Drawdown Minimization

One measure of riskiness of an investment is “drawdown”, defined, most often in the asset management space, as the decline in net asset value from a historic high point. Mathematically, if the net asset value is denoted by Vt , t ≥ 0, then the current “peak-to-trough” drawdown is given by Dt = Vt − max0≤u≤t Vu. The maximum drawdown, max0≤u≤t Du, is a statistic that the CFTC forces managed futures advisors to disclose, and so many investment advisors and managers implicitly face drawdown constraints in setting their investment strategies. Hedge funds, for example, implicitly face drawdown constraints in that many multiperiod hedge fund contracts reflect investor preferences related to the maximum drop in a fund’s asset value from the previous peak. These often include a high-water-mark provision that sets the strike price of each period’s incentive fee equal to the all-time high of fund value (see Hedge Funds). Another measure of riskiness that is related but often confused terminologically with drawdown is that of “shortfall”, which is simply the gap, or loss level of the current value from the initial or some other given value. This value could be constant but more often is determined by a stochastic exogenous or endogenous benchmark. For example, the shortfall with respect to the endogenous benchmark of the running maximum is the drawdown. Since drawdown and shortfall are essentially equivalent in single period models, the research on the topic reviewed in this article is focused on multiperiod and, in particular, on continuous-time models pioneered in [16] and [17] where optimal portfolio rules are derived by solving a multiperiod portfolio optimization problem. The minimization of short-fall probability in a single-period model dates back to [18] and [21]. See Value-at-Risk; Expected Shortfall for work on portfolio selection with drawdown constraints in single period mean-variance models. In the continuous time framework, there is an implicit nonnegativity constraint on wealth, which is one form of a shortfall constraint (see Merton Problem). The models reviewed here differ in their assumptions regarding investment horizons (finite or infinite), constraints (fixed or stochastic benchmark), stochastic processes (diffusion with and without jumps), as well as objective function (purely probabilistic or expected utility based). In general, without transactions costs, the incorporation of drawdown constraints induces a portfolio insurance strategy: specifically, in the stationary stochastic model case, the strategy is that of a constant proportions portfolio insurance (CPPI) with different “floor” levels determined by the horizon and the objective (see Transaction Costs for portfolio optimization with transaction costs). In this case, the risky asset price is assumed to follow a geometric Brownian motion with drift μ+ r , and diffusion coefficient σ 2 where r is the rate of return on cash. Hence, as is standard (see Merton Problem), the dynamics of the investor’s wealth portfolio are given by