Drawdown and Its Linearity
In this article, we first discuss the conditional drawdown at risk (CDaR), its special cases — maximum drawdown and average drawdown — and then move to the optimization problem for the return–CDaR efficient frontier as discussed by Uryasev et al. 1
Assumption
This approach concentrates on the portfolio equity curve over a particular sample path (historical, or the most probable future sample path) — i.e. we define a sample-path risk function rather than a risk measure over a set of sample paths. It makes no assumption about the underlying probability distribution, which unlocks various practical applications.
CDaR
For some value of the confidence parameter \alpha, the \alpha-CDaR is defined as the mean of the worst (1-\alpha) \times 100\% drawdowns experienced over some period of time on a sample path.
The CDaR risk function is based on the CVaR concept and can be viewed as a modification of CVaR to the case where the loss function is defined as a drawdown. Similarly to CVaR — the average of all losses greater than or equal to \alpha-VaR — CDaR is the average of all drawdowns greater than or equal to the \alpha drawdown-at-risk (\zeta_\alpha), such that a low CVaR ensures a low VaR.
Risk functions are parameterized by the portfolio weight matrix W; T is the total time period considered for the calculation — the sample-path time horizon.
CDaR_\alpha(W) = \frac{1}{(1- \alpha)T} \int_{\Omega_\alpha} D(W,t) \,dt \tag{1}
where \Omega(W) = \{t \in [0,T] : D(W,t) \geq \zeta_\alpha(W)\}.
By definition, with a specified probability \alpha, the \alpha-DaR (\zeta_\alpha) of a portfolio is the lowest amount such that the probability the drawdown will not exceed \zeta_\alpha is greater than or equal to \alpha.
We consider two risk functions that are special cases of CDaR — maximum drawdown (MaxDD) and average drawdown (AvDD) — which are limiting cases as \alpha \to 1 and \alpha \to 0 respectively:
MaxDD(W) = \max_{t \in [0,T]} D(W,t) \tag{2}
AvDD(W) = \frac{1}{T} \int_{0}^{T} D(W,t) \,dt \tag{3}
Linear optimization
MaxDD
We consider a modification of the classic return–risk efficient-frontier optimization: maximize return while constraining maximum drawdown to a pre-defined level.
\max_{W}{W^TR}
constrained on
MaxDD(W) \leq v_1 C
where R is the return matrix R = [r_1, r_2, r_3, \dots, r_\gamma], \gamma is all constituents, and v_1 is the percentage of deployed capital C allowed to be in drawdown.
Maximum drawdown is not linear in the parameter W. It can be made linear using an auxiliary variable, as follows:
MaxDD(W) : \max_{k \in (0,T)} DD_k \leq v_1 C \tag{4}
MaxDD(W) : \max_{k \in (0,T)} \Big( \big(\max_{j \in (0,k)} W^TR_j\big) - W^TR_k \Big) \leq v_1 C \tag{5}
Converting the first maximum function in Equation 5, using the property of maximum, applies a constraint on each day to be less than the desired drawdown:
MaxDD(W): \big(\max_{j \in (0,k)} W^TR_j\big) - W^TR_k \leq v_1 C \quad \forall\ k \in [0,T] \tag{6}
Converting the second max function:
MaxDD(W) : u_k - W^TR_k \leq v_1 C \quad \forall\ k \in [0,T] \tag{7}
where u_k = \max_{j \in (0,k)} (W^TR_j) is an auxiliary variable.
Defining u_k in Equation 7: u_k is equal to a maximum function, hence it’s an increasing function.
u_k \geq u_{k-1} \quad \forall\ k \in [0,T],\ \text{with } u_0 = 0 \tag{8}
Also, u_k should be equal to or greater than W^TR_j:
u_k \geq W^TR_j \quad \forall\ j \in [0,k],\ k \in [0,T] \tag{9}
W_{min} \leq W \leq W_{max} \tag{10}
So the optimization is converted to a convex optimization problem with a linear performance function and a piecewise-linear convex constraint, given by maximizing return subject to Equation 6 through Equation 10.
CDaR
\max_{W}{W^TR}
constrained on
CDaR_\alpha(W) \leq v_2 C
Rewriting Equation 1:
CDaR_\alpha(W) = \min_{\zeta}\Big[\zeta+\frac{\sum_{k \in (0,T)}[DD_k-\zeta]^+}{(1-\alpha)T}\Big] \tag{11}
where the function [x]^+ is defined as \max(0,x).
The CDaR constraint function can be linearized as follows:
\zeta+\frac{\sum_{k \in (0,T)}z_k}{(1-\alpha)T} \tag{12}
Defining z_k:
z_k = [DD_k-\zeta]^+
Since z_k is the positive part:
z_k \geq DD_k - \zeta \quad \forall\ k \in [0,T] \tag{13}
Using Equation 7, DD_k can be written in terms of u_k:
z_k \geq u_k - W^TR_k - \zeta \quad \forall\ k \in [0,T] \tag{14}
And since z_k is always positive:
z_k \geq 0 \tag{15}
Similarly to MaxDD, the optimization problem converts to maximizing return subject to Equation 14, Equation 15, and Equation 8 through Equation 10.
An important feature of this formulation is that it does not involve the threshold function \zeta_\alpha directly. At the optimal solution, the variables W and \zeta_\alpha together give the optimal portfolio and the corresponding value of the threshold function — see 2 for further discussion.
Footnotes
Alexei Chekhlov, Stanislav Uryasev, Michael Zabarankin. Portfolio Optimization With Drawdown Constraints. ise.ufl.edu/uryasev↩︎
R. Tyrrell Rockafellar, Stanislav Uryasev. Optimization of Conditional Value-at-Risk. ise.ufl.edu/uryasev↩︎