Synergy effect of cooperative investment

Abstract

Cooperative investment consists of two problems: finding an optimal cooperative investment strategy and fairly dividing investment outcome among participating agents. In general, the two problems cannot be solved separately. It is known that when agents' preferences are represented by mean-deviation functionals, sharing of optimal portfolio creates instruments that, on the one hand, satisfy individual risk preferences but, on the other hand, are not replicable on an incomplete market, so that each agent is strictly better off in participating in cooperative investment than investing alone. This synergy effect is shown to hold when agents' acceptance sets are represented by cash-invariant utility functions in the case of multiperiod investment with an arbitrary feasible investment set. In this case, a set of all Pareto-optimal allocations is characterized, and an equilibrium-based method for selecting a "fair" Pareto-optimal allocation is suggested. It is also shown that if exists, the "fair" allocation belongs to the core of the corresponding cooperative game. The equilibrium-based method is then extended to the case of arbitrary utility functions. The obtained results are demonstrated in a multiperiod cooperative investment problem with investors imposing drawdown constraints on investment strategies.

On the solution uniqueness in portfolio optimization and risk analysis

“…The utility of investor i is U i (Z i ), where U i is some utility function and Z i the random wealth of agent i at the investment horizon. Grechuk and Zabarankin (2017) show that, under some mild conditions on U i , cooperative investment is strictly preferable for all agents compared to their optimal individual investment strategies. In the cooperative investment problem, the coalition's preferences can be represented by a cooperative utility function U * .…”

“…The general problem of cooperative investment can be formulated as follows, see Grechuk and Zabarankin (2017). Let F ⊂ L 2 (Ω) be a feasible set, representing rates of return from feasible investment oppor-tunities on the market without a riskless asset:…”

“…In various problems in economics and finance, including capital allocation (Kalkbrener, 2005), risk sharing (Filipović and Kupper, 2008), cooperative investment (Grechuk and Zabarankin, 2017), inverse portfolio problem (Bertsimas et al, 2012), and generalized Black-Litterman model (Palczewski and Palczewski, 2018), it is important to be able to identify a unique solution. We show that a solution to many of such problems can be expressed in an explicit way using a sub-gradient of some convex function f : R N → R at some point X ∈ R N , and the solution is unique if and only if f is differentiable at X.…”

“…, Z m ) such that X * = m i=1 Z i . There are usually infinitely many Pareto-optimal wealth allocations, but Grechuk and Zabarankin (2017) defined an allocation which can be considered as "fair". The issue is that this "fair" allocation is, in general, non-unique, as we demonstrate in this paper.…”

“…Secondly, a unique optimal portfolio corresponds to many risk identifiers and, consequently, there are many Pareto-optimal sharing arrangements in cooperative investment, which is obviously highly inconvenient in practice. It is also suprising as this possibility was only inferred from the general convexity theory and treated as an unlikely and inconvenient case that is not of prime importance, see Grechuk and Zabarankin (2017). The third consequence is related to the extension of the Black-Litterman model to arbitrary distributions and deviation measures (Palczewski and Palczewski, 2018).…”