SlideShare una empresa de Scribd logo
1 de 70
Descargar para leer sin conexión
Probability Distributions


    Istanbul Bilgi University
    FEC 512 Financial Econometrics-I
    Asst. Prof. Dr. Orhan Erdem
Some Common Probability
     Distributions
                 Probability
                Distributions

  Discrete                                   Continuous
 Probability                                  Probability
Distributions                                Distributions

                                                    Normal
 Binomial
                                                    Uniform
 Poisson
                                                    Lognormal

                                                                Lecture 3-2
                FEC 512 Probability Distributions
Discrete Probability Distribution
   Example random variable X = total number of
Experiment: toss 2 coins,
tails in two tosses.

                                   Probability distribution
                                 X         Probability
                T                 0                        0.25
                                  1                        0.50
      T
                                  2                        0.25

      T         T
                                                                  Lecture 3-3
                       FEC 512 Probability Distributions
The Binomial Distribution
                 Probability
                Distributions

  Discrete
 Probability
Distributions

 Binomial

 Poisson



                                                    Lecture 3-4
                FEC 512 Probability Distributions
The Binomial Distribution
Characteristics of the Binomial Distribution:
  A trial has only two possible outcomes – “success”
  or “failure”
  There is a fixed number, n, of identical trials
  The trials of the experiment are independent of
  each other
  The probability of a success, p, remains constant
  from trial to trial
  If p represents the probability of a success, then
  (1-p) = q is the probability of a failure
                                                    Lecture 3-5
                FEC 512 Probability Distributions
Binomial Distribution Settings

A manufacturing plant labels items as either
defective or acceptable
A firm bidding for a contract will either get the
contract or not
A marketing research firm receives survey
responses of “yes I will buy” or “no I will not”
New job applicants either accept the offer or
reject it
                                                     Lecture 3-6
                 FEC 512 Probability Distributions
Counting Rule for Combinations

A combination is an outcome of an experiment
where x objects are selected from a group of n
objects
                    n!
            C=   n
                 x
               x! (n − x )!
        where:
                     n! =n(n - 1)(n - 2) . . . (2)(1)
                     x! = x(x - 1)(x - 2) . . . (2)(1)
                     0! = 1    (by definition)


                                                          Lecture 3-7
                      FEC 512 Probability Distributions
Binomial Distribution Formula
                          n!      x n−x
             P(x) =               pq
                    x ! (n − x )!

P(x) = probability of x successes in n trials,
       with probability of success p on each trial               Example: Flip a coin four
                                                                  times, let x = # heads:
 x = number of ‘successes’ in sample,
                                                                           n=4
      (x = 0, 1, 2, ..., n)
                                                                          p = 0.5
 p = probability of “success” per trial
                                                                      q = (1 - .5) = .5
 q = probability of “failure” = (1 – p)
 n = number of trials (sample size)                                   x = 0, 1, 2, 3, 4

                                                                                    Lecture 3-8
                             FEC 512 Probability Distributions
Binomial Distribution
 The shape of the binomial distribution depends on
 the values of p and n      P(X) n = 5 p = 0.1
 Mean                                  .6
                                       .4
                                       .2
   Here, n = 5 and p = .1
                                        0                                       X
                                                0        1   2   3   4   5


                                                         n = 5 p = 0.5
                                            P(X)
                                       .6
                                       .4
   Here, n = 5 and p = .5              .2
                                                                                X
                                        0
                                                0        1   2   3   4   5
                                                                             Lecture 3-9
                     FEC 512 Probability Distributions
Binomial Distribution
Characteristics
Mean
                     µ = E(x) = np
 Variance and Standard
 Deviation
                            σ = npq
                2


                             σ = npq
Where   n = sample size
        p = probability of success
        q = (1 – p) = probability of failure

                                                            Lecture 3-10
                        FEC 512 Probability Distributions
Binomial Characteristics
      Examples

Mean = (5)(.1) = 0.5
 µ = np                                                  n = 5 p = 0.1
                                            P(X)
                                       .6
                                       .4
 σ = npq = (5)(.1)(1 − .1)             .2
          = 0.6708                      0                                        X
                                                0        1   2   3   4   5


 µ = np = (5)(.5) = 2.5                                  n = 5 p = 0.5
                                            P(X)
                                       .6
                                       .4
 σ = npq = (5)(.5)(1 − .5)             .2
          = 1.118                                                                X
                                        0
                                                0        1   2   3   4   5
                                                                             Lecture 3-11
                     FEC 512 Probability Distributions
A binomial tree of asset prices
     “Example”
   We wish to know the value of an asset after two time periods.
   Each of the time periods the asset may rise (a success) with a probability of 0.5
   or it may fall (a failure) with a probability of 0.5.
   Assume asset price movement in one time period is independent of that in the
   other time period.

                                                                    Su2
                      Su                                           (60.50)
                     (55)
                                                             Sud=Sdu
                                                               (49.5)
S=50
                        Sd
                                                                    Sd2
                       (45)
                                                                   (40.50)   T2
   T0                   T1

                                                                                  Lecture 3-12
                               FEC 512 Probability Distributions
A binomial tree of asset prices
    “Example”
    If the asset had previously risen by a factor u, it would either rise again by u to
    Su2 or would fall by d to Sud.
    If the asset had previoulsy fallen to Sd, it could rise by u to Sud or fall further
    to Sd2
    Suppose that u=1.1, d=0.9, and S=50
    Hence the expected value can be calculated as:

           µ = (60.50 * 0.25) + (49.50 * 0.50) + (40.50 * 0.25) = 50

    The variance is

 σ2 = (60.5 – 50)2 * 0.25 + (49.50 – 50)2 * 0.50 + (40.5 – 50)2 * 0.25 = 50.25




                                                                              Lecture 3-13
                              FEC 512 Probability Distributions
The Poisson Distribution
                 Probability
                Distributions

  Discrete
 Probability
Distributions

 Binomial

 Poisson



                                                    Lecture 3-14
                FEC 512 Probability Distributions
The Poisson Distribution
  To use binomial distribution, we must be able
  to count the # successes and failures. Although
  in many situations you may be able to count #
  successes, you often cannot count # failures.
  Example: An emergency call center could
  easily count the # calls its unit respond to in 1
  hour, but how could it determine how many
  calls it didnt receive?




                                                      Lecture 3-15
                  FEC 512 Probability Distributions
Characteristics of the Poisson Distribution

   The outcomes of interest are rare relative to the
   possible outcomes
   The average number of outcomes of interest per
   time or space interval is λ
   The number of outcomes of interest are random,
   and the occurrence of one outcome does not
   influence the chances of another outcome of
   interest
   The probability of that an outcome of interest
   occurs in a given segment is the same for all
   segments
                                                       Lecture 3-16
                   FEC 512 Probability Distributions
Poisson Distribution Formula

                                                       − λt
                     ( λt ) e               x
            P( x ) =
                           x!
where:
  t = size of the segment of interest
  x = number of successes in segment of interest
  λ = expected number of successes in a segment of unit size
  e = base of the natural logarithm system (2.71828...)



                                                              Lecture 3-17
                   FEC 512 Probability Distributions
Poisson Distribution (continued)

Ex. Page requests arrive at a                        0,18
                                                     0,17
  webserver at an average rate                       0,16
  of 5 every second. If the                          0,15
  number of requests in a                            0,14
                                                     0,13
  second has a Poisson                               0,12
  distribution, find the probability                 0,11
                                                     0,10
  that 15 requests will be made               p(X=x) 0,09
                                                     0,08
  in a given second.                                 0,07
                 e −5 515                            0,06
P ( X = 15 )   =          = 0.00016                  0,05
                                                     0,04
                   15!                               0,03
                                                     0,02
                                                     0,01
                                                     0,00
Here is what the distribution                               1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
  function for the above example                                        x=number of page requests in a second
  looks like
                                                                                                            Lecture 3-18
                             FEC 512 Probability Distributions
Poisson Distribution Characteristics

Mean
                              µ = λt
 Variance and Standard
 Deviation
                             σ = λt
                 2


                             σ = λt
        λ = number of successes in a segment of unit size
where
        t = the size of the segment of interest

                                                            Lecture 3-19
                      FEC 512 Probability Distributions
Graph of Poisson Probabilities

                             0.70

 Graphically:                0.60

λ = .05 and t = 100          0.50

           λt =              0.40


                      P(x)
    X      0.50              0.30

    0     0.6065
                             0.20
    1     0.3033
                             0.10
    2     0.0758
                             0.00
    3     0.0126                      0         1        2         3       4   5   6          7

    4     0.0016                                                       x
    5     0.0002
                                          P(x = 2) = .0758
    6     0.0000
    7     0.0000
                                                                                       Lecture 3-20
                               FEC 512 Probability Distributions
Poisson Distribution Shape

                  The shape of the Poisson Distribution
                  depends on the parameters λ and t:
                       λt = 0.50                                                             λt = 3.0
       0.70                                                           0.25


       0.60
                                                                      0.20
       0.50

                                                                      0.15
       0.40




                                                               P(x)
P(x)




       0.30                                                           0.10

       0.20
                                                                      0.05
       0.10

