18. Copulas Flashcards

1
Q

Define a copula

A
  • 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
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2
Q

What is the invariance property

A
  • It is determined by relative order of observations and not the exact shape of the marginal distributions
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3
Q

List the properties of a copula

A
  • 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
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4
Q

State Sklar’s theorem

A

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

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5
Q

Describe a Survival copula

A

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)

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6
Q

What’s the purpose of the coefficients of tail dependence

A

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

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7
Q

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

A
  • 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
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8
Q

List the 3 main categories of copulas

A
  1. Fundamental
  2. Archimedian
  3. Implicit
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9
Q

Outline the 3 Frechet-Hoefding copulas

A

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

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10
Q

Give examples of Archimedian copulas

A
  • Gumbel
  • Frank
  • Clayton
  • Generalised Clayton
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11
Q

Outline the properties of implicit copulas

A
  • 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
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12
Q

How can the parameters be found?

A
  • MLE
  • Parametrisation based on rank correlation
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