ERM Chapter 18

Question 1

What is the marginal distribution?

Answer 1

The individual distribution of each of the risk factors in isolation.

Question 2

What is the difference between joint distributions and copulas?

Answer 2

The joint distribution expresses the dependence of interrelated factors on one another implicitly, whilst copious allows for this dependence explicitly.

Question 3

What is a copula?

Answer 3

Expresses a multivariate cumulative distribution function in terms of the individual marginal cumulative functions.

P(X < x, Y < y) = Fx,y(x, y) = Cx,y[Fx(x), Fy(y)]

The key benefit of copulas is the deconstruction of a joint distribution of a set of variables into components that allows each component to be adjusted independently of the others.

Question 4

What are the basic properties of copulas?

Answer 4

1. Increasing the range of values for the individual variables must increase the joint probability of observing a combination within that range:

C(u1, u2, …, uN) is an increasing function of each input variable.

2. If we integrate out all other variables (by setting their CDF’s to the maximum value of 1 so as to include all possible values), we should just have the marginal distribution of variable i.

C(1, 1, u3, …, 1) = u3

3. A valid probability is produced by the copulas function for any valid combination of the parameters:

(Insert photo of formula here)

Question 5

What is Sklar’s theorem?

Answer 5

• let F be a joint distribution function with marginal cumulative distribution functions F1, …, FN then Sklar’s theorem states that there exists a copula, C, such that for all x1, …, xN E (-infinity, infinity)
  F(x1, …, xN) = C(F1(x1), …, FN(xN))
• if the marginal cumulative distribution functions are continuous, then C is unique.
• conversely, if C is a copula and F1, …, FN are univariate cumulative distribution functions, then the function F defined above is a joint cumulative distribution function with marginal distribution functions F1, …, FN.

Question 6

Outline the formula for discrete copulas.

Answer 6

Method 1:
1/(1+T) <= F(x,y) <= T/(1+T)

Method 2:
1/2T <= F(x,y) <= (T-1/2)/T

Question 7

What is the survival copula?

Answer 7

F^(x,y) = P[X > x, Y > y] = C^[F^x(X), F^y(Y)],
where F^x(X) = 1 - Fx(X), F^y(Y) = 1 - Fy(Y)

C^(1-u, 1-v) = 1 - u - v - C(u, v)

The survival copula expresses the joint probability in terms of marginal survival probabilities

Question 8

What is the difference between concordance or association and dependence?

Answer 8

Concordance or association does not imply that one directly influences the other. For example, both might be dependent upon a third variable.

Question 9

What are the axioms for a good measure of concordance?

Answer 9

• completeness of domain: Mx,y defined for all values of x and y, with x and y being continuous
• symmetry: Mx,y = My,x
• coherence: if C(Fx(x), Fy(y)) >= C(Fw(w), Fz(z)), then Mx,y >= Mw,z
• unit range: -1 <= Mx,y <= 1
• interdependence: if X and Y are independent then Mx,y = 0
• covergence

Question 10

Name the three main types of copulas.

Answer 10

1. fundamental copulas
2. explicit copulas
3. implicit copulas

Question 11

What are the three types of fundamental copulas?

Answer 11

1. independence copula - no upper or lower tail dependence as variables are independent, thus dependence = 0.
2. co-monotonicity copula - represents perfect positive dependence between the variables, thus dependence = 1.
3. counter-monotonicity copula - represents perfect negative dependence between the two variables, so any tail dependence between the variables will only manifest itself when the variables are at opposite ends i.e. one is high and one is low. There will be no special relationship when both are low or both are high.

Question 12

What are the Frechet-Hoffding bounds?

Answer 12

In the bivariate case the co-montonicity and counter-monotonicity copulas represent the extremes of the possible levels of association between variables. They are the upper and lower boundaries for all copulas - known as the Frechet-Hoffding bounds.

Question 13

Outline the characteristics of generator functions.

Answer 13

• w: [0, 1] -> [0, infinity)
• valid generator functions are continuous and strictly decreasing functions
• the pseudo-inverse of w with the domain [0, infinity] is defined to be:
  w-1(x) = w-1(x), 0

Question 14

Outline Archimedean copulas.

Answer 14

- Four types of copulas, being:
> Gumbel copula
> Frank copula
> Clayton copula
> Generalised Clayton copula

• c(u1, …, uN) = w-1[sum(w(ui))]
• to find the copula function, set the generator function equal to k and solve for u, where u will equal the pseudo-inverse of w

A: - relatively simple to use. In particular, they are closed-form probability distributions and avoid the need for integration.

D: - small number of parameters involved (<3) means their application to heterogeneous groups of variables is limited.

Question 15

Outline the Gumbel copula.

Answer 15

• generator function:

(-ln(u))^a, 1<=a

Question 16

Outline the Frank copula.

Answer 16

• generator function:
• ln[(e^(-au)-1)/(e^-a)-1)]
• somewhat limited by its characteristics (no tail dependency, symmetric form) but is considered when modelling joint- and last-survivor annuities and exchange rate movements.

Question 17

Outline the Clayton copula.

Answer 17

• generator function:
  1/a x (u^(-a) - 1)
• the absence of upper tail dependency, but potential lower tail dependency makes it suitable for modelling situation where associations increase for extreme lower values, but not for extreme higher values. e.g. modelling returns from a portfolio of investments, where negative returns are likely to occur simultaneously on a number of investments.

Question 18

Outline the generalised Clayton copula.

Answer 18

• generator function:
  1/(a^B) x (u^(-a) - 1)^B
• when B = 1, generalised Clayton copula becomes the standard Clayton copula
• flexible in it’s application, as it has both upper and lower tail dependencies which can be adjusted independently using the two parameters.

Question 19

What are the two types of implicit copulas?

Answer 19

1. Normal (Gaussian) copula

2. Student’s t-copula

Question 20

Outline the Gaussian copula.

Answer 20

• a correlation matrix will need to be specified with ones on the leading diagonal
• if the correlation matrix is the identity matrix, the Gaussian copula becomes the independence copula
• if the correlation matrix consists entirely of ones, then the Gaussian copula becomes the minimum (co-monotonicity) copula
• if N = 2 and p = -1, then the Gaussian copula becomes the maximum (counter-monotonicity) copula

D: - lack of tail dependency
- the fact it is defined by a single parameter

Question 21

Outline the Student’s t-copula.

Answer 21

• overcomes the key disadvantage of the Gaussian copula as it has both upper and lower tail dependencies
• the smaller the value of y, the greater the level of association at the extremes of the marginals
• the two-parameter copula enables the degree of dependence at the extremes to be controlled independently of the correlation matrix by varying the number of DOF’s
• if the correlation matrix is the identity matrix, the Student’s t-copula does NOT become the independence copula as random variables from an uncorrelated multivariate t-distribution are not independent
• if the correlation consists entirely of ones, the Student’s t-copula becomes the minimum (co-monotonicity) copula
• if y is infinity then the Student’s t-copula approaches the Gaussian copula