                                                                      0.00
       0.00
                                                                             1   2   3   4    5   6       7   8   9   10   11    12
              0    1    2   3       4   5    6      7

                                                                                                      x
                                x



                                                                                                                  Lecture 3-21
                                            FEC 512 Probability Distributions
The Normal Distribution
     Probability
    Distributions

                                 Continuous
                                  Probability
                                 Distributions

                                        Normal

                                        Uniform

                                        Lognormal
                                                    Lecture 3-22
    FEC 512 Probability Distributions
The Normal Distribution
 The distribution whose pdf is given by

                          ( x−µ )2   
                         −                          f(x)
                 1                   
                             2σ 2
       f ( x) =                      
                     e
                2π σ
                                                                   σ
 ‘Location is determined by                                                           x
the mean, µ                                                    µ
  Spread is determined by the
standard deviation, σ



                                                                       Lecture 3-23
                           FEC 512 Probability Distributions
The Normal Distribution

                                            f(x)
 ‘Bell Shaped’
 Symmetrical
                                                         σ
The random variable has                                                     x
an infinite theoretical                              µ
range: + ∞ to − ∞




                                                             Lecture 3-24
                 FEC 512 Probability Distributions
Many Normal Distributions




By varying the parameters µ and σ, we obtain
        different normal distributions

                                                    Lecture 3-25
                FEC 512 Probability Distributions
The Normal Distribution Shape


 f(x)   Changing µ shifts the
        distribution left or right.

                                            Changing σ increases
                                            or decreases the
                             σ              spread.


                       µ                              x

                                                           Lecture 3-26
             FEC 512 Probability Distributions
Finding Normal Probabilities

Probability is the
       Probability is measured                 by the area
area under the
curve! under the curve
     f(x)
                                               P (a ≤ x ≤ b)




                             a            b             x

                                                               Lecture 3-27
                  FEC 512 Probability Distributions
Probability as
Area Under the Curve
 The total area under the curve is 1.0, and the curve is
 symmetric, so half is above the mean, half is below

f(x) P( −∞ < x < µ) = 0.5
                                                   P(µ < x < ∞ ) = 0.5



                       0.5            0.5

                                                              x
                                 µ
               P(−∞ < x < ∞) = 1.0
                                                                   Lecture 3-28
                   FEC 512 Probability Distributions
Empirical Rules

What can we say about the distribution of values
around the mean? There are some general rules:
    f(x)

                                    µ ± 1σ encloses about
                                          68% of x’s
                     σ          σ



                                                       x
               µ−1σ µ µ+1
                −         +1σ
                          +1
                   68.26%
                                                           Lecture 3-29
                  FEC 512 Probability Distributions
The Empirical Rule
                                                                   (continued)

     µ ± 2σ covers about 95% of x’s
     µ ± 3σ covers about 99.7% of x’s



                                                     3σ       3σ
   2σ       2σ
        µ                  x                              µ                         x

     95.44%                                           99.72%


                                                                     Lecture 3-30
                 FEC 512 Probability Distributions
The Standard Normal Distribution
 Also known as the “z” distribution
 Mean is defined to be 0
 Standard Deviation is 1
       f(z)

                              1
                                                   z
                         0
Values above the mean have positive z-values,
values below the mean have negative z-values
                                                       Lecture 3-31
               FEC 512 Probability Distributions
Transformation to the Standard
 Normal Distribution
Any normal distribution (with any mean and
standard deviation combination) can be
transformed into the standard normal
distribution (z)

              x −µ
           z=
                σ

                                                Lecture 3-32
            FEC 512 Probability Distributions
Example

If x is distributed normally with mean of
100 and standard deviation of 50, the z
value for x = 200 is
        x − µ 200 − 100
     z=      =          = 2.0
          σ      50

This says that x = 200 is two standard
deviations (2 increments of 50 units) above
the mean of 100.
                                                 Lecture 3-33
             FEC 512 Probability Distributions
Comparing x and z units

                                                    µ = 100
                                                    σ = 50



                    100                       200   x
                     0                        2.0   z
Note that the distribution is the same, only the
scale has changed. We can express the problem in
original units (x) or in standardized units (z)
                                                              Lecture 3-34
                FEC 512 Probability Distributions
The Standard Normal Table

   The Standard Normal table in the
 textbooks gives the probability from the
 mean (zero) up to a desired value for z


                                                               .4772

Example:
P(0 < z < 2.00) = .4772
                                                                       z
                                                        0   2.00

                                                                           Lecture 3-35
                    FEC 512 Probability Distributions
The Standard Normal Table
                                                               (continued)
                              The column gives the value
                              of z to the second decimal
                              point
                z            0.00       0.01        0.02   …

                0.1
The row
shows the       0.2
                 .
                 .
value of z to                                   The value within the
                 .
the first                                       table gives the
                              .4772
                2.0
decimal point                                   probability from z = 0
                                                up to the desired z
                2.0
  P(0 < z < 2.00) = .4772
                                                value
                                                                 Lecture 3-36
                    FEC 512 Probability Distributions
Z Table example

  Suppose x is normal with mean 8.0 and
  standard deviation 5.0. Find P(8 < x < 8.6)

Calculate z-values:

    x −µ 8 −8
 z=     =     =0
      σ    5
                                                               8 8.6              x
   x − µ 8.6 − 8                                               0 0.12             Z
z=      =        = 0.12
     σ      5                                              P(8 < x < 8.6)
                                                          = P(0 < z < 0.12)
                                                                            Lecture 3-37
                      FEC 512 Probability Distributions
Z Table example
                                                                     (continued)

Suppose x is normal with mean 8.0 and
standard deviation 5.0. Find P(8 < x < 8.6)

              µ=8                                                   µ=0
              σ=5                                                   σ=1


                       x                                                      z
      8 8.6                                                0 0.12

 P(8 < x < 8.6)                                       P(0 < z < 0.12)


                                                                        Lecture 3-38
                  FEC 512 Probability Distributions
Solution: Finding P(0 < z < 0.12)

Standard Normal Probability                                  P(8 < x < 8.6)
Table (Portion)                                             = P(0 < z < 0.12)
 z   .00     .01   .02                                             .0478
0.0 .0000 .0040 .0080

0.1 .0398 .0438 .0478
0.2 .0793 .0832 .0871
                                                                           Z
                                                         0.00
0.3 .1179 .1217 .1255
                                                            0.12

                                                                       Lecture 3-39
                     FEC 512 Probability Distributions
Lower Tail Probabilities

 Suppose x is normal with mean 8.0
 and standard deviation 5.0.
 Now Find P(7.4 < x < 8)




                                                  Z
                                   8.0
                             7.4
                                                      Lecture 3-40
              FEC 512 Probability Distributions
Lower Tail Probabilities
                                                                 (continued)

 Now Find P(7.4 < x < 8)…

   The Normal distribution is
 symmetric, so we use the                        .0478
 same table even if z-values
 are negative:
 P(7.4 < x < 8)
 = P(-0.12 < z < 0)
                                                                  Z
 = .0478                                                   8.0
                                                     7.4

                                                                   Lecture 3-41
                      FEC 512 Probability Distributions
Distributions of Portfolio Returns
Example. Assume that the stock index in a country has an annual return distribution
that is normal with µ = 0.15 and σ = 0.30. What is probability that in a given year the
stock index will exceed an annual return of 100%? What is the probability that the
     index
will produce a negative return in a given year?

We first need to transform the normal variable into a standard normal.
     X − 0.15 1.00 − 0.15
z=           =            = 2.83
      0.30       0.30
Looking up 2.83 in the normal table, we find that F(z) is 0.9977. So 1 – F(z) = 0.0023.
For finding the probability of a negative return, the transformation yields
     X − 0.15 0 − 0.15
z=           =         = −0.50
      0.30      0.30
This time we are interested in F(z), which is 0.3085.



                                                                              Lecture 3-42
                                   FEC 512 Probability Distributions
Linear Combinations of Two Normal
Random Variables
 Let X~N(µX,σX2 ) and Y~N(µY,σY2 ) and
 σXY=cov(X,Y).
 If Z=aX+bY where a,b are constants, then
 Z~N(µZ,σZ2 ) where
  µZ=a µX +b µY
 σZ2=a2 σX2 +b2 σY2 +2abσXY=a2 σX2 +b2 σY2
 +2abσX σYρ



                                                    Lecture 3-43
                FEC 512 Probability Distributions
Kurtosis and Skewness of Normal
Distribution
 The skewness of a normal distribution is 0. Why?
 The kurtosis of a normal distribution is 3. Hence 3
 is a benchmark value for tail thickness of a bell-
 shaped distribution.
 If kurt(X)>3, the dist. has thicker tails than norm.
 dist.
 If kurt(X)<3, the dist. has thinner tails than norm.
 dist.



