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This paper studies optimal dynamic portfolios for investors concerned with the performance of their portfolios relative to a benchmark. Assuming that asset returns follow a multi-linear factor model similar to the structure of Ross (1976) [Ross, S., 1976. The arbitrage theory of the capital asset pricing model. Journal of Economic Theory, 13, 342–360] and that portfolio managers adopt a mean tracking error analysis similar to that of Roll (1992) [Roll, R., 1992. A mean/variance analysis of tracking error. Journal of Portfolio Management, 18, 13–22], we develop a dynamic model of active portfolio management maximizing risk adjusted excess return over a selected benchmark. Unlike the case of constant proportional portfolios for standard utility maximization, our optimal portfolio policy is state dependent, being a function of time to investment horizon, the return on the benchmark portfolio, and the return on the investment portfolio. We define a dynamic performance measure which relates portfolio’s return to its risk sensitivity. Abnormal returns at each point in time are quantified as the difference between the realized and the model-fitted returns. Risk sensitivity is estimated through a dynamic matching that minimizes the total fitted error of portfolio returns. For illustration, we analyze eight representative mutual funds in the U.S. market and show how this model can be used in practice.

Professional fund managers are frequently judged by their ex post excess returns relative to a prescribed benchmark. Most money managers adopt an optimal strategy that maximizes an expected excess return adjusted by the tracking error relative to the benchmark; see, Roll (1992). This is a sensible investment approach because fund sponsors wisely expect their investment portfolios to maintain a performance level that is close to a desired benchmark. We analyze an optimal dynamic portfolio and asset allocation policy for investors who are concerned about the performances of their portfolios relative to that of a given benchmark. Maximizing the expected utility of the excess return over a chosen benchmark is sometimes referred to as active portfolio management, while passive portfolio management just establishes a portfolio that possibly tracks the chosen benchmark; see, Roll, 1992 and Sharpe, 1964. There are many professional and institutional investors who follow this benchmark-oriented procedure. For example, many equity mutual funds take the S&P 500 Index as a benchmark and try to beat it. Some bond funds try to exceed the performance of Lehman Brothers Bond Index. For an analysis of active portfolio management in a static setting, see Grinold and Kahn (2000). In the standard utility maximization with constant relative risk aversion (CRRA), the optimal policies are all constant proportion portfolio allocation strategies. The portfolio is continuously rebalanced so as to always keep a constant proportion of the total fund value in the various asset classes, regardless of the level of the fund. Although such policies have a variety of optimality properties for the ordinary portfolio problem and are used in asset allocation practice (see, Perold and Sharpe, 1988 and Black and Perold, 1992), some investors are reluctant to use constant proportion strategies in the belief that their expectations suggest that varying weights would be more profitable. By maximizing the probability that the investment fund achieves a certain performance goal before falling below a predetermined shortfall relative to the benchmark, Browne (2000) relates the optimal portfolio policy to a state variable, the ratio of the level of investment portfolio to the benchmark portfolio, which leads to an analytical solution in a complete market setting. Managers of actively managed mutual funds are interested in shifting the investment policy with changes of returns on both their investment portfolios and the benchmark portfolio from time to time. Academic researchers define this market activity as market timing; see, Becker et al., 1999, Coggin et al., 1993 and Ferson and Warther, 1996, and Treynor and Mazuy (1966). This paper addresses this issue for a general incomplete market where the investor is allowed to invest in a large number of stocks which may include all the individual components of equity indices. In a static setting, all efficient portfolios can be obtained from the market portfolio by using leverage, assuming normally distributed returns or quadratic utilities. However, as shown here, this is generally not true in a dynamic setting except for the very special case that the market portfolio is equivalent to a leveraged growth optimum portfolio. Consistent with the standard risk/return tradeoff, the objective of the optimization model is to maximize the expected differential returns on the investment portfolio and the benchmark adjusted by its quadratic variation over the investment horizon. This objective is intuitive and easily understood; it has a simple model structure and has been used widely in the practice of portfolio management. Based on this setting, we derive an analytical solution assuming that the model parameters are time varying. The optimal portfolio is a linear combination of the riskless asset, the growth optimal portfolio and the benchmark portfolio. Existing models have more or less ignored the activities in shifting the portfolio weights during the investment horizon. Dynamic optimal portfolios require portfolio managers to reformulate their portfolios given new observations. Especially for those mutual funds whose performances are tied to a selected benchmark, updating of a portfolio over time appears to be even more appropriate. A very common question often arises as how to define managers’ portfolio strategies, without observing their risk sensitivity. Our first step is to infer the risk sensitivity of portfolio managers by utilizing the outcome of the portfolio returns over time. Estimation of risk sensitivity is based on dynamic matching. In other words, portfolios’ abnormal returns are measured as the difference between the actual outcomes of the portfolio return and the model’s implied return. The estimated risk sensitivity is then defined as the minimizer of the total deviation of portfolio returns from the model’s implied returns. Subsequently, the performance measure is defined as the expected value of the discounted abnormal returns over the investment horizon. Foster and Stutzer (2002) examined performance and risk aversion of funds with benchmarks using a large deviations approach. The paper proceeds as follows. Section 2 discusses the setting of the financial market model for asset prices and the formulation of the problem to be studied. Section 3 presents the solutions for the optimal investment policies and the optimal portfolio returns over time. Section 4 discusses a dynamic performance measure with risk sensitivity estimated through dynamic matching. Section 5 presents the empirical analysis of the model and its implication for portfolio management. Section 6 concludes.

This paper studied a dynamic model of portfolio management process related to the problem of outperforming a benchmark. Assuming managers adopt a benchmarking procedure for investment, the optimal portfolio policy is state dependent, being a function of time to the investment horizon, the return on the benchmark, and the return on the investment portfolio itself. We obtained an explicit formula for calculating the optimal portfolio weights in a continuous time setting. Based on risk sensitivities, we also studied a model for portfolio performance evaluation. Risk sensitivities are estimated using a dynamic matching approach which minimizes the total error between the actual returns and the returns implied by the model. Empirical analysis shows that there is a strong relation between the level of performance and a manger’s risk sensitivity. We briefly discussed how this model can be used for evaluating portfolio performance. We selected a sample of the U.S. funds to illustrate the implementation of our model. The fitted portfolio value is very close to reality with only a small margin of difference.