probability concepts key concepts are described probability rules are introduced expected values,...
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PROBABILITY CONCEPTS
Key concepts are described Probability rules are introduced Expected values, standard deviation,
covariance and correlation for individual portfolio returns are explained
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Probability TerminologyRandom variable: uncertain number
Outcome: realization of random variable
Event: set of one or more outcomes
Mutually exclusive: cannot both happen
Exhaustive: set of events includes all possible outcomes
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Two Properties of Probability
Probability of an event, P(Ei), is between 0 and 1
0 ≤ P(Ei) ≤ 1
For a set of events that are mutually exclusive and exhaustive, the sum of probabilities is 1
ΣP(Ei) = 1
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Types of Probability
Empirical: based on analysis of data
Subjective: based on personal perception
A priori: based on reasoning, not experience
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Odds For or Against
Probability that a horse will win a race = 20%
Odds for: 0.20 / (1 – 0.20) = 1/4
= one-to-four
Odds against: (1 – 0.20) / 0.20 = 4/1
= four-to-one
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Conditional vs. UnconditionalTwo types of probability:
Unconditional: P(A), the probability of an eventregardless of the outcomes of other events, e.g., probability market will be up for the day
Conditional: P(A|B), the probability of A given that B has occurred, e.g., probability that the market will be up for the day, given that the Fed raises interest rates
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Joint ProbabilityThe probability that both of two events will occur is their joint probability
Example using conditional probability:
P (interest rates will increase) = P(I) = 40%
P (recession given a rate increase) = P(R|I) = 70%
Probability of a recession and an increase in rates,
P(RI) = P(R|I) × P(I) = 0.7 × 0.4 = 28%
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Probability that at Least One of Two Events Will OccurP(A or B) = P(A) + P(B) – P(AB)
We must subtract the joint probability P(AB)
Don’t doublecount P(AB)
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Addition Rule ExampleP(I) = prob. of rising interest rates is 40%
P(R) = prob. of recession is 34%
Joint probability P(RI) = 0.28 (calculated earlier)
Probability of either rising interest rates or recession
= P(R or I) = P(R) + P(I) – P(RI)
= 0.34 + 0.40 – 0.28 = 0.46
For mutually exclusive events the
joint probability P(AB) = 0 so:
P(A or B) = P(A) + P(B)
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Joint Probability of any Number of Independent EventsDependent events: knowing the outcome of one
tells you something about the probability of the other
Independent events: occurrence of one event does not influence the occurrence of the other. For the joint probability of independent events, just multiply
Example: Flipping a fair coin, P (heads) = 50% The probability of 3 heads in succession is:
0.5 × 0.5 × 0.5 =0.53 = 0.125 or 12.5%
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Calculating Unconditional ProbabilityGiven:
P (interest rate increase) = P(I) = 0.4
P (no interest rate increase) = P(IC) = 1 – 0.4 = 0.6
P (Recession | Increase) = P(R|I) = 0.70
P (Recession | No Increase) = P(R|IC) = 0.10
What is the (unconditional) probability of recession?
P(R) = P(R|I) × P(I) + P(R|IC) × P(IC)
= 0.70 × 0.40 + 0.10 × 0.60 = 0.34
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EPS = $1.80Prob = 18%
An Investment Tree
EPS = $1.70Prob = 42%
EPS = $1.30Prob = 24%
EPS = $1.00Prob = 16%
Expected EPS = $1.51
Prob of good economy
60%
Prob of poor economy40%
30%
70%
60%
40%Prob of poor stock performance
Prob of good stock performance
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Expected Value using Total Probability
Using the probabilities from the Tree:
Expected(EPS) = $1.51
= .18(1.80) + .42(1.70) + .24(1.30) + .16(1.00)
Conditional Expectations of EPS:
E(EPS)|good economy =
.30(1.80) + .70(1.70) = $1.73
E(EPS)|poor economy =
.60(1.30) + .40(1.00) = $1.18
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CovarianceCovariance: A measure of how two variables move
together Values range from minus infinity to positive infinity Units of covariance difficult to interpret Covariance positive when the two variables tend to be
above (below) their expected values at the same time
For each observation, multiply each probability times the product of the two random variables’
deviations from their means and sum them
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Correlation Correlation: A standardized measure of the linear
relationship between two variables
Values range from +1, perfect positive correlation
to –1, perfect negative correlation r is sample correlation coefficient ρ is population correlation coefficient
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CorrelationExample: The covariance between two assets is 0.0046, σA = 0.0623 and σB = 0.0991. What is the correlation between the two assets (ρA,B)?
