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Procedía Computer Science 31 (2014) 577 - 586

Information Technology and Quantitative Management (ITQM 2014)

Research on Impact of Moral Hazard on Individual Credit Risk

Shuai Lia, Yang Yanga, Zhou Zongfanga*

_aSchool of Economics and Management, UESTC, ChengDu, 610054, China_

Abstract

Moral hazard is a prominent issue in personal behaviour of credit risk management. This paper considered the bank loan interest rates as the control variables. It analysed the relationship between the default probability and the probability of the occurrence of moral hazard of individuals and the impact of moral hazard to individual credit risk mechanism from the theoretical level. The results show that: (1) the bank loan rates have strong effects on both the moral hazard of personal loan and credit risk. (2) The relationship of the probability of the occurrence of moral hazard of individuals and the default rate of individuals is generally non-linear. However, there is a constraint interval of the default rate. When that interval is determined, the probability of the occurrence of moral hazard of individuals and the default rate of individuals has linear relationship, and the bigger the interval is, the smaller probability of the occurrence of moral hazard is.

© 2014Publishedby ElsevierB.V.This is anopenaccess article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of the Organizing Committee of ITQM 2014. Keywords: Personal Credit; Moral Hazard; Probability of Default; Loan Interest Rates

1. Introduction

In recent years, with the rapid development of personal loans of China's commercial banks, personal loan becomes the hot spot of financial market competition with tremendous market potential. Currently, commercial banks all take the development of personal credit business as an important component of development strategy.

* Corresponding author. Tel. +086-138-8089-0605. E-mail address: y_yang@aliyun.com.

1877-0509 © 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of the Organizing Committee of ITQM 2014. doi: 10.1016/j.procs.2014.05.304

However, both in theoretical research and management tools and methods, domestic commercial banks consumer credit risk management is lagging behind the state, which seriously hindered the development of domestic consumer credit. Therefore, to strengthen the individual credit risk assessment and management is on urgent. Large number of cases shows: direct causes of personal credit default are mainly from two aspects: firstly, the borrower's investment or management failure, which make them lose the repayment ability; secondly, the borrowers do not follow the loan agreement, and use the bank loans for other purposes, or the lack of willingness to repay.

Current researches on credit risk are mainly about metrics and predictive probability of default, such as the classical theory based on option structural model assuming that an event of default as a simplified model of random events 2, as well as a variety of statistically based and AI measurement methods 3. But there are few quantitative research for consumer credit customer credit risk, Huang Huizhong, Zhou Zongfang (2010) for the BP neural network in the personal credit evaluation model application defects, an improved LMBP algorithm and ILMBP personal credit evaluation model is applied 4; Li Jianping, etc. (2004) apply support vector machine (SVM) in commercial bank personal credit evaluation methods of the study, to obtain a better prediction results 5; Yang Yu, Shi Xiuhong, etc. (2009) established a bilateral antibody probability model based on Artificial Immune mechanisms, and with the logistic regression model were compared 6; Yang Yang, Zhou Zongfang, etc. (2012) for the enterprise Group's credit risk and the credit risk of moral interactions have been studied 7 . So far, research on the interrelationship of moral hazard and the credit risk of personal loan is not found.

This paper analyses the individual credit risk and the credit risk between moral influence mechanisms from the theoretical level. Section 2 of this paper is the underlying assumptions; Section 3 gives the measure formula for the probability of success of investment projects A; Section 4 and 5, respectively, discussed the lending rate on consumer credit and the credit risk of moral hazard effects, and based on this, discusses the transmission of the moral hazard of personal credit to credit risk and the reaction of the credit risk against moral hazard.

