Portfolio Problems With Transaction Costs Based On CVaR

Portfolio selection is one of the critical problems of modern finance theory. The mainproblem which portfolio solves is how to distribute some amount of capital into differentkinds of assets to make sure that the return is the maximum under the condition that therisk is below some scalar or the risk is the minimum under the condition that the return isgreater than the given constant. On the other hand, transaction costs are popular kinds offrictions in the real market. The effect of transaction costs to the portfolio optimization isdominant. It will lead to invalid portfolio if transaction costs are neglected and transactioncosts also have a great effect to the portfolio selection. This paper considers transactioncosts into portfolio problems based on CVaR(Conditional Value at Risk), and gains resultsas follows:In Chapter 2, there lists some propaedeutics mostly about the definition of CVaRand some of its characteristics widely used.In Chapter 3, transaction costs are considered into the process of portfolio based onthe document 40 and the mean-CVaR model with transaction costs is constructed. Themodel is changed into a stochastic convex programming with expected function in theconstraint by an auxiliary function substituting for the CVaR function and the changeis proved to be equivalent to the formal problem. When the return vector of the assetsis continuously distributed, Monte Carlo method is used to change the stochastic pro-gramming into the deterministic programming. Then using the smoothing technique andthe linearity technique to deal with the model's non-smooth part, the model is translatedinto two problems: the non-linear programming problem and the linear programming.Numerical tests are performed and we find that compared with the smoothing methodthe linearity method has the advantage of small error but also has disadvantages suchas the scale of the linear programming increases with the sampling number, the non-zeronumber of the assets is so small that can't give enough information to the investor andso on. Numerical tests also compare the results when the transaction costs are from zeroto gradually increasing value and demonstrate that the selection, the return and the riskof the portfolio modify a lot after the transaction costs are contained in the model.In Chapter 4, portfolio model with transaction costs is constructed, which is a singleobjective and considers minimizing sum of the risk and the minus return under the con- straint of the total money unchangeably. When the random vector is discrete distribution,the model is changed into a linear two stage problem of recourse by using the auxiliaryfunction and the linearity technique. Based on the idea of Benders decomposition andthe L-shaped method, we solve the problem. Numerical tests are executed in two aspects.The effect of transaction costs to the portfolio in this model is analyzed by changing thetransaction costs. The proportions of the assets and the return, have modified very much.Numerical tests also give the graph of the relationship between the confidence level andthe risk.Through the analysis of the above two different model, results can be found that, inthe small scale, the effect of transaction costs to the portfolio optimization is dominantand the return and the proportions of the assets of the portfolio both have a large amountof change.