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Probability 702341 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 702341 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 702341 QA in Finance/ Ch 3 Probability in Finance The classical approach to probability P(A) No. of outcomes associated with the event Total number of outcomes 702341 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 No. of Z occurences P(Z) No. of exp eriments 702341 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 702341 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 ??? 702341 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} AΛ B = { } = Ø A and B is MUTUALLY EXCLUSIVE 702341 QA in Finance/ Ch 3 Probability in Finance Mutually exclusive A B 702341 QA in Finance/ Ch 3 Probability in Finance The addition rule applied to nonmutually exclusive events P(A or B) = P(A) + P(B) – P(A and B) A 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 702341 QA in Finance/ Ch 3 Probability in Finance The multiplication rule applied to nonindependent 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? 702341 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 P( Ak / B) P( Ak ).P( B / Ak ) iN ( P( A ).P( B / A )) i 1 i i 702341 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. 702341 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 702341 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=1 Probability that Z=r 0 PX j 1 1 2 3 4 5 6 1/6 1/6 1/6 1/6 1/6 1/6 PX 1 j 702341 QA in Finance/ Ch 3 Probability in Finance Discrete probability distribution Event: Toss two coins T T T Count the number of tails Probability Distribution Values Probability 0 1/4 = .25 1 2/4 = .50 2 1/4 = .25 T 702341 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 … % 702341 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) f ( X )dX 1 702341 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 X X jP X j j e.g.: Toss 2 coins, count the number of tails, compute X jP X j j 0 2.5 1.5 2 .25 1 702341 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 2 E X X j P X j 2 2 – e.g. Toss two coins, count number of tails, compute variance X j P X j 2 2 0 1 .25 1 1 .5 2 1 .25 .5 2 2 2 702341 QA in Finance/ Ch 3 Probability in Finance Computing the Mean for Investment Returns Return per $1,000 for two types of investments P(XiYi) Economic condition Investment Dow Jones fund X Growth Stock Y .2 Recession -$100 -$200 .5 Stable Economy + 100 + 50 .3 Expanding Economy + 250 + 350 E X X 100.2 100.5 250.3 $105 E Y Y 200.2 50.5 350 .3 $90 702341 QA in Finance/ Ch 3 Probability in Finance Computing the Mean for Investment Returns Return per $1,000 for two types of investments Investment Economic condition Dow Jones fund X Growth Stock Y P(XiYi) .2 Recession -$100 -$200 .5 Stable Economy + 100 + 50 .3 Expanding Economy + 250 + 350 100 105 .2 100 105 .5 250 105 .3 2 2 X 2 2 X 121.35 14, 725 200 90 .2 50 90 .5 350 90 .3 2 2 Y 37,900 2 Y 194.68 2 702341 QA in Finance/ Ch 3 Probability in Finance Probability Distribution Important probability distributions in finance Discrete: BINOMIAL POISSON Continuous: NORMAL LOG NORMAL 702341 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 702341 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 702341 QA in Finance/ Ch 3 Probability in Finance Binomial probability distribution Su2 j=2 Sud = Sdu j=1 Sd2 j=0 Su S Sd J = number of success 702341 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 0.25 2. there are two ways to achieving an outcome 702341 QA in Finance/ Ch 3 Probability in Finance Binomial probability distribution Combination rule n! n X X P X p 1 p X ! n X ! P X : probability of X successes given n and p X : number of "successes" in sample X 0,1, , n p : the probability of each "success" n : sample size or the number of binomial trials 702341 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 702341 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 702341 QA in Finance/ Ch 3 Probability in Finance A binomial tree of asset prices 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 Su2 Su Sud = Sdu S=50 Sd suppose: u= 1.10, d = 1/1.10 T0 Sd2 T1 T2 702341 QA in Finance/ Ch 3 Probability in Finance A binomial tree of asset prices Su2 (60.50) Su (55) Sud = Sdu (50) S=50 Su (45.45) T0 T1 Sd2 (41.32) T2 702341 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) = 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 702341 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 702341 QA in Finance/ Ch 3 Probability in Finance The Poisson distribution e P X X! P X : probability of X "successes" given X X : number of "successes" per unit : expected (average) number of "successes" e : 2.71828 (base of natural logs) 702341 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. e3.6 3.64 P X .1912 4! 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 ??? 702341 QA in Finance/ Ch 3 Probability in Finance The normal distribution Most important continuous probability distribution Bell shaped f(X) Symmetrical Mean, median and mode are equal X f ( X )dX 1 Mean Median Mode 702341 QA in Finance/ Ch 3 Probability in Finance The normal distribution f X 1 e 1 2 2 X 2 2 f X : density of random variable X 3.14159; e 2.71828 : population mean : population standard deviation X : value of random variable X 702341 QA in Finance/ Ch 3 Probability in Finance The normal distribution Probability is the area under the curve! P c X d ? f(X) c d X 702341 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 !!! 702341 QA in Finance/ Ch 3 Probability in Finance Standardizing example Z X 6.2 5 0.12 10 Standardized Normal Distribution Normal Distribution 10 5 Z 1 6.2 X Z 0 0.12 Z 702341 QA in Finance/ Ch 3 Probability in Finance P 2.9 X 7.1 .1664 Z X 2.9 5 .21 10 Z X 7.1 5 .21 10 Standardized Normal Distribution Normal Distribution 10 Z 1 .0832 .0832 2.9 5 7.1 X 0.21 Z 0 0.21 Z 702341 QA in Finance/ Ch 3 Probability in Finance P 2.9 X 7.1 .1664(continued) Cumulative Standardized Normal Distribution Table (Portion) Z .00 .01 Z 0 Z 1 .02 .5832 0.0 .5000 .5040 .5080 0.1 .5398 .5438 .5478 0.2 .5793 .5832 .5871 0.3 .6179 .6217 .6255 0 Z = 0.21 702341 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% 702341 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. 702341 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. 702341 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 ? 702341 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. 100% E (r ) 702341 QA in Finance/ Ch 3 Probability in Finance The lognormal distribution Mathematical trick !!! Distribution of Vt/Vo is Lognormal Distribution of ln(Vt/Vo) is Normal Vt P ln V0 Vt P V0 0 Vt /V0 ln( Vt / V0 ) 702341 QA in Finance/ Ch 3 Probability in Finance