Feasible portfolios under tracking error, β, α and utility constraints

  • Received November 27, 2017;
    Accepted January 22, 2018;
    Published February 20, 2018
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  • Article Info
    Volume 15 2018, Issue #1, pp. 141-153
  • Cited by
    4 articles

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The investment nous of active managers is judged on their ability to outperform specified benchmarks while complying with strict constraints on, for example, tracking errors, β and Value at Risk. Tracking error constraints give rise to a tracking error frontier – an ellipse in risk/return space which encloses theoretically possible (but not necessarily efficient) portfolios. The β frontier is a parabola in risk/return space and defines the threshold of portfolios subject to a specified β requirement. An α - TE frontier is similarly shaped: portfolios on this frontier have a specified TE for a maximum TE. Utility and associated risk aversion have also been explored for constrained portfolios. This paper contributes by establishing the impossibility of satisfying more than two constraints simultaneously and explores the behavior of these constraints on the maximum risk-adjusted return portfolio (defined arbitrarily here as the optimal portfolio).

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    • Figure 1. Positions of portfolios P0 and P1 on the efficient frontier and the gain , G = rp − rb the fund manager’s outperformance target
    • Figure 2. TE frontier, TE -constrained portfolio and constant TE frontier (with TE = 5%). (a) shows the naïve portfolio: excess return is maximised for a given TE constraint. (b) shows Jorion’s (2003) suggestion: observe constraints from (a), but restrict
    • Figure 3. (a) Position of β frontier for β = 0.9, 1.0 and 1.1 and (b) maximum and minimum β values for changing tracking errors
    • Figure 4. The α-TE frontier for various levels of α. Other frontiers are shown for comparison. Levels of α are indicated on the graph. TE = 5%, rf = 2%
    • Figure 5. (a) Utility function tangential to the maximum Sharpe ratio portfolio on the constant TE frontier and (b) θ as a function of tracking error and risk-free rate
    • Figure 6. (a) Maximum Sharpe ratio as a function of tracking error and risk-free rate and (b) utility function (risk aversion) as a function of tracking error and portfolio risk
    • Table 1. Properties of portfolios 0 and 1 in terms of a, b and c