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By Dr. Jatin Thukral
Quantitative Analysis
CFA Level I
Outline
• Probability Concepts• Common Probability Distributions• Sampling and Estimation• Hypothesis Testing
Probability Concepts
• A random variable is an uncertain quantity• An outcome is an observed value of a random
variable• An event is a single outcome or a set of
outcomes• Mutually exclusive and exhaustive events are
those that include all possible outcomes
Defining Properties of Probability
• The probability of occurrence of any event Ei is between 0 and 1
0 ≤ P(Ei) ≤ 1
• If a set of event E1 ,E2 , E3 , …. , EN is mutually exclusive and exhaustive, then
∑P(Ei) = 1
Types of Probabilities
• Empirical probability: based on historical data
• A priori probability: based on reasoning
• Subjective probability: based on personal judgment
Probability in terms of Odds
• Alternate way of expressing probabilities
• The odds that an event with probability 0.2 will occur are 0.2/(1-0.2) = 0.25
Unconditional and Conditional Probabilities
• Unconditional Probability: The marginal probability that an event occurs regardless of any past or future event
• Conditional Probability: The probability of any event given that another event(s) has occurred or will occur
An example illustrating unconditional, conditional and joint probabilities
Rules of Probability• Multiplication Rule:
Joint Probability P(AB) = P(A|B) P(B)
• Addition Rule: P( A or B) = P(A) + P(B) – P(AB)
• Total Probability Rule: P(A) = P(A|B1) P(B1) + P(A|B2) P(B2)
+ ………. + P(A|BN) P(BN) where B1, B2 ,…, BN is a mutually exclusive and exhaustive
set of outcomes
Applications of Probability Rules
• Multiplication rule is used to compute joint probability
• Addition rule is used to determine probability that at least one of two event occurs
• Joint probability of any number of independent events is calculated as
P(ABCD)=P(A) P(B) P(C) P(D)
Independent Events
• Independent Events: Occurrence of one event has no influence on occurrence of other events
P(A|B) = P(A)
Expected Value
Expected value is the weighted average of the possible outcomes of a random variable, where the weights are the probabilities that the outcomes will occur
Diagramming an investment problem using a tree diagram
Covariance
• Covariance is a measure of how two assets move together
• Variance is covariance of an asset with itself• Covariance can range from negative infinity to
positive infinity
Correlation Coefficient
• Correlation is covariance divided by product of standard deviations
• Measures the strength of linear relationship between two random variables
• Has no units and ranges from -1 to +1• If Corr(Ri,Rj)=1, the random variables have perfect
positive correlation
Portfolio Expected Value and Variance
• Portfolio Expected Value
• Portfolio Variance
Bayes’ Formula
• Bayes’ formula is used to update a given set of prior probabilities for a given event in response to the arrival of new information
Labeling
• There are n items and each can receive one of k different labels
• The number that receive label 1 is n1, that receive label 2 is n2 and so on, such that
• The total number of ways that the labels can be assigned is
Combination Formula
• A special case of labeling where k=2• n1=r and n2=n-r• Choosing r items from a set of n items• nCr is the number of possible ways of selecting r
items from n items when order of selection is not important
Permutation Formula
• nPr is the number of possible ways to select r items from a set of n items when the order of selection is important
Outline
• Probability Concepts• Common Probability Distributions• Sampling and Estimation• Hypothesis Testing
Probability Distribution
• A probability distribution describes the probabilities of all the possible outcomes for a random variable
• A discrete random variable is one for which number of possible outcomes can be counted
• A continuous random variable has infinite possible outcomes
Probability Function
• A probability function p(x) specifies the probability that a random variable is equal to a specific value x
• The two key properties are 0 ≤ p(x) ≤ 1 ∑ p(x) = 1
Probability Density Function (pdf) and Cumulative Distribution Function (cdf)
• A pdf is a function f(x) that generates the probability that outcomes of a continuous distribution lie within a particular range
• A cdf defines the probability that a random variable X takes on a value less than a specific value x
Pdf and Cdf of Normal Distribution
Discrete Uniform Random Variable and a Binomial Random Variable
• A discrete uniform random variable: probabilities for all possible outcomes are equal
• A binomial random variable is the number of “successes” in a given number of trials when outcome can be either “success” or “failure”
Binomial Random Variable
• The probability of success p is constant for each trial and the trials are independent
• For number of trials = 1, the binomial random variable is called Bernoulli random variable
• The expected value of a Binomial variable is
Stock Price Movements as Binomial Tree
Continuous Uniform Distribution
• Outcomes can only occur between some lower limit a and some upper limit b
• P(X < a) = P(X > b) = 0• P(x1 ≤ X ≤ x2) = (x2 – x1) / (b – a)
Normal Distribution
• Completely described by mean µ and variance σ2, stated as X ~ N(µ,σ2)
• Skewness = 0, that is P(X ≤ µ) = P(µ ≤ X) = 0.5• Kurtosis = 3• A linear combination of normally distributed
random variable is also normally distributed• Probabilities of outcomes further above and
below from mean get smaller but never zero
Normal Distribution
Multivariate Distributions
• Specifies the probabilities associated with a group of random variables
• Meaningful only if behavior of one random variable depends on behavior of others
• Multivariate normal distribution is completely defined by three set of parameters
Confidence Interval
• A confidence interval is a range of values around the expected outcome within which we expect the actual outcome to be some specified percentage of the time
Standard Normal Distribution
• A standard normal distribution is a normal distribution that has been standardized so that it has mean zero and standard deviation 1
• Standardization is the process of converting an observed value for a random variable to its z-value
Using the z-table• Consider an EPS distributed with µ=6,σ =2.
