Copulas

Question 1

Explain the concept of copulas

Answer 1

Competing approach to using multivariate distributions. They are tools for modelling dependence of several random variables and describing their interrelation

Question 2

Why are copulas popular among acturaies

Answer 2

Generally we prefer to model one variable at a time and then combine them rather than fitting a model across multiple variable

Question 3

How is a random variable full described? and then how are multiple random variables fully desribed?

Answer 3

By its cdf or the marginal distributions as it will be called.

We obtain a full description of multiple RVs using their marginals and the type of interrelation between them

Question 4

If RVs are independent what is their joint distribution function

Answer 4

Product of their cdfs

Question 5

Whats important fact for copula theory of cdfs and Uniformity

Answer 5

When one applies a cdf to the value of a variable those cdfs are uniformly distributed. If U~0,1 and F is a cdf then the probability the generalized inverse <x is equal to the cdf of x

Question 6

What is a d dimensional copula

Answer 6

D dimensional copula is a function which is cumulative distribution function with uniform marginals. C(u)=C(u1,…,ud)

Question 7

What are the properties of a copula function

Answer 7

1. CDFs are always icnreasing - C is increasing
2. Marginal in component i is obtained by setting C(1,….,ui,1,1,…,1) = ui

Question 8

What si the main function of copulas for finance and give an example of use in insurance

Answer 8

Main purpose is they allow us to examine tail behaviour and dependence. Probability X exceeds q given that Y does for example.
Ex: in motor insurance probability that health costs will be above C given the car damage was X

Question 9

Explain Sklars theorem

Answer 9

Consider a cdf F with marginals F1….Fd - there exists a copula such that F(x1,…xd)=C(F1(x1),….,Fd(xd)) for all Xi. If Fi is continuous then C is unique otherwise C is uniquely determined only on the range of cdf Fi.

Question 10

Are copulas applicable to discrete and continuous?

Answer 10

Not really natural for discrete distributions

Question 11

Does a copula have a density

Answer 11

Yes since it produces a cdf uniformly distributed. You can find the density by differentiation

Question 12

What is the independence copula

Answer 12

Case of no dependence - Copula is just the product of Ui’s

Question 13

What is an elliptical distribution

Answer 13

An elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution.

Question 14

What is the two dimensional gaussian copula the alternative for

Answer 14

The bivariate normal approach

Question 15

Why is normal a special case for elliptical distributions

Answer 15

Independence is equivalent to roe =0

Question 16

What is a comprehensive copula - How does it apply to gaussian

Answer 16

Copula varies by only one parameter - Roe or correlation. Gaussian copula has this quality - parameterised entirely based on roe

Question 17

Compare the gaussian copula to the t copula

Answer 17

Student t has heavier tails than normal so generates a more flexible surface. The extreme cases are modelled more pronouced due to tail dependence in t copula. In addition to correlation t copula has free parameter of meaning more complex shapes can be created

Question 18

Name two archimedian copulas

Answer 18

Clayton and Gumbell

Question 19

What is the gumbel copula

Answer 19

Copula with upper tail dependence with zero lower tail dependence

Question 20

What is the Clayton copula

Answer 20

Copula with lower tail dependence with zero upper tail dependence

Question 21

When it the independence copula achieved from gumbel or clayton

Answer 21

Gumbel - When theta = 1
Clayton - when theta goes to 0 (limit)

Question 22

Name three measures of dependence we stduy

Answer 22

Linear correlation
Rank correlation
Coefficients of tail dependence

Question 23

Be aware of correlations do not imply…

Answer 23

Small correlations do not imply small dependence always especially in the tails.

Question 24

Describe the process of rank correlation

Answer 24

Order observations, rank them, do for all variables, calculate the correlation fo ranks, big advanatge is its scale invariant

Question 25

On a graph of a copula where is upper tail dependence and lower tail dependence illustrated?

Answer 25

Right upper corner - upper tail
Left lower corner - lower tail

Question 26

What does upper tail dependence mean consider RVs U1, U2 and copula C

Answer 26

Upper tail dependence means with large values of U1, U2 large values are expected

Question 27

What are the values of Lamda u or l showing dependence in tails

Answer 27

Lamdau>0 : Have upper tail dependence
Lamdau=0 : No upper tail dependence
Lamdal>0 : Have lower tail dependence
Lamdal=0 : No lower tail dependence

Question 28

Clayton copula LamdaL and Lamda u

Answer 28

Lamda L = 2^(-1/theta) Lamda u=0

Question 29

Gumbel copula LamdaL and Lamda u

Answer 29

Lamda L =0 Lamda u = 2-2^(1/theta)

Question 30

Student t copula LamdaL and Lamda u

Answer 30

Lamda l = Lamda u = (2t(dof v+1)(-SQRT((V+1)(1-p)/(1+p)))

Where p is correlation
V is free parameter