2. Assumptions

To simplify the discussion, assume that individual's credit funds are invested in whether a more risky investment Project A or a less risky business Projects B. A project is characterized with high-risk and high-return, while Project B is characterized with low-risk, low-return. Clearly, the Project A cannot apply for personal loans from banks, while Project B will receive bank loans. Without loss of generality, we make the following assumptions:

Hypothesis 1: The expected return of invest funds x in investment Project A is xSA , if the revenue of Project A is insufficient to repay the bank loan principal and interest, then Project A is deemed bad investment. The probability of Project A being a successful investment is denoted Pa, the distribution function of 5A is , its density function is ^

Hypothesis 2: The expected return of investing funds x in operating project B is % which is randomly distributed in [0, xSb ] 8, the conditional distribution functions and conditional density function, respectively, are F(r|x) and f (r|x) , where xSb is the upper bound of the revenue of Project B. If the revenue of Project B is insufficient to repay the principal and interest, then Project B is deemed failed. Hypothesis 3: Once the investment fails, the consumer credit default occurs. In accordance to reality, the paper also made the following hypothesis:

Hypothesis 4: The game between banks and individuals are dominant individuals, that banks do not know the information that Project A is high-risk.

Hypothesis 5: When the loan interest rate is sufficiently high, the probability of default personal credit loans rise with interest rates.

3. Probability of Default Personal Loans Based on Moral Hazard

3.1. Unsecured Assets

Once moral hazard occurs, according to Merton (1974) model, we assume that the expected return of investing funds x in Project A is V= xSA, which follows a geometric Brownian motion, i.e. dV = PAVdt + oVdW (1)

In this assumption, the conditions probability Pa of Project A being successful depends on the expected profit of Project A and the principal and interest to pay after loan maturity, then the condition probability of Project A being failure is as follows.

1 - Pa = Pr{Vt < Dt\Vo = V} = Pr{lnVt < ln Dt||Vo = V} (2)

Among them, Vt and Dt respectively are the expected return of Project A and the principal and interest to pay at t time (when loan is due), Vo is the initial expected return of Project A.

According to formula (1), the expected return of Project A at t time is as follows

ln Vt = ln V +

t + t (3)

Among them, ^ indicates the risk-free rate. Fore ~ N(0,1), then

1 - PA = pr

Under the normal distribution assumption, the conditional probability of Project A being success is as follows:

Pa = 1

ln V +

Among them O (□) indicates the cumulative probability function of the standard normal distribution.

According to Hypothesis 3, when Project A fails, the probability of consumer credit default can be measured by the following formula.

Pd =4>

ln V +

3.2. Secured Assets

When asset-backed is considered, presume that the market value of the guaranty is Mt at t time (when the loan is due), then the sum of the expected return V of Project A and the value of guaranty M exceed the sum of the principal and interest of the loanDt, Project A is success, and the probability of it is

PA=Pr{ k= Vt+M-Dt >0} (7)

So, when pledge of assets is considered, the default probability function of consumer credit is:

w = f (V), k = V +M - Dt, w e[0,1] (8)

In the figure below, w demonstrates the probability of default, when the personal mortgage is invested in Project A.

Fig.1. Personal mortgage default probability function Apparently, fk) is decreasing function of k.. Considering the factor of Loss Aversion [8-9], when k> 0, the magnitude of individuals to reduce the probability of default caused by k rise a unit is smaller than the magnitude of individuals to increase the probability of default caused by k decline a unit. Therefore, when k> 0, fk) is a concave function, denoted as g(k), when k> 0, according to Hypothesis 3, defaults is inevitable, so fk)=1.

Based on the analysis above, we can construct a default probability function when a personal mortgage loan is invested in Project A:

fg(k), k > 0

f (k) = rW (9)

[ 1, k < 0

Among them, g(k) is a concave function, which can be set according to specific condition. Therefore, we get the method to measure the default rate of a personal mortgage loan, when personal mortgage loan is invested in Project A, in the case that pledge of assets is considered.

