Developing Studies On The Coherence Of Risk Measures

Markowitz issued a breakthrough paper "Portfolio Selection" in 1952, which opened a modern financial study area. This paper presented a mean-variance analysis model (M-V model), establishing the investment portfolio choice theory under uncertain conditions. Markowitz firstly quantified investment risk as the variance about an asset's expected return, also served as a prelude to the modern financial risk measurement study. Because of the pioneering work, Markowitz in 1990 won the Nobel Prize for Economics. M-V model was the method to solve risk-return issue, which is the most fundamental problem in financial investment domain. The M-V model known as the "the first revolution of Wall Street" became the main research topic on financial investment theory and the important decision-making tool for practice. Today, the M-V model still occupies a dominant position in the modern portfolio theory and practice.From the variance (Markowitz, 1952) to the Value at Risk (J. P. Morgan, 1994), the quest for a more reasonable risk measure has never stopped. One of the most important recent achievements in the financial risk area is the theory of coherent risk measures proposed by Artzner, Delbaen, Eber and Heath (1997, 1999). Artzner et al. (1997, 1999) proposed to do for risk what Euclid and others had done for geometry: they postulated a set of risk-measure axioms– the axioms of coherency– and began to work out their exertion. Their study makes us aware of a big gap existing between the mainstream risk measurement methods using now and reasonable risk measures and also makes the academic research of risk measures in recent years a hotspot.Based on the pioneering work by Artzner et al. (1997, 1999), this dissertation makes a further in-depth study of the consistency of risk measure on their axioms (Subadditivity, Translation Invariance and Positive Homogeneity) and an in-depth exploration in the following areas: the risk diversification mechanism, the consistency of risk measure from the view of equity investors and the nonlinear-shape risk perception. Finally, the new mean-variance model of S-shape risk perception is made, and the original model's abilities of explanation are expanded. All of these provide new perspectives and new methods on the research of risk measurement and portfolio selection. The main content about this dissertation is summarized in the following:1. This dissertation, firstly, introduces different definition of risk by scholars and elaborates on the causes for diversity of Risk Measures. It then connects all the present kinds of risk measures and finds out how risk measures has developed based on the general law for people's perception and behavior. It also introduces some new progress on the research of risk measures. Subsequently, it puts forward the basic thinking line and general frame of its research, trying to put various risk measures in one general frame and form. It is useful for us to grasp the research method of risk measures in nature and improve risk management techniques.2. Based on the Euclid Space of the multifactor model, we study the quantitative relation between the sum of the risk value for separate assets and that for the portfolio which is different from the prior study in the subadditive character of risk measures. Under the assumptions of normal distributions for all risk factors, we deduce the equation between the risk value for separate assets and the portfolio and claime that coherent measures of risk have the same coefficient of risk diversification. It is demonstrated that the optimal solution of the M-V Model is also the same optimal solution of the portfolio problem to any coherent risk measure. This dissertation provides a method to study the risk, measured by coherent measures and the reciprocity mechanism among separate assets.3. Based on the fact of that different actors in financial markets are acting differently and have different roles, the actors'risk can be classified into Equity Risk and Risk of Creditor's Rights. Then the Quasi-Coherent Risk Measure based on option pricing theory from the view of equity investors is defined to measure Equity Risk. Under this definition, Translation Invariance and Shift Invariance are united by the Quasi-Coherent Risk Measure properties of Translation and Incorruptibility. The Quasi-Coherent Risk Measure which is an insurance style is an extension of the Coherence Risk Measure in Artzner et al. (1997, 1999) paper which is a bail style.4. Based on the new view-angle of risk exposure & risk perception, a cluster of S-shape curves are constructed by using trigonometric functions. Thereupon, traditional research method on risk measures is expanded and complicated no-linear risk perception can be measured. Finally, the portfolio model of S-shape risk perception and an empirical study are made. The results indicate that different risk perception between institutional and individual investors under the same investment chances induces different investment behaviors. The primary innovations of the dissertation are summarized in the following: Innovation 1Based on the Euclid Space of the multifactor model and the Positivie Homogeniety of coherent risk measure, we study the quantitative relation between the sum of the risk value for separate assets and the risk value for the portfolio which is different from the prior study in the subadditive character of risk measures. The research provides new perspectives and new methods on studying the risk, measured by coherent measures, reciprocity mechanism among separate assets. Finally the following results are gotten.Result1: Under the assumptions of independent identical normal distributions for all risk factors, we deduce the linear equation between the risk value for separate assets and the portfolio and claime that coherent measures of risk have the same coefficient of risk diversification. The coefficient of risk diversification meets the rule of vector addition in the Euclidean Space. The formula of synthesis risk of different assets is similar with that of force vectors. The addition formula of different assets risks measured by the coherent risk measure is similar with that of the standard deviations. The portfolio risk can be expressed as one function on all of the single asset risk.Result2: Under the assumptions of independent identical normal distributions for all risk factors, it is also demonstrated that the optimal solution of the M-V model is also the same optimal solution of the portfolio problem where the risk is measured by any coherent risk measure.Innovation 2Based on the fact of that different actors in financial markets are acting differently and have different roles, a new risk classification has been proposed, so as to give more groundwork for next research. Then the Quasi-Coherent Risk Measure based on option pricing theory from the view of equity investors is defined to measure Equity Risk. Under this definition, Translation Invariance and Shift Invariance are united by the Quasi-Coherent Risk Measure properties——Translation and Incorruptibility.The Quasi-Coherent Risk Measures have all properties that the Coherent Risk Measures (Artzner et al., 1997 and 1999) and the Insurance Premium Measures (Robert Jarrow, 2002), the Deviation Risk Measures (Rockafellar et al., 2006)have. And accordingly some properties of the Quasi-Coherent Risk Measures have been further improved and extended. It will be able to better describe the reality of the real world of human equity risks.The Quasi-Coherent Risk Measure which is an insurance style is an extension of the Coherence Risk Measure in Artzner et al. (1997, 1999) paper which is a bail style.Innovation 3Based on the deficiency of the traditional financial theory and enlightened by the financial theory research, one new method about measing investors'risk which based on the three variables (loss, possibility of loss and risk exposure), that extends the traditional method which based on the dual variables (loss, possibility of loss), is put forward from the perspective of the investors'risk feeling and thus highlights the risk exposure measurement study's status. Thus the M-V Model under the new risk feeling type is established.It approaches the risk from a new perspective and provides new idea and new method for the construction of a more complicated and delicate risk feeling. It is an in-depth study and further expansion on the Positive Homogeneity of the consistency risk measure. In nature, the Positive Homogeneity of the consistency risk measure depicts investors'risk feeling on terms of linearity.The new portfolio model under new risk feelings, in addition to the explanation ability of the M-V model, can also interpret the economic phenomenon that facing to the same investment opportunities investors have different investment behaviors because different personality of the risk. The M-V model is a special case of the new model. This dissertation is supported by Key Projects of National Natural Science Foundation of China (No. 70331001).