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Review of Statistics

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CHAPTER 3 Review of Statistics INTRODUCTION The creation of histograms and probability distributions from empirical data. The statistical parameters used to ... – PowerPoint PPT presentation

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Title: Review of Statistics


1
CHAPTER 3
  • Review of Statistics

2
INTRODUCTION
  • The creation of histograms and probability
    distributions from empirical data.
  • The statistical parameters used to describe
    the distribution of losses mean, standard
    deviation, skew, and kurtosis.
  • Examples of market-risk and credit-risk loss
    distributions to give an understanding of the
    practical problems that we face.
  • The idealized distributions that are used
    to describe risk the Normal and Beta
    probability distributions.

3
Construction of Probability Densities from
Historical Data
  • Two Examples
  • The daily return rates of U.S. SP 500 stock
    index
  • The daily return rates of Taiwan company Acer
    2353

4
Use 2-year data (near 500 daily return rates
data) to simulate the underlying distribution of
return rates of our portfolio Rt, for t1 to 500
Assume the return rate in the next trading day is
drawn from the same distribution Rt, for
t501,502,
If we assume the return rate follows the normal
distribution, then the potential loss can be
presented by standard error
Standard error, s
Standard error, s
5
(1)If we assume the return rate follows the
normal distribution, then the potential loss can
be presented by standard error (2) The P return
ratelt-2.33Xs1 The P return
ratelt-1.96Xs2.5 The P return
ratelt-1.645Xs5 (3) If we assume the initial
investment amount is 100,000, the loss of
gt100,000X 2.33Xs in the next day will have 1
probability of occurrences
Standard error, s (0.94)
Standard error, s
6
DESCRIPTIVE STATISTICS MEAN, STANDARD DEVIATION,
SKEW, AND KURTOSIS
  • Mean
  • Standard Deviation

7
DESCRIPTIVE STATISTICS MEAN, STANDARD DEVIATION,
SKEW, AND KURTOSIS
  • Skew
  • Kurtosis

8
The Normal Distribution
  • The Noemal distribution is also known as the
    Gaussian distribution or Bell curve.
  • It is the distribution most commonly used to
    describe the random changes in market-risk
    factors, such as exchange rates, interest rates,
    and equity prices.
  • This distribution is very common in nature
    because of the Central Limit Theorem, which
    states that if a large amount of independent,
    identically distributed, random numbers are
    added together, the outcome will tend to be
    Normally distributed

9
The Normal Distribution
  • The equation for the Normal distribution is as
    follows

10
Comparison of Normal Distribution with Actual
Data
11
Comparison of Normal Distribution with Actual
Data

12
The Solutions for Non-Normality
  • Historical simulation method
  • Student t setting
  • Stochastic volatility settings
  • Jump diffusion models
  • Extreme value theory (EVT)

13
The Log-Normal Distribution
  • The Log-normal distribution is useful for
    describing variables which cannot have a
    negative value, such as interest rates and
    stock prices.
  • If the variable has a Log-normal distribution,
    then the log of the variable will have a Normal
    distribution
  • If x Log-Normal
  • Then Log(x) Normal

14
The Log-Normal Distribution
  • Conversely, if you have a variable that is
    Normally distributed, and you want to produce a
    variable that has a Log-normal distribution,
    take the exponential of the Normal variable
  • If z Normal
  • Then ez Log-Normal

15
The Log-Normal Distribution
16
The Beta Distribution
  • The Beta distribution is useful in describing
    credit-risk losses, which are typically highly
    skewed.
  • The formula for the Beta distribution is quite
    complex however, it is available in most
    spreadsheet applications.

17
The Beta Distribution
  • As with the Normal distribution, it only requires
    two parameters (in this case called a and ß) to
    define the shape.
  • a and ß are functions of the desired mean and
    standard deviation of the distribution they are
    calculated as follows

18
CORRELATION AND COVARIANCE
  • So far, we have been discussing the statistics of
    isolated variables, such as the change in the
    equity prices.
  • We also need to describe the extent to which
    two variables move together, e g, the changes
    m equity prices and changes in interest rates.

19
CORRELATION AND COVARIANCE
  • If two random variables show a pattern of
    tending to increase at the same time, then they
    are said to have a positive correlation.
  • If one tends to decrease when the other
    increases, they have a negative correlation
  • If they are completely independent, and there
    is no relationship between the movement of x
    and y, they are said to have zero correlation.

20
CORRELATION AND COVARIANCE
  • The, quantification of correlation starts with
    covariance.
  • The covariance of two variables can be thought
    of as an extension from calculating the
    variance for a single variable.
  • Earlier, we defined the variance as follows

21
CORRELATION AND COVARIANCE
22
CORRELATION AND COVARIANCE
  • The covariance between the variables is
    calculated by multiplying the variables together
    at each observation

23
CORRELATION AND COVARIANCE
  • The correlation is defined by normalizing the
    covariance with respect to the individual
    variances

24
THE STATISTICS FOR A SUM OF NUMBERS.
  • In risk measurement, we are often interested In
    finding the statistics for a result which is the
    sum of many variables
  • For example, the loss on a portfolio is the sum
    of the losses on the individual instruments
  • Similarity, the trading loss over a year is the
    sum of the losses on the individual days
  • Let us consider an example in which y is the sum
    of two random numbers, x1 and x2

25
THE STATISTICS FOR A SUM OF NUMBERS.
26
THE STATISTICS FOR A SUM OF NUMBERS.
27
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28
THE STATISTICS FOR A SUM OF NUMBERS.
  • One particularly useful application of this
    equation is when the correlation between the
    variables is zero
  • This assumption is commonly made for day-to-day
    changes m market variables.
  • If we make this assumption then the variance of
    the loss over multiple days is simply the sum of
    the variances for each day

29
THE STATISTICS FOR A SUM OF NUMBERS.
30
BASIC MATRIX OPERATIONS
  • When there are many variables, the normal
    algebraic expressions become cumbersome.
  • An alternative way of writing these expressions
    is in matrix form.
  • Matrices are just representations of the
    parameter in an equation

31
BASIC MATRIX OPERATIONS
  • You may have used matrices m physics to represent
    distances m multiple dimensions, e g, m the x, y,
    and z coordinates.
  • In risk, matrices are commonly used to represent
    weights on different risk factors, such as
    interest rates, equities, FX, and commodity
    prices

32
BASIC MATRIX OPERATIONS
  • For example, we could say that the value of an
    equity portfolio was the sum of the number (n) of
    each equity multiplied by the value (v) of each

33
BASIC MATRIX OPERATIONS
34
BASIC MATRIX OPERATIONS
35
BASIC MATRIX OPERATIONS
36
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