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2009 FRM培训课程课堂讲义

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    2009 FRM培训课程课堂讲义

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    Module 1

    • Quantitative Analysis
    • Quantitative Analysis
    • Quantitative Fundamentals
    • Statistical Properties and Forecasting of Correlation, Covariance, and Volatility
    • Monte Carlo Simulation and Extreme Value Theory

     Quantitative Fundamentals

    Probability Terminology
    Random 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
    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
    Conditional vs. Unconditional
    Two types of probability:

    Unconditional: P(A), the probability of an event
    regardless 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
    Joint Probability
    The 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%
    Probability that at Least 1 of 2 Events Will Occur
    P(A or B) = P(A) + P(B) – P(AB)
    We must subtract the joint probability P(AB)
    Addition Rule Example
    P(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)
    Joint Probability of any Number of Independent Events
    Dependent 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%
    Additive Properties
    Three theorems on expectation:
    If c is any constant, then E(cX) = cE(X)
    If X and Y are any random variables, then  E(X + Y) = E(X) + E(Y)
    If X and Y are independent random variables,  then E(XY) = E(X)E(Y)
    Variance
    Four theorems on variance:
    Var(X) = E[(X - µ)2] = E(X2) – [E(X)]2, where µ = E(X)

    If c is any constant, Var(cX) = c2Var(X)

    The quantity E(X-a)2 is a minimum when a = µ = E(X)

    If X and Y are independent random variables, then
    Var(X + Y) = Var(X) + Var(Y)
    Var(X – Y) = Var(X) + Var(Y)
    Covariance
    Covariance: 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
    Correlation
    Correlation: A standardized measure of the linear relationship between two variables
    Correlation
    Example: 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)?
    Joint Probability Function
    Bayes’ Formula
    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?
    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?
    Choosing r Objects from n Objects
    When order does matter, we use the

    permutation formula:
    Calculator Solutions
    How many ways to choose 3 from 5, order doesn’t matter? 5 → 2nd → nCr → 3 → =  10
    Discrete and Continuous Probability Distributions
    A probability distribution gives the probabilities of all possible outcomes for a random variable
    A discrete distribution has a finite number of possible outcomes
    A continuous distribution has an infinite number of possible outcomes

    The number of days next week on which it will rain is a discrete random variable that can take on the values {0,1,2,3,4,5,6,7}
    The amount of rain that will fall next week is a continuous random variable
    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
    Discrete Uniform
    A discrete uniform distribution has a finite number of possible outcomes, all of which are equally likely

      e.g. p(x) = 0.2 for X = {1,2,3,4,5}
     p(2) = 20%  F(3) = 60%  Prob (2  ≤  X  ≤  5) = 80%
    Binomial Tree
    Two possible outcomes each period, up or down

    Prob (up move) + Prob (down move) = 1
    Up factor (U) > 1  Down factor (D) = 1/U

    Example:
    Beginning stock price (S0) = $20
    Prob up = 60%         Prob down = 40%
    Up factor = 1.12    Down factor 1/1.12
    A Binomial Tree for Stock Price
    Continuous Uniform Distribution
    Probability distributed evenly over an interval
    e.g. continuous uniform over the interval 2 to 10

    P(X < 2) = 0     P(X > 10) = 0

    P(3 ≤ X ≤ 5) = (5 - 3)/(10 - 2) = 2/8 = 25%
    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
    Standard Normal Distribution
    A normal distribution that has been standardized so that mean = 0 and standard deviation = 1
    To standardize a random variable, calculate the z-value
    Subtract the mean (so mean = 0) and divide by standard deviation (so σ = 1)
    Calculating Probabilities Using the Standard Normal Distribution
    Calculating Probabilities Using the Standard Normal Distribution
    Calculating Probabilities Using the Standard Normal Distribution

    Poisson Distribution
    Sample Question
    Assume bond defaults are represented by a Poisson distribution and that the probability of default is 2%.  For a bond portfolio of 20 bonds, what is the probability of 0 defaults?
    λ = p × n = (0.02)(20) = 0.4
    Binomial vs. Normal Distribution
    Normal vs. Poisson Distribution
       Given that there is a relationship between the binomial and the normal distribution, and between the Poisson and the binomial distribution, one would also expect a relationship between the Poisson and the normal distribution.

