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Correlations and Copulas. Measures of Dependence. The risk can be split into two parts: the individual risks and the dependence structure between them. Measures of dependence include: Correlation Rank Correlation Coefficient Tail Dependence Association. Correlation and Covariance.

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measures of dependence
Measures of Dependence
  • The risk can be split into two parts:
    • the individual risks and
    • the dependence structure between them
  • Measures of dependence include:
    • Correlation
    • Rank Correlation
    • Coefficient Tail Dependence
    • Association
correlation and covariance
Correlation and Covariance
  • The coefficient of correlation between two variables X and Y is defined as
  • The covariance is

E(YX)−E(Y)E(X)

independence
Independence
  • X and Y are independent if the knowledge of one does not affect the probability distribution for the other

where denotes the probability density function

correlation pitfalls
Correlation Pitfalls

A correlation of 0 is not equivalent to independence

If (X, Y ) are jointly normal, Corr(X,Y ) = 0 implies independence of X and Y

In general this is not true: even perfectly related RVs can have zero correlation:

types of dependence
Types of Dependence

E(Y)

E(Y)

X

X

(a)

(b)

E(Y)

X

(c)

correlation pitfalls cont
Correlation Pitfalls (cont.)

Correlation is invariant under linear transformations, but not under general transformations:

Example, two log-normal RVs have a different correlation than the underlying normal RVs

A small correlation does not imply a small degree of dependency.

stylized facts of correlations
Stylized Facts of Correlations

Correlation clustering:

periods of high (low) correlation are likely to be followed by periods of high (low) correlation

Asymmetry and co-movement with volatility:

high volatility in falling markets goes hand in hand with a strong increase in correlation, but this is not the case for rising markets

This reduces opportunities for diversification in stock-market declines.

monitoring correlation between two variables x and y
Monitoring Correlation Between Two Variables X and Y

Define xi=(Xi−Xi-1)/Xi-1 and yi=(Yi−Yi-1)/Yi-1

Also

varx,n: daily variance of X calculated on day n-1

vary,n: daily variance of Y calculated on day n-1

covn: covariance calculated on day n-1

The correlation is

covariance
Covariance
  • The covariance on day n is

E(xnyn)−E(xn)E(yn)

  • It is usually approximated as E(xnyn)
correlation for multivariate case
Correlation for Multivariate Case
  • If X is m-dimensional and Y n-dimensional then
  • Cov(X,Y) is given by the m×n-matrix with entries Cov(Xi, Yj )
  •  = Cov(X,Y) is called covariance matrix
positive finite definite condition
Positive Finite Definite Condition

A variance-covariance matrix, , is internally consistent if the positive semi-definite condition

holds for all vectors w

example
Example

The variance covariance matrix

is not internally consistent. When w=[1,1,-1] the condition for positive semidefinite is not satisfied.

correlation as a measure of dependence
Correlation as a Measure of Dependence
  • Correlation as a measure of dependence fully determines the dependence structure for normal distributions and, more generally, elliptical distributions while it fails to do so outside this class.
  • Even within this class correlation has to be handled with care: while a correlation of zero for multivariate normally distributed RVs implies independence, a correlation of zero for, say, t-distributed rvs does not imply independence
multivariate normal distribution
Multivariate Normal Distribution
  • Fairly easy to handle
  • A variance-covariance matrix defines the variances of and correlations between variables
  • To be internally consistent a variance-covariance matrix must be positive semidefinite
bivariate normal pdf
Bivariate Normal PDF

Probability density function of a bivariate normal distribution:

x and y bivariate normal
X and Y Bivariate Normal
  • Conditional on the value of X, Y is normal with mean

and standard deviation where X, Y, X, and Y are the unconditional means and SDs of X and Y and xy is the coefficient of correlation between X and Y

generating random samples for monte carlo simulation
Generating Random Samples for Monte Carlo Simulation
  • =NORMSINV(RAND()) gives a random sample from a normal distribution in Excel
  • For a multivariate normal distribution a method known as Cholesky’s decomposition can be used to generate random samples
factor models
Factor Models
  • When there are N variables, Vi (i = 1, 2,..N), in a multivariate normal distribution there are N(N−1)/2 correlations
  • We can reduce the number of correlation parameters that have to be estimated with a factor model
one factor model continued
If Ui have standard normal distributions we can set

where the common factor F and the idiosyncratic component Zi have independent standard normal distributions

Correlation between Uiand Ujis ai aj

One-Factor Model continued
copulas
Copulas

A powerful concept to aggregate the risks — the copula function — has beenintroduced in finance by Embrechts, McNeil, and Straumann [1999,2000]

A copula is a function that links univariate marginal distributions to the full multivariate distribution

This function is the joint distribution function of N standard uniform random variables.

copulas1
Copulas
  • • The dependence relationship between two random variables X and Y is obscured by the marginal densities of X and Y
  • • One can think of the copula density as the density that filters or extracts the marginal information from the joint distribution of X and Y.
  • • To describe, study and measure statistical dependence between random variables X and Y one may study the copula densities.
  • Vice versa, to build a joint distribution between two random variables X ~G() and Y~H(), one may construct first the copula on [0,1]2 and utilize the inverse transformation and
  • G-1() and H-1().
cumulative density function theorem
Cumulative Density Function Theorem

Let X be a continuous random variable with distribution function F()

Let Y be a transformation of X such that

Y=F(X).

The distribution of Y is uniform on [0,1].

sklar s 1959 theorem the bivariate case
Sklar’s (1959) Theorem- The Bivariate Case

X, Y are continuous random variables such that

X  ~G(·), Y  ~ H(·)

G(·), H(·): Cumulative distribution functions – cdf’s

Create the mapping of X  into X such that X=G(X ) then X has a Uniform distribution on [0,1] This mapping is called the probability integral transformation e.g. Nelsen (1999).

