Copulas A copula is the joint distribution of random variables U 1 , - PowerPoint PPT Presentation

ST 810-006 Statistics and Financial Risk Copulas A copula is the joint distribution of random variables U 1 , U 2 , . . . , U p , each of which is marginally uniformly distributed as U (0 , 1). The term copula is also used for the joint cumulative distribution function of such a distribution, C ( u 1 , u 2 , . . . , u p ) = P ( U 1 ≤ u 1 , U 2 ≤ u 2 , . . . , U p ≤ u p ) . 1 / 39

ST 810-006 Statistics and Financial Risk Examples If U 1 , U 2 , . . . , U p are independent , C ( u 1 , u 2 , . . . , u p ) = u 1 × u 2 × · · · × u p . If they are completely dependent ( U 1 = U 2 = · · · = U p with probability 1), C ( u 1 , u 2 , . . . , u p ) = min ( u 1 , u 2 , . . . , u p ) 2 / 39

ST 810-006 Statistics and Financial Risk Sklar’s Theorem Copulas are important because of Sklar’s Theorem: For any random variables X 1 , X 2 , . . . , X p with joint c.d.f. F ( x 1 , x 2 , . . . , x p ) = P ( X 1 ≤ x 1 , X 2 ≤ x 2 , . . . , X p ≤ x p ) and marginal c.d.f.s F j ( x ) = P ( X j ≤ x ) , j = 1 , 2 , . . . , p , there exists a copula such that F ( x 1 , x 2 , . . . , x p ) = C [ F 1 ( x 1 ) , F 2 ( x 2 ) , . . . , F p ( x p )] . If each F j ( x ) is continuous, C is unique. 3 / 39

ST 810-006 Statistics and Financial Risk That is, we can describe the joint distribution of X 1 , X 2 , . . . , X p by the marginal distributions F j ( x ) and the copula C . The copula (Latin: link) links the marginal distributions together to form the joint distribution. From a modeling perspective, Sklar’s Theorem allows us to separate the modeling of the marginal distributions F j ( x ) from the dependence structure, which is expressed in C . 4 / 39

ST 810-006 Statistics and Financial Risk The proof is simple in the case that all F j ( x ) are continuous, because in this case each has an inverse function F − 1 ( · ) such j that F − 1 � � ( u ) = u , for all 0 ≤ u ≤ 1. F j j If U j = F j ( X j ), then U j ∼ U (0 , 1): � X j ≤ F − 1 � P ( U j ≤ u ) = P [ F j ( X j ) ≤ u ] = P ( u ) j F − 1 � � = F j ( u ) = u . j Write C for the c.d.f. of this copula; then F ( x 1 , x 2 , . . . , x p ) = P ( X 1 ≤ x 1 , X 2 ≤ x 2 , . . . , X p ≤ x p ) = P [ U 1 ≤ F 1 ( x 1 ) , U 2 ≤ F 2 ( x 2 ) , . . . , U p ≤ F p ( x p )] = C [ F 1 ( x 1 ) , F 2 ( x 2 ) , . . . , F p ( x p )] . 5 / 39

ST 810-006 Statistics and Financial Risk Copula Density When F ( · ) and C ( · ) are differentiable, the equation F ( x 1 , x 2 , . . . , x p ) = C [ F 1 ( x 1 ) , F 2 ( x 2 ) , . . . , F p ( x p )] for the joint cumulative distribution function implies that the joint probability density function (pdf) satisfies f ( x 1 , x 2 , . . . , x p ) f 1 ( x 1 ) f 2 ( x 2 ) . . . f p ( x p ) = c [ F 1 ( x 1 ) , F 2 ( x 2 ) , . . . , F p ( x p )] where c ( · ) is the pdf of the copula distribution: ∂ p c ( u 1 , u 2 , . . . , u p ) = C ( u 1 , u 2 , . . . , u p ) . ∂ u 1 ∂ u 2 . . . ∂ u p 6 / 39

ST 810-006 Statistics and Financial Risk That is, the copula pdf is the ratio of the joint pdf to what it would have been under independence. So we can also interpret the copula as the adjustment that we need to make to convert the independence pdf into the joint pdf. 7 / 39

ST 810-006 Statistics and Financial Risk Classes of Copulas An Archimedean copula has the form � ψ − 1 ( u 1 ) + ψ − 1 ( u 2 ) + · · · + ψ − 1 ( u p ) � C ( u 1 , u 2 , . . . , u p ) = ψ for an appropriate generator ψ ( · ). The Clayton, Frank, Gumbel, and Joe copulas are Archimedean. Each has a single parameter that controls the degree of dependence. 8 / 39

ST 810-006 Statistics and Financial Risk A copula can be extracted from any joint distribution F ( · ): F − 1 1 ( u 1 ) , F − 1 2 ( u 2 ) , . . . F − 1 � � C ( u 1 , u 2 , . . . , u p ) = F p ( u p ) Unlike the Pearson (or linear ) correlation, the copula is invariant under monotone increasing transformations of the random variables. If F ( · ) is a multivariate normal distribution N p ( µ , Σ ), then C ( · ) is a Gaussian copula. If the location or scale of the distribution is changed, the copula does not change, so conventionally µ = 0 and Σ = R , a correlation matrix. 9 / 39

ST 810-006 Statistics and Financial Risk Gaussian copula for p = 2 , ρ = 0 . 5 Gaussian copula density, rho = 0.5 1.0 0.5 4 2.5 2 0.8 1.5 0.6 1 0.4 1.5 0.2 1 2 2.5 0.5 0.0 3 0.0 0.2 0.4 0.6 0.8 1.0 10 / 39

