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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 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.
Construction of Probability
Densities from Historical Data
• Two Examples
– The daily return rates of U.S. S&P 500 stock
index
– The daily return rates of Taiwan company:
Acer 2353
Distribution of Return Rate for U.S. Market
0.35
Use 2-year data (near
500 daily return rates
data) to simulate the
underlying distribution of
return rates of our
portfolio
Assume the return
rate in the next
trading day is drawn
from the same
distribution
0.3
0.25
Rt, for t=1 to 500
Rt, for t=501,502, …
0.2
If we assume the return
rate follows the normal
distribution, then the
potential loss can be
presented by standard
error
0.15
Standard0.1
error, σ
Standard
error, σ
0.05
0
-4
-3
-2
-1
0
1
2
3
4
Distribution of Return Rate for U.S. Market
(1)If we assume the return rate
follows the normal distribution, then the potential
loss can be presented by standard error
0.35
(2) The P[ return rate<-2.33Xσ]=1%
The P[ return rate<-1.96Xσ]=2.5% 0.3
The P[ return rate<-1.645Xσ]=5%
0.25
(3) If we assume the initial investment amount is 100,000, the loss
of ”>100,000X 2.33Xσ” in the next day0.2
will have 1% probability of occurrences
0.15
Standard0.1
error, σ
Standard
error, σ
(0.94%)
0.05
0
-4
-3
-2
-1
0
1
2
3
4
DESCRIPTIVE STATISTICS: MEAN,
STANDARD DEVIATION, SKEW, AND
KURTOSIS
• Mean
• Standard Deviation
DESCRIPTIVE STATISTICS: MEAN,
STANDARD DEVIATION, SKEW, AND
KURTOSIS
• Skew
• Kurtosis
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
The Normal Distribution
• The equation for the Normal
distribution is as follows:
Comparison of Normal
Distribution with Actual Data
(a)PDF of Dow Jones Index Return Shock: Linear Model
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-5 -4.7 -4.4 -4.1 -3.8 -3.5 -3.2 -2.9 -2.6 -2.3
-2 -1.7 -1.4 -1.1 -0.8 -0.5 -0.2 0.12 0.42 0.72 1.02 1.32 1.62 1.92 2.22 2.52 2.82 3.12 3.42 3.72 4.02 4.32 4.62 4.92
Comparison of Normal
Distribution with Actual Data
Table 1. Skewness, Kurtosis, and 1%, 2.5%, 5% Critical Values for Returns
Shocks of Various Indices
Statistics Coefficients
Dow Jones
FCI
FTSE
Nikkei
Skewness Coefficients (N=0)
-2.26
2.20
-0.53
0.17
Kurtosis Coefficients (N=3)
58.21
157.14
22.92
18.03
1% Left-tailed Critical Value (N= -2.33)
-2.43
-2.46
-2.49
-2.78
2.5% Left-tailed Critical Value (N= -1.96)
-1.90
-1.69
-1.87
-2.10
5% Left-tailed Critical Value (N= -1.65)
-1.45
-1.26
-1.46
-1.55
1% Right-tailed Critical Value (N=2.33)
2.43
2.24
2.32
2.82
2.5% Right-tailed Critical Value (N=1.96)
1.92
1.48
1.75
1.97
5% Right-tailed Critical Value (N=1.65)
1.44
1.15
1.36
1.42
Number of Observations
4838
4758
3801
5045
The Solutions for Non-Normality
Historical simulation method
Student t setting
Stochastic volatility settings
Jump diffusion models
Extreme value theory (EVT)
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
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
The Log-Normal Distribution
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.
The Beta Distribution
• As with the Normal distribution, it only
requires two parameters (in this case
called α and β) to define the shape.
• α and β are functions of the desired mean
and standard deviation of the distribution;
they are calculated as follows:
 2 (1   )


2

 2 (1   ) 2

 (   1)
2

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.
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.
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:
CORRELATION AND
COVARIANCE
CORRELATION AND
COVARIANCE
• The covariance between the variables is
calculated by multiplying the variables
together at each observation:
CORRELATION AND
COVARIANCE
• The correlation is defined by normalizing
the covariance with respect to the
individual variances:
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
THE STATISTICS FOR A SUM OF
NUMBERS.
THE STATISTICS FOR A SUM OF
NUMBERS.
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:
THE STATISTICS FOR A SUM OF
NUMBERS.
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
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
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:
BASIC MATRIX OPERATIONS
BASIC MATRIX OPERATIONS
BASIC MATRIX OPERATIONS