Any bivariate joint distribution of (X ,Y ) can be transformed to a bivariate copula (X,Y)={G(X ), H(Y )} –Sklar (1959).

Thus, a bivariate copula is a bivariate distribution with uniform marginal disturbutions (marginals).

copula mathematical definition
CopulaMathematical Definition
  • A n-dimensional copula C is a function which is a cumulative distribution function with uniform marginals:
  • The condition that C is a distribution function leads to the following properties
    • As cdfs are always increasing is increasing in each component ui.
    • The marginal component is obtained by setting uj = 1 for all j i and it must be uniformly distributed,
    • For ai<bi the probability

must be non-negative

an example
An Example

Let Si be the value of Stock i. Let Vpf be the value of a portfolio

5% Value-at-Risk of a Portfolio is defined as follows:

Gaussian Copulas have been used to model dependence between (S1, S2, …..,Sn)

copulas derived from distributions
Copulas Derived from Distributions
  • Typical multivariate distributions describe important dependence structures. The copulas derived can be derived from distributions.
  • The multivariate normal distribution will lead to the Gaussian copula.
  • The multivariate Student t-distribution leads to the t-copula.
gaussian copula models
Gaussian Copula Models:
  • Suppose we wish to define a correlation structure between two variable V1 and V2 that do not have normal distributions
  • We transform the variable V1 to a new variable U1 that has a standard normal distribution on a “percentile-to-percentile” basis.
  • We transform the variable V2 to a new variable U2 that has a standard normal distribution on a “percentile-to-percentile” basis.
  • U1 and U2 are assumed to have a bivariate normal distribution
the correlation structure between the v s is defined by that between the u s

V

V

2

1

One

-

to

-

one

mappings

-6

-4

-2

0

-6

-4

-2

0

2

4

6

U

2

U

1

Correlation

Assumption

The Correlation Structure Between the V’s is Defined by that Between the U’s

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

V

V

2

1

One

-

to

-

one

mappings

-6

-4

-2

0

2

2

4

4

6

6

-6

-4

-2

0

2

4

6

U

2

U

1

Correlation

Assumption

v 1 mapping to u 1
V1 Mapping to U1

Use function NORMINV in Excel to get values in for U1

v 2 mapping to u 2
V2 Mapping to U2

Use function NORMINV in Excel to get values in for U2

example of calculation of joint cumulative distribution
Example of Calculation of Joint Cumulative Distribution
  • Probability that V1 and V2 are both less than 0.2 is the probability that U1 < −0.84 and U2 < −1.41
  • When copula correlation is 0.5 this is

M( −0.84, −1.41, 0.5) = 0.043

where M is the cumulative distribution function for the bivariate normal distribution

gaussian copula algebraic relationship
Gaussian Copula – algebraic relationship
  • Let G1 and G2 be the cumulative marginal probability distributions of V1 and V2
  • Map V1 = v1 to U1 = u1 so that
  • Map V2 = v2 to U2 = u2 so that
  •  is the cumulative normal distribution function
gaussian copula algebraic relationship1
Gaussian Copula – algebraic relationship
  • U1 and U2 are assumed to be bivariate normal
  • The two-dimensional Gaussian copula

where  is the 22 matrix with 1 on the diagonal and correlation coefficient  otherwise. denotes the cdf for a bivariate normal distribution with zero mean and covariance matrix .

  • This representation is equivalent to
multivariate gaussian copula
Multivariate Gaussian Copula
  • We can similarly define a correlation structure between V1, V2,…Vn
  • We transform each variable Vito a new variable Ui that has a standard normal distribution on a “percentile-to-percentile” basis.
  • The U’s are assumed to have a multivariate normal distribution
factor copula model
Factor Copula Model

In a factor copula model the correlation structure between the U’s is generated by assuming one or more factors.

credit default correlation
Credit Default Correlation
  • The credit default correlation between two companies is a measure of their tendency to default at about the same time
  • Default correlation is important in risk management when analyzing the benefits of credit risk diversification
  • It is also important in the valuation of some credit derivatives
model for loan portfolio
Model for Loan Portfolio
  • We map the time to default for company i, Ti, to a new variable Ui and assume

where F and the Zi have independent standard normal distributions

  • The copula correlation is  =a2
  • Define Qi as the cumulative probability distribution of Ti
  • Prob(Ui<U) = Prob(Ti<T) when N(U) = Qi(T)
analysis
Analysis
  • To analyze the model we
    • Calculate the probability that, conditional on the value of F, Ui is less than some value U
    • This is the same as the probability that Ti is less that T where T and U are the same percentiles of their distributions
    • And
    • This is also Prob(Ti<T|F)
analysis cont
Analysis (cont.)

This leads to

where PD is the probability of default in time T

the model continued
The Model continued
  • The worst case default rate for portfolio for a time horizon of T and a confidence limit of X is
  • The VaR for this time horizon and confidence limit is

where L is loan principal and R is recovery rate

appendix 3 gaussian copula with student t distribution
Appendix 3: Gaussian Copula with Student t Distribution
  • Sample U1 and U2 from a bivariate normal distribution with the given correlation .
  • Convert each sample into a variable with a Student t-distribution on a percentile-to-percentile basis.
  • Suppose that U1 is in cell C1. The Excel function TINV gives a “two-tail” inverse of the t-distribution. An Excel instruction for determining V1 is therefore

=IF(NORMSDIST(C1)<0.5,TINV(2*NORMSDIST(C1),df),TINV(2*(1-NORMSDIST(C1)),df))

where df stands for degrees of freedom parameter

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