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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. SP 500 stock

index - The daily return rates of Taiwan company Acer

2353

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

(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

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

Comparison of Normal Distribution with Actual

Data

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 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

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.

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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

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