4. Effect of Moral Hazard

4.1 Personal moral hazard

If individual borrow money xb from a bank with interest rate r, and invest the money in operational Project B, according to Hypothesis 2, the expected return when Project B is success is

Vb (xb ) = f(1 Jr-Xb (1 + r)]f (t|xb )dz (10)

*xb (l+r)

Among them, P=xbSb indicates the upper bound of the return of investing funds xb in Project B. Step calculate formula (10), we get

VB(xB) = lp- (r + l)xB]F(0\xB) - f F(r\xB)dT (11)

I JxB(l + r) I

According to the nature of distribution function, F(^XB ) = 1, so formula (11) can be noted as

Vb(Xb) = P- (r +1)Xb -i/?(1 )F(t|xb)dr (12)

Jxb (1+r)

If an individual borrow money xb from a bank for Project B and invest them in Project A, according to Hypothesis 1, the expected return of Project A is

va ( x b ) _ pa [xb^a xb (1 + r)] (13)

If moral hazard exist in personal credit, then when the expected return of Project A is far higher than that of Project B, the individual will move the fund of Project B into Project A, when the following formula holds, the personal credit will have moral hazard.

xb (1 + r)(1 - Pa ) - f (1 ) F(r^ )dr]

Jx„ (1+r ) 1

c. JxB (1+r) " 1

SA >-P-—--(14)

Among them, y>1 is the personal risk appetite factor, which reflects the extent how the expected return of

Project A is higher than Project B when both of them are successful. If an individual is risk appetite type, when

^>1, as long as the expected return of Project A is higher than Project B when success, personal moral hazard

will occur; If the individual is risk averse, then only when ^is large enough, moral hazard will occur. Furthermore, the right part of formula (15) in denoted as:

y[fi - xB (1 + r)(1 - Pa ) - f ) F(r|xB )dr]

(1+ r)

xB (1+r) = H (r) (15)

Then the probability of personal credit occurrence of moral hazard is: Pm = 1 -W(H (r)) (16)

Among them, ^ (• ) is the distribution function of SA . Derivate formula (15) on r, then

^^ = r[F(xb (1 + r)|xb ) - (1 - PA )] (17)

According to Hypothesis 2, F(xB (1 + r)|xB ) is the probability of operating Project B to fail and the probability of Project B to fail ought to be less than that of Project A

1" Pa > F(xb (1 + r)|xb ) (18)

SH(r) _

So-< U . In other worlds, with the increase in bank lending rates r, H(r) decreases

monotonically, and according to formula (16) and the character of distribution function, Pm increases monotonically. Thus we can get the following conclusions:

Conclusion 1: If there is moral hazard of consumer credit, consumer credit probability of occurrence of moral hazard increases monotonically along with the bank loans interest rates increasing.

4.2. Individual credit risk

If an individual borrows money from a bank at interest rate r, then the probability of consumer credit default that the bank facing is

PD = Pm (1 - Pa ) + (1 - P^ F(x(1 + r)|x) (19)

Among them, the first item indicates the probability of Project A to fail, when moral hazard happens to consumer credit; the second item indicates the probability of Project B to fail, when moral hazard do not

happen to consumer credit.

According to formula (19), the default probability of consumer credit Pd do not always monotonically increase along with the probability of moral hazard of consumer credit increasing, but depend on the revenue distribution of Pa and Project B®. Especially, when the probability of moral hazard of consumer credit is 0, the default rate of consumer credit equals that of Project B to fail.

The following discussion is about the relation between bank lending rates and probability of personal credit default PD .

We may assume that the probability of default Pd is a continuous function of r, according to Conclusion 1, when the lending interest rate r reach the minimum value(denoted as ro), the probability of moral hazard of consumer credit is minimum, therefore, when r = r0, Pm^O which means that the default rate of consumer credit Pd approximately equals to the probability of Project B to fail F (x (1 + r) |x) . According to the characteristic of distribution function, there is a certain sufficiently small positive half-neighborhood of ro O£=[ro> ro+£], in which £is a sufficiently small positive number, when reOs, default probability increases along with the increasing of r, then.

