Closely related to CVaR-optimal portfolios are portfolio optimization problems that try to achieve weight solutions with respect to the portfolio’s draw-down. This kind of optimization was proposed by Chekhlov et al. (2000, 2005) and Pardalos et al. (2004).
A comparison between draw-down optimal portfolios and other risk-measure-related optimization techniques is made in Krokhmal et al. (2002). The task of finding optimal portfolio allocations with respect to draw-down is of considerable interest to asset managers. In an attempt to limit the chances of business termination, asset managers are eager to avoid either large portfolio draw-downs and/or draw-downs over an extended period of time. A client might be tempted to withdraw his mandate when this occurs, resulting in the loss of management fees. This point has been exemplified by Chekhlov et al. (2005) for a commodity trading adviser.
The draw-down of a portfolio at timetis defined as the difference between the maximum uncompounded portfolio value prior totand its value att. More formally, denote byW(𝝎,t) =y′t𝝎the uncompounded portfolio value at timet, with𝝎the port- folio weights for theNassets included in it andyt the cumulated returns. Then the draw-down,𝔻(𝝎,t), is defined as
𝔻(𝝎,t) = max
0≤𝜏≤t{W(𝝎, 𝜏)} −W(𝝎,t). (12.30)
k k The authors cited above deduced three functional risk measures from this
definition: maximum draw-down (MaxDD), average draw-down (AvDD), and conditional draw-down at risk (CDaR). Similar to CVaR, CDaR is dependent on the chosen confidence level𝛼, so that this concept constitutes a family of risk functions.
The definition of CDaR is akin to CVaR—see (12.27)—for a data set in the time interval[0,T]:
CDaR(𝝎)𝛼= min
𝜁
{
𝜁+ 1
(1−𝛼)T ∫
T 0
[𝔻(𝝎,t) −𝜁]+ }
dt, (12.31)
where𝜁 is a threshold value for the draw-downs, such that only(1−𝛼)T observa- tions exceed this value. If the number of observations is not an integer, then𝜁is the weighted average of draw-downs that strictly exceed this threshold value and its next lower bounded value, hence a computation similar to CVaR is utilized, but now CDaR is a risk functional and not a risk measure as in the case of CVaR, and it is expressed in nominal rather than in return space. The limiting cases of this family of risk functions are MaxDD and AvDD. For 𝛼→1, CDaR approaches the maximum draw-down:
CDaR(𝝎)𝛼→1=MaxDD(𝝎) =max0≤t≤T{𝔻(𝝎,t)}. The AvDD results for𝛼=0, that is, CDaR(𝝎)𝛼=0=AvDD(𝝎) =1∕T∫0T𝔻(𝝎,t)dt.
With respect to the optimization of a portfolio’s draw-down, these risk func- tionals can be implemented as inequality constraints for a fixed share of wealth at risk. For instance, one could require that the MaxDD is at most 10% of the capital:
MaxDD(𝝎)≤𝜈1C, where 0≤𝜈1≤1 andCis the available capital/wealth. Similarly, this draw-down constraint can be specified for AvDD, AvDD(𝝎)≤𝜈2C, and CDaR, CDaR(𝝎)≤𝜈3C, as can a linear combination of these three risk functionals, with 0≤𝜈1, 𝜈2, 𝜈3 ≤1. The portfolio optimization is then expressed in discrete terms and the objective is defined as maximizing the annualized average portfolio return:
R(𝝎) = 1
dCy′T𝝎, (12.32)
where dis the number of years in the time interval[0,T]. More precisely, the fol- lowing three linear program formulations are stated in Chekhlov et al. (2005) for optimizing a portfolio such that its MaxDD, AvDD, or CDaR is bounded by a certain wealth fraction. The task of maximizing the average annualized portfolio return with respect to limiting the maximum draw-down is given by
PMaxDD=arg max
𝜔∈Ω,u∈ℝR(𝝎) = 1 dCy′T𝝎, uk−y′k𝝎≤𝜈1C,
uk≥y′k𝝎, (12.33)
uk≥uk−1, u0=0,
k k whereudenotes a(T+1×1)vector of slack variables in the program formulation,
that is, the maximum portfolio values up to time periodkwith 1≤k≤T.
When the portfolio is optimized with respect to limiting the average draw-down, only the first set of inequality constraints needs to be replaced with the discrete ana- logue of the mean draw-down expressed in continuous time as stated above, leading to
PAvDD=arg max
𝜔∈Ω,u∈ℝR(𝝎) = 1 dCy′T𝝎, 1
T
∑T k=1
(uk−y′k𝝎)≤𝜈2C,
uk≥y′k𝝎, (12.34)
uk≥uk−1, u0=0.
The setup for the CDaR linear program is slightly more cumbersome compared to the previous two. In addition to the slack variables,u, that represent the maximum portfolio wealth levels, two additional auxiliary variables need to be introduced: first, the threshold draw-down value𝜁 dependent on the confidence level𝛼; and second, the(T×1)vectorzrepresenting the weak threshold exceedances. Hence, the linear programming problem is given by
PCDaR= arg max
𝜔∈Ω,u∈ℝ,z∈ℝ,𝜁∈ℝR(𝝎) = 1 dCy′T𝝎,
𝜁+ 1
(1−𝛼)T
∑T k=1
zk≤𝜈3C,
zk≥uk−y′k𝝎−𝜁, (12.35)
zk≥0, uk≥y′k𝝎, uk≥uk−1, u0=0.
These linear programs can be amended by further restrictions on𝝎such as budget, non-negativity, and/or box constraints.
A word of caution concerning these concepts is in order. The MaxDD as well as the AvDD might capture or reflect portfolio losses inadequately. Recall that historic return trajectories are employed in the linear program. Therefore, the portfolio allo- cation that obeys a maximum draw-down constraint can be severely impacted by a single worst draw-down. On the other hand, the average draw-down concept is incon- clusive about the size of the maximum portfolio’s draw-down, but this can become quite an important issue in terms of risk management and control. The CDaR measure
k k circumvents these pitfalls to some extent, but all measures assume that the historic
path dependencies will somehow prevail in the subsequent periods.