What is probability that EPS ≥ 9.70?• The z-value of EPS = 9.70 is
• From z-table, F(1.85)=0.9678• F(EPS > 9.70) = 1-0.9678 = 0.0322
Shortfall Risk and Safety-First Ratio• Shortfall risk is probability that return will fall
below a target value over a given time period• Roy’s safety-first criterion states that optimal
portfolio minimizes the probability that return falls below a minimum threshold level, i.e.,
• If returns are normally distributed, then
Lognormal Distributions
• The lognormal distribution is generated by the function ex where x is normally distributed
Monte Carlo Simulations
• Technique based on repeated generation of one or more factors that affect security values
• A probability distribution is assigned to each factor
• Each set of randomly generated factors is used with a pricing model to value the security
• Limitations: fairly complex; results no better than the assumptions on factor distributions
Historical Simulation
• Based on actual changes in risk factors over some time period
• Advantage of using the actual distribution of risk factors and not estimations
• Limitation compared to Monte Carlo simulation is that can’t answer “What if” questions
Outline
• Probability Concepts• Common Probability Distributions• Sampling and Estimation• Hypothesis Testing
Sampling• Simple random sampling: selecting a sample such
that each item has same chance of being selected• Systematic sampling: selecting every mth item• Sampling error: difference between sample statistic
(mean, variance) and its population parameter• Sampling distribution of a sample statistic is a
probability distribution of all possible sample statistics computed from a set of equal-sized samples
Stratified Random Sampling
• Population is separated into smaller groups based on one or more distinguishing characteristics
• From each group, or stratum, a random sample is taken and results are pooled
• Size of sample from each stratum depends on size of stratum relative to population
• Often used in bond indexing
Time-Series and Cross-Sectional Data
• Times-series data: observations taken over a period of time at specific and equally spaced time intervals
• Cross-sectional data: observations taken at a single point in time
Central Limit Theorem
• For simple random samples of size n from a population with mean µ and variance σ2, the sampling distribution of sample mean approaches N(µ,σ2/ n)
• Extremely useful: Normal distribution is easy to apply to hypothesis testing and construct confidence intervals
• Inference can be made irrespective of population distribution if n>30
Standard Error of Sample Mean
• It is the standard deviation of the distribution of the sample means
Point Estimates and Confidence Interval
• Point estimates are single values used to estimate population parameters. For ex.
