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Dynamic Mean-LPM and Mean CVAR Portfolio Optimization in Continuous-Time

Pub Date

2014.2.14

Abstract

Instead of controlling symmetric risks measured by central moments of investment return or terminal wealth, more and more portfolio models have shifed their focus to manage ‘asymmetric’ downside risks that the investment return is below certain threshold(对投资组合的关注重点从对称风险转移到了非对称风险)，Among the existing downside risk measures, the lower-partial moments (LPM) and conditonal value-at-risk(CVaR) are probably most promising. In this paper we investigate the dynamic mean-LPM and mean-CVaR portfolio optimization problems in continuous-time, while the current literature has only witnessed their static versions. Our contributions are two-fold, in both building up tractable formulations and deriving corresponding analytical solutions.（推导出动态情况下的模型和解析解），By imposing a limit funding level on the terminal wealth, we conquer the ill-posedness exhibited in the class of mean-downside risk portfolio models. The limit funding level not only enables us to solve both dynamic mean-LPM and mean-CVaR portfolio optimization problems, but also offers a flexiability to tame the aggressiveness of the portfolio policies generated from such mean-downside risk models. More specifically, for a general market setting, we prove the existence and uniqueness of the Lagrangian multipliers, which is a key step in applying the martingale approach. Moreover, for situations where the opportunity set of the market setting is deterministic, we derive analytic portfolio policies for both dynamic mean-LPM and mean-CVaR.

Introduction

使用方差度量风险考虑了均值收益的两侧不确定性，因此学者提出了考虑给定目标的下行风险的度量，其中以Fishbburn¹提出的LPM测度最为著名.

The LPM enables us to represent a general form of downside risk measures with two parameters, the benchmark level $\gamma$, which is set by the investor himself, and the order of the order of the moments, $q$, which represents the risk attitude of the investor. Due to the freedom offered by different combinations of the pair $q$ and $\gamma$, we can adopt LPM to pursue different investment goals in portfolio optimization.

在LPM模型中，设置$q=0$等效于Roy提出的safety-first准则，设置$q=1$为expected regret(ER)风险测度，设置$q=2$表示指定目标下的半离差(semideviation)或者当$\gamma$设置为末期期望财富时的半方差(semivariance). Konno等人证明了在投资组合管理中LPM表现出的卓越性能，Zhu等人在LPM风险测度下建立了robust portfolio.

The VaR, defined as the threshold point with a specified exceeding probability of great loss, becomes popular in the financial industry since the mid 90s. However, VaR fails to satisfy the axiomatic system of coherent risk measures proposed by Aetzner et al. More critically, the non-convexity of VaR leads to some difficulty in solving the corresponding portfolio optimization problem. On the other hand, the conditional Value-at-Risk (CVaR), also known as expected shortfall, is defined as the expected value of the loss exceeding the VaR. CVaR possesses several good properties, such as convexity, monotonicity and homogeneity.(CVaR具有的一些比较好的性质). Rockafellar and Uryasev prove that CVaR can be computed by solving an auxiliary linear programming problem in which the VaR needs not to be known in advance.

目前CVaR已经广泛应用于衍生品投资组合，信用风险优化和鲁棒投资组合管理中. 上述文献中建立的模型均为静态模型，本质上是buy-and-hold，从长期投资的角度来看，这种策略是不合适的，过去几十年间，很多随机规划的方法被提出用于求解mean-cvar问题，随机规划模型同时使用了时间离散化和状态离散化，但是这种方法的计算成本很高，一般只能进行两阶段或者三阶段的计算，在动态均值-风险模型中，目前最为成熟的方法是动态MV模型优化. 由于方差项的不可分离性导致动态规划下建立MV模型发展停滞了80年. 通过嵌入结构(embedding scheme), Li和Ng²推导出离散时间状态下的动态MV策略以及Zhou和Li³推导出连续时间下的动态MV策略，近年来，动态MV模型中的时间一致性(time consistency)成为了研究热点⁴，尽管从表面上看mean-downside risk模型似乎是动态MV模型的自然扩展，但是Jin等人⁵证明了在连续时间下，一般的mean-downside risk portfolio优化模型的最优解是难以得到的.

Jin et al. show that a general class of mean-downside risk portfolio optimization models under a continuous-time setting is ill-posed in the sense that the optimal value cannot be achieved.

Chiu等人⁶在safety-first criteria下求解了dynamic asset-liability management问题，本文采用类似的解决方案，通过设置问题上界的方法求解

we adopt a similar solution idea to attach to this class of problems an upper limit on the funding level of the terminal wealth.

这种设置的上界相当于设计一个变量去控制投资者的贪心程度，在求出mean-LPM和mean-CVaR的数值解之前，本文先论证了理论基础，当市场机会集为确定时(market opportunity set is deterministic)，进一步推导出了半解析（semi-analytical）投资组合策略.

The dynamic mean-LPM portfolio policy demonstrates very distinct features when compared with the dynamic MV portfolio policy. When the market condition is good, the mean-LPM investor tends to invest more aggressively in risky assets when compared to an MV investor. When the market condition is in the medium state, the mean-LPM investor prefers to allocate more wealth in the risky asset. However, when the market condition is in a bad state, the mean-LPM investor allocates again more wealth in the risky assets than MV investor. this phenomena can be regarded as the gambling effect of dynamic mean-LPM investors.

论文中使用的符号表如下

SymbolsMeanings$1_{\mathcal{B}}$示性函数$A^{\prime }$矩阵$A$的转置$a_{+}$$a$的非负部分，$a_{+}=a{1}_{a\ge 0}$$(a{)}_{+}^q$$(a{)}_{+}$的$q$次方变量$X$的标准正态分布的cdf函数$\Phi (y):=\mathbb{P}(X\le y)=\frac{1}{\sqrt{2\pi }}{\int }_{\infty }^y\exp (-\frac{s^2}{2})ds$

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Mean-risk analysis with risk associated with below-target returns ↩︎

Optimal Dynamic Portfolio Selection Multiperiod Mean-Variance Formulation ↩︎

continuous-time mean variance portfolio selection_a stochastic LQ framework ↩︎

dynamic mean variance asset allocation ↩︎

Continuous-time mean-risk portfolio selection ↩︎

Roy’s safety-first portfolio principle in financial risk ↩︎