Pricing Optimization Under CvaR Criteria

Managing risk is extremely important in both business practice and finance management.Before managing risk,we should first pay attention to the risk measurement criterion.Different risk preferences correspond to different criterion choices.In the context of the current pandemic,inflation,geopolitical and energy crises,both the supply chain management in manufacturing and trading industries,as well as the credit risk management in financial industry,are facing great challenges,and more inclined to focus on tail risk,to face the future unknown economic fluctuations and competition.Therefore,the pattern of the tail risk is a key point in the study of risk measurement.This paper focuses on the common tail risk measurement criterion CVaR(Conditional value at risk).In view of its better mathematical properties and more practical measurement significance,it has been gradually improved and applied in academia and industry since it was proposed at the end of the 20 th century.The CVaR criterion is applicable to any area where risk needs to be measured,especially management and finance.Based on CVaR criterion,this paper restudies inventory pricing and interest rate pricing from the perspectives of risk aversion preference and risk control.The inventory pricing model faces the uncertainty of demand,while the interest rate pricing model faces the uncertainty of customer’s refusal of loan,default or early payment after loan.For the former,this paper uses the operational statistics method and maxmin method respectively to make CVaR optimization decision for inventory pricing model under parametric demand model and non-parametric demand model.For the latter,this paper analyzes its optimization properties and tries to use real loan data to solve and analyze the expectation-CVaR model to make interest rate pricing decisions.From expectation criterion to CVaR criterion(expectation is a special case when CVaR confidence level is 0),the CVaR objective function or CVaR risk constraint in inventory pricing and interest rate pricing is derived and optimized based on the good properties of CVaR.By using the auxiliary function introduced,CVaR is easy to be derived when it is used as the objective function,and easy to be transformed into linear constraint combination when it is used as the constraint function.By analyzing CVaR objective function and CVaR constraint function,this paper puts a new perspective into the inventory pricing management and interest rate pricing management.In the first part,the inventory pricing problem under parametric demand model is studied,and the operational statistics theory in newsvendor pricing model is combined with quantile regression estimation,so as to develop the operational statistics method suitable for general demand model.Operational statistics method is the improvement on the traditional estimation method,the traditional estimation method is first to maximize profit function to get the optimal solution of inventory quantity and price,and then use the classical statistical method such as least-squares estimation,maximum likelihood estimation and moment estimation to construct the inventory quantity and price,while the operational statistics methods combine parameter estimation with optimization of profits.This method first constructs the set of operational statistics,and then finds the estimator that maximizes the prior expected profit in the set.This statistic is better than the classical statistical estimator and can increase the expected profit.At present,operational statistics method is mostly used in inventory model,and recently it has been applied to inventory pricing model in literature,but the assumption that demand follows exponential distribution makes its conclusion difficult to be applied to other distributions.In this paper,the operational statistics method is introduced into the newsvendor pricing model,and the distribution of demand is not assumed.Since it is observed that the expression of optimal inventory is a quantile of the random demand,based on the quantile regression estimation of inventory,a uniform set of operational statistics is constructed,which is not affected by the form of demand distribution.Based on the CVaR optimization criterion,the optimal inventory statistic which is also unaffected by the confidence level is obtained.Specifically,this part first combines CVaR with operational statistics method based on the definition and properties of CVaR,and explains how can this method improve inventory estimation in newsvendor problem.Second,the newsvendor pricing model for inventory and price decisions under CVaR criteria is detailed,and the data-driven estimation of optimal inventory and price is introduced.Finally,the operational statistical methodology framework is described and applied to inventory estimation in newsvendor pricing problem with demand following general distribution under CVaR criterion.In the second part,the inventory pricing problem under the non-parametric demand model is studied.The max-min method in extended from the newsvendor model to the newsvendor pricing model,and from the expectation criterion to the CVaR criterion.The unimodal property of CVaR objective function with respect to inventory and price is analyzed.Max-min method is applicable to inventory pricing optimization management under incomplete information,when just the expectation and variance of demand are known,to maximize the lower bound on the expected profit.It’s a trade-off between maximizing the expected profit and the worst-case demand distribution in a distribution with the same variance of the mean,which is maximizing the worst-case expected profit.In this paper,the price is regarded as an endogenous variable,and the max-min method is extended from the inventory problem to the inventory pricing problem,without assuming the distribution function of the demand model or the specific form of the demand function,but in a wide range of requirements model(IPE multiplicative demand model and IPE additive demand model).This paper proves the unimodal properties of objective function,derives the expressions of optimal inventory and price,and analyzes their sensitivity to the risk aversion level.The above conclusion extends the existing newsvendor pricing theory greatly in terms of measurement criterion,demand model,uniqueness of optimal solutions and its ability to be written into concrete expressions.Specifically,this section first reviews the lower bound of max-min newsvendor pricing model.Secondly,the max-min newsvendor pricing problem under CVaR criterion is analyzed,the lower bound of CVaR is constructed,the solutions of optimal inventory and price are derived,and their sensitivity to confidence level is analyzed.Thirdly,the analysis of max-min newsvendor pricing problem is extended to additive demand model,and it is shown that the unimodal property of the lower bound still holds under the additive demand model with IPE property.Finally,numerical experiments are conducted to evaluate the performance of max-min newsvendor pricing model under CVaR criterion.The third part studies the loan interest rate pricing model based on CVaR,integrates realistic factors and survival analysis theory to construct a comprehensive and complete interest rate pricing model,including acceptance probability,default event,early repayment event,loss given default,customer segmentation,and Cox risk model for estimating default and early repayment probability.Two ways of thinking about the calculation of expected profit are given,and the CVaR constraint function is developed in one of the ways.The expection-CVaR interest rate pricing model is constructed,and the corresponding profits and risks of different interest rates in different customer groups are analyzed with real data.Firstly,it introduces the common repayment methods of loans and the basic model of interest rate pricing based on the repayment method of equal principal and interest.The basic model considers the acceptance probability,default situation and customer segmentation.On the basis of this model,CVaR risk constraints are added to construct the expect-CVaR model,and its properties are analyzed.The optimization conditions of the expected profit objective function are given,and the CVaR constraints are linearized.Further,this part will focus on the construction of a complete interest rate pricing model of equal principal and interest repayment,this model will also consider the actual situation of the causes of early repayment events,loss given default(LGD),using Cox survival analysis model fitting the probability of default and early repayment,construct expectation-CVaR model.And the profits and risks of each customer segmentation under different interest rates are empirically analyzed with real data.