ST 810-006 Statistics and Financial Risk Gaussian copula for p = 2 , ρ = 0 . 5 Gaussian copula density, rho = 0.5 cop u u 11 / 39

ST 810-006 Statistics and Financial Risk If F ( · ) is a multivariate t -distribution t ν ( µ , Σ ), then C ( · ) is a t -copula. Again, conventionally µ = 0 and Σ = R , a correlation matrix. But note that when X ∼ t ν ( µ , Σ ): if ν > 1 then E ( X ) = µ ; if ν > 2 then � � ν Var( X ) = Σ . ν − 2 � ν − 2 Y = ν X has the standardized multivariate t -distribution. 12 / 39

ST 810-006 Statistics and Financial Risk t-copula for p = 2 , ν = 5 , ρ = 0 . 5 t copula density, rho = 0.5 df = 5 1.0 5 3 2 0.8 0.6 1 0.4 0.2 2 3 1 5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 13 / 39

ST 810-006 Statistics and Financial Risk t-copula for p = 2 , ν = 5 , ρ = 0 . 5 t copula density, rho = 0.5 df = 5 cop u u 14 / 39

ST 810-006 Statistics and Financial Risk When R = I , the multivariate normal distribution is that of independent standard normal variables, and the copula has a constant density. But the t -copula still shows dependence. 15 / 39

ST 810-006 Statistics and Financial Risk t-copula for p = 2 , ν = 5 , ρ = 0 t copula density, rho = 0 df = 5 1.0 1 0.8 1.2 1.2 0.8 0.6 0.4 0.8 0.8 0.2 1 1 1.2 2 . 1 0.8 1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 16 / 39

ST 810-006 Statistics and Financial Risk t-copula for p = 2 , ν = 5 , ρ = 0 t copula density, rho = 0 df = 5 cop u u 17 / 39

ST 810-006 Statistics and Financial Risk Application to Portfolio Risk Consider a portfolio of p corporate bonds. Ignore interest rate risk, and write the value of bond i at time t when its rating is ρ as B i ( t , ρ ). If the rating of bond i at time t is ρ i ( t ), the portfolio value is � V ( t ) = B i [ t , ρ i ( t )] . i The distribution of the change in value from t to t + δ t can be calculated from the joint conditional distribution of ρ 1 ( t + δ t ) , ρ 2 ( t + δ t ) , . . . , ρ p ( t + δ t ) . 18 / 39

ST 810-006 Statistics and Financial Risk Historical data give plausible values for the marginal transition probabilities P ρ,ρ ′ = P [ ρ i ( t + δ t ) = ρ ′ | ρ i ( t ) = ρ ] . We can write ρ i ( t + δ t ) as a function of a uniform variable U i : ρ i ( t + δ t ) = ρ ′ , where ρ ′ satisfies P [ ρ i ( t + δ t ) < ρ ′ | ρ i ( t )] < U i ≤ P [ ρ i ( t + δ t ) ≤ ρ ′ | ρ i ( t )] . That is, we break the interval (0 , 1) into subintervals with endpoints at the cumulative probabilities P [ ρ i ( t + δ t ) ≤ ρ ′ | ρ i ( t )], and find the subinterval containing U i . 19 / 39

ST 810-006 Statistics and Financial Risk To model the joint transition probabilities, we need a joint distribution for U 1 , U 2 , . . . , U p . Independence is not plausible: if the bonds were all at the same initial rating, independence implies a multinomial distribution for the counts in the various ratings at a later time, and historical data show considerable over-dispersion relative to the multinomial. We therefore need a copula ; the question is, which one? 20 / 39

ST 810-006 Statistics and Financial Risk The Gaussian Copula Modeling and simulation of dependent transitions is usually described in terms of Gaussian random variables instead of uniform random variables. Write Z i = Φ − 1 ( U i ), so Z i ∼ N (0 , 1), and break the whole real line ( −∞ , ∞ ) into subintervals at the Gaussian quantiles corresponding to the cumulative probabilities, Φ − 1 { P [ ρ i ( t + δ t ) ≤ ρ ′ | ρ i ( t )] } . Then Z i falls into one of the new subintervals iff U i falls into the corresponding subinterval of (0 , 1), and we can determine the new rating as a function of Z i by finding the subinterval containing Z i . 21 / 39

ST 810-006 Statistics and Financial Risk The question of what copula to use now becomes: how to model the joint distribution of Z 1 , Z 2 , . . . , Z p , given that each of them is marginally N (0 , 1). The obvious choice is the multivariate normal distribution N p ( 0 , R ) with zero mean vector and with dispersion matrix equal to some correlation matrix R . This is equivalent to using the Gaussian copula defined by U i = Φ ( Z i ) , i = 1 , 2 , . . . , p , where the Z ’s have this multivariate normal distribution. 22 / 39

ST 810-006 Statistics and Financial Risk Choice of R The Gaussian copula for p = 2 has only the single parameter ρ . When p > 2, the 1 2 p ( p − 1) off-diagonal correlations must be specified, constrained by the non-negative definiteness of R . When the variables correspond to entities with appropriate structure, a factor model may be used, such as in the Moody’s capital model: Z i = λ 1 ζ G + λ 2 ζ R i + λ 3 ζ I i + λ 4 ζ F , i where λ 2 1 + λ 2 2 + λ 2 3 + λ 2 4 = 1, and the ζ s are independent standard normal variables. So each Z i is N (0 , 1), and the correlation of Z i and Z j depends on which ζ s they share. 23 / 39