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Synergy effect of cooperative investment

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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.

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Notes

  1. In Wilson (1968), a group of agents participating in cooperative investment is called a syndicate.

  2. Given two instruments with random payoffs \(X_1\) and \(X_2\), agent i prefers \(X_1\) if and only if \(U_i(X_1)>U_i(X_2)\).

  3. For extension to the non-monotone case, see Acciaio (2007).

  4. Mean-deviation functionals combine two quantities: expected value and general deviation measure of a random variable.

  5. This is the so-called “common belief” assumption.

  6. Two r.v.’s \(X:\Omega \rightarrow {\mathbb {R}}\) and \(Y:\Omega \rightarrow {\mathbb {R}}\) are said to be comonotone, if there exists a set \(A\subset \Omega \) such that \(P[A]=1\) and \((X(\omega _1)-X(\omega _2))(Y(\omega _1)-Y(\omega _2))\geqslant 0\) for all \(\omega _1, \omega _2 \in A\).

  7. A probability space is atomless, if there exists an r.v. with continuous cumulative distribution function.

  8. Namely, this holds if \(\Omega \) is atomless and \({\mathcal {L}}^2(\Omega )\) is separable.

  9. Formally, we first introduce a probability measure \({\mathbb {P}}_{\mathcal {T}}\) on \({\mathcal {T}}\) such that \({\mathbb {P}}_{\mathcal {T}}(t_i)=1/k\), \(i=1,\ldots ,k\), if \({\mathcal {T}}=\{t_1,t_2,\ldots ,t_k=T\}\), and \({\mathbb {P}}_{\mathcal {T}}(S)=l(S)/T\) if \({\mathcal {T}}=[0,T]\), where S is any measurable subset of [0, T] with Lebesgue measure l(S), and then introduce a probability measure on \(\Omega \times {\mathcal {T}}\) as a produce measure of \({\mathbb {P}}\) and \({\mathbb {P}}_{\mathcal {T}}\).

  10. See Theorem T25 in Chapter II in Dellacherie and Meyer (1978).

  11. Filipovic and Kupper (2008) considered no optimal cooperative investment.

  12. \(U_i\) has no local maxima if for all \(X\in {\mathcal {X}}\) and \(A\subset {\mathcal {X}}\) such that A is open and \(X\in A\), there exists \(X'\in A\) such that \(U_i(X')>U_i(X)\).

  13. In fact, Theorem 1 in Dana et al. (1997) proves the existence of quasiequilibrium, i.e. a pair \((\mathbf{Z},P^*)\) such that (i) \(\mathbf{Z}\) is \(\mathbf{Q}\)-feasible, (ii) \(P^*(Z_i)=P^*(Q_i)\), \(i\in I\), and (iii) \(P^*(X)\geqslant P^*(Q_i)\), whenever \(U^*_i(X)>U^*_i(Z_i)\), \(i\in I\). A quasiequilibrium is a \(\mathbf{Q}\)-equilibrium if \(P^*(Z_i)>\inf \nolimits _{X\in {\mathcal {X}}_i}P^*(X)\), \(i\in I\).

  14. (1) “invest,” “invest”; (2) “invest,” “not invest”; (3) “not invest,” “invest”; (4) “not invest,” “not invest”; (5) “invest,” and then if “win,” “invest” (if “lose,” “not invest”); and (6) “invest,” and then if “lose,” “invest” (if “win,” “not invest”).

References

  • Aase, K. (2002). Perspectives of risk sharing. Scandinavian Actuarial Journal, 2, 73–128.

    Article  Google Scholar 

  • Acciaio, B. (2007). Optimal risk sharing with non-monotone monetary functionals. Finance and Stochastics, 11(2), 267–289.

    Article  Google Scholar 

  • Becker, R. (1991). Fundamental theorems of welfare economics in infinite dimensional commodity spaces. In M. Ali Khan & N. Yannelis (Eds.), Equilibrium theory in infinite dimensional spaces. Studies in economic theory, 1 (pp. 124–175). Berlin: Springer.

    Chapter  Google Scholar 

  • Burgert, C., & Ruschendorf, L. (2008). Allocation of risks and equilibrium in markets with finitely many traders. Insurance: Mathematics and Economics, 42(1), 177–188.

    Google Scholar 

  • Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., & Montrucchio, L. (2011). Risk measures: Rationality and diversification. Mathematical Finance, 21, 743–774.

    Google Scholar 

  • Chekhlov, A., Uryasev, S., & Zabarankin, M. (2003). Portfolio optimization with drawdown constraints. In B. Scherer (Ed.), Asset and liability management tools (pp. 263–278). London: Risk Books.

    Google Scholar 

  • Chekhlov, A., Uryasev, S., & Zabarankin, M. (2005). Drawdown measure in portfolio optimization. International Journal of Theoretical and Applied Finance, 8(1), 13–58.

    Article  Google Scholar 

  • Cheng, H. (1991). Asset market equilibrium in infinite dimensional complete markets. Journal of Mathematical Economic, 20, 137–152.

    Article  Google Scholar 

  • Cvitanic, J., & Karatzas, I. (1995). On portfolio optimization under “drawdown” constraints. IMA Lecture Notes in Mathematics & Applications, 65, 77–88.

    Google Scholar 

  • Dana, R.-A. (2011). Comonotonicity, efficient risk-sharing and equilibria in markets with short-selling for concave law-invariant utilities. Journal of Mathematical Economics, 47(3), 328–335.

    Article  Google Scholar 

  • Dana, R.-A., Le Van, C., & Magnien, F. (1997). General equilibrium in asset markets with or without short-selling. Journal of Mathematical Analysis and Applications, 206(2), 567–588.