                                                       Lecture 3-44
                   FEC 512 Probability Distributions
The Uniform Distribution
      Probability
     Distributions

                                  Continuous
                                   Probability
                                  Distributions

                                         Normal

                                         Uniform

                                         Lognormal
                                                     Lecture 3-45
     FEC 512 Probability Distributions
The Uniform Distribution


   The uniform distribution is a
   probability distribution that has
   equal probabilities for all possible
   outcomes of the random variable



                                                   Lecture 3-46
               FEC 512 Probability Distributions
The Uniform Distribution                                      (continued)

   The Continuous Uniform Distribution:

                         1
                                            if a ≤ x ≤ b
                        b−a
          f(x) =
                            0                  otherwise
      where
        f(x) = value of the density function at any x value
        a = lower limit of the interval
        b = upper limit of the interval

                                                               Lecture 3-47
                       FEC 512 Probability Distributions
Uniform Distribution

  Example: Uniform Probability Distribution
           Over the range 2 ≤ x ≤ 6:
               1
      f(x) = 6 - 2 = .25 for 2 ≤ x ≤ 6

        f(x)
       .25


                                                       x
                2                                  6

                                                           Lecture 3-48
               FEC 512 Probability Distributions
The Lognormal Distribution
       Probability
      Distributions

                                   Continuous
                                    Probability
                                   Distributions

                                          Normal

                                          Uniform

                                          Lognormal
                                                      Lecture 3-49
      FEC 512 Probability Distributions
The Lognormal Distribution

 Let Z ~N(µ,σ2), and X=eZ r.v. X is said to be log-
 normally distributed with parameters µ and σ2
                  lnX~N(µ,σ2)
 or
 In other words, X is lognormal if its “ln” is normally
 distributed.
 If X,Y is lognormally distributed, their linear
 combination(i.e. Portfolio of two stocks) may not be
 lognormal.



                                                        Lecture 3-50
                    FEC 512 Probability Distributions
Lecture 3-51
FEC 512 Probability Distributions
The median of X is eµ, and the expected value of X
              σ2
         µ+
is                 . The expectation is larger than the
              2
     e

median because the lognormal distribution is right-
skewed, and the skew. is more extreme with larger
values of σ.




                                                               Lecture 3-52
                           FEC 512 Probability Distributions
Example

 Let rt = ln( Pt / Pt −1 )     denote the log-return on an
 asset and assume that rt ~ N(µ,σ2). Let R = ( P P P )      −              t −1
                                                                t
                                                            t
                                                                    t −1

 denote the simple monthly return, since we know
                                       e rt = 1 + Rt . Since rt is
         rt = ln(1 + Rt )
 that                          and
 normally distributed e t = 1 + Rt is log-normally
                            r
                           .............
 distributed.




                                                                                  Lecture 3-53
                        FEC 512 Probability Distributions
Sample Moments

 Above we introduced the four statistical moments
 mean,variance, skewness, kurtosis.
 Given a pdf, we are able to calculate these stat.
 Moments according to the fomulae.
 In practical applications however, we are faced with
 the situation that we observe realizations of a
 pdf(e.g. The daily return of the IMKB-100 index over
 the last year), but we do not know the distribution
 that generates these returns. So taking expectation
 is impossible
 But having the observations x1,…,xn, we can try to
 estimate the “true moments” out of the sample.
 These estimates are called sample moments.

                                                       Lecture 3-54
                   FEC 512 Probability Distributions
Mean (Arithmetic Average)
The Mean is the arithmetic average of data
values
 Sample mean                             n = Sample Size
                    n

                  ∑x               x1 + x 2 + L + x n
                             i
            x=                   =
                   i =1
                        n                   n




                                                           Lecture 3-55
                 FEC 512 Probability Distributions
Variation

   Measures of variation give information
   on the spread or variability of the
   data values.




                                      Same center,
                                   different variation

                                                         Lecture 3-56
               FEC 512 Probability Distributions
Variance
 Average of squared deviations of values from the
 mean
   Sample variance:                                       n

                                                      ∑ (x            − x)   2
                                                                  i
                                         s2 =             i =1
                                                                 n -1
   Sample standard deviation:
                                                           n

                                                          ∑ (x        − x)   2
                                                                  i
                                          s=              i=1
                                                                 n -1
                                                                                 Lecture 3-57
                      FEC 512 Probability Distributions
Calculation Example:
              Sample Standard Deviation
Sample
Data (Xi) :    10        12        14          15         17      18     18   24
                       n=8                    Mean = x = 16

       (10 − x ) 2 + (12 − x ) 2 + (14 − x ) 2 + L + (24 − x ) 2
 s=
                                  n −1

      (10 − 16)        + (12 − 16)            + (14 − 16)             + L + (24 − 16)
                   2                      2                       2                        2
  =
                                               8 −1

      126
  =            =       4.2426
       7
                                                                                   Lecture 3-58
                              FEC 512 Probability Distributions
Comparing Standard Deviations

  Data A
                                                                       Mean = 15.5
                                                                        s = 3.338
  11   12   13   14   15    16      17      18      19         20 21


  Data B
                                                                       Mean = 15.5
                                                                        s = .9258
  11   12   13   14   15    16      17      18      19         20 21


       Data C
                                                                       Mean = 15.5
                                                                        s = 4.57
  11   12   13   14   15    16      17      18      19         20 21


                                                                            Lecture 3-59
                           FEC 512 Probability Distributions
Coefficient of Variation

 Measures relative variation
 Always in percentage (%)
 Shows variation relative to mean
 Is used to compare two or more sets of data
 measured in different units
               Sample C.V.
                 s
            CV =   ⋅ 100%
                 x
                  
                                                    Lecture 3-60
                FEC 512 Probability Distributions
Comparing Coefficient
of Variation
Stock A:
  Average price last year = $50
  Standard deviation = $5
          s          $5
    CVA =   ⋅ 100% =     ⋅ 100% = 10%
          x          $50
                                                  Both stocks
                                                    have the same
Stock B:                                            standard
                                                    deviation, but
  Average price last year = $100                    stock B is less
                                                    variable relative
  Standard deviation = $5                           to its price
          s           $5
    CVB =   ⋅ 100% =      ⋅ 100% = 5%
          x          $100
          
                                                          Lecture 3-61
                FEC 512 Probability Distributions
Sharpe Ratio

 Let Dt=Rt-Rf where Rf is the riskfree rate of
 return.                  T
            T

                                 ∑
         1
     D = ∑ Dt                 (Dt − D) 2
         T t =1
                   σD =  t =1

                                          T −1
 Sharpe Ratio is
                    (D)
             S=
                     σD
 Reading: “The Sharpe Ratio”, William F. Sharpe
                                                       Lecture 3-62
                   FEC 512 Probability Distributions
Skewness

 The moment coefficient of skewness is derived by
 calculating the third moment about the mean and
 dividing by the cube of standard deviation :

                           (                )
                                    
                      ∑ X −X
                                                3

                                    
                                    
                        n −1
                                    
                                        3
                                   2
                     ∑(                     )
                               X −X 
                
                                     
                               n −1
                                     
                                     
                                                        Lecture 3-63
                    FEC 512 Probability Distributions
Shape of a Distribution

            Describes how data is distributed
            Symmetric or skewed


                                                                        Right-Skewed
                                      Symmetric
    Left-Skewed




                                 Mean = Median                                 Median < Mean
Mean < Median
 (Longer tail extends to left)                                         (Longer tail extends to right)
                                                                                         Lecture 3-64
                                   FEC 512 Probability Distributions
Kurtosis

 Skewness indicates the degree of symmetry
 in the frequency distribution
 Kurtosis indicates the peakedness of that
 distribution




                                                   Lecture 3-65
               FEC 512 Probability Distributions
Kurtosis (continued)

              ∑(X − X )
                                 4



                   n −1
                                         4
                                    
              ∑(X − X )
                                 2
                                    
                                    
                    n −1
                                    
                                    




                                                   Lecture 3-66
               FEC 512 Probability Distributions
About the Probability Distribution of
Returns
      The assumption that period returns(e.g. Daily, monthly,
      annually) are normally distributed is inconsistent with the
      limited liability feature of most financial instruments, R≥-1.
      There are a number of empirical facts about return distributions
         While normal distribution is perfectly symmetric about its mean,
 1.
         dailt stock returns are frequently skewed to the right. And few of
         them are skewed to the left.
         The sample daily return distributions for many individuals stocks
 2.
         exhibit “excess kurtosis” or “fat tails”. i.e. There is more
         probability in the tails than would be justified by normal
         distribution. The extent of this excess kurtosis diminishes
         substantially, however , when monthly data is used.




                                                                    Lecture 3-67
                            FEC 512 Probability Distributions
Emprical Return Distributions: IMKB-100
Daily
    500
                                                      Series: XU100RETURNS
                                                      Sample 1/04/1993 12/29/2004
    400                                               Observations 2968

                                                      Mean           0.002654
    300                                               Median         0.001996
                                                      Maximum        0.194510
                                                      Minimum       -0.181093
    200                                               Std. Dev.      0.031281
                                                      Skewness       0.139399
                                                      Kurtosis       6.290761
    100
                                                      Jarque-Bera   1348.812
                                                      Probability   0.000000
      0
          -0.1   0.0          0.1          0.2




                                                                                    Lecture 3-68
                  FEC 512 Probability Distributions
Emprical Return Distributions: IMKB-100
Monthly
  24
                                                    Series: XU100RETURNS
                                                    Sample 1993M01 2004M12
  20
                                                    Observations 144

  16                                                Mean           5.899510
                                                    Median         5.043156
                                                    Maximum        79.78386
  12
                                                    Minimum       -39.03413
                                                    Std. Dev.      17.21817
   8                                                Skewness       0.903614
                                                    Kurtosis       5.585146
   4
                                                    Jarque-Bera   59.69430
                                                    Probability   0.000000
   0
       -40   -20   0   20   40      60       80



 Pay attention to Skewness and Kurtosis.