• Confidence interval estimates result in a range of values within which the actual parameter will lie with probability 1-α. Here, α is called the level of significance and 1-α is called degree of confidence
Construction of Confidence Intervals
point estimate ± (reliability factor X standard error)
Where
Point estimate = vale of a sample statisticReliability factor = a number depending on the
sampling distribution of estimation and probability that estimate falls within confidence interval
Standard error = standard error of the point estimate
Desirable Properties of an Estimate
• Unbiased: Expected value of the estimator is equal to the parameter you are trying to estimate
• Efficient: Variance of sampling distribution is smaller than all the other unbiased estimators
• Consistent: Accuracy of parameter estimate increases as the sample size increases
Student’s t-distribution
• Bell-shaped distribution, symmetric about mean• Appropriate distribution to use when
constructing confidence intervals based on small samples (n<30) from populations with unknown variance and a normal (or approx.) distribution
• Also appropriate when variance is unknown and sample size is large enough that CLT will ensure that sampling distribution is approx. normal
Properties of Student’s t-distribution
• Symmetrical about mean• Defined by a single parameter, the degrees of
freedom, equal to n – 1• More probability in tails than normal
distribution• As degrees of freedom get larger it
approaches standard normal distribution• Fatter tails => difficult to reject null hypothesis
Distributions for Different Degrees of Freedom
Calculating Confidence Interval
• Population is normal distributed with known variance
Calculating Confidence Interval
• Population is normal distributed with unknown variance
Constructing Confidence Intervals
• Unlike standard normal distribution, the reliability factor for t-distribution depends on sample size, so we have to look for reliability factors from table
• Owing to fatter tails of the t-distribution, confidence intervals using t-distribution are wider/more conservative
Selecting the Appropriate Test Statistic
Biases in Estimation Based on Samples
• Data-Mining Bias: Repetitive search for data patterns results in data-mining
• Sample Selection Bias: Some data is systematically excluded because of lack of availability
• Survivorship Bias: Most common bias, only surviving data is considered
• Look-Ahead Bias: A relationship is tested based on data not available on test date
• Time-Period Bias: Period over which data is gathered is too short or too long
Outline
• Probability Concepts• Common Probability Distributions• Sampling and Estimation• Hypothesis Testing
Hypothesis Testing Procedure
Null Hypothesis and Alternative Hypothesis
• The null hypothesis H0 is the hypothesis that the researcher wants to reject. For ex,
• The alternate hypothesis Ha is what is concluded if there is sufficient evidence to reject the null hypothesis
Two-Tailed Tests of Hypotheses
• A two-tailed test for population mean is
• The general decision rule for a two-tailed test is
Two-Tailed Test Using Standard Normal(z) Distribution
• The decision rule for a two-tailed z-test at α=0.05 is
Example of Two-Tailed Test
• Data gathered on daily returns for 250 days; Mean daily return=0.1%, Sample Std. Dev. =0.25%. Belief: Mean Daily Return ≠ 0
• Null Hypothesis:• Decision Rule:
• Standard error = 0.0025/√250• Test Statistic = (0.001 – 0)/(0.0025/√250) = 6.33• Since 6.33 > 1.96, reject the null hypothesis
One-Tailed Hypothesis Test
• The null and alternative hypotheses are either• The null and alternative hypotheses are either
• Using Standard Normal(z) Distribution
Test Statistic
• Standard Error when population σ is known:
• Standard Error when population σ is not known:
Critical Value and Decision Rule
• A test statistic is a random variable that may follow one of several distributions: the t-distribution, the z-distribution, the chi-square distribution and the F-distribution
• The critical value for the appropriate test statistic depends on its distribution
• The decision rule: if the test statistic is (greater, less than) X, reject the null
Type I and Type II Errors
• Type I error: Rejection of H0 when it is actually true
• Type II error: Failure to reject H0 when it is false• The significance level is the probability of making a
Type I error and is designated by α• Power of a test is the probability of correctly
rejecting H0 when it is false:• Therefore, Power of a test = 1 – P(Type II error)
Type I and Type II Errors
Relation Between Confidence Intervals and Hypothesis Tests
• A confidence interval is determined as
• The above expression can be rewritten as
p-value and Hypothesis Testing
• The p-value is the smallest level of significance for which H0 can be rejected assuming it is true
• For example, Signif. level =95%,
Test statistic =2.3
Population mean: when to use t-test?• Use t-test when the population variance is
unknown and either n≥30 or n<30 but distribution is approx. normal
• A t-statistic with n-1 degrees of freedom is computed as
Population mean: when to use z-test?• Use z-test when population is normally
distributed and variance is known• A z-statistic is computed as
Population mean: when to use z-test?• Also, use z-test when sample size is large and
variance is unknown• Corresponding z-statistic is computed as
Critical z-Values
Equality Test of Two Population Means
• Three possible hypotheses structures
• Two possible t-statistics• Unknown variances but assumed equal• Unknown variances assumed unequal
Unknown Variances Assumed Equal
Unknown Variances Assumed Uequal
Paired Comparisons Test
• Test whether the means of the difference between observations for two samples are different
• Two-sided Test
• One-sided Test
t-statistic for Paired Comparisons Test
Test Variance of Normally-Distributed Population
• Chi-square test– Two-tailed test
– One-tailed test
Decision Rule for a Two-Tailed Chi-Square Test
Chi-Square Test Statistic
• The chi-square test statistic with n-1 degrees of freedom is computed as
Equality of Variance of Two Normally Distributed Populations
• F-distributed test statistic– Two-tailed test
– One-tailed test
• F-statistic computed as
F-test
• Always put larger variance in numerator• n1 – 1 and n2 – 1 are degrees of freedom used
to identify critical value from the F-table
Parametric and Nonparametric Tests
• Parametric Tests: Rely on assumptions regarding the distribution of the population and are specific to population parameters
• Nonparametric Tests: Either do not consider a particular population parameter or have few assumptions about the population that is sampled