Correlation, Diversification

Thus far we have concerned ourselves with the method to compute a VaR 99.9 capital requirement for a single UoM without any regard for dependencies with other UoMs. The types of dependencies that may exist in LDA modeling are discussed in Aue and Kalkbrener (2006) [10] and Frachot et al (2004) [71]. In the LDA framework, dependencies among events and losses may occur in numerous way. In Aue and Kalkbrener (2006) [10], we learn of intra-(within) and inter-(between) cell correlation that may arise.

intra-cell

• dependence between frequency of lossesNk within a cell,

• dependence between frequency of losses Nk and the severity of losses Xk,Nk

within a cell,

• dependence between severity of lossesXk,1, Xk,2, . . . , Xk,Nk within a cell,

inter-cell

• dependence between the frequency distributions N1, . . . , Nm in different cells,

• dependence between the loss distributions S1, . . . , Sm in different cells.

There are some combinations that are difficult to model due to a lack of data or they violate certain principles. For example, for loss events occurring within a cell that do not occur independently, the frequency distribution can not be Poisson. In addition, from Frachotet al [71] we learn that severity correlation is difficult to tackle under the

LDA framework. Severity between cells may be evident (e.g. internal fraud losses are high when external fraud losses are high); however a basic feature of actuarial models assume that losses within a cell are jointly independent. Hence it would be difficult to assume severity independence within each cell and severity correlation between cells. While Aue and Kalkbrener (2006) [10] decided to model inter-cell dependencies between frequencies for Deutsche Bank, most banks choose to model dependencies on the aggregate loss level as stated by the OCC (2012) [95].

Having focused the attention to dependencies between aggregate losses, the most relevant way to incorporate this relationship is through a copula. What is needed first though is to define a marginal distribution. We do this by first defining the joint cumulative probability distribution function of X and Y as

F(a, b) = P{X ≤a, Y ≤b}, − ∞< a, b <∞. (2.30) Then the distribution of X can be obtained from the joint distribution of X and Y as

FX(a) = P{X ≤a} = P{X ≤a, Y <∞} = P lim b→∞{X ≤a, Y ≤b} = lim b→∞P n X ≤a, Y ≤b} = lim b→∞F(a, b) = F(a,∞). (2.31)

The CDF obtained is referred to as the marginal distribution which is the CDF with- out reference to other values of other variables. We have seen this already as the individual aggregate loss distributions characterizing each UoM. This is the required element for the copula. Formally we may then define the copula as taken from Panjer (2006) [101].

Definition 2.4.6. (Copula) A d-variate copula C is the joint distribution func- tion of d Uniform (0,1) random variables. If the d random variables are listed as

U1, U2, . . . , Ud, then the C may be written as

C(u1, . . . , ud) =P(U1 ≤u1, . . . , Ud≤ud,). (2.32)

Considering continuous random variables X1, X2, . . . , Xn with distribution functions

F1, F2, . . . , Fd respectively with joint distribution given by F, then the probability

integral transform F1(X1), F2(X2), . . . , Fd(Xd) are each distributed as Uniform (0,1)

random variables. Hence copulas can be seen to be joint distribution functions of Uni- form (0,1) random variables. Thus a copula evaluated at F1(x1), F2(x2), . . . , Fd(xd)

can be written as C F1(x1), . . . , Fd(xd) =P U1 ≤F1(x1), . . . , Ud≤Fd(xd) . (2.33)

With the inverse (or quantile function) defined as

F−1(u) = inf

x {Fj(x)≥u}, (2.34)

the copula evaluated at F1(x1), F2(x2), . . . , Fd(xd) can be rewritten as

C F1(x1), . . . , Fd(xd) = P F−1(U1)≤x1, . . . , Fd−1(Ud)≤xd = P(X1 ≤x1, . . . , Xd≤xd) = F(x1, . . . , xd). (2.35)

Sklar’s theorem (1959) states in a formal way that for any joint distribution function

F, there is a unique copulaC that satisfies (2.35). Conversely, for any copula C and any distribution function F1(x1), . . . , Fd(xd)

, the function C F1(x1), . . . , Fd(xd)

is a joint distribution function with marginals F1(x1), . . . , Fd(xd)

There are two main reason why we choose to integrate copulas into our talk of AMA modeling

• risk can be split into two parts: the individual risk and the dependence structure between them,

• a dependence structure may be defined without reference to the modeling spec- ification of individual risks.

While the discussion of dependence is of great importance, the questions surrounding a best practice is another area of research in itself. For example, choosing the “right” copula to model the dependencies unfortunately has no obvious answer according to Embrechts (2009) [59]. For example, one encounters selection from such choices as

• Gaussian/Normal copula,

• Student t copula (with associated degrees of freedom), • Gumbel copula,

• Clayton copula, • Frank copula.

What this equates to is banks who seek to report a diversified operational risk capital requirement may favour such a copula that reduces capital. For instance from Em- brechts (2012) [60], we see that joint tail dependence is a copula property, regardless what the marginals are. In addition, under asymptotic independence joint extremes are very rare. This is shown in Figure 2.2. We point out that the top-right quadrant under the Gumbel copula realizes greater joint tail dependence as shown by an in- creased number of hits. This is contrasted to the Gaussian copula that has less joint realizations in the top-right quadrant.

Figure 2.2: 1000 random variates from two distributions with identical Gamma(3,1) marginal distributions and identical correlation ρ = 0.7, but different dependence structures.

Hence the dependence structure directly explains the value that diversification granted

V aRdiversif ied = (1−D)V aR, 0< D < 1, (2.36)

where often D is sought to be anywhere from [0.1, 0.3] or translated into a 10-30% reduction in diversified capital as pointed out by Embrechts (2012) [60]. We have seen this before in (2.15) where now we use a copula to perform the aggregation and perhaps receive a diversification benefit.