                                                                              Lecture 3-69
                            FEC 512 Probability Distributions
Value At Risk




                                                    Lecture 3-70
                FEC 512 Probability Distributions

Más contenido relacionado

La actualidad más candente

Lesson 24: Optimization II
Lesson 24: Optimization IILesson 24: Optimization II
Lesson 24: Optimization IIMatthew Leingang
 
Confidence Intervals––Exact Intervals, Jackknife, and Bootstrap
Confidence Intervals––Exact Intervals, Jackknife, and BootstrapConfidence Intervals––Exact Intervals, Jackknife, and Bootstrap
Confidence Intervals––Exact Intervals, Jackknife, and BootstrapFrancesco Casalegno
 
04 random-variables-probability-distributionsrv
04 random-variables-probability-distributionsrv04 random-variables-probability-distributionsrv
04 random-variables-probability-distributionsrvPooja Sakhla
 
Lesson 17: Indeterminate forms and l'Hôpital's Rule (slides)
Lesson 17: Indeterminate forms and l'Hôpital's Rule (slides)Lesson 17: Indeterminate forms and l'Hôpital's Rule (slides)
Lesson 17: Indeterminate forms and l'Hôpital's Rule (slides)Matthew Leingang
 
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
 
Lesson 23: Antiderivatives (slides)
Lesson 23: Antiderivatives (slides)Lesson 23: Antiderivatives (slides)
Lesson 23: Antiderivatives (slides)Matthew Leingang
 
Ssp notes
Ssp notesSsp notes
Ssp notesbalu902
 
A Geometric Note on a Type of Multiple Testing-07-24-2015
A Geometric Note on a Type of Multiple Testing-07-24-2015A Geometric Note on a Type of Multiple Testing-07-24-2015
A Geometric Note on a Type of Multiple Testing-07-24-2015Junfeng Liu
 
The jackknife and bootstrap
The jackknife and bootstrapThe jackknife and bootstrap
The jackknife and bootstrapPaul Gardner
 
Lesson 23: Antiderivatives
Lesson 23: AntiderivativesLesson 23: Antiderivatives
Lesson 23: AntiderivativesMatthew Leingang
 
Statistics Presentation week 6
Statistics Presentation week 6Statistics Presentation week 6
Statistics Presentation week 6krookroo
 
Lesson 23: Antiderivatives
Lesson 23: AntiderivativesLesson 23: Antiderivatives
Lesson 23: AntiderivativesMatthew Leingang
 

La actualidad más candente (20)

Lesson 24: Optimization II
Lesson 24: Optimization IILesson 24: Optimization II
Lesson 24: Optimization II
 
Confidence Intervals––Exact Intervals, Jackknife, and Bootstrap
Confidence Intervals––Exact Intervals, Jackknife, and BootstrapConfidence Intervals––Exact Intervals, Jackknife, and Bootstrap
Confidence Intervals––Exact Intervals, Jackknife, and Bootstrap
 
04 random-variables-probability-distributionsrv
04 random-variables-probability-distributionsrv04 random-variables-probability-distributionsrv
04 random-variables-probability-distributionsrv
 
Paper06
Paper06Paper06
Paper06
 
Lesson 17: Indeterminate forms and l'Hôpital's Rule (slides)
Lesson 17: Indeterminate forms and l'Hôpital's Rule (slides)Lesson 17: Indeterminate forms and l'Hôpital's Rule (slides)
Lesson 17: Indeterminate forms and l'Hôpital's Rule (slides)
 
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)
 
Lesson 23: Antiderivatives (slides)
Lesson 23: Antiderivatives (slides)Lesson 23: Antiderivatives (slides)
Lesson 23: Antiderivatives (slides)
 
Prob distros
Prob distrosProb distros
Prob distros
 
Ssp notes
Ssp notesSsp notes
Ssp notes
 
A Geometric Note on a Type of Multiple Testing-07-24-2015
A Geometric Note on a Type of Multiple Testing-07-24-2015A Geometric Note on a Type of Multiple Testing-07-24-2015
A Geometric Note on a Type of Multiple Testing-07-24-2015
 
2019 PMED Spring Course - SMARTs-Part II - Eric Laber, April 10, 2019
2019 PMED Spring Course - SMARTs-Part II - Eric Laber, April 10, 2019 2019 PMED Spring Course - SMARTs-Part II - Eric Laber, April 10, 2019
2019 PMED Spring Course - SMARTs-Part II - Eric Laber, April 10, 2019
 
Notes 6-2
Notes 6-2Notes 6-2
Notes 6-2
 
The jackknife and bootstrap
The jackknife and bootstrapThe jackknife and bootstrap
The jackknife and bootstrap
 
Raices primitivas
Raices primitivasRaices primitivas
Raices primitivas
 
2019 PMED Spring Course - Introduction to Nonsmooth Inference - Eric Laber, A...
2019 PMED Spring Course - Introduction to Nonsmooth Inference - Eric Laber, A...2019 PMED Spring Course - Introduction to Nonsmooth Inference - Eric Laber, A...
2019 PMED Spring Course - Introduction to Nonsmooth Inference - Eric Laber, A...
 
Lesson 23: Antiderivatives
Lesson 23: AntiderivativesLesson 23: Antiderivatives
Lesson 23: Antiderivatives
 
1 - Linear Regression
1 - Linear Regression1 - Linear Regression
1 - Linear Regression
 
Midterm I Review
Midterm I ReviewMidterm I Review
Midterm I Review
 
Statistics Presentation week 6
Statistics Presentation week 6Statistics Presentation week 6
Statistics Presentation week 6
 
Lesson 23: Antiderivatives
Lesson 23: AntiderivativesLesson 23: Antiderivatives
Lesson 23: Antiderivatives
 

Destacado

Chap04 basic probability
Chap04 basic probabilityChap04 basic probability
Chap04 basic probabilityFathia Baroroh
 
securitization+musyarakah+murabahah+and+ijarah
securitization+musyarakah+murabahah+and+ijarahsecuritization+musyarakah+murabahah+and+ijarah
securitization+musyarakah+murabahah+and+ijarahmandalina landy
 
Chapter 05
Chapter 05Chapter 05
Chapter 05bmcfad01
 
Bba 3274 qm week 4 decision analysis
Bba 3274 qm week 4 decision analysisBba 3274 qm week 4 decision analysis
Bba 3274 qm week 4 decision analysisStephen Ong
 
Poisson distribution
Poisson distributionPoisson distribution
Poisson distributionStudent
 
Forecasting And Decision Making
Forecasting And Decision MakingForecasting And Decision Making
Forecasting And Decision MakingVikash Rathour
 
Chapter 06
Chapter 06Chapter 06
Chapter 06bmcfad01
 
Probability distribution 2
Probability distribution 2Probability distribution 2
Probability distribution 2Nilanjan Bhaumik
 
Chapter 12
Chapter 12Chapter 12
Chapter 12bmcfad01
 
Chapter 07
Chapter 07Chapter 07
Chapter 07bmcfad01
 
PROJECT REPORT 2(2014)TATA PROJECT
PROJECT REPORT 2(2014)TATA PROJECTPROJECT REPORT 2(2014)TATA PROJECT
PROJECT REPORT 2(2014)TATA PROJECTNilanjan Bhaumik
 
Discrete Probability Distributions
Discrete Probability DistributionsDiscrete Probability Distributions
Discrete Probability Distributionsmandalina landy
 
7. binomial distribution
7. binomial distribution7. binomial distribution
7. binomial distributionKaran Kukreja
 
Forecasting Techniques
Forecasting TechniquesForecasting Techniques
Forecasting Techniquesguest865c0e0c
 

Destacado (16)

Chap04 basic probability
Chap04 basic probabilityChap04 basic probability
Chap04 basic probability
 
securitization+musyarakah+murabahah+and+ijarah
securitization+musyarakah+murabahah+and+ijarahsecuritization+musyarakah+murabahah+and+ijarah
securitization+musyarakah+murabahah+and+ijarah
 
Heizer mod d
Heizer mod dHeizer mod d
Heizer mod d
 
Chapter 05
Chapter 05Chapter 05
Chapter 05
 
Bba 3274 qm week 4 decision analysis
Bba 3274 qm week 4 decision analysisBba 3274 qm week 4 decision analysis
Bba 3274 qm week 4 decision analysis
 
Poisson distribution
Poisson distributionPoisson distribution
Poisson distribution
 
Forecasting And Decision Making
Forecasting And Decision MakingForecasting And Decision Making
Forecasting And Decision Making
 
Chapter 06
Chapter 06Chapter 06
Chapter 06
 
Probability distribution 2
Probability distribution 2Probability distribution 2
Probability distribution 2
 
Chapter 12
Chapter 12Chapter 12
Chapter 12
 
Measures of dispersion
Measures of dispersionMeasures of dispersion
Measures of dispersion
 