    Article  Google Scholar 

  • Dellacherie, C., & Meyer, P. A. (1978). Probabilities and potential. New York: North-Holland Pub. Co.

    Google Scholar 

  • Filipovic, D., & Kupper, M. (2008). Equilibrium prices for monetary utility functions. International Journal of Theoretical and Applied Finance, 11, 325–343.

    Article  Google Scholar 

  • Follmer, H., & Schied, A. (2011). Stochastic finance (3rd ed.). Berlin: de Gruyter.

    Book  Google Scholar 

  • Gilboa, I., Samuelson, L., & Schmeidler, D. (2014). No-betting-Pareto dominance. Econometrica, 82, 1405–1442.

    Article  Google Scholar 

  • Grechuk, B. (2015). The center of a convex set and capital allocation. European Journal of Operational Research, 243(2), 628–636.

    Article  Google Scholar 

  • Grechuk, B., Molyboha, A., & Zabarankin, M. (2012). Mean-deviation analysis in the theory of choice. Risk Analysis, 32(8), 1277–1292.

    Article  Google Scholar 

  • Grechuk, B., Molyboha, A., & Zabarankin, M. (2013). Cooperative games with general deviation measures. Mathematical Finance, 23(2), 339–365.

    Article  Google Scholar 

  • Grechuk, B., & Zabarankin, M. (2012). Optimal risk sharing with general deviation measures. Annals of Operations Research, 200(1), 9–21.

    Article  Google Scholar 

  • Grechuk, B., & Zabarankin, M. (2014). Inverse portfolio problem with mean-deviation model. European Journal of Operational Research, 234(2), 481–490.

    Article  Google Scholar 

  • Grechuk, B., & Zabarankin, M. (2015). Inverse portfolio problem with coherent risk measures. European Journal of Operational Research. doi:10.1016/j.ejor.2015.09.050.

  • Grossman, S. J., & Zhou, Z. (1993). Optimal investment strategies for controlling drawdowns. Mathematical Finance, 3(3), 241–276.

    Article  Google Scholar 

  • Jouini, E., Schachermayer, W., & Touzi, N. (2008). Optimal risk sharing for law invariant monetary utility functions. Mathematical Finance, 18(2), 269–292.

    Article  Google Scholar 

  • Kiesel, S., & Ruschendorf, L. (2014). Optimal risk allocation for convex risk functionals in general risk domains. Statistics & Risk Modeling, 31(3–4), 335–365.

    Google Scholar 

  • Kurdila, A., & Zabarankin, M. (2005). Convex functional analysis. Basel: Birkhäuser.

    Google Scholar 

  • Li, D., & Ng, W.-L. (2000). Optimal dynamic portfolio selection: Multiperiod mean–variance formulation. Mathematical Finance, 10, 387–406.

    Article  Google Scholar 

  • Li, D., & Zhou, X. Y. (2000). Continuous-time mean–variance portfolio selection: A stochastic LQ framework. Applied Mathematics and Optimization, 42, 19–33.

    Article  Google Scholar 

  • Ludkovski, M., & Ruschendorf, L. (2008). On comonotonicity of Pareto optimal risk sharing. Statistics and Probability Letters, 78(10), 1181–1188.

    Article  Google Scholar 

  • Luthi, H., & Doege, J. (2005). Convex risk measures for portfolio optimization and concepts of flexibility. Mathematical Programming, Series B, 104((2–3)), 541–559.

    Article  Google Scholar 

  • Markowitz, H. M. (1952). Portfolio selection. Journal of Finance, 7(1), 77–91.

    Google Scholar 

  • Muthuraman, K., & Kumar, S. (2006). Multidimensional portfolio optimization with proportional transaction costs. Mathematical Finance, 16(2), 301–335.

    Article  Google Scholar 

  • Osborne, J., & Rubinstein, A. (1994). A course in game theory. Cambridge, MA: MIT Press.

    Google Scholar 

  • Rockafellar, R. T., Uryasev, S., & Zabarankin, M. (2006). Generalized deviations in risk analysis. Finance & Stochastics, 10(1), 51–74.

    Article  Google Scholar 

  • Rockafellar, R. T., Uryasev, S., & Zabarankin, M. (2006). Optimality conditions in portfolio analysis with general deviation measures. Mathematical Programming, 108(2–3), 515–540.

    Article  Google Scholar 

  • Rockafellar, R. T., Uryasev, S., & Zabarankin, M. (2006). Master funds in portfolio analysis with general deviation measures. The Journal of Banking and Finance, 30(2), 743–777.

    Article  Google Scholar 

  • Wilson, R. (1968). The theory of syndicates. Econometrica, 36(1), 119–132.

    Article  Google Scholar 

  • Xia, J. (2004). Multi-agent investment in incomplete markets. Finance and Stochastics, 8(2), 241–259.

    Article  Google Scholar 

  • Zalinescu, C. (2002). Convex analysis in general vector spaces. Singapore: World Scientific.

    Book  Google Scholar 

  • Zabarankin, M., Pavlikov, K., & Uryasev, S. (2014). Capital asset pricing model (CAPM) with drawdown measure. European Journal of Operational Research, 234(2), 508–517.

    Article  Google Scholar 

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Acknowledgments

The authors are grateful to the referees for their comments and suggestions, which helped to improve the quality of the paper. The first author also thanks the University of Leicester for granting him the academic study leave to conduct this research.

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Authors and Affiliations

  1. Department of Mathematics, University of Leicester, Leicester, UK

    Bogdan Grechuk

  2. Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ, USA

    Michael Zabarankin

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Correspondence to Michael Zabarankin.

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Grechuk, B., Zabarankin, M. Synergy effect of cooperative investment. Ann Oper Res 249, 409–431 (2017). https://doi.org/10.1007/s10479-015-2051-x

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