بهينه‌سازي استوار سبد مالي با استفاده از رويكرد ارزش در معرض خطر شرطي موزون

Robust Optimization of the Portfolio Selection Problem using Weighted Conditional Value At-Risk Approach

در اين نوشتار مسيله‌ي انتخاب سبد مالي با استفاده از رويكرد بهينه‌سازي استوار مورد بررسي قرار گرفته است. بدين‌منظور، ارزش در معرض خطر شرطي موزون (WCVaR) ــ كه تركيبي است از چند ارزش در معرض خطر شرطي با سطوح اطمينان مختلف ــ به‌عنوان معيار بهينه‌سازي در نظر گرفته شده است. ارزش در معرض خطر شرطي موزون (WCVaR) يك مدل خطي است كه از لحاظ محاسباتي كارايي بالايي دارد. مهم‌ترين ويژگي اين مدل توليد جواب‌هايي با خصوصيت چيرگي تصادفي است. توسعه‌ي صورت گرفته در اين مدل، در نظر گرفتن داده‌هاي غير‌قطعي است و بازده‌هاي انتظاري نيز غير‌قطعي‌اند؛ از اين رو براي بررسي داده‌ها از رويكرد استوار استفاده مي‌شود. افزون براين، براي توسعه‌ي مدل ارايه شده در اين مقاله از برخي محدوديت‌ها براي حفظ تنوع‌پذيري نيز استفاده شده است.

In this paper, we develop a robust model for the portfolio selection problem under uncertainty conditions. We construct a robust model, whose risk measure is weighted conditional value at-risk (WCVaR). WCVaR is a combination of the conditional value at-risk (CVaR) with several tolerance levels. We use combinations of the conditional value at-risk (CVaR) measures to get some approximations of the tail Gini’s mean difference, with the advantage of being computationally much simpler than the Gini’s measure itself. The studied model is SSD consistent and LP computable. In stochastic dominance, uncertain returns (modeled as random variable) are compared by the point wise comparison of some performance functions constructed from their distribution functions. The first performance function is defined as the right-continuous cumulative distribution function, and it defines first degree stochastic dominance (FSD). The second function is derived from the first, and it defines second degree stochastic dominance (SSD). WCVaR is a safety measure with uncertain returns. To handle the parameter uncertainty problem, there is a recent research trend in the development of new robust optimization approaches. Traditional optimization methods require full knowledge of parameters to allow transformation to a stochastic program. From full information, assumptions of parameters following specific known distributions can sometimes be too strong and their validity criticized. The last approach of robust optimization proposes a robust formulation that is linear, applicable, deterministically solvable, and extendable to discrete optimization without the loss of feasibility of solution. Another advantage of the model is its ease in controlling the level of conservatism. In this paper, we apply this approach and develop the robust weighted conditional value at-risk in a portfolio selection problem. We also show the performance of robust optimization in the flexibility of financial markets.