Chapter 07
Chapter 07Chapter 07
Chapter 07
 
PROJECT REPORT 2(2014)TATA PROJECT
PROJECT REPORT 2(2014)TATA PROJECTPROJECT REPORT 2(2014)TATA PROJECT
PROJECT REPORT 2(2014)TATA PROJECT
 
Discrete Probability Distributions
Discrete Probability DistributionsDiscrete Probability Distributions
Discrete Probability Distributions
 
7. binomial distribution
7. binomial distribution7. binomial distribution
7. binomial distribution
 
Forecasting Techniques
Forecasting TechniquesForecasting Techniques
Forecasting Techniques
 

Similar a FEC 512.03

Chap04 discrete random variables and probability distribution
Chap04 discrete random variables and probability distributionChap04 discrete random variables and probability distribution
Chap04 discrete random variables and probability distributionJudianto Nugroho
 
Discrete probability distributions
Discrete probability distributionsDiscrete probability distributions
Discrete probability distributionsCikgu Marzuqi
 
Basics of probability in statistical simulation and stochastic programming
Basics of probability in statistical simulation and stochastic programmingBasics of probability in statistical simulation and stochastic programming
Basics of probability in statistical simulation and stochastic programmingSSA KPI
 
PROBABILITY_DISTRIBUTION.pptx
PROBABILITY_DISTRIBUTION.pptxPROBABILITY_DISTRIBUTION.pptx
PROBABILITY_DISTRIBUTION.pptxshrutisingh143670
 
Testing for mixtures by seeking components
Testing for mixtures by seeking componentsTesting for mixtures by seeking components
Testing for mixtures by seeking componentsChristian Robert
 
從 VAE 走向深度學習新理論
從 VAE 走向深度學習新理論從 VAE 走向深度學習新理論
從 VAE 走向深度學習新理論岳華 杜
 
vdocuments.mx_chapter-5-probability-distributions-56a36d9fddc1e.ppt
vdocuments.mx_chapter-5-probability-distributions-56a36d9fddc1e.pptvdocuments.mx_chapter-5-probability-distributions-56a36d9fddc1e.ppt
vdocuments.mx_chapter-5-probability-distributions-56a36d9fddc1e.pptCharlesElquimeGalapo
 
Dissertation Defense: The Physics of DNA, RNA, and RNA-like polymers
Dissertation Defense: The Physics of DNA, RNA, and RNA-like polymersDissertation Defense: The Physics of DNA, RNA, and RNA-like polymers
Dissertation Defense: The Physics of DNA, RNA, and RNA-like polymersLi Tai Fang
 
random variable and distribution
random variable and distributionrandom variable and distribution
random variable and distributionlovemucheca
 
group4-randomvariableanddistribution-151014015655-lva1-app6891 (1).pdf
group4-randomvariableanddistribution-151014015655-lva1-app6891 (1).pdfgroup4-randomvariableanddistribution-151014015655-lva1-app6891 (1).pdf
group4-randomvariableanddistribution-151014015655-lva1-app6891 (1).pdfPedhaBabu
 
group4-randomvariableanddistribution-151014015655-lva1-app6891.pdf
group4-randomvariableanddistribution-151014015655-lva1-app6891.pdfgroup4-randomvariableanddistribution-151014015655-lva1-app6891.pdf
group4-randomvariableanddistribution-151014015655-lva1-app6891.pdfAliceRivera13
 
Big Data Analysis
Big Data AnalysisBig Data Analysis
Big Data AnalysisNBER
 

Similar a FEC 512.03 (20)

Chap04 discrete random variables and probability distribution
Chap04 discrete random variables and probability distributionChap04 discrete random variables and probability distribution
Chap04 discrete random variables and probability distribution
 
Binomial probability distributions
Binomial probability distributions  Binomial probability distributions
Binomial probability distributions
 
Discrete probability distributions
Discrete probability distributionsDiscrete probability distributions
Discrete probability distributions
 
Pro dist
Pro distPro dist
Pro dist
 
Basics of probability in statistical simulation and stochastic programming
Basics of probability in statistical simulation and stochastic programmingBasics of probability in statistical simulation and stochastic programming
Basics of probability in statistical simulation and stochastic programming
 
Random Variables
Random VariablesRandom Variables
Random Variables
 
PROBABILITY_DISTRIBUTION.pptx
PROBABILITY_DISTRIBUTION.pptxPROBABILITY_DISTRIBUTION.pptx
PROBABILITY_DISTRIBUTION.pptx
 
Binomial lecture
Binomial lectureBinomial lecture
Binomial lecture
 
lecture4.pdf
lecture4.pdflecture4.pdf
lecture4.pdf
 
Testing for mixtures by seeking components
Testing for mixtures by seeking componentsTesting for mixtures by seeking components
Testing for mixtures by seeking components
 
1630 the binomial distribution
1630 the binomial distribution1630 the binomial distribution
1630 the binomial distribution
 
Binomial distribution
Binomial distributionBinomial distribution
Binomial distribution
 
從 VAE 走向深度學習新理論
從 VAE 走向深度學習新理論從 VAE 走向深度學習新理論
從 VAE 走向深度學習新理論
 
vdocuments.mx_chapter-5-probability-distributions-56a36d9fddc1e.ppt
vdocuments.mx_chapter-5-probability-distributions-56a36d9fddc1e.pptvdocuments.mx_chapter-5-probability-distributions-56a36d9fddc1e.ppt
vdocuments.mx_chapter-5-probability-distributions-56a36d9fddc1e.ppt
 
Statistical Distributions
Statistical DistributionsStatistical Distributions
Statistical Distributions
 
Dissertation Defense: The Physics of DNA, RNA, and RNA-like polymers
Dissertation Defense: The Physics of DNA, RNA, and RNA-like polymersDissertation Defense: The Physics of DNA, RNA, and RNA-like polymers
Dissertation Defense: The Physics of DNA, RNA, and RNA-like polymers
 
random variable and distribution
random variable and distributionrandom variable and distribution
random variable and distribution
 
group4-randomvariableanddistribution-151014015655-lva1-app6891 (1).pdf
group4-randomvariableanddistribution-151014015655-lva1-app6891 (1).pdfgroup4-randomvariableanddistribution-151014015655-lva1-app6891 (1).pdf
group4-randomvariableanddistribution-151014015655-lva1-app6891 (1).pdf
 
group4-randomvariableanddistribution-151014015655-lva1-app6891.pdf
group4-randomvariableanddistribution-151014015655-lva1-app6891.pdfgroup4-randomvariableanddistribution-151014015655-lva1-app6891.pdf
group4-randomvariableanddistribution-151014015655-lva1-app6891.pdf
 
Big Data Analysis
Big Data AnalysisBig Data Analysis
Big Data Analysis
 

Más de Orhan Erdem

Más de Orhan Erdem (8)

2009 Financial Crisis
2009 Financial Crisis2009 Financial Crisis
2009 Financial Crisis
 
VOB
VOBVOB
VOB
 
FEC 512.07
FEC 512.07FEC 512.07
FEC 512.07
 
C.R.
C.R.C.R.
C.R.
 
FEC Seminar: C.R.
FEC Seminar: C.R.FEC Seminar: C.R.
FEC Seminar: C.R.
 
FEC 512.05
FEC 512.05FEC 512.05
FEC 512.05
 
FEC 512.04
FEC 512.04FEC 512.04
FEC 512.04
 
FEC 512.01
FEC 512.01FEC 512.01
FEC 512.01
 

Último

IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdf
IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdfIaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdf
IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdfDaniel Santiago Silva Capera
 
Linked Data in Production: Moving Beyond Ontologies
Linked Data in Production: Moving Beyond OntologiesLinked Data in Production: Moving Beyond Ontologies
Linked Data in Production: Moving Beyond OntologiesDavid Newbury
 
20230202 - Introduction to tis-py
20230202 - Introduction to tis-py20230202 - Introduction to tis-py
20230202 - Introduction to tis-pyJamie (Taka) Wang
 
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve DecarbonizationUsing IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve DecarbonizationIES VE
 
UiPath Solutions Management Preview - Northern CA Chapter - March 22.pdf
UiPath Solutions Management Preview - Northern CA Chapter - March 22.pdfUiPath Solutions Management Preview - Northern CA Chapter - March 22.pdf
UiPath Solutions Management Preview - Northern CA Chapter - March 22.pdfDianaGray10
 
UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6DianaGray10
 
The Data Metaverse: Unpacking the Roles, Use Cases, and Tech Trends in Data a...
The Data Metaverse: Unpacking the Roles, Use Cases, and Tech Trends in Data a...The Data Metaverse: Unpacking the Roles, Use Cases, and Tech Trends in Data a...
The Data Metaverse: Unpacking the Roles, Use Cases, and Tech Trends in Data a...Aggregage
 
UiPath Studio Web workshop series - Day 8
UiPath Studio Web workshop series - Day 8UiPath Studio Web workshop series - Day 8
UiPath Studio Web workshop series - Day 8DianaGray10
 
Designing A Time bound resource download URL
Designing A Time bound resource download URLDesigning A Time bound resource download URL
Designing A Time bound resource download URLRuncy Oommen
 
Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024SkyPlanner
 
UiPath Platform: The Backend Engine Powering Your Automation - Session 1
UiPath Platform: The Backend Engine Powering Your Automation - Session 1UiPath Platform: The Backend Engine Powering Your Automation - Session 1
UiPath Platform: The Backend Engine Powering Your Automation - Session 1DianaGray10
 
AI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just MinutesAI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just MinutesMd Hossain Ali
 
UiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation DevelopersUiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation DevelopersUiPathCommunity
 
Nanopower In Semiconductor Industry.pdf
Nanopower  In Semiconductor Industry.pdfNanopower  In Semiconductor Industry.pdf
Nanopower In Semiconductor Industry.pdfPedro Manuel
 
Igniting Next Level Productivity with AI-Infused Data Integration Workflows
Igniting Next Level Productivity with AI-Infused Data Integration WorkflowsIgniting Next Level Productivity with AI-Infused Data Integration Workflows
Igniting Next Level Productivity with AI-Infused Data Integration WorkflowsSafe Software
 
Building AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptxBuilding AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptxUdaiappa Ramachandran
 
Introduction to Matsuo Laboratory (ENG).pptx
Introduction to Matsuo Laboratory (ENG).pptxIntroduction to Matsuo Laboratory (ENG).pptx
Introduction to Matsuo Laboratory (ENG).pptxMatsuo Lab
 
COMPUTER 10: Lesson 7 - File Storage and Online Collaboration
COMPUTER 10: Lesson 7 - File Storage and Online CollaborationCOMPUTER 10: Lesson 7 - File Storage and Online Collaboration
COMPUTER 10: Lesson 7 - File Storage and Online Collaborationbruanjhuli
 
NIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopNIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopBachir Benyammi
 

Último (20)

IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdf
IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdfIaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdf
IaC & GitOps in a Nutshell - a FridayInANuthshell Episode.pdf
 
Linked Data in Production: Moving Beyond Ontologies
Linked Data in Production: Moving Beyond OntologiesLinked Data in Production: Moving Beyond Ontologies
Linked Data in Production: Moving Beyond Ontologies
 
20230202 - Introduction to tis-py
20230202 - Introduction to tis-py20230202 - Introduction to tis-py
20230202 - Introduction to tis-py
 
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve DecarbonizationUsing IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
Using IESVE for Loads, Sizing and Heat Pump Modeling to Achieve Decarbonization
 
UiPath Solutions Management Preview - Northern CA Chapter - March 22.pdf
UiPath Solutions Management Preview - Northern CA Chapter - March 22.pdfUiPath Solutions Management Preview - Northern CA Chapter - March 22.pdf
UiPath Solutions Management Preview - Northern CA Chapter - March 22.pdf
 
UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6
 
The Data Metaverse: Unpacking the Roles, Use Cases, and Tech Trends in Data a...
The Data Metaverse: Unpacking the Roles, Use Cases, and Tech Trends in Data a...The Data Metaverse: Unpacking the Roles, Use Cases, and Tech Trends in Data a...
The Data Metaverse: Unpacking the Roles, Use Cases, and Tech Trends in Data a...
 
UiPath Studio Web workshop series - Day 8
UiPath Studio Web workshop series - Day 8UiPath Studio Web workshop series - Day 8
UiPath Studio Web workshop series - Day 8
 
Designing A Time bound resource download URL
Designing A Time bound resource download URLDesigning A Time bound resource download URL
Designing A Time bound resource download URL
 
Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024
 
UiPath Platform: The Backend Engine Powering Your Automation - Session 1
UiPath Platform: The Backend Engine Powering Your Automation - Session 1UiPath Platform: The Backend Engine Powering Your Automation - Session 1
UiPath Platform: The Backend Engine Powering Your Automation - Session 1
 
AI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just MinutesAI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just Minutes
 
UiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation DevelopersUiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation Developers
 
Nanopower In Semiconductor Industry.pdf
Nanopower  In Semiconductor Industry.pdfNanopower  In Semiconductor Industry.pdf
Nanopower In Semiconductor Industry.pdf
 
20230104 - machine vision
20230104 - machine vision20230104 - machine vision
20230104 - machine vision
 
Igniting Next Level Productivity with AI-Infused Data Integration Workflows
Igniting Next Level Productivity with AI-Infused Data Integration WorkflowsIgniting Next Level Productivity with AI-Infused Data Integration Workflows
Igniting Next Level Productivity with AI-Infused Data Integration Workflows
 
Building AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptxBuilding AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptx
 
Introduction to Matsuo Laboratory (ENG).pptx
Introduction to Matsuo Laboratory (ENG).pptxIntroduction to Matsuo Laboratory (ENG).pptx
Introduction to Matsuo Laboratory (ENG).pptx
 
COMPUTER 10: Lesson 7 - File Storage and Online Collaboration
COMPUTER 10: Lesson 7 - File Storage and Online CollaborationCOMPUTER 10: Lesson 7 - File Storage and Online Collaboration
COMPUTER 10: Lesson 7 - File Storage and Online Collaboration
 
NIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopNIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 Workshop
 

