18. Copulas Flashcards
Define a copula
- Joint distribution function that takes marginal probabilities as it’s arguments. In N-dimensions, its expressed as:
C(u1,u2…,un) = P(U1<=u1, …., Un<=un)
Where ui=F(Xi), ie individual cumulative distribution functions each lying between [0 and 1] - It is determined by relative order of observations and not the exact shape of the marginal distributions which is the invariance property
What is the invariance property
- It is determined by relative order of observations and not the exact shape of the marginal distributions
List the properties of a copula
- Must be an increasing function of its inputs
- If values of all but one of the marginal CDFs are equal to 1, then the copula is equal to the value of the remaining marginal CDF
- Copula must always return a non-negative probability
State Sklar’s theorem
If F is a joint CDF and F_1,…,F_N are marginal CDFs,then there exists a copula such that
for all x_1,…x_N∈[-∞,∞]:
F_(x_1,…,x_N ) (x_1,…,x_N )=C[F_(x_1 ) (x_1 ),….,F_(x_N (x_N ) )]
Furthermore, if marginal distributions are continuous then copula is unique
Describe a Survival copula
Every copula has a survival copula that expresses joint survival probabilities in terms of marginal probabilities:
F ̅(x,y)=F[X>x,Y>y]=C ̅(F ̅(x),F ̅(y))
where F ̅(x)=1-F(x)and F ̅(y)=1-F(y)
Link between C and C ̅:
C ̅(1-u,1-v)=1-u-v+C(u,v)
What’s the purpose of the coefficients of tail dependence
Describe how marginal distributions are related or move together at extreme ends of distribution
Outline why the coefficients of tail dependence are important to quantify risk exposure
- It can be used to describe the joint concentration of risk that the organisation is concerned about.
- It can be used to describe the risks occurring at extreme but low probability events to ensure that company understand how risks will interact and hence
- … better prepare for it by holding sufficient capital.
- For example, considering the asset returns on equities and money markets during a recession as they may not be independent at the extremes where diversification fails.
- Presence of upper and lower tail dependence helps decide on which copula is most appropriate to model risk.
- For example, if level of risk is higher at extreme values then a copula with upper and lower tail dependence can be considered.
- If only at negative extremes then copula with lower tail dependence and vice versa
List the 3 main categories of copulas
- Fundamental
- Archimedian
- Implicit
Outline the 3 Frechet-Hoefding copulas
Independent- zero dependence
Minimum- full + dependency
Maximum- full - dependency
Give examples of Archimedian copulas
- Gumbel
- Frank
- Clayton
- Generalised Clayton
Outline the properties of implicit copulas
- Based on well known multivariate distributions but no simple closed-form expression exists
- E.g. Normal amd t-copula
- T more flexible in level of tail dependency
- Smaller the gamma, the greater the level of dependency
- As gamma tends to infinity copula tends to normal
- Combining t-distributions using t-copula produces multivariate std t-distribution
How can the parameters be found?
- MLE
- Parametrisation based on rank correlation