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**Static fund separation of long-term investments.**
*(English)*
Zbl 1331.91164

Summary: This paper proves a class of static fund separation theorems, valid for investors with a long horizon and constant relative risk aversion, and with stochastic investment opportunities. An optimal portfolio decomposes as a constant mix of a few preference-free funds, which are common to all investors. The weight in each fund is a constant that may depend on an investor’s risk aversion, but not on the state variable, which changes over time. Vice versa, the composition of each fund may depend on the state, but not on the risk aversion, since a fund appears in the portfolios of different investors. We prove these results for two classes of models with a single state variable, and several assets with constant correlations with the state. In the linear class, the state is an Ornstein-Uhlenbeck process, risk premia are affine in the state, while volatilities and the interest rate are constant. In the square root class, the state follows a square root diffusion, expected returns and the interest rate are affine in the state, while volatilities are linear in the square root of the state.

### MSC:

91G10 | Portfolio theory |

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\textit{P. Guasoni} and \textit{S. Robertson}, Math. Finance 25, No. 4, 789--826 (2015; Zbl 1331.91164)

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### References:

[1] | Barberis, Investing for the Long Run When Returns Are Predictable, J. Finance 55 (1) pp 225– (2000) |

[2] | Bensoussan, Optimal Consumption and Portfolio Decisions with Partially Observed Real Prices, Math. Finance 19 (2) pp 215– (2009) · Zbl 1168.91375 |

[3] | Brandt, Estimating Portfolio and Consumption Choice: A Conditional Euler Equations Approach, J. Finance 54 (5) pp 1609– (1999) |

[4] | Campbell, Predicting Excess Stock Returns Out of Sample: Can Anything Beat the Historical Average?, Rev. Finan c. 21 (4) pp 1509– (2008) |

[5] | Cass, The Structure of Investor Preferences and Asset Returns, and Separability in Portfolio Allocation: A Contribution to the Pure Theory of Mutual Funds, J. Econ. Theory 2 pp 122– (1970) · Zbl 06522379 |

[6] | Chamberlain, Asset Pricing in Multiperiod Securities Markets, Econometrica 56 (6) pp 1283– (1988) · Zbl 0657.90012 |

[7] | Cheridito, Equivalent and Absolutely Continuous Measure Changes for Jump-Diffusion Processes, Ann. Appl. Probab. 15 (3) pp 1713– (2005) · Zbl 1082.60034 |

[8] | Cochrane, The Dog that Did Not Bark: A Defense of Return Predictability, Rev. Financ. Stud. 21 (4) pp 1533– (2008) |

[9] | Cox, A Continuous-Time Portfolio Turnpike Theorem, J. Econ. Dyn. Control 16 (3-4) pp 491– (1992) · Zbl 0760.90006 |

[10] | Dybvig, Portfolio Turnpikes, Rev. Financ. Stud. 12 (1) (1999) |

[11] | Feller, Two Singular Diffusion Problems, Ann. Math. 54 (2) pp 173– (1951) · Zbl 0045.04901 |

[12] | Glasserman, Monte Carlo Methods in Financial Engineering: Applications of Mathematics (New York) (2004) · Zbl 1038.91045 |

[13] | Guasoni , P. C. Kardaras S. Robertson H. Xing 2011 Abstract, Classic, and Explicit Turnpikes · Zbl 1303.91157 |

[14] | Guasoni, Portfolios and Risk Premia for the Long Run, Ann. Appl. Probab. 22 (1) pp 239– (2012) · Zbl 1247.91172 |

[15] | Hakansson, Convergence to Isoelastic Utility and Policy in Multiperiod Portfolio Choice, J. Financ. Econ. 1 pp 201– (1974) |

[16] | Hille, On Laguerre’s Series, Second Note, Proc. Natl. Acad. Sci. U.S.A. 12 (4) pp 265– (1926) · JFM 52.0281.02 |

[17] | Huang, Turnpike Behavior of Long-Term Investments, Finance Stoch. 3 (1) pp 15– (1999) · Zbl 0924.90017 |

[18] | Huberman, Portfolio Turnpike Theorems, Risk Aversion, and Regularly Varying Utility Functions, Econometrica 51 (5) pp 1345– (1983) · Zbl 0521.90014 |

[19] | Khanna, A Generalization of the Mutual Fund Theorem, Finance Stoch. 3 (2) pp 167– (1999) · Zbl 0926.91022 |

[20] | Kim, Dynamic Nonmyopic Portfolio Behavior, Rev. Financ. Stud. 9 (1) pp 141– (1996) |

[21] | Kramkov, The Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Markets, Ann. Appl. Probab. 9 (3) pp 904– (1999) · Zbl 0967.91017 |

[22] | Lebedev, Special Functions and Their Applications (1972) |

[23] | Leland, Mathematical Models in Investment and Finance (1972) |

[24] | Magnus, Formulas and Theorems for the Special Functions of Mathematical Physics (1966) |

[25] | Merton, An Intertemporal Capital Asset Pricing Model, Econometrica 41 (5) pp 867– (1973) · Zbl 0283.90003 |

[26] | Miklavčič, Applied Functional Analysis and Partial Differential Equations (1998) |

[27] | Pan, The Jump-Risk Premia Implicit in Options: Evidence from an Integrated Time-Series Study, J. Financ. Econ. 63 (1) pp 3– (2002) |

[28] | Reed, Methods of Modern Mathematical Physics. I. Functional Analysis (1972) · Zbl 0242.46001 |

[29] | Ross, Mutual Fund Separation in Financial Theory-The Separating Distributions, J. Econom. Theory 17 (2) pp 254– (1978) · Zbl 0379.90010 |

[30] | Schachermayer, In Which Financial Markets Do Mutual Fund Theorems Hold True?, Finance Stoch. 13 (1) pp 49– (2009) · Zbl 1199.91279 |

[31] | Tobin, Liquidity Preference as Behavior Towards Risk, Rev. Econ. Stud. 25 (2) pp 65– (1958) |

[32] | Wachter, Portfolio and Consumption Decisions under Mean-Reverting Returns: An Exact Solution for Complete Markets, J. Financ. Quant. Anal. 37 (1) pp 63– (2002) |

[33] | Welch, A Comprehensive Look at the Empirical Performance of Equity Premium Prediction, Rev. Financ. Stud. 21 (4) pp 1455– (2008) |

[34] | Xia, Learning about Predictability: The Effects of Parameter Uncertainty on Dynamic Asset Allocation, J. Finance 56 (1) pp 205– (2001) |

[35] | Zariphopoulou, A Solution Approach to Valuation with Unhedgeable Risks, Finance Stoch. 5 (1) pp 61– (2001) · Zbl 0977.93081 |

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