# Review of Statistics - PowerPoint PPT Presentation

PPT – Review of Statistics PowerPoint presentation | free to download - id: 6a6fde-NTI0Z The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
Title:

## Review of Statistics

Description:

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

Number of Views:43
Avg rating:3.0/5.0
Slides: 37
Provided by: accNckuEd
Category:
Tags:
Transcript and Presenter's Notes

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

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
(No Transcript)
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
(No Transcript)