       As it turns out, the Poisson distribution approaches the normal distribution as:
    Lognormal Distribution
    Statistics
    Descriptive statistics describe the properties of a large data set
    Inferential statistics uses a sample from a population to make probabilistic statements about the characteristics of a population
    A population is a complete set of outcomes
    A sample is a subset drawn from a population
    A Histogram
    A Frequency Polygon
    Population and Sample Means
    Population and sample means have different symbols but are both arithmetic means
    Median
    Midpoint of a data set, half above and half below

    With an odd number of observations
       2, 5, 7, 11, 14 Median = 7

    With an even number of observations, median is the average of the 2 middle observations
      3, 9, 10, 20  Median = (9 + 10)/2 = 9.5

    Less affected by extreme values than the mean
    Mode
    Value occurring most frequently in a data set
      2, 4, 5, 5, 7, 8, 8, 8, 10, 12 
      Mode = 8


    Data sets can have more than 1 mode (bimodal, trimodal, etc.)
    Geometric Mean
     Geometric mean is used to calculate compound growth rates
    If the returns are constant over time, geometric mean equals arithmetic mean
    The greater the variability of returns over time, the more the arithmetic mean will exceed the geometric mean
    Geometric Mean
    Population Variance and Std. Deviation
    Variance is the average
    of the squared deviations
    from the mean
    Standard deviation is
    the square root of
    variance
    Population Variance (σ2)
    Example:
     Returns on 4 stocks: 15%, –5%, 12%, 22%

     Population Mean (µ)= 11%
    Population Standard Deviation

     Variance: (σ 2)= 98.5
    Sample Variance (s2) and Sample Standard Deviation (s)
    Key difference between calculation of σ2 and s2 is that the sum of the squared deviations for s2 is divided by n – 1 instead of n
    Chebyshev’s Inequality
    Specifies the minimum percentage of observations that lie within k standard deviations of the mean; applies to any distribution 
    Skewness
    Skew measures the degree to which a distribution lacks symmetry
    A symmetrical distribution has skew = 0
    Positive Skew = Right Skew
    Positive skew has outliers in the right tail 
    Skew absolute values > 0.5 are significant 
    Mean is most affected by outliers
    Negative Skew = Left Skew
    Negative skew has outliers in the left tail
    Again, mean is most affected by outliers
    Kurtosis
    Measures the degree to which a distribution is more or less peaked than a normal distribution
    Leptokurtic (kurtosis > 3) is more peaked with fatter tails (more extreme outliers)

    Kurtosis for a normal distribution is 3.0

    Excess kurtosis is kurtosis minus 3

    Excess kurtosis is zero for a normal distribution

    Excess kurtosis greater than 1.0 in absolute value is considered significant
    Sampling
    To make inferences about the parameters of a population, we will use a sample
    A simple random sample is one where every population member has an equal chance of being selected
    A sampling distribution is the distribution of sample statistics for repeated samples of size n
    Sampling error is the difference between a sample statistic and true population parameter

    Stratified Random Sampling

    1. Create subgroups from population based on important characteristics, e.g. identify bonds according to: callable, ratings, maturity, coupon
    2. Select samples from each subgroup in proportion to the size of the subgroup

    Used to construct bond portfolios to match a bond index or to construct a sample that has certain characteristics in common with the underlying population Central Limit Theorem
    For any population with mean µ and variance σ2,  as the size of a random sample gets large, the distribution of sample means approaches a normal dist. with mean µ and variance σ2 /n
    Allows us to make inferences about and construct confidence intervals for population means based on sample means
    Standard Error of the Sample Mean
    Standard error of sample mean is the standard deviation of the distribution of sample means.

    更多

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