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Abstract
Portfolios in which all assets contribute equally to the conditional value-at-risk (CVaR) represent an interesting variation of the popular risk parity investment strategy. This paper considers the use of convex optimization to find long-only equal risk contribution (ERC) portfolios for CVaR given a set of equally likely scenarios of asset returns. We provide second-order conic and non-linear formulations of the problem, which yields an ERC portfolio when CVaR is both positive and differentiable at the optimal solution. We identify sufficient conditions for differentiability and develop a heuristic that obtains an approximate ERC portfolio when the conditions are not satisfied. Computational tests show that the approach performs well compared to non-convex formulations that have been proposed in the literature.
Acknowledgements
We thank our colleague, Ian Iscoe, for helpful discussions and the anonymous reviewers, whose comments improved an earlier version of the paper.
Disclosure statement
No potential conflict of interest was reported by the authors.
Notes
1 This is evident, for example, in Bridgewater’s All Weather Fund, launched in 1996 as the first risk parity fund (Podolsky et al. Citation2012). Note that ERC indexes do not necessarily follow this rule. For example, members of the FTSE Global Equal Risk Contribution Index Series are constructed by equalizing the risk contributions of large cap constituents of the underlying index, provided that they have a sufficient trading history, while the remaining constituents are included at their market capitalization weights (e.g. FTSE Russell Citation2016).
2 Investment gains correspond to negative values of X. We use this convention to maintain consistency between the signs of risk measures and the values of the random variable.
3 Rockafellar and Uryasev (Citation2002) define if
(so that the conditional expectation is well-defined) and
otherwise, which is equivalent to (Equation1
(1) ).
4 Instead of (Equation5(5) ), some prior studies have approximated
and then used
to compute CVaR contributions. While this correctly obtains
and
, when
, it ignores the scenarios in
and incorrectly yields
and
.
5 Portfolios y and yield identical position weights for any
.
6 Problem (Equation10(10) ) was proposed by Maillard et al. (Citation2010) but several alternative formulations are possible. Notably,
can be replaced by
in the objective function (Mausser and Romanko (Citation2014) give a second-order conic programming formulation of this problem) or one can minimize
with
as discussed, for example, in Spinu (Citation2013) or Bai et al. (Citation2016).
7 Since the goal is only to find the ERC position weights , the value
of the optimal portfolio in problem (Equation10
(10) ), which depends on the constant b, is unimportant.
8 This has been proposed, more generally, in the context of risk budgeting by Bruder and Roncalli (Citation2012) and Haugh et al. (Citation2017). However, neither paper considers discrete distributions.
9 While Cesarone and Colucci (Citation2018) propose to solve problem (Equation11(11) ) for scenario-based CVaR, they do not address the issue of non-differentiability.
10 We round to four decimal digits in our numerical experiments.
11 Note that LSCP minimizes , not
. In this case,
while the minimum is
at
. However, w corresponds to
in LSCP and
exceeds
.
12 Note that can be interpreted as a set of conditional scenario probabilities given the occurrence of some event
, i.e.
. However,
cannot depend only on the portfolio loss
, i.e. the intuitive choice
is inappropriate when
. See Tasche (Citation2002) for details.
13 Tables and report the weight of asset 1 as returned by IPOPT upon termination, which may not correspond to the best solution found when the algorithm fails to converge. In fact, for Distribution II, a solution with error 0.0321 was encountered for all initial points in table .
14 While an actual ERC portfolio typically would comprise a number of different asset classes, rather than only equities, the chosen portfolio is sufficient to evaluate the computational aspects of the problem. The selected stocks exhibit a significant number of negative pairwise correlations, as might be expected to occur among the asset classes in an ERC portfolio.