Copulas Flashcards
Explain the concept of copulas
Competing approach to using multivariate distributions. They are tools for modelling dependence of several random variables and describing their interrelation
Why are copulas popular among acturaies
Generally we prefer to model one variable at a time and then combine them rather than fitting a model across multiple variable
How is a random variable full described? and then how are multiple random variables fully desribed?
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
If RVs are independent what is their joint distribution function
Product of their cdfs
Whats important fact for copula theory of cdfs and Uniformity
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
What is a d dimensional copula
D dimensional copula is a function which is cumulative distribution function with uniform marginals. C(u)=C(u1,…,ud)
What are the properties of a copula function
- CDFs are always icnreasing - C is increasing
- Marginal in component i is obtained by setting C(1,….,ui,1,1,…,1) = ui
What si the main function of copulas for finance and give an example of use in insurance
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
Explain Sklars theorem
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.
Are copulas applicable to discrete and continuous?
Not really natural for discrete distributions
Does a copula have a density
Yes since it produces a cdf uniformly distributed. You can find the density by differentiation
What is the independence copula
Case of no dependence - Copula is just the product of Ui’s
What is an elliptical distribution
An elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution.
What is the two dimensional gaussian copula the alternative for
The bivariate normal approach
Why is normal a special case for elliptical distributions
Independence is equivalent to roe =0
What is a comprehensive copula - How does it apply to gaussian
Copula varies by only one parameter - Roe or correlation. Gaussian copula has this quality - parameterised entirely based on roe
Compare the gaussian copula to the t copula
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
Name two archimedian copulas
Clayton and Gumbell
What is the gumbel copula
Copula with upper tail dependence with zero lower tail dependence
What is the Clayton copula
Copula with lower tail dependence with zero upper tail dependence
When it the independence copula achieved from gumbel or clayton
Gumbel - When theta = 1
Clayton - when theta goes to 0 (limit)
Name three measures of dependence we stduy
Linear correlation
Rank correlation
Coefficients of tail dependence
Be aware of correlations do not imply…
Small correlations do not imply small dependence always especially in the tails.
Describe the process of rank correlation
Order observations, rank them, do for all variables, calculate the correlation fo ranks, big advanatge is its scale invariant
On a graph of a copula where is upper tail dependence and lower tail dependence illustrated?
Right upper corner - upper tail
Left lower corner - lower tail
What does upper tail dependence mean consider RVs U1, U2 and copula C
Upper tail dependence means with large values of U1, U2 large values are expected
What are the values of Lamda u or l showing dependence in tails
Lamdau>0 : Have upper tail dependence
Lamdau=0 : No upper tail dependence
Lamdal>0 : Have lower tail dependence
Lamdal=0 : No lower tail dependence
Clayton copula LamdaL and Lamda u
Lamda L = 2^(-1/theta) Lamda u=0
Gumbel copula LamdaL and Lamda u
Lamda L =0 Lamda u = 2-2^(1/theta)
Student t copula LamdaL and Lamda u
Lamda l = Lamda u = (2t(dof v+1)(-SQRT((V+1)(1-p)/(1+p)))
Where p is correlation
V is free parameter
