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 i

ndex– The daily return rates of Taiwan company: Ac

er 2353

Distribution of Return Rate for U.S. Market

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-4 -3 -2 -1 0 1 2 3 4

Use 2-year data (near 500 daily return rates data) to simulate the underlying distribution of return rates of our portfolio

Rt, for t=1 to 500

Assume the return rate in the next trading day is drawn from the same distribution

Rt, for t=501,502, …

Standard error, σ

Standard error, σ

If we assume the return rate follows the normal distribution, then the potential loss can be presented by standard error

Distribution of Return Rate for U.S. Market

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-4 -3 -2 -1 0 1 2 3 4

Standard error, σ (0.94%)

Standard error, σ

(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 rate<-2.33Xσ]=1%

The P[ return rate<-1.96Xσ]=2.5%

The P[ return rate<-1.645Xσ]=5%

(3) If we assume the initial investment amount is 100,000, the loss of ”>100,000X 2.33Xσ” in the next day will have 1% probability of occurrences

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:

(a)PDF of Dow Jones Index Return Shock: Linear Model

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-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

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 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:

)1()1(

)1(

2

22

2

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