FEC 512.03

  • 1. Probability Distributions Istanbul Bilgi University FEC 512 Financial Econometrics-I Asst. Prof. Dr. Orhan Erdem
  • 2. Some Common Probability Distributions Probability Distributions Discrete Continuous Probability Probability Distributions Distributions Normal Binomial Uniform Poisson Lognormal Lecture 3-2 FEC 512 Probability Distributions
  • 3. Discrete Probability Distribution Example random variable X = total number of Experiment: toss 2 coins, tails in two tosses. Probability distribution X Probability T 0 0.25 1 0.50 T 2 0.25 T T Lecture 3-3 FEC 512 Probability Distributions
  • 4. The Binomial Distribution Probability Distributions Discrete Probability Distributions Binomial Poisson Lecture 3-4 FEC 512 Probability Distributions
  • 5. The Binomial Distribution Characteristics of the Binomial Distribution: A trial has only two possible outcomes – “success” or “failure” There is a fixed number, n, of identical trials The trials of the experiment are independent of each other The probability of a success, p, remains constant from trial to trial If p represents the probability of a success, then (1-p) = q is the probability of a failure Lecture 3-5 FEC 512 Probability Distributions
  • 6. Binomial Distribution Settings A manufacturing plant labels items as either defective or acceptable A firm bidding for a contract will either get the contract or not A marketing research firm receives survey responses of “yes I will buy” or “no I will not” New job applicants either accept the offer or reject it Lecture 3-6 FEC 512 Probability Distributions
  • 7. Counting Rule for Combinations A combination is an outcome of an experiment where x objects are selected from a group of n objects n! C= n x x! (n − x )! where: n! =n(n - 1)(n - 2) . . . (2)(1) x! = x(x - 1)(x - 2) . . . (2)(1) 0! = 1 (by definition) Lecture 3-7 FEC 512 Probability Distributions
  • 8. Binomial Distribution Formula n! x n−x P(x) = pq x ! (n − x )! P(x) = probability of x successes in n trials, with probability of success p on each trial Example: Flip a coin four times, let x = # heads: x = number of ‘successes’ in sample, n=4 (x = 0, 1, 2, ..., n) p = 0.5 p = probability of “success” per trial q = (1 - .5) = .5 q = probability of “failure” = (1 – p) n = number of trials (sample size) x = 0, 1, 2, 3, 4 Lecture 3-8 FEC 512 Probability Distributions
  • 9. Binomial Distribution The shape of the binomial distribution depends on the values of p and n P(X) n = 5 p = 0.1 Mean .6 .4 .2 Here, n = 5 and p = .1 0 X 0 1 2 3 4 5 n = 5 p = 0.5 P(X) .6 .4 Here, n = 5 and p = .5 .2 X 0 0 1 2 3 4 5 Lecture 3-9 FEC 512 Probability Distributions
  • 10. Binomial Distribution Characteristics Mean µ = E(x) = np Variance and Standard Deviation σ = npq 2 σ = npq Where n = sample size p = probability of success q = (1 – p) = probability of failure Lecture 3-10 FEC 512 Probability Distributions
  • 11. Binomial Characteristics Examples Mean = (5)(.1) = 0.5 µ = np n = 5 p = 0.1 P(X) .6 .4 σ = npq = (5)(.1)(1 − .1) .2 = 0.6708 0 X 0 1 2 3 4 5 µ = np = (5)(.5) = 2.5 n = 5 p = 0.5 P(X) .6 .4 σ = npq = (5)(.5)(1 − .5) .2 = 1.118 X 0 0 1 2 3 4 5 Lecture 3-11 FEC 512 Probability Distributions
  • 12. A binomial tree of asset prices “Example” We wish to know the value of an asset after two time periods. Each of the time periods the asset may rise (a success) with a probability of 0.5 or it may fall (a failure) with a probability of 0.5. Assume asset price movement in one time period is independent of that in the other time period. Su2 Su (60.50) (55) Sud=Sdu (49.5) S=50 Sd Sd2 (45) (40.50) T2 T0 T1 Lecture 3-12 FEC 512 Probability Distributions
  • 13. A binomial tree of asset prices “Example” If the asset had previously risen by a factor u, it would either rise again by u to Su2 or would fall by d to Sud. If the asset had previoulsy fallen to Sd, it could rise by u to Sud or fall further to Sd2 Suppose that u=1.1, d=0.9, and S=50 Hence the expected value can be calculated as: µ = (60.50 * 0.25) + (49.50 * 0.50) + (40.50 * 0.25) = 50 The variance is σ2 = (60.5 – 50)2 * 0.25 + (49.50 – 50)2 * 0.50 + (40.5 – 50)2 * 0.25 = 50.25 Lecture 3-13 FEC 512 Probability Distributions
  • 14. The Poisson Distribution Probability Distributions Discrete Probability Distributions Binomial Poisson Lecture 3-14 FEC 512 Probability Distributions
  • 15. The Poisson Distribution To use binomial distribution, we must be able to count the # successes and failures. Although in many situations you may be able to count # successes, you often cannot count # failures. Example: An emergency call center could easily count the # calls its unit respond to in 1 hour, but how could it determine how many calls it didnt receive? Lecture 3-15 FEC 512 Probability Distributions
  • 16. Characteristics of the Poisson Distribution The outcomes of interest are rare relative to the possible outcomes The average number of outcomes of interest per time or space interval is λ The number of outcomes of interest are random, and the occurrence of one outcome does not influence the chances of another outcome of interest The probability of that an outcome of interest occurs in a given segment is the same for all segments Lecture 3-16 FEC 512 Probability Distributions
  • 17. Poisson Distribution Formula − λt ( λt ) e x P( x ) = x! where: t = size of the segment of interest x = number of successes in segment of interest λ = expected number of successes in a segment of unit size e = base of the natural logarithm system (2.71828...) Lecture 3-17 FEC 512 Probability Distributions
  • 18. Poisson Distribution (continued) Ex. Page requests arrive at a 0,18 0,17 webserver at an average rate 0,16 of 5 every second. If the 0,15 number of requests in a 0,14 0,13 second has a Poisson 0,12 distribution, find the probability 0,11 0,10 that 15 requests will be made p(X=x) 0,09 0,08 in a given second. 0,07 e −5 515 0,06 P ( X = 15 ) = = 0.00016 0,05 0,04 15! 0,03 0,02 0,01 0,00 Here is what the distribution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 function for the above example x=number of page requests in a second looks like Lecture 3-18 FEC 512 Probability Distributions
  • 19. Poisson Distribution Characteristics Mean µ = λt Variance and Standard Deviation σ = λt 2 σ = λt λ = number of successes in a segment of unit size where t = the size of the segment of interest Lecture 3-19 FEC 512 Probability Distributions
  • 20. Graph of Poisson Probabilities 0.70 Graphically: 0.60 λ = .05 and t = 100 0.50 λt = 0.40 P(x) X 0.50 0.30 0 0.6065 0.20 1 0.3033 0.10 2 0.0758 0.00 3 0.0126 0 1 2 3 4 5 6 7 4 0.0016 x 5 0.0002 P(x = 2) = .0758 6 0.0000 7 0.0000 Lecture 3-20 FEC 512 Probability Distributions
  • 21. Poisson Distribution Shape The shape of the Poisson Distribution depends on the parameters λ and t: λt = 0.50 λt = 3.0 0.70 0.25 0.60 0.20 0.50 0.15 0.40 P(x) P(x) 0.30 0.10 0.20 0.05 0.10 0.00 0.00 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 x x Lecture 3-21 FEC 512 Probability Distributions
  • 22. The Normal Distribution Probability Distributions Continuous Probability Distributions Normal Uniform Lognormal Lecture 3-22 FEC 512 Probability Distributions
  • 23. The Normal Distribution The distribution whose pdf is given by  ( x−µ )2  −  f(x) 1   2σ 2 f ( x) =   e 2π σ σ ‘Location is determined by x the mean, µ µ Spread is determined by the standard deviation, σ Lecture 3-23 FEC 512 Probability Distributions
  • 24. The Normal Distribution f(x) ‘Bell Shaped’ Symmetrical σ The random variable has x an infinite theoretical µ range: + ∞ to − ∞ Lecture 3-24 FEC 512 Probability Distributions
  • 25. Many Normal Distributions By varying the parameters µ and σ, we obtain different normal distributions Lecture 3-25 FEC 512 Probability Distributions
  • 26. The Normal Distribution Shape f(x) Changing µ shifts the distribution left or right. Changing σ increases or decreases the σ spread. µ x Lecture 3-26 FEC 512 Probability Distributions
  • 27. Finding Normal Probabilities Probability is the Probability is measured by the area area under the curve! under the curve f(x) P (a ≤ x ≤ b) a b x Lecture 3-27 FEC 512 Probability Distributions
  • 28. Probability as Area Under the Curve The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below f(x) P( −∞ < x < µ) = 0.5 P(µ < x < ∞ ) = 0.5 0.5 0.5 x µ P(−∞ < x < ∞) = 1.0 Lecture 3-28 FEC 512 Probability Distributions
  • 29. Empirical Rules What can we say about the distribution of values around the mean? There are some general rules: f(x) µ ± 1σ encloses about 68% of x’s σ σ x µ−1σ µ µ+1 − +1σ +1 68.26% Lecture 3-29 FEC 512 Probability Distributions
  • 30. The Empirical Rule (continued) µ ± 2σ covers about 95% of x’s µ ± 3σ covers about 99.7% of x’s 3σ 3σ 2σ 2σ µ x µ x 95.44% 99.72% Lecture 3-30 FEC 512 Probability Distributions
  • 31. The Standard Normal Distribution Also known as the “z” distribution Mean is defined to be 0 Standard Deviation is 1 f(z) 1 z 0 Values above the mean have positive z-values, values below the mean have negative z-values Lecture 3-31 FEC 512 Probability Distributions
  • 32. Transformation to the Standard Normal Distribution Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normal distribution (z) x −µ z= σ Lecture 3-32 FEC 512 Probability Distributions
  • 33. Example If x is distributed normally with mean of 100 and standard deviation of 50, the z value for x = 200 is x − µ 200 − 100 z= = = 2.0 σ 50 This says that x = 200 is two standard deviations (2 increments of 50 units) above the mean of 100. Lecture 3-33 FEC 512 Probability Distributions
  • 34. Comparing x and z units µ = 100 σ = 50 100 200 x 0 2.0 z Note that the distribution is the same, only the scale has changed. We can express the problem in original units (x) or in standardized units (z) Lecture 3-34 FEC 512 Probability Distributions
  • 35. The Standard Normal Table The Standard Normal table in the textbooks gives the probability from the mean (zero) up to a desired value for z .4772 Example: P(0 < z < 2.00) = .4772 z 0 2.00 Lecture 3-35 FEC 512 Probability Distributions
