On Data-Driven Drawdown Control
with Restart Mechanism in Trading

To remedy the equal riskiness on the dispersion risks, many surrogate risk measures are proposed to pay attention to the downside risks. This includes Value at Risk (VaR); see Jorion (2000), Conditional Value at Risk (CVaR); see Rockafellar and Uryasev (2000), absolute drawdown; see Magdon-Ismail and Atiya (2004); Hayes (2006), conditional expected drawdown (CED), the tail mean of maximum drawdown distributions, see Goldberg and Mahmoud (2017), and the more general coherent risk that axiomatize the risk measures; see Luenberger (2013); Shapiro et al. (2021). See also Korn et al. (2022) for an empirical study of comparing various drawdown-based risk metrics. In this paper, we focus on a more practical drawdown measure, the maximum percentage drawdown, the maximum percentage drops in wealth over time, as the risk measure.

Different types of drawdown and methodologies are studied extensively in the existing literature. For example, optimal drawdown control problem in a continuous-time setting are studied in Grossman and Zhou (1993); Cvitanic and Karatzas (1994); Chekhlov et al. (2005); Malekpour and Barmish (2012, 2013). See also Boyd et al. (2017) for a study on multiperiod portfolio optimization problems involving drawdown as a constraint in a discrete-time setting. A recent study that uses deep reinforcement learning to address practical drawdown issues can be found in Wu et al. (2022).

Among all the existing papers, the prior works in Hsieh and Barmish (2017a, b) are the most closely related papers, where the key result, the so-called drawdown modulation lemma, is proved. Roughly speaking, it shows a necessary and sufficient conditions for a broad class of control policies that guarantees an almost sure maximum percentage drawdown protection for all time. However, Hsieh and Barmish (2017a) indicate that, in practice, when the prespecified drawdown limit is approaching, the trading policy may behave like a stop-loss order, and the trading may be stopped; see also Hsieh (2022a) for a study on a class of affine policies with a stop-loss order. To this end, we extend the drawdown modulation policy with a novel restart mechanism to remedy the stop-loss phenomenon.

This paper extends the existing drawdown modulation policy with a novel restart mechanism. The preliminaries are provided in Section 2. We formulate an optimal drawdown control problem for a two-asset portfolio with one risky and one riskless asset. Then we extend the existing drawdown modulation theory so that the riskless asset is explicitly involved; see Lemma 3.1. The necessary and sufficient conditions for a broad class of control policy which we call the drawdown modulation policy are provided. Then, the modulation policy with a restart mechanism is discussed in Section 3. The idea of the restart is simple: When the percentage drawdown up-to-date is close to the prespecified drawdown limit, we restart the trades with an updated policy. We also provide numerical examples involving ETF and cryptocurrency historical price data to support our findings; see Section 4.

Fix an integer $N>0.$ For stage $k=0,1,\dots,N$, we let $S(k)>0$ denote the prices of the underlying financial asset at stage $k$. The associated per-period returns are given by $X(k):=\frac{S(k+1)-S(k)}{S(k)}$ and the returns are assumed to be bounded, i.e., $X_{\min}\leq X(k)\leq X_{\max}$ with $X_{\min}$ and $X_{\max}$ being points in the support, denoted by $\mathcal{X}$, and satisfying $-1<X_{\min}<0<X_{\max}.$ For the money market asset, e.g., bond or bank account, we use $r_{f}(k)$ to denote the interest rate at stage $k$.

Beginning at some initial account value $V(0)>0$, consider a portfolio consisting of two assets, with one being risky and the other being a riskless asset with interest rate $r_{f}(k)\in[0,X_{\max}]$ for all $k$ almost surely. For stage $k=0,1,\dots$, we let $V(k)$ denote the account value at stage $k$. Then the evolution of the account value dynamics is described by the stochastic recursion

Given a prespecified drawdown limit $d_{\max}\in(0,1)$, we focus on conditions on selecting a policy $u(k)$ under which satisfaction of the constraint $d(k)\leq d_{\max}$ is assured for all $k$ with probability one where $d(k)$ is the percentage drawdown up to date $k$, which is defined below.

Consistent with Hsieh and Barmish (2017a), fix the prespecified drawdown limit $d_{\max}\in(0,1)$. With the aid of Lemma 3.1, one can readily obtain a class of policy functions $u(k)$ expressed as a linear time-varying feedback controller parameterized by a gain $\gamma$, leading to the satisfaction of the drawdown specification. Specifically, we express $u(k)$ in the feedback form

with $K(k):=\gamma M(k)$ and

Equation (1) is called the drawdown modulation policy, which is parameterized by the two parameters $(\gamma,d_{\max})$.

