Optimal portfolio allocations with tracking error volatility and stochastic hedging constraints

Abstract

The performance of mutual fund or pension fund managers is often evaluated by comparing the returns of managed portfolios with those of a benchmark. As most portfolio managers use dynamic rules for rebalancing their portfolios, we use a dynamic framework to study the optimization of the tracking error–return trade-off. Following these observations, we assume that the manager minimizes the tracking error under an expected return goal (or, equivalently, maximizes the information ratio). Moreover, we assume that he/she complies with a stochastic hedging constraint whereby the terminal value of the portfolio is (almost surely) higher than a given stochastic payoff. This general setting includes the case of a minimum wealth level at the horizon date and the case of a performance constraint on terminal wealth as measured by the benchmark (i.e. terminal portfolio wealth should be at least equal to a given proportion of the index). When the manager cares about absolute returns and relative returns as well, the risk–return trade-off acquires an extra dimension since risk comprises two components. This extra risk dimension substantially modifies the characteristics of portfolio strategies. The optimal solutions involve pricing and duplication of spread options. Optimal terminal wealth profiles are derived in a general setting, and optimal strategies are determined when security prices follow geometric Brownian motions and interest rates remain constant. A numerical example illustrates the type of strategies generated by the model.

Keywords:

• Asset management
• Asset allocation
• Continuous time finance
• Portfolio optimization
• Stochastic hedging constraints
• Tracking error

JEL Classification::

• G1
• G11

Acknowledgement

The Institute for Quantitative Investment Research provided financial support to the first two authors for this research.

Notes

†Clarke et al. (1994) argue that the tracking error model should be developed from the concept of aversion to regret (regret comes when the portfolio deviates from the benchmark in the wrong direction). Jorion (2003) shows that additional constraints can mitigate the inefficiency of benchmarking. Another way to rationalize benchmarking, as argued by Wagner (2001), is to acknowledge the fact that there might be a lack of information on the assessment of the return distributions of the traded assets. Rudolf et al. (1999) show that linear models (as opposed to quadratic models) for minimizing tracking errors are consistent with expected utility maximization.

‡It has been proved by Goetzmann et al. (2007) that IR can be manipulated. These manipulation strategies can be either static or dynamic, but the effects of dynamic manipulation schemes are often more dramatic.

§Bajeux-Besnainou et al. (2001) provided a theoretical explanation of the Canner et al. (1997) puzzle. As a key argument to solve the puzzle, they use the ability of investors to continuously manage their positions.

†Besides, El Karoui et al. (2005) express the portfolio optimal value in a general form and are not interested in the actual duplication of the put. Since the put is not actually traded, the actual implementation of the portfolio strategy involves its duplication.

‡For simplicity, we do not consider intermediary dividend payments. In practice, many fund managers measure their performance relative to a total return benchmark with dividends reinvested (e.g. MSCI and FTSE for equity). Taking dividends into account is a straightforward technical exercise.

§Since the quadratic program is not scale invariant, the risk tolerance parameter a must be defined as a proportion of the initial wealth W ₀: when W ₀ is different from 1, in order to obtain decision rules independent of the initial wealth W ₀, the objective function is written

¶Some technical ‘feasibility’ conditions have to be satisfied for this optimization program to have a solution. In particular, the stochastic payoff C_T should be attainable through an initial investment equal to or less than $1(E(C_T /G_T )≤1), otherwise this hedging constraint could never be met with an initial investment of $1.

†The empirical IR is directly linked to the t-statistics of the test alpha > 0.

‡This TEV constraint may be imposed by the funds prospectus. Besides, the benchmarked funds are classified by their empirical TEV and excessive deviations may cause declassification. For instance, funds tracking a benchmark are divided into three classes by the IOSCO (International Organization of Securities Commissions) and by regulators, for instance the French AMF (Autorité des Marchés Financiers): Indexed (TEV≤1%), tilted (TEV around 2%) or active funds (with a TEV often specified in the prospectus). Once a fund is in a given class, the TEV is constrained by the regulator or/and by the risk manager.

§Empirical and theoretical arguments have been formulated in favor of the IR as a performance measure:

•	the empirical IR is linked to the t-statistics of the test α > 0 (α is defined with respect to the benchmark);
•	the Sharpe ratio of a portfolio (with respect to the usually unobservable tangent portfolio) is equal to the sum of its IR and of the Sharpe ratio of the benchmark (the latter is usually suboptimal, since the benchmark is presumably inefficient, which justifies the active manager's attempt to outperform it). IR maximization thus implies Sharpe ratio maximization.

†However, performance fees are not necessarily linear (convex schemes exist) and hedge funds are free to apply asymmetric schemes. Nevertheless, the majority of performance fees are linear affine in excess returns and the literature on optimal contracting has focused on such schemes.

‡This argument has recently been discussed and studied extensively in the literature. In particular, Carpenter (2000) shows that a manager compensated with an asymmetric fee does not always take excessive risk. He shows that a manager compensated with a call option on the assets under management may either take dramatic risks (when her option is out of the money), or be surprisingly conservative (when her option is in the money).

§It solves min E((X_T )²), s.t. E(X_T /G_T )=1.

¶We are assuming here that the initial wealth W ₀ equals $1, which allows us to use parameter a instead of aW ₀. In this case, the dimension of the parameter a is a $ amount (see footnote 6).

⊥Note that such a mean–variance program is not ‘time consistent’, because its solution at time 0 does ‘not match’ the solution at t > 0 (see, for instance, Li and Ng (2000), Lim and Zhou (2002) and Basak and Chabakauri (2009a,b)). However, the solution at time 0 (‘pre-committed’ mean–variance optimization) coincides with the solution of the quadratic program (P) defined in the sequel, which is ‘time consistent’, and which is the optimization program considered throughout this paper.

†Cox and Huang (1989) first introduced the option-like decomposition of the optimal strategy in the particular case of positive wealth constraints.

‡In this case, the benchmark I_T must be attainable through a buy and hold strategy (which is the case, for instance, for value weighted indices).

§Calling P ₀ the price of the put at date 0, the initial investment in the unconstrained strategy is 1–P ₀.

†The TEV constraint mitigates this incentive.

‡The solution and the results are very similar to those of this paper.

§Since the market portfolio has by definition a zero weight in the index I, ‘compatibility’ conditions require .

†Note that these momentum or contrarian strategies are defined, depending on the authors, either according to variations of weights as done here, or according to variations of $ amounts invested in the funds when the values of these funds vary.