Mathematical Models Of Loan Guarantees With Investment And Hedging-based Utility Risk Measure

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.