Proof. Since the drawdown modulation policy satisfies Lemma 3.1, it assures $d(k)\leq d_{\max}$ for all $k$ with probability one. Therefore, it follows that

Having obtained the drawdown modulation policy (1), a natural question arises of how to select an “optimal” $\gamma.$ To this end, we define the total return up to terminal stage $N$ as a ratio

where the subscript of $R_{\gamma}(\cdot)$ is used to emphasize the dependence on the gain $\gamma$. Define $J(\gamma):=\mathbb{E}[R_{\gamma}(N)]$. Then, we consider a multiperiod drawdown-based stochastic optimization problem

subject to

It is readily verified that

Therefore, we rewrite the problem as the following equivalent form

In the sequel, we shall use $\gamma^{*}$ to denote a maximizer of the optimization problem above. In practice, if one view that the optimum $\gamma^{*}$ obtained may be too aggressive, a practical but arguably suboptimal way is to introduce an additional fraction, call it $\alpha$, that is used to shrink the investment size; see Maclean et al. (2010) for a similar idea. That is, instead of working with $\gamma^{*}$, one may work with $\alpha\gamma^{*}$ where $\alpha\in(0,1]$.

As mentioned in Section 1, while the derived drawdown modulation policy provides almost sure drawdown protection, it may incur a stop-loss behavior. To remedy this, we now introduce a restart mechanism into the modulation policy. Specifically, let $\varepsilon\in(0,d_{\max})$ be a prespecified threshold parameter. Then we set the threshold for restarting the trade by

If, at some stage $k=k_{0}$, Inequality (3) is satisfied, the trading is restarted by re-initializing $d(k_{0}):=0$ and reset the time-varying gain function $K(k)$ of the modulation policy $u(k)=K(k)V(k)$ at that stage $k=k_{0}$ as

where $\alpha e^{-k_{0}/N}$ represents a forgetting factor with fraction $\alpha\in(0,1]$ mentioned previously. Then we continue the trade until the next restart stage or to the terminal stage $N$.

Consider the Vanguard Total World Stock Index Fund ETF (Ticker: VT)²²2VT invests in both foreign and U.S. stocks; hence it can be viewed as a good representative of the global stock market to be the risky asset covering a one-year in-sample period for optimization from January 02, 2019 to January 02, 2020, and the out-of-sample period from January 02, 2020 to September 20, 2022, which contains a total of $N=684$ trading days; see Figure 2 where the cyan colored trajectory is used for in-sample optimization and the blue colored trajectory is used for the out-of-sample trading test. It should be noted that due to the COVID-19 pandemic, the considered prices covering the 2020 stock market crash from February 20, 2020 to April 7, 2020, and a recovery period after the crash. Thus, we view this dataset as an excellent back-testing case for our proposed drawdown modulation policy with the restart.

Without loss of generality, consider the initial account value to be $V(0):=\$1$. To implement the drawdown modulation policy with restart, we set $d_{\max}:=0.1$ and restart threshold $\varepsilon:=d_{\max}/10$. With the data collected in the in-sample period, the corresponding feasible set is $\Gamma\approx[-30.79,\;34.9]$. Then we numerically solve the optimization problem (3.2) via Monte-Carlo simulations. It follows that any $\gamma\in(8.5,34.9)$ share an almost identical optimal value when the cash-financing condition is imposed. For the sake of risk-averseness, we pick the infimum of the candidates, i.e., $\gamma^{*}:=8.5$. Using this $\gamma$, we obtain the drawdown modulation policy $u^{*}(k)=\gamma^{*}M(k)V(k).$

The corresponding trading performance is shown in Figure 2, where the green dots indicate that the trade is restarted. In the same figure, we also compare it with the standard buy-and-hold strategy³³3Here, we mean buy and hold on the risky asset VT with $K(k)=1$. and the modulation policy without restart. We see clearly that the modulation policy with restart leads to superior performance. Some key performance metrics, including maximum drawdown and cumulative rate returns, and $N$-period Sharpe ratio⁴⁴4The per-period Sharpe ratio is $SR:=\frac{\mu-r_{f}}{\sigma}$ where $\mu$ is the per-period sample mean return, $\sigma$ is the per-period sample standard deviation, and $r_{f}$ is the per-period riskless return. are reported in Table 1.

As a second example, we consider a portfolio consisting of cryptocurrency BTC-USD and a riskless asset. The BTC-USD asset covers the same in-sample and out-of-sample periods described in Example 4.1. From January 02, 2020 to September 20, 2022, it has a total of $N=993$ trading days⁵⁵5It is worth noting that the cryptocurrency is typically traded at 24 hours a day, seven days a week. Therefore, it has longer tradings days than that trades with VT in Section 4.1.; see Figure 3. The corresponding feasible set for $\gamma$ is $\Gamma=[-5.761,\;7.083]$.

Take $d_{\max}:=0.2$ and restart threshold $\varepsilon:=d_{\max}/10$. By solving the optimal drawdown control problem (3.2), we obtain $\gamma^{*}\approx 5.138$; see Figure 4 for $J(\gamma)$ versus $\gamma\in\Gamma$. Note that the $J(\gamma)$ is clearly not concave for $\gamma\in\Gamma$. To consider the volatile nature of cryptocurrency and unforeseen estimation error, we consider using a fractional $\gamma^{*}$ by $\alpha\gamma^{*}$ with $\alpha=1/2$. That is, $u^{*}(k)=K^{*}(k)V(k)$ with $K^{*}(k)=\frac{\gamma^{*}}{2}M(k)V(k)$. The trading performances using drawdown modulation policy with and without restart, and buy-and-hold strategy are shown in Figure 5, where the green dots indicate that the trades were restarted. Some performance metrics are summarized in Table 2 where we see that the modulation with restart provides the highest Sharpe ratio among all the other strategies.