A Study On Conditional Risk At Value Models

Value-at-Risk (VaR) is a wide-used, comprehensible and calculable method for risk management in financial field. Now, VaR has achieved its great success in practice. With respect to a specified probability level α , the VaR loss value of a portfolio is the lowest amount y such that, with probability α, the loss will not exceed y. The concept of Conditional Value at Risk (CVaR) was presented mainly to solve the problem that VaR is not sub-additive. With a specified probability α, the CVaR loss value is the conditional expectation of losses above the VaR. CVaR overcomes several limitations of VaR and has good properties, especially its good computability. If the object function is CVaR, we can get CVaR loss value by using linear optimization technique. At the same time, we also can obtain the VaR value. Since CVaR is a relatively new area for risk management, there are several issues to be solved in both theory and practice.Now, the main research on CVaR focuses on single loss and continuous type of the loss function. It consists of VaR loss value and CVaR loss value with a specified probability. The main problem is to find the portfolio to achieve the minimal VaR loss value and the minimal CVaR loss value, with the key being to prove an equality theorem, i.e., the solution of this problem equal to the solution of another problem which is more easier to be solved.This paper studies CVaR models with discrete type of loss, multiple losses and over multiple periods based on above-mentioned model. Moreover, we show that the optimization problem for solving CVaR is equivalent to another optimization problem, which is easier to be solved. Concretely, we study the CVaR model for single loss with discrete type of loss, CVaR model for multiple losses with discrete type of losses, CVaR model for multiple losses with continuous type of losses and CVaR model over multiple periods. There are six chapters in this paper. In Chapter 1, we introduce the concepts concerning the VaR model. Then, in Chapter 2, we give the status of the research for VaR and CVaR. In Chapter 3 to Chapter 6, we study the CVaR models for single loss with discrete type of loss, the CVaR model for multiple losses with discrete type of losses, the CVaR model for multiple losses with continuous type of losses and the CVaR model over multiple periods, respectively.The four aspects of our results are as follows.1. We study the CVaR model for single loss with discrete type of loss. First, wesuppose the random vector representing the uncertainties that affect the loss is discrete type, and its value is finite. Here, the main result of CVaR model is not equal to that of CVaR model for single loss with continuous type of loss. We get the corresponding a -VaR and a -FCVaR through solving another optimization problem, which is easier to be solved. Then, we use the above model and method to analyze some examples for portfolio selection. Finally, we discuss the application to hedging.2. We study the CVaR model for multiple losses with discrete type of losses, where the random vectors representing uncertainties that affect the loss are multiple with discrete type of losses. First, we consider the CVaR model for multiple losses of multiple a -VaR. Giving the confidence level for every loss function, we introduce the concept of the vector a -CVaR, the vector a -CVaR, the vector a -FCVaR and the corresponding multi-objective CVaR optimization problems, and show that the a -VaR and a -FCVaR can get from solving another optimization problems, which is easier to be solved. Then, we consider the CVaR model with multiple losses based on weights. We introduce the concept of a -CVaR, a -CVaR, a -FCVaR based on the given weights, and show that the a -VaR and a -FCVaR can be obtained by solving other optimization problems, which are easier to be solved. Furthermore, we discuss the CVaR model of minimal a -VaR with a specified confidence level based on the given weights. Finally, we use the above models and methods to analyze some examples for portfolio selection and hedging with a -VaR and a -FCVaR.3. We study the CVaR model for multiple losses with continuous type of losses, where the random vectors representing uncertainties that affect the losses are multiple with continuous type. First, giving the confidence level for each loss function, we introduce the concept of vector a -VaR, vector a -CVaR and the corresponding multi-objective CVaR optimization problems. We show that the vector a -VaR and the vector a -CVaR are equivalent to the results of another single objective optimization problem, which is easier to be solved. Then, we consider the CVaR model for multiple losses with a specified probability based on the weights. We introduce the concept of a -VaR, a -CVaR and the corresponding multi-objective CVaR optimization problems. We also show that the or-VaR and a -CVaR can be obtained by solving another optimization problem, which is easier to be solved. Furthermore, we study the CVaR model of the minimal a -VaR and the maximal a -VaR with a specified probability based on weights. Finally, we use the above models and methods to analyze portfolio selection problems and hedging problems.4. We study the CVaR model over multiple periods. First, we consider the casewhere the state transference over periods is deterministic. Under the given confidence levels for each period, we introduce the concept of a -VaR, a -CVaR and the corresponding multi-objective CVaR optimization problem over the whole process. We show that the CVaR optimization problem can be expressed by a set of recursive equations. Moreover, we transform the set of equations into another set of recursive equations that have better properties. Then, under a given confidence level of the whole process, we introduce the concept of a -VaR, a -CVaR and the corresponding multi-objective CVaR optimization problem based on weights. Under some conditions, we show that the CVaR optimization problem can be transformed into a single objective optimization program. Finally, we consider the CVaR model over multiple periods under Markov state transference. We present and study three optimization value functions of CVaR for infinite horizons, finite horizons and terminal loss, respectively.