18. Copulas

Question 1

Define a copula

Answer 1

• 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

Question 2

What is the invariance property

Answer 2

• It is determined by relative order of observations and not the exact shape of the marginal distributions

Question 3

List the properties of a copula

Answer 3

• 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

Question 4

State Sklar’s theorem

Answer 4

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

Question 5

Describe a Survival copula

Answer 5

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)

Question 6

What’s the purpose of the coefficients of tail dependence

Answer 6

Describe how marginal distributions are related or move together at extreme ends of distribution

Question 7

Outline why the coefficients of tail dependence are important to quantify risk exposure

Answer 7

• 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

Question 8

List the 3 main categories of copulas

Answer 8

1. Fundamental
2. Archimedian
3. Implicit

Question 9

Outline the 3 Frechet-Hoefding copulas

Answer 9

Independent- zero dependence
Minimum- full + dependency
Maximum- full - dependency

Question 10

Give examples of Archimedian copulas

Answer 10

• Gumbel
• Frank
• Clayton
• Generalised Clayton

Question 11

Outline the properties of implicit copulas

Answer 11

• 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

Question 12

How can the parameters be found?

Answer 12

• MLE
• Parametrisation based on rank correlation