Interest Rate And Credit Limit Optimization,and The Impact Of Risk Attitudes On The Decisions

Interest rate and credit limit are two core features of credit products,which not only bring uncertainty to the profit of each unit loan,but also affect the borrowing needs of borrowers,and thus play a vital role in optimizing lenders’ profit.Traditional credit risk analysis focuses on developing credit score models to classify borrowers,and credit decisions are also based on the credit risk.Credit risk is an important indicator that affects lenders’ profit,but it cannot be completely equivalent to the profit objective.Moreover,some practical cases show that there is still considerable room for improving interest rate and credit limit decisions.Therefore,researches on credit decisions based on profit optimization has received growing attention.Besides,a phenomenon in the credit market is noteworthy,that is the heterogeneity of interest rates and credit limits.Even if the borrower’s risk characteristics and other characteristics are controlled,this heterogeneity can be found.Heterogeneity may come from the supply-side of credit or the demand-side of credit.This paper found in the literature that the risk attitudes of lenders will affect theirs decisions.Therefore,the paper tries to study how risk attitudes will affect interest rates and credit limit decisions.The research of the paper mainly focuses on the optimization of interest rate and credit limit.First of all,an optimization model of interest rate and credit limit is established,and the optimization theory and the sensitivity of optimal decisions are presented.Secondly,the impact of risk averse on the optimal interest rate is analyzed.The paper studies this impact by comparing risk averse interest rate and risk neutral interest rate.During the comparison,adverse selection is incorporated into the analysis,and the paper finds that owing to kinds of factors,risk averse interest rate may be higher or lower.Finally,two common risk attitudes,risk averse and loss averse,are considered to investigate their effect on the optimal credit limit.And the paper supplements the whole study by considering the effect of borrowing needs on the optimal decisions.The content and main conclusions of the paper are as follows:(1)The paper studies the simple loan,whose principal and interest rate are paid when loan is matured,and establishes an expected profit model of interest rate and credit limit.Considering the impact of the probability of acceptance in the model,this paper studies the optimal decisions in these two kinds of models,with or without acceptance probability.Optimality conditions are derived,and the effect of model parameters on optimal decisions is discussed.When acceptance is not included in the model,the optimal interest rate is higher,and the optimal credit limit is lower,since the model ignores the fact that high interest rate is not attractive,and high credit limit is prefered.Regardless of whether the model includes the acceptance probability,the optimal credit limit decreases with loss given default(LGD)and riskfree rate.When acceptance probability is not included,risk-free rate does not affect the optimal interest rate,while LGD reduces the optimal interest rate.When the acceptance is considered,the optimal interest rate is decreasing with risk-free rate,and the effect of LGD is ambiguous.In the absence of acceptance probability,the effect of credit limit and interest rate on each other is consistently negative.Optimal credit limit is decreasing with interest rate,and optimal interest rate is decreasing with credit limit.However,with the acceptance probability,the relationship between interest rate and credit limit is hard to determine.Based on the above analysis results,this paper also proposes algorithms that optimize the interest rate and credit limit simultaneously.According to the theory of the block coordinate descent,the algorithms are convergent,which is also illustrated by the numerical example.(2)The paper compares the risk averse optimal interest rate and the risk neutral optimal interest rate.The optimality conditions of the commonly used interest rate optimization model and the sensitivity analysis of the optimal interest rate are given firstly.This part fills the research gap and lays the foundation for the subsequent analysis.When comparing the two interest rates,firstly,this paper considers the simpler case that the acceptance probability is not included and finds that the risk averse rate is lower than the risk neutral rate.Then,when the acceptance probability is included,the paper adopts the analysis method in the literature,and discusses the interest rate comparison under two scenarios,one is that there is no adverse selection at all,that is the default/repayment probability is constant with respect to the interest rate.The other is a situation in which there is adverse selection,that is the repayment probability decreases with the interest rate.Comparing the two scenarios,the paper finds that,first,when adverse selection is strong,that is,when the repayment probability falls rapidly with the rising interest rate,the optimal interest rate of the risk averse lender should be lower.Second,lenders should set a higher interest rate when the borrower’s risk of default is relatively high.Finally,when those two factors are at a relatively low level,the interest rate of lenders with high level risk aversion should be higher,and if the risk aversion is low,the lenders’ optimal interest rate is lower.Moreover,the existence of adverse selection makes it more stringent for lenders to choose a higher interest rate,therefor lender is more likely to set a lower interest rate.(3)The paper analyzes the impact of risk attitude on the optimal credit limit.This paper considers two common risk attitudes: risk aversion and loss aversion.The optimal credit limit of risk averse lender,regardless of whether the acceptance probability is included,must be lower than the risk neutral one.But only when the acceptance probability is not included,the optimal credit limit decreases with the degree of risk aversion.After considering the acceptance probability,the effect of the degree of risk aversion on the optimal credit limit is ambiguous.The risk averse credit limit decreases with LGD,and if the coefficient of relative risk aversion is low,the risk averse credit limit also decreases with risk-free rate.Meanwhile,regardless of the acceptance probability,the loss averse credit limit is lower than the risk neutral one,and it decreases with the degree of loss aversion.Moreover,the loss averse credit limit is negatively related to LGD and risk-free rate.(4)The impact of borrowing demand is studied.Including the acceptance probability,a complex condition is required to determine that the interest rate will increase with the change of borrowing needs.When the acceptance probability is not considered in the model,the condition is relatively simple.If the mixed partial derivative of the repayment probability is less than 0,and the coefficient of relative risk aversion of utility function is relatively high,it can be determined that the interest rate increases with the change of borrowing demand.This paper not only complements the research on the optimization of interest rate and credit limit,but also provides some theoretical support to understand and investigate some phenomena in credit market.The main contributions of this paper are as follows: an expected profit model for the simultaneous optimization of interest rate and credit limit is established,and optimality conditions are given.The mechanism and process of how interest rates are affected by lenders’ risk attitudes and adverse selection are given.The effect of risk attitudes on the optimal credit limit is determined.The impact of borrowing demand on the optimal decisions is discussed.