Further Extensions

1. VaR bounds for portfolios in the presence of dependence information

The Rearrangement Algorithm (RA) makes it possible to obtain approximations for sharp VaR upper and lower bounds of portfolios in which the marginal distributions are known but not their interdependence (unconstrained bounds); see [1]. The gap between the upper and lower VaR bound is typically very high and can only be reduced by using dependence information. The following directions to incorporate dependence information, and to get approximations for constrained VaR bounds have been recently explored:

• in [2] one assumes that besides knowledge of the marginal distributions als the joint distribution is known on a subset of the support. The authors apply the RA in an appropriate way to obtain approximations for sharp VaR bounds.

• in [3] one assumes that besides knowledge of the marginal distributions also an upper bound on the variance of the portfolio sum is known.  The authors develop an extended version of the RA (ERA) that makes it possible to obtain approximations for sharp VaR bounds. Further developments are [4] where information on higher order moments is exploited and [10] where independence information and variance information is combined.

• in [1] and [5] one assumes that besides knowledge of the marginal distributions also some pairwise distributions are known. For certain types of portfolios one is able to obtain sharp VaR bounds explicitly. 

2. Improvements of the RA

The RA operates on d*n matrix and amounts to rearranging (swapping) values within the n columns such that one gets a minimum maximal row sum (or maximum minimal row sum). i.e., the d row sums are as constant as possible. To achieve this objective, the RA puts the elements in a given column in opposite order of the values that are obtained by taking the sum of all other other columns. This is repeated until all columns are oppositely ordered to the sum of all others.  

• in [3] one observes that minimizing the variance among the row sums requires that for any decomposition of {1, 2, . . . , n} into two disjoint sets A and B, the sum of columns with index in A must be oppositely ordered to the sum of the remaining columns (with index in B). This observation is also consistent with the notion of ∑-countermonotonicity introduced in [8] and gives rise to the Block RA (Remark 4.1 of [3], Section 2 in [6]). As compared to the RA, the Block RA (labeled as BRA) makes it possible to further decrease the variance among the row sums and to obtain faster convergence rates (with respect to a suitable stopping rule). See [6].

• In [11] a selection rule based on the idea of variance equalization to find the disjoint sets in an optimal way. It is shown that this version of block RA (labeled as BRAVE) can be seen as a generalization of the Karmarkar-Karp difference algorithm for solving the number partitioning problem. When applied to the number partitioning problem, it is shown that BRAVE outperforms the Karmarkar-Karp differencing algorithm.

• In [6] a stochastic Block RA is designed using the Markov Chain theory: it converges to the global minimum of the expectation of a convex function of a sum of dependent variables in a finite amount of time.

• In [4] it is shown that for some particular type of matrices a suitable application of the RA allows to obtain after n-1 steps (rearrangements) the situation in which  all columns are oppositely ordered to the sum of all others. This development makes it possible to get approximations for the minimum variance of a sum of (scaled) Bernoulli distributions in a fast way. 

• In [9] the focus is on the original RA. Specifically, on deals with numerical aspects, run-time improvements, meaningful default parameters, tips for practitioners, application (and corresponding challenges) in high dimensions and more. The paper and accompanying implementations in the R package qrmtools (jointly with Kurt Hornik from R Core) and vignette show the major improvements in the inhomogeneous case, but also stress the numerical challenges of computing (worst or best) VaR in the homogeneous case. 

References

[1] Embrechts, P., Puccetti, G. and L. Rüschendorf (2013). Model uncertainty and VaR aggregation. J. Bank. Financ., 37(8), 2750-2764. paper - post-print (updated) version

[2] Bernard, C. and S. Vanduffel (2014). A New Approach to Assessing Model Risk in High Dimensions. J. Bank. Financ., 58, September 2015, 166-178. paper

[3] Bernard, C., L. Rüschendorf and S. Vanduffel (2013). Value-at-Risk bounds with variance constraints, Forthcoming in Journal of Risk and Insurance,preprint

[4] Bernard, C., Rüschendorf, L., Vanduffel, S. and J. Yao  (2014). How Robust is the Value-at-Riskof Credit Risk Portfolios? European Journal of Finance, 23(6), 507-534. preprint

[5] Embrechts, P. and Puccetti, G., (2009). Bounds for the sum of dependent risks having overlapping marginals. Journal of Multivariate Analysis.,  101(1), 177–190.  paper

[6] Bernard, C. and D. Mc Leish (2016). Algorithms for Finding Copulas Minimizing Convex Functions of Sums, Asia Pac. J. Oper. Res., 33. 

[7] Lee, W., and J. Y. Ahn (2014). On the multidimensional extension of countermonotonicity and its applications. Insurance: Mathematics and Economics, 56, 68–79. paper

[8] Puccetti, G., and R. Wang (2014). General extremal dependence concepts. Statistical Science 30(4), 485-517. 

[9] Hofert, M., Memartoluie, A., Saunders, D. and T. Wirjanto. Improved Algorithms for Computing Worst Value-at-Risk: Numerical Challenges and the Adaptive Rearrangement Algorithm. Forthcoming in Statistics and Risk Modeling. preprint

[10] Puccetti, G., Rüschendorf, L., Small, D. and S. Vanduffel (2015). Reduction of Value-at-Risk bounds via independence and variance information, Forthcoming in Scandinavian Actuarial Journal,preprint

[11] Boudt, K., Jakobsons, E. and S. Vanduffel (2015). Block Rearranging Elements within Matrix Columns to Minimize the Variability of the Row Sums, Forthcoming in 4OR - a quarterly journal of operations research, preprint