Downside Risk Management In A Risk-taker's Perspective

Markowitz proposed to use the variance as the risk measure, based on which he build the mean-variance portfolio optimization model. The variance is easy to use, and the mean-variance model is widely used in practice due to its simplicity. The variance is a two-sided risk measure (TRM), which is a reasonable risk measure from the perspective of market supervisor, because both the extremal downward and upward changes are not propitious to the development of financial markets.However in the perspective of the risk-takers, the variance as a TRM is not a reasonable risk measure, because the risk takers are more concerned about the loss they will suffer due to taking the risk. Actually after Markowitz, Roy published the second seminal paper on portfolio theory, in which he proposed the safety-first principle. Roy stated that an investor will set some target rate of return or disaster level, and wish to minimize the possibility of portfolio return below the target level. Roy was the first scholar to measure risk as the downside loss. Mao showed the rationality of Roy's theory by empirical investigation. During the summer of 1969, Mao interviewed eight medium and large companies in several industries. In his findings of the survey Mao pointed out that in a practical investor's view, the risk is usually related to his target rate of return, and the downside risk, i.e., the negative deviation from the target return is the investor's concerns. Kahneman and Tversky, the founder of the behavioral finance theory, proposed the well-known prospect theory, which extended the view of Mao according to the psychological experiments. They stated that"gains and losses are defined relative to some neutral reference point"and"it is not so much that people hate uncertainty-but rather, they hate losing".Therefore for those risk-takers such as the institutions and individual investors, they are more concerned about the downside risk. This thesis studies the risk management problems the risk-takers are faced with, including the risk-taking constraint of future's holder, the risk-taking premiums of closed-end fund's holder, the optimal risk transferring of insured in insurance market, and the optimal risk diversification of stock investors in stock markets. We consider risk management problems stated above in different markets, taking into account of data availability and empirical importance.The main contents and conclusions of the thesis are summarized as follows:(1) An overview of risk measurement theory and their applications is presented in the first part. According to different risk-measuring purposes, the risk measures are classified into three types: (i) The first type of risk measures focuses on how to quantify the risk feeling. The deviation risk measures such as variance, and other perceived risk measures can be classified into this class. (ii) The second type of risk measures tries to quantify the risk prices, which include zero utility premium principle and Wang premium principle. (iii) The third type of risk measures is concerned about how to quantify the risk exposures. Both VaR and CVaR belong to the third type. Meanwhile, there are some connections between different types of risk measures, which we also have a brief introduction. Risks can be transferred or be diversified. The former is the subject of optimal insurance theory, and the latter is what optimal portfolio theory focuses on. So we have a brief review of the optimal insurance and optimal portfolio problems.(2) We propose the target-based generalized coherent risk measure. Based on the coherent risk measures and insurance risk measures, we incorporate the target into the definition of risk, and propose the generalized coherent risk measure. It can be proved that the generalized coherent risk measures also have some good properties, and are consistent with the investors'risk preferences. The standardized LPM and the put option premium as two intuitive risk measures also belong to the class of generalized coherent risk measures.(3) We propose the risk constraint model of the risk-takers. The risk-taker will be faced with a bankruptcy when the risk he takes exceeds his tolerance. Because of leverage trading in future market, the trader will be faced a severer risk when market goes against his speculation. Moreover the availability of future price data convince us to use the futures market as an example to illustrate the risk constraint model the future traders will be faced with. We use the POT (Peaks Over Threshold) model to fit the tail distribution of financial time series, and use VaR and CVaR as risk measures to set an appropriate margin level. The risk constraint model is also applicable to such risk-takers as banks and insurance companies.(4) We test whether the discount of closed-end fund(CEF) represents the downside risk premium which is offered to the investors, and examine the impact of lifeboat clause on discount rate using quantitative methods. The existence of CEF discount discourages the investors from holding CEF, who has to bear additional risk that the discount rate becomes high. CEF companies all over the world have tried many ways to provide risk compensation for investors so as to narrow the discounts. The data availability also convenience our research of relationship between CEF discounts and risk premium. More specifically, in this paper, based on the work of Merton, we use a put option premium to measure the performance of the closed-end fund's manager, and the downside risk the investors is suffered. The empirical analyses suggest that the discount of closed-end fund can represents the downside risk premium. In August 2008, Dacheng Fund Company issued a closed-end fund, which involves a lifeboat clause. We study the valuation method of the lifeboat clause, and its impact on the discount rate of the closed-end fund.(5) We show the equivalence between the optimal insurance strategy for an individual under an intertemporal equilibrium and the put option. Under some well-accepted assumptions, we prove that the optimal insurance strategy for an individual under an intertemporal equilibrium actually is equivalent to buying a put option, which is written on his holding asset with a proper strike price.(6) We solve the optimal insurance problem under the insurer's risk constraint. The insurer will be faced with risk exposure after he sells an insurance contract. In previous literature, little attention was paid to the controlling of the insurer's risk exposure. In this thesis, we consider the optimal insurance problem under the constraint that the insurer wishes to control his VaR, expected loss, or maximal loss below some prespecified level. Meanwhile we consider the measures the insurance companies can take to control their insurance loss exposures.(7) We solve the mean-LPM model using the predictive data. Since the risk-takers are more concerned about the loss below some level, we use the LPM as the risk measure. Moreover, people make their investing decisions on the basis of the predictive data, instead of the historical data. We use the AR(1)-GJR(1,1)-POT model to fit the marginal distribution of each asset, and use t-Copula function to model the dependence structure between assets. After getting the multivariate joint distributions, we use the Monte-Carlo technique to generate scenarios of asset returns. We make comparisons between different portfolio optimization model under different risk measures, including the variance and lower partial moment, and different data, including the historical data and predictive data. The empirical analyses show the advantage of the model using predictive data and LPM risk measure.The main innovations of this thesis include:(1) We propose the target-based generalized coherent risk measureBased on their seminal paper of Artzner et al.(1999) and Jarrow(2002), we incorporate the target into the definition of risk measure, and propose the generalized coherent risk measure. It can be proved that the generalized coherent risk measures also have some good properties, and are consistent with the investors'risk preferences. We also show the standardized LPM and the put option premium belong to the class of generalized coherent risk measures.(2) We propose a new method to choose an appropriate threshold in POT modelThe POT model is a widely-used method to model the extreme events, in which the choice of threshold is an important parameter. Based on the traditional Hill plot, we propose a quantitative method to choose an appropriate threshold, which can make full use of sample data, thus overcoming the shortcoming of bootstrapping MSE method.(3) We show the equivalence between the optimal insurance for an individual under an intertemporal equilibrium and the put optionDifferent from the existing literatures, we determine the insurance premium via the intertemporal equilibrium condition. Under some well-accepted assumptions, we prove the optimal insurance for an individual under an intertemporal equilibrium is equivalent to buying a put option, which is written on his/her holding asset with a proper strike price.(4) We solve the optimal insurance problem under the insurer's risk constraintThe insurer will be faced with risk exposure after he sells an insurance contract. In previous literature on optimal insurance problem, little attention was paid to the controlling of the insurer's risk exposure. In this thesis, we solve the optimal insurance problem under the constraint the insurer wishes to control the VaR, expected loss or maximal loss below a prespecified level.