Optimal Investment Under a Hard Drawdown Constraint
Under what model can a portfolio respect a running-maximum wealth floor?
Grossman and Zhou solve an investment problem with a running-maximum wealth floor inside a continuous-time Brownian model and a specified utility objective. Their result relies on those distributional assumptions; a discrete strategy with gaps needs separate pathwise testing.
The Grossman-Zhou policy is derived for a continuous-time Brownian market with a running-maximum floor and stated utility.
Evidence map
| Aspect | Finding |
|---|---|
| What it is | Grossman and Zhou study a continuous-time investment problem in which wealth is kept above a fixed fraction of its running maximum. |
| Key result / formula | The constraint is W_t at least alpha times M_t, where M_t is the highest wealth reached by time t and alpha lies between zero and one. |
| Why it matters for backtesting | The paper supplies a precise hypothesis for a sizing experiment: compare a rule based on distance to the running high-water-mark floor with a fixed-fraction rule at matched average exposure and the same transaction costs. |
What it is
Under their Brownian-market model and constant relative risk aversion utility, the policy changes risky exposure as the distance to that floor changes.
Key result / formula
The authors' optimal risky investment under the stated utility and market assumptions is proportional to the surplus W_t minus alpha M_t. This is a high-water-mark floor, which moves upward after new highs. The result is derived inside a continuous-price, continuously adjustable model; it is not a guarantee that a discrete backtest or live account can maintain the same floor. A gap through the limit, delayed execution, financing costs or a maximum order size can break the model-to-market transfer.
Why it matters for backtesting
This note does not establish that Stochastly currently offers that sizing rule as a runnable node. If it is implemented, test the actual order path and report how often the simulated floor is crossed under bar gaps or intrabar uncertainty. Estimate the return distribution only on training data, freeze the sizing parameters before the next period, and report both missed upside and observed drawdown. The theoretical optimum cannot be transferred to trading from an in-sample equity curve alone.
For a discrete trading strategy, the practical comparison is between observed wealth paths and a separately specified floor rule. Sampling only daily closes can miss an intraday breach. Report the monitoring frequency and the gap between continuous-time assumptions and the available observations.
Source
Grossman & Zhou, "Optimal Investment Strategies for Controlling Drawdowns", Mathematical Finance 3(3), 1993, 241-276. Primary source