in Finance The lognormal distribution  Modern portfolio theory assumes that investment return are normally distributed random variable. Is that true ?

QA in Finance/ Ch 3 Probability in Finance The lognormal distribution  However, this is not true, investment return can only take on values between -100% and % which are not symmetrically distributed, but skewed.

QA in Finance/ Ch 3 Probability in Finance The lognormal distribution  Mathematical trick !!! Distribution of Vt/Vo is LognormalDistribution of ln(Vt/Vo) is Normal