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Expected Value, Variance, and Standard Deviation (probability model)
Expected Value: E(X) = ΣP(xi)xi
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Expected Value, Variance, and Standard Deviation (probability model)
Variance: σ2(X) = ΣP(xi)[xi – E(X)]2
Standard deviation: square root of σ2 = 0.1136
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Portfolio Expected Return
Expected return on a portfolio is a weighted average of the expected returns on the assets in the portfolio
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Portfolio Variance and Standard Deviation
Portfolio variance also uses the weight of the assets in the portfolio
Portfolio standard deviation is the square root of the variance
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Joint Probability FunctionReturns RB= 40% RB= 20% RB= 0% E(RB)=18%
RA= 20% 0.15
RA= 15% 0.60
RA= 4% 0.25
E(RA)=13%
CovAB= 0.15 (.20 - .13) (.40 - .18) +
0.6 (.15 - .13) (.20 - .18) +
0.25 (.04 - .13) (0 - .18) = 0.0066
Probabilities
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Good earnings (G)
Bayes’ Formula
Poor earnings (P)
Prob. of interest rate cut (C)
60%
40%
70%
30%
20%
80%
Good earnings (G)
Poor earnings (P)
Prob. of no interest rate cut
42%
18%
8%
32%
Prob (C|G) = 42/(42 + 8) = 84%
Prob (C|G) = [Prob(G|C) × Prob(C)]/Prob(G)
Prob (C/G) = (70% * 60% )/ (42% +8%)
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Factorial for Labeling
Out of 10 stocks, 5 will be rated buy, 3 will be rated hold, and 2 will be rated sell. How many ways are there to do this?
10!
5! 3! 2!2,520
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Choosing r Objects from n Objects
When order does not matter and with just 2 possible labels, we can use the combination formula (binomial formula)
Example: You have 5 stocks and want to place orders to sell 3 of them. How many different combinations of 3 stocks are there?
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Choosing r Objects from n Objects
When order does matter, we use the
permutation formula:
You have 5 stocks and want to sell 3, one at a time. The order of the stock sales matters. How many ways are there to choose the 3 stocks to sell in order?
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Calculator Solutions: nCr and nPr
How many ways to choose 3 from 5, order doesn’t matter? 5 → 2nd → nCr → 3 → = 10
How many ways to choose 3 from 5, order does matter? 5 → 2nd → nPr → 3 → = 60
Functions only on BAII Plus (and Professional)
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Probability Functions A probability function, p(x), gives the probability
that a discrete random variable will take on the value x
e.g. p(x) = x/15 for X = {1,2,3,4,5}→ p(3) = 20% A probability density function (pdf), f(x) can be
used to evaluate the probability that a continuous random variable with take on a value within a range
A cumulative distribution function (cdf), F(x), gives the probability that a random variable will be less than or equal to a given value
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Properties of Normal Distribution
Completely described by mean and variance Symmetric about the mean (skewness = 0) Kurtosis (a measure of peakedness) = 3 Linear combination of normally distributed random
variables is also normally distributed Probabilities decrease further from the mean, but the
tails go on forever
Multivariate normal: more than one r.v., correlation between their outcomes
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Confidence Interval: Normal DistributionConfidence interval: a range of values around an expected outcome within which we expect the actual outcome to occur some specified percent of the time.
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Confidence Interval: Normal Distribution
Example: The mean annual return (normally distributed) on a portfolio over many years is 11%, and the standard deviation of returns is 8%. A 95% confidence interval on next year’s return is 11% + (1.96)(8%) = –4.7% to 26.7%
90% confidence interval = X ± 1.65s
95% confidence interval = X ± 1.96s
99% confidence interval = X ± 2.58s
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Standard Normal DistributionA normal distribution that has been standardized
so that mean = 0 and standard deviation = 1To standardize a random variable, calculate the
z-valueSubtract the mean (so mean = 0) and divide by
standard deviation (so σ = 1)
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Calculating Probabilities Using the Standard Normal DistributionExample 1: The EPS for a large sample of firms is normally distributed and has µ = $4.00 and σ = $1.50. Find the probability of a value being lower than $3.70.
3.70 is 0.20 standard deviations below the mean of 4.00.
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Calculating Probabilities Using the Standard Normal DistributionExample 1 cont.: Here we need to find the area under the curve to the left of the z-value of –0.20.
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Excerpt from a Table of Cumulative Probabilities for a Standard Normal Distribution
Calculating Probabilities Using the Standard Normal Distribution
For negative z-value calculate 1 – table value
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Calculating Probabilities Using the Standard Normal Distribution
Find the area to the left of z-value + 0.20: From the table this is 0.5793
Since the distribution is symmetric, for negative values we take 1 minus the table value
Probability of values less than $3.70 is 1 – 0.5793 = 42.07%
With a z-table for negatives, F(-0.20) = 0.4207