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Portfolio drawdown control is a crucial but difficult aspect of investment management. Grossman and Zhou [1993] presented a framework in which the optimal risk level of a portfolio can be determined with the objective of maximizing the portfolio's long-term performance while controlling its drawdown within a preset limit. Other research in this area includes Cvitanic and Karatzas [1995], Klass and Nowicki [2005], and Yang and Zhong [2011], Similar to the setup in the Black-Scholes' option pricing model, Grossman and Zhou assumed that the required rebalance between the risky asset and cash can be executed continuously with no transaction costs. They also introduced the notion of "decayed drawdown" where portfolio drawdown is gradually "forgiven" or "discounted" over time at a certain decay rate. A higher decay rate would give rise to a higher long-term growth rate but at the expense of having less control of the absolute drawdown of the portfolio. The resulting optimal policy is to somewhat delay the risk reduction depending on the decay rate applied. A higher decay rate would result in increased delay in risk reduction and hence force the portfolio to have more "negative gamma" as drawdown approaches its limit, causing more aggressive de-risking and rerisking. In practice with transaction costs, the prescribed policy becomes difficult to implement in the vicinity of the drawdown limit. The notion of "decayed drawdown" is central to the flexibility and benefit of the Grossman-Zhou framework as well as to the difficulty experienced in its implementation. We will first review this notion, then we will present the two main results of this article and a summary of other risk-taking rules as special cases of the more general GrossmanZhou framework.
Our first result highlights the difficulty of implementing the Grossman-Zhou policy in the presence of transaction costs. We show that in a discrete-time setting with proportional transaction costs, drawdown control according to the Grossman-Zhou policy can make the expected return of the portfolio turn negative at a certain level of drawdown, become more negative as drawdown deepens, and approach negative infinity as the portfolio approaches its drawdown limit (assuming non-zero decay rate). For large enough drawdowns, the combination of the steepness of the Grossman-Zhou policy and transaction costs would push the portfolio toward its drawdown limit...