The concept presented in this section can be viewed as a synthesis of the approaches presented in the previous two sections. In an MDP portfolio an asset allocation is determined which yields the greatest diversification, by definition. However, this ap- proach is based on the symmetric correlations between the assets. As already pointed out in Chapter 9, the Pearson correlation coefficient only measures the dependence between two random variables correctly if these are jointly normally (elliptically) distributed. Furthermore, covariance/correlation takes deviations below and above the mean into account, but a risk-averse (long-only) investor seeks positive returns (the upside or right tail of the distribution) and only considers negative returns (the downside) as risk. This point was made by Markowitz (1952) when he noted the ap- propriateness of the lower partial second moment as a dispersion measure. In that sense, the lower tail dependence coefficient measures the strength of the relation- ship between asset returns when both are extremely negative at the same time, and is
k k therefore a downside risk measure, as is VaR or CVaR. Dependent on the underlying
copula assumption (e.g., the EVT copula and/or the empirical copula), the value of the TDC also depends on the count of ranked observations used in its calculation, which is akin to the confidence level of VaR/CVaR. For Archimedean copulae the TDC only depends on the copula parameter. In this section the TDC for a distributional copula, such as the Student’stcopula, is discarded because it is a dependence measure for the lower and the upper tail and, hence, does not fit well in the task of risk management (see Section 9.3). In the following, a brief account of the non-parametric estimation of the TDC is given, followed by a description of how optimal portfolio solutions can be derived from it. A general account and synopsis of tail dependence is provided, for instance, in Coles et al. (1999) and Heffernan (2000).
A synopsis of the non-parametric TDC estimators is provided in Dobri´c and Schmid (2005), Frahm et al. (2005), and Schmidt and Stadtmüller (2006). Let (X,Y)denote the percentage losses of two investments. The lower tail dependence coefficient, 𝜆L, between these two random variables is invariant with respect to strictly increasing transformations applied to (X,Y) and does not depend on the marginal distributions of the assets’ returns. It is solely a function of the assumed copula. The joint distribution function ofXandYis given by
FX,Y(x,y) =P(X≤x,Y ≤y)for(x,y) ∈ℝ2. (11.16) This bivariate distribution can be stated equivalently in terms of the copula,C, as
FX,Y(x,y) =C(FX(x),FY(y)), (11.17) whereFX(x)andFY(y)are the marginal distributions ofXandY, respectively. The copula is the joint distribution of the marginal distribution functions: C=P(U≤ u,V≤𝑣), withU=FX(x)andV =FY(y), and therefore maps from[0,1]2into[0,1].
If the limit exists, then the lower tail dependence coefficient is defined as 𝜆L= lim
u→0
C(u,u)
u . (11.18)
This limit can be interpreted as a conditional probability, and as such the lower tail dependence coefficient is bounded in the interval[0,1]. The bounds are realized for an independence copula (𝜆L=0) and a co-monotonic copula (𝜆L=1), respectively.
The non-parametric estimators for𝜆Lare derived from the empirical copula. For a given sample ofNobservation pairs(X1,Y1),…,(XN,YN)with corresponding order statisticsX(1)≤X(2)≤ã ã ã≤X(N)andY(1)≤Y(2)≤ã ã ã≤Y(N), the empirical copula is defined as
CN (i
N, j N
)
= 1 N
∑N
l=1
I(Xl≤X(i)∧Yl≤Y(j)) (11.19) withi,j=1,…,N;Iis the indicator function, which takes a value of 1 if the condition stated in the parentheses is true. By definition,CNtakes a value of zero fori=j=0.
In the literature cited above, three consistent and asymptotically unbiased estimators for 𝜆L are provided, which all depend on a threshold parameterk(i.e.,
k k the number of order statistics considered). The first estimator, 𝜆(1)L (N,k), is an
approximation of the derivative of𝜆Lwith respect touby the slope of a secant in the neighborhood of it (note thatuis written ask∕N):
𝜆(1)L (N,k) = [k
N ]−1
⋅CN
(k N, k
N )
. (11.20)
The second estimator is based on the slope coefficient of a simple affine linear regression between the values of the copula as regressand and the tail probabilities i∕N,i=1,…,k, as the regressor. It is defined as
𝜆(2)L (N,k) = [ k
∑
i=1
(i N
)2]−1
⋅
∑k
i=1
[i
N ⋅CN(i N, i
N )]
. (11.21)
A third means of estimating𝜆L non-parametrically is derived from a mixture of the co-monotonic and independence copulae. Here the lower tail dependence coef- ficient is the weighting parameter between these two copulae. The estimator is then defined as
𝜆(3)L (N,k) =
∑k i=1
( CN(
i N,Ni)
− (i
N
)2) ((
i N
)
− (i
N
)2)
∑k i=1
(
i
N −
(i N
)2)2 . (11.22)
Of crucial importance for estimating the lower tail dependence coefficient is an appropriate selection of k. This parameter is akin to the threshold value for the peaks-over-threshold method in the field of EVT and hence the trade-off between bias and variance is applicable here, too. Choosingktoo small will result in an imprecise estimate, and too high in a biased estimate. The conditions stated above of consistency and unbiasedness are met ifk∼√
N, as shown in Dobri´c and Schmid (2005).
The information on the size of lower tail dependence between assets can be uti- lized in several ways. For instance, a straightforward application would be to replace Pearson’s correlation matrix—employed in determining the allocation for a most di- versified portfolio—with the lower tail dependence coefficients, whereby the main diagonal elements are set equal to one. After rescaling the weight vectors by the assets’ volatilities, one would obtain a “most diversified/minimum tail-dependent”
portfolio. Alternatively, if one ranked the TDCs between the constituent assets of a benchmark and the benchmark itself, one could employ this information to select the financial instruments that are the least dependent with extreme losses in the bench- mark, but independent with respect to the upside. It should be clear that this asset selection will yield a different outcome than following a low-𝛽strategy. Even though the present account has focused on the lower tail dependence, the approach sketched above for selecting constituent assets could also be based on the difference between the upper and lower tail dependence. These two TDCs could be retrieved from the Gumbel and Clayton copula, respectively. This procedure is akin to selecting assets
k k according to their risk–reward ratio. However, it should be stressed that using tail
dependence as a criterion in portfolio optimization should always be accompanied by an additional measure of risk for the assets. A low value of tail dependence is non-informative with respect to the asset risk in terms of either its volatility or its downside risk.