On Inefficiency of Markowitz-Style Investment Strategies When Drawdown is Important

The focal point of this paper is the issue of "drawdown" which arises in recursive betting scenarios and related applications in the stock market. Roughly speaking, drawdown is understood to mean drops in wealth over time from peaks to subsequent lows. Motivated by the fact that this issue is of paramount concern to conservative investors, we dispense with the classical variance as the risk metric and work with drawdown and mean return as the risk-reward pair. In this setting, the main results in this paper address the so-called "efficiency" of linear time-invariant (LTI) investment feedback strategies which correspond to Markowitz-style schemes in the finance literature. Our analysis begins with the following principle which is widely used in finance: Given two investment opportunities, if one of them has higher risk and lower return, it will be deemed to be inefficient or strictly dominated and generally rejected in the marketplace. In this framework, with risk-reward pair as described above, our main result is that classical Markowitz-style strategies are inefficient. To establish this, we use a new investment strategy which involves a time-varying linear feedback block K(k), called the drawdown modulator. Using this instead of the original LTI feedback block K in the Markowitz scheme, the desired domination is obtained. As a bonus, it is also seen that the modulator assures a worst-case level of drawdown protection with probability one.Comment: This paper has been published in Proceedings of 56th IEEE Conference on Decision and Control (CDC) 201

Kelly Betting Can Be Too Conservative

Kelly betting is a prescription for optimal resource allocation among a set of gambles which are typically repeated in an independent and identically distributed manner. In this setting, there is a large body of literature which includes arguments that the theory often leads to bets which are "too aggressive" with respect to various risk metrics. To remedy this problem, many papers include prescriptions for scaling down the bet size. Such schemes are referred to as Fractional Kelly Betting. In this paper, we take the opposite tack. That is, we show that in many cases, the theoretical Kelly-based results may lead to bets which are "too conservative" rather than too aggressive. To make this argument, we consider a random vector X with its assumed probability distribution and draw m samples to obtain an empirically-derived counterpart Xhat. Subsequently, we derive and compare the resulting Kelly bets for both X and Xhat with consideration of sample size m as part of the analysis. This leads to identification of many cases which have the following salient feature: The resulting bet size using the true theoretical distribution for X is much smaller than that for Xhat. If instead the bet is based on empirical data, "golden" opportunities are identified which are essentially rejected when the purely theoretical model is used. To formalize these ideas, we provide a result which we call the Restricted Betting Theorem. An extreme case of the theorem is obtained when X has unbounded support. In this situation, using X, the Kelly theory can lead to no betting at all.Comment: Accepted in 2016 IEEE 55th Conference on Decision and Control (CDC

On the Benefit of Nonlinear Control for Robust Logarithmic Growth: Coin Flipping Games as a Demonstration Case

The takeoff point for this letter is the voluminous body of literature addressing recursive betting games with expected logarithmic growth of wealth being the performance criterion. Whereas almost all existing papers involve use of linear feedback, the use of nonlinear control is conspicuously absent. This is epitomized by the large subset of this literature dealing with Kelly Betting. With this as the high-level motivation, we study the potential for use of nonlinear control in this framework. To this end, we consider a “demonstration case” which is one of the simplest scenarios encountered in this line of research: repeated flips of a biased coin with probability of heads p , and even-money payoff on each flip. First, we formulate a new robust nonlinear control problem which we believe is both simple to understand and apropos for dealing with concerns about distributional robustness; i.e., instead of assuming that p is perfectly known as in the case of the classical Kelly formulation, we begin with a bounding set P⊆[0,1] for this probability. Then, we provide a theorem, our main result, which gives a closed-form description of the optimal robust nonlinear controller and a corollary which establishes that it robustly outperforms linear controllers such as those found in the literature. A second, less significant, contribution of this letter bears upon the computability of our solution. For an n-flip game, whereas an admissible controller has 2n−1 parameters, at the optimum only O( n2 ) of them turn out to be distinct. Finally, it is noted that the initial assumptions on payoffs and the use of the uniform distribution on p are made mainly for simplicity of the exposition and compliance with length requirements for a Letter. Accordingly, this letter also includes a new Section with a discussion indicating how these assumptions can be relaxed

Robustness of systems with uncertainties in the input

In B. R. Barmish (IEEE Trans. Automat. ControlAC-22, No. 7 (1977) 123, 124; AC-24, No. 6 (1979), 921-926) and B. R. Barmish and Y. H. Lin ("Proceedings of the 7th IFAC World Congress, Helsinki 1978") a new notion of "robustness" was defined for a class of dynamical systems having uncertainty in the input-output relationship. This paper generalizes the results in the above-mentioned references in two fundamental ways: (i) We make significantly less restrictive hypotheses about the manner in which the uncertain parameters enter the system model. Unlike the multiplicative structure assumed in previous work, we study a far more general class of nonlinear integral flows, (ii) We remove the restriction that the admissible input set be compact. The appropriate notion to investigate in this framework is seen to be that of approximate robustness. Roughly speaking, an approximately robust system is one for which the output can be guaranteed to lie "[var epsilon]-close" to a prespecified set at some future time T > 0. This guarantee must hold for all admissible (possibly time-varying) variations in the values of the uncertain parameters. The principal result of this paper is a necessary and sufficient condition for approximate robustness. To "test" this condition, one must solve a finite-dimensional optimization problem over a compact domain, the unit simplex. Such a result is tantamount to a major reduction in the complexity of the problem; i.e., the original robustness problem which is infinite-dimensional admits a finite-dimensional parameterization. It is also shown how this theory specializes to the existing theory of Barmish and Barmish and Lin under the imposition of additional assumptions. A number of illustrative examples and special cases are presented. A detailed computer implementation of the theory is also discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/24205/1/0000464.pd

Training as Usual: Can Therapist Behavior Change After Reading a Manual and Attending a Brief Workship on Cognitive Behavioral Therapy for Youth Anxiety?

There exists an ongoing movement to transport empirically supported treatments (ESTs), developed and evaluated in research clinics, to service providing clinics. ESTs refer to psychological interventions that have been evaluated scientifically (e.g., randomized controlled trial; RCT) and satisfy the Chambless and Hollon (1998) criteria (Kendall & Beidas, 2007). Dissemination research encompasses both dissemination (purposeful distribution of relevant information and materials to clinicians) and implementation (adoption and integration of EST in clinical practice) of ESTs (Lomas, 1993). However, for a variety of reasons (Addis & Krasnow, 2000; Riley, Schuman, Forman-Hoffman, Mihm, Applegate, & Asif, 2007), resistance to dissemination and implementation exists. We focus on training therapists in ESTs (i.e., dissemination). Thus, a key question arises: Do current training efforts practice in the community (i.e., reading a manual and attending a brief training workshop) effectively influence therapist behavior in those who are naïve to fundamental principles of an EST