  • 36. The Standard Normal Table (continued) The column gives the value of z to the second decimal point z 0.00 0.01 0.02 … 0.1 The row shows the 0.2 . . value of z to The value within the . the first table gives the .4772 2.0 decimal point probability from z = 0 up to the desired z 2.0 P(0 < z < 2.00) = .4772 value Lecture 3-36 FEC 512 Probability Distributions
  • 37. Z Table example Suppose x is normal with mean 8.0 and standard deviation 5.0. Find P(8 < x < 8.6) Calculate z-values: x −µ 8 −8 z= = =0 σ 5 8 8.6 x x − µ 8.6 − 8 0 0.12 Z z= = = 0.12 σ 5 P(8 < x < 8.6) = P(0 < z < 0.12) Lecture 3-37 FEC 512 Probability Distributions
  • 38. Z Table example (continued) Suppose x is normal with mean 8.0 and standard deviation 5.0. Find P(8 < x < 8.6) µ=8 µ=0 σ=5 σ=1 x z 8 8.6 0 0.12 P(8 < x < 8.6) P(0 < z < 0.12) Lecture 3-38 FEC 512 Probability Distributions
  • 39. Solution: Finding P(0 < z < 0.12) Standard Normal Probability P(8 < x < 8.6) Table (Portion) = P(0 < z < 0.12) z .00 .01 .02 .0478 0.0 .0000 .0040 .0080 0.1 .0398 .0438 .0478 0.2 .0793 .0832 .0871 Z 0.00 0.3 .1179 .1217 .1255 0.12 Lecture 3-39 FEC 512 Probability Distributions
  • 40. Lower Tail Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(7.4 < x < 8) Z 8.0 7.4 Lecture 3-40 FEC 512 Probability Distributions
  • 41. Lower Tail Probabilities (continued) Now Find P(7.4 < x < 8)… The Normal distribution is symmetric, so we use the .0478 same table even if z-values are negative: P(7.4 < x < 8) = P(-0.12 < z < 0) Z = .0478 8.0 7.4 Lecture 3-41 FEC 512 Probability Distributions
  • 42. Distributions of Portfolio Returns Example. Assume that the stock index in a country has an annual return distribution that is normal with µ = 0.15 and σ = 0.30. What is probability that in a given year the stock index will exceed an annual return of 100%? What is the probability that the index will produce a negative return in a given year? We first need to transform the normal variable into a standard normal. X − 0.15 1.00 − 0.15 z= = = 2.83 0.30 0.30 Looking up 2.83 in the normal table, we find that F(z) is 0.9977. So 1 – F(z) = 0.0023. For finding the probability of a negative return, the transformation yields X − 0.15 0 − 0.15 z= = = −0.50 0.30 0.30 This time we are interested in F(z), which is 0.3085. Lecture 3-42 FEC 512 Probability Distributions
  • 43. Linear Combinations of Two Normal Random Variables Let X~N(µX,σX2 ) and Y~N(µY,σY2 ) and σXY=cov(X,Y). If Z=aX+bY where a,b are constants, then Z~N(µZ,σZ2 ) where µZ=a µX +b µY σZ2=a2 σX2 +b2 σY2 +2abσXY=a2 σX2 +b2 σY2 +2abσX σYρ Lecture 3-43 FEC 512 Probability Distributions
  • 44. Kurtosis and Skewness of Normal Distribution The skewness of a normal distribution is 0. Why? The kurtosis of a normal distribution is 3. Hence 3 is a benchmark value for tail thickness of a bell- shaped distribution. If kurt(X)>3, the dist. has thicker tails than norm. dist. If kurt(X)<3, the dist. has thinner tails than norm. dist. Lecture 3-44 FEC 512 Probability Distributions
  • 45. The Uniform Distribution Probability Distributions Continuous Probability Distributions Normal Uniform Lognormal Lecture 3-45 FEC 512 Probability Distributions
  • 46. The Uniform Distribution The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable Lecture 3-46 FEC 512 Probability Distributions
  • 47. The Uniform Distribution (continued) The Continuous Uniform Distribution: 1 if a ≤ x ≤ b b−a f(x) = 0 otherwise where f(x) = value of the density function at any x value a = lower limit of the interval b = upper limit of the interval Lecture 3-47 FEC 512 Probability Distributions
  • 48. Uniform Distribution Example: Uniform Probability Distribution Over the range 2 ≤ x ≤ 6: 1 f(x) = 6 - 2 = .25 for 2 ≤ x ≤ 6 f(x) .25 x 2 6 Lecture 3-48 FEC 512 Probability Distributions
  • 49. The Lognormal Distribution Probability Distributions Continuous Probability Distributions Normal Uniform Lognormal Lecture 3-49 FEC 512 Probability Distributions
  • 50. The Lognormal Distribution Let Z ~N(µ,σ2), and X=eZ r.v. X is said to be log- normally distributed with parameters µ and σ2 lnX~N(µ,σ2) or In other words, X is lognormal if its “ln” is normally distributed. If X,Y is lognormally distributed, their linear combination(i.e. Portfolio of two stocks) may not be lognormal. Lecture 3-50 FEC 512 Probability Distributions
  • 51. Lecture 3-51 FEC 512 Probability Distributions
  • 52. The median of X is eµ, and the expected value of X σ2 µ+ is . The expectation is larger than the 2 e median because the lognormal distribution is right- skewed, and the skew. is more extreme with larger values of σ. Lecture 3-52 FEC 512 Probability Distributions
  • 53. Example Let rt = ln( Pt / Pt −1 ) denote the log-return on an asset and assume that rt ~ N(µ,σ2). Let R = ( P P P ) − t −1 t t t −1 denote the simple monthly return, since we know e rt = 1 + Rt . Since rt is rt = ln(1 + Rt ) that and normally distributed e t = 1 + Rt is log-normally r ............. distributed. Lecture 3-53 FEC 512 Probability Distributions
  • 54. Sample Moments Above we introduced the four statistical moments mean,variance, skewness, kurtosis. Given a pdf, we are able to calculate these stat. Moments according to the fomulae. In practical applications however, we are faced with the situation that we observe realizations of a pdf(e.g. The daily return of the IMKB-100 index over the last year), but we do not know the distribution that generates these returns. So taking expectation is impossible But having the observations x1,…,xn, we can try to estimate the “true moments” out of the sample. These estimates are called sample moments. Lecture 3-54 FEC 512 Probability Distributions
  • 55. Mean (Arithmetic Average) The Mean is the arithmetic average of data values Sample mean n = Sample Size n ∑x x1 + x 2 + L + x n i x= = i =1 n n Lecture 3-55 FEC 512 Probability Distributions
  • 56. Variation Measures of variation give information on the spread or variability of the data values. Same center, different variation Lecture 3-56 FEC 512 Probability Distributions
  • 57. Variance Average of squared deviations of values from the mean Sample variance: n ∑ (x − x) 2 i s2 = i =1 n -1 Sample standard deviation: n ∑ (x − x) 2 i s= i=1 n -1 Lecture 3-57 FEC 512 Probability Distributions
  • 58. Calculation Example: Sample Standard Deviation Sample Data (Xi) : 10 12 14 15 17 18 18 24 n=8 Mean = x = 16 (10 − x ) 2 + (12 − x ) 2 + (14 − x ) 2 + L + (24 − x ) 2 s= n −1 (10 − 16) + (12 − 16) + (14 − 16) + L + (24 − 16) 2 2 2 2 = 8 −1 126 = = 4.2426 7 Lecture 3-58 FEC 512 Probability Distributions
  • 59. Comparing Standard Deviations Data A Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21 Data B Mean = 15.5 s = .9258 11 12 13 14 15 16 17 18 19 20 21 Data C Mean = 15.5 s = 4.57 11 12 13 14 15 16 17 18 19 20 21 Lecture 3-59 FEC 512 Probability Distributions
  • 60. Coefficient of Variation Measures relative variation Always in percentage (%) Shows variation relative to mean Is used to compare two or more sets of data measured in different units Sample C.V. s CV =   ⋅ 100% x   Lecture 3-60 FEC 512 Probability Distributions
  • 61. Comparing Coefficient of Variation Stock A: Average price last year = $50 Standard deviation = $5 s $5 CVA =   ⋅ 100% = ⋅ 100% = 10% x $50  Both stocks have the same Stock B: standard deviation, but Average price last year = $100 stock B is less variable relative Standard deviation = $5 to its price s $5 CVB =   ⋅ 100% = ⋅ 100% = 5% x $100  Lecture 3-61 FEC 512 Probability Distributions
  • 62. Sharpe Ratio Let Dt=Rt-Rf where Rf is the riskfree rate of return. T T ∑ 1 D = ∑ Dt (Dt − D) 2 T t =1 σD = t =1 T −1 Sharpe Ratio is (D) S= σD Reading: “The Sharpe Ratio”, William F. Sharpe Lecture 3-62 FEC 512 Probability Distributions
  • 63. Skewness The moment coefficient of skewness is derived by calculating the third moment about the mean and dividing by the cube of standard deviation : ( )   ∑ X −X 3     n −1   3  2 ∑( ) X −X     n −1     Lecture 3-63 FEC 512 Probability Distributions
  • 64. Shape of a Distribution Describes how data is distributed Symmetric or skewed Right-Skewed Symmetric Left-Skewed Mean = Median Median < Mean Mean < Median (Longer tail extends to left) (Longer tail extends to right) Lecture 3-64 FEC 512 Probability Distributions
  • 65. Kurtosis Skewness indicates the degree of symmetry in the frequency distribution Kurtosis indicates the peakedness of that distribution Lecture 3-65 FEC 512 Probability Distributions
  • 66. Kurtosis (continued) ∑(X − X ) 4 n −1 4   ∑(X − X ) 2     n −1     Lecture 3-66 FEC 512 Probability Distributions
  • 67. About the Probability Distribution of Returns The assumption that period returns(e.g. Daily, monthly, annually) are normally distributed is inconsistent with the limited liability feature of most financial instruments, R≥-1. There are a number of empirical facts about return distributions While normal distribution is perfectly symmetric about its mean, 1. dailt stock returns are frequently skewed to the right. And few of them are skewed to the left. The sample daily return distributions for many individuals stocks 2. exhibit “excess kurtosis” or “fat tails”. i.e. There is more probability in the tails than would be justified by normal distribution. The extent of this excess kurtosis diminishes substantially, however , when monthly data is used. Lecture 3-67 FEC 512 Probability Distributions
  • 68. Emprical Return Distributions: IMKB-100 Daily 500 Series: XU100RETURNS Sample 1/04/1993 12/29/2004 400 Observations 2968 Mean 0.002654 300 Median 0.001996 Maximum 0.194510 Minimum -0.181093 200 Std. Dev. 0.031281 Skewness 0.139399 Kurtosis 6.290761 100 Jarque-Bera 1348.812 Probability 0.000000 0 -0.1 0.0 0.1 0.2 Lecture 3-68 FEC 512 Probability Distributions
  • 69. Emprical Return Distributions: IMKB-100 Monthly 24 Series: XU100RETURNS Sample 1993M01 2004M12 20 Observations 144 16 Mean 5.899510 Median 5.043156 Maximum 79.78386 12 Minimum -39.03413 Std. Dev. 17.21817 8 Skewness 0.903614 Kurtosis 5.585146 4 Jarque-Bera 59.69430 Probability 0.000000 0 -40 -20 0 20 40 60 80 Pay attention to Skewness and Kurtosis. Lecture 3-69 FEC 512 Probability Distributions
  • 70. Value At Risk Lecture 3-70 FEC 512 Probability Distributions