投保联动及主观风险测度数学模型

    解决中小企业“融资难，融资贵”的问题是当前数学和金融交叉研究的一大热点，同时，投资和风险是讨论金融问题时离不开的两大切入点。本学位论文综合运用数学方法和金融经济学理论，从担保投资和风险度量的两个角度协助中小企业融资。首先，从多个角度分析新型的担保模式——``投保联动”，建立了用以量化的、符合经济学规律的数学模型。同时，在担保市场是完全竞争的假设下，研究了股权、债权以及相应衍生品的风险中性定价，对最优担保合同进行了定量分析，提出了担保过程中的股权回购数学模型。最后，从风险对冲的角度，提出了一种新的、个性化的效用风险测度。
    
    受到新冠疫情下的经济波动对担保行业影响的启发，从经济环境的角度出发并结合实际的担保业务创新，在理论上研究一种新兴的担保模式——投保联动数学模型。现有的投保联动模式中，一个企业家由于没有足够的抵押品从银行借到足够的资金来投资一个项目，于是与第三方担保公司达成了一项协议。担保公司为债务提供完全担保，并承诺为未来的扩张性投资提供资金。由于这一完全担保，借款是没有风险的，因而银行愿意以最低的利率贷款给企业家。作为回报，企业家在借款时向担保公司支付一笔固定费用，并在扩张性投资发生时转让给担保公司一部分股权。假设企业现金流分别服从双指数跳扩散数学模型和马尔科夫区制转换数学模型，定量地分析经济形势交替变化对担保合同和企业投融资决策的影响。与传统的关于担保的研究形成鲜明对比的是，在这两类不同模型假设下，担保费用在投资之前就已经制定了计算方法。基于这样的模型，本文通过求解一系列的最优化问题和概率问题，解析地给出了公平担保定价公式、企业最优破产选择、最优投资时机和最优融资选择。
    
    在完成投保联动基本数学模型构架及计算求解后，本文从实际出发，将关注点转向增长型投资的优化计算问题。分别在增长期权属于出资方和经营者的条件下，沿用传统的方法分析了固定担保费用的数额和担保公司最终股份所有权的大小之间的公平关系。同时，为确定两种不同的投资方案的时机和定价提供了详细的数值解法，并给出了可靠的模拟计算结果。通过数值实验发现，如果担保公司拥有增长型投资的决策权，那么该项目会比企业家选择增长投资时机更有价值，将控制权转让给担保公司可以缓解投资不足的问题。 
    
    基于实际的经济活动，本文还考虑了含有股权回购的投保联动担保合同，建立了一个新的数学模型来描述担保成本、两阶段投资方案和股份回购方案的定价和最优停时问题。 本文发现，存在一个使公司价值最大化的最佳担保成本组合，其中固定担保费率约为1\%或2\%，与中国政府对某些行业的建议一致。如果回购是由企业家而不是担保公司发起的，公司价值会更高，而且协商回购会更早发生。强制回购的时间越晚，或者协商回购的时间越早，公司价值就越高。
    
    最后，考虑到在解决中小企业融资问题时，只考虑投资而不考虑风险是不够全面的，而现有的风险度量标准主要是为那些大到不能倒闭的金融机构制定的，本文提出了一种新的主观风险测度——基于对冲的效用风险测度（HBU）。这是为需要对金融产品进行全面风险评估的个人投资者定制的新颖风险度量方法。本文从数学上严格地证明了HBU是一个凸的风险测度。进一步严格地证明了如果效用具有恒定的相对风险厌恶指数，HBU也是一致的风险测度。粗略地说，HBU是广义效用无差别价格的相反数，它取决于投资者的效用函数和他们可以使用的用来对冲风险的金融工具。本文发现并证明了HBU的数学性质，同时通过提供的两个应用举例进一步解释了该主观风险测度的科学性和合理性。

Through a combination of mathematical methods and financial economics theory, the credit guarantee with investment as a new quantitative and economically consistent mathematical model of the guarantee model is developed in this thesis. Under the assumption that the guarantee market is perfectly competitive, the risk-neutral pricing of equity, debt and corresponding derivatives is investigated, the optimal guarantee contract is quantitatively analysed, and a mathematical model of equity buyback in the guarantee process is proposed. Finally, a new and personalised measure of utility risk from a risk hedging perspective is presented. First, inspired by the impact of economic fluctuations on the guarantee industry caused by COVID-19, the credit guarantee with investment mathematical model is studied theoretically from the perspective of the economic environment and based on actual guarantee business innovations. In the existing credit guarantee with investment model, an entrepreneur, who does not have enough collateral to borrow enough money from a bank to invest in a project, enters into an agreement with a third-party guarantee company. The guarantee company fully guarantees the debt and promises to fund future expansionary investments. As a result of this full guarantee, the borrowing is risk-free and thus the bank is willing to lend to the entrepreneur at the lowest possible interest rate. In return, the entrepreneur pays a fixed fee to the guarantee company when borrowing money and transfers a certain portion of the equity to the guarantee company when the expansionary investment occurs. On the basis of the assumption that corporate cash flows obey the double exponential jump diffusion model and the Markov regime switching model, the impact of alternating changes in economic conditions and corporate financing decisions are quantitatively analyzed. In contrast to the traditional studies on guarantees, under the assumptions of these two different types of models, the calculation method of guarantee fees is developed before the investment is made. Based on this model and by solving a series of optimization and probability problems, the fair guarantee scheme, the optimal bankruptcy option, the optimal stopping time and the optimal financing option are given analytically in this thesis. Having completed the construction and calculation of the basic mathematical model of the credit guarantee with investment, the focus of this thesis is turned to the problem of optimising the calculation of growth investments. Following the traditional approach, the fair relationship between the amount of the fixed guarantee fee and the eventual size of the guarantee company’s share ownership is analysed in this thesis. Numerical experiments reveal that the project is more valuable if the guarantee company has the right to make decisions about the timing of the growth investment. Transferring control to the guarantee company can alleviate the problem of underinvestment. Based on actual economic activity, the credit guarantee with investment guarantee contracts containing equity buybacks are also considered in this thesis, and a new mathematical model is developed to describe the pricing and optimal stopping time problems for guarantee costs, two-stage investment options and share buyback options. It is found that there exists an optimal combination of guarantee costs that maximises the value of the firm, where the fixed guarantee rate is around 1% or 2%, which is quite consistent with the Chinese government’s recommendations for certain industries. Firm value is higher if buyback is initiated by the entrepreneur rather than the guarantee company, and negotiated buybacks will occur sooner. The later the mandatory buyback occurs, or the earlier the negotiated buyback, the higher the value of the company will be. Finally, considering that existing risk measures are mainly developed for too-bigto-fail financial institutions, the hedge-based utility risk measure (HBU) is presented in this thesis as a new subjective risk measure. This is a novel risk measure tailored for individual investors who need a comprehensive risk assessment of financial products. The HBU is strictly mathematically proven to be a convex risk measure. If utility has a constant relative risk aversion index, it is rigorously shown that HBU is also a consistent risk measure. Roughly speaking, HBU is the opposite of the generalised utility-free price, which depends on the claimant’s utility and the hedging instruments available to them. The mathematical properties of HBU is discovered and demonstrated in this thesis, and the two application examples provided explain the scientific and rational nature of this risk measure.

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