> 0, r e [r0,r0 +e) (20)

Derivate formula (19) at r : dP dP

-D = [1 - Pa - F(x (1 + r)\x)]-M - (1 - Pm )xf (x(1 + r)|x) (21)

Furthermore, according to hypothesis 2, for any 1- PA e [0,1] ,there is certain r = r g Os , making

1 - PA = F(x(1 + r)|x), so according to formula (21): dPDi

-H=r < 0 (22)

Then according to Hypothesis 5, there is r > r , making dPD I

-H=~ > 0 (23)

. ~ dPD | , ~

Therefore, there is r e (r,r ), making- r=r = 0 , in other words, when r = r ^ (r, , the default

dr 1 "

probability Pd reaches the minimum. It can be seen that the lending interest rate r0 that making the probability Pd of moral hazard of consumer credit minimum is not the lending interest rate that make the default

probability minimum.

Furthermore, from formula (20), (21), and (23), we can see that the default probability of consumer credit PD is monotonically increasing at r0, but PD first increases monotonically then decreases monotonically and then increase again, along with the increasing of the lending interest rate r. That means there is a lending interest rate r and r which respectively make the consumer credit default probability reaching the maximum and minimum. And r is in the interval (ro, r ). In other words, along with the lending interest rate r increasing, the default probability of consumer credit first increase and then reaches the maximum at r = r and then decrease, reaching the minimum atr = r .

Corollary 1: When moral hazard of consumer credit exists, the lending interest rate r that makes the

consumer credit reaching minimum is bigger than the lending interest rate r that makes the maximum.

Definition 1: If the probability of consumer credit default increases with lending rate r, then r is a lending rates which increases according to personal credit defaults probability; otherwise, if the individual credit default probability increases with lower lending rates (re (r , r )), then we say r is a lending rates which declines according to personal credit defaults probability.

The Corollary 1 showed that: when the moral hazard of personal credit exists, with the lower interest rates on bank loans, consumer credit default probability Pd do not a monotonically decline, lower lending rates do not necessarily reduce the risk of default of consumer credit. In other words, the bank can not lower lending rates to control consumer credit default risk.

4.3. Consumer's moral hazard

According to Conclusion 1, with the increase in bank lending rates r , probability of moral hazard of personal credit Pm monotonically increase, but default probability of consumer credit Pd is not monotonically increasing. Therefore we get the following important conclusions:

Conclusion 2: The relation between default probability and the probability of moral hazard of consumer credit is neither non-monotonic nor non-linearity. And there is exogenous lending rate r and r , respectively making individual credit default probability reaches a maximum and minimum.

Corollary 2: With the probability of moral hazard of consumer credit increases monotonically, the default rate of personal credit, first to reach the maximum and then reach the minimum.

Furthermore, according to formula(14)~(16), the smaller the gap between Pa5a and 5b is, the bigger H(r) is, the smaller the probability of moral hazard of personal credit Pm=1- 0(H (r)) is. Therefore, the transfer

mechanism of the moral hazard of personal credit to credit risk is as follows: (1) If the probability of moral hazard of the individual credit probability is 0, then the probability of default on consumer credit is equal to the probability of failure of Project B. (2) If the expected return of Project A is slightly larger than the return upper bound of Project B, then the probability of moral hazard of consumer credit is low. Once moral hazard occur in consumer credit, then the default probability of consumer credit will show a monotonic increase trends, along with the loan interest rate rising.(3) If the expected return of Project A is far larger than the return upper bound of Project B, then the probability of moral hazard of consumer credit is high. And the personal credit default rate will first decline and then rise, along with the rising of the lending interest rate.

5. Impact of Moral Hazard

The following discussion is about the counterproductive of individual credit risk to moral hazard. Denote

0 = f(x(1 + r)|x) , which is the probability of failure of Project B. According to formula (16), then

Pd = PM (1" PA ) + (1" PM> PM = Q- Dp (24)

Because the probability of failure of Project A is higher than that of Project B, the denominator of the formula is positive. Apparently, Pd e [d, 1-Pa](the default rate of personal credit is ranged between the probability of failure of Project A and the probability of failure of Project B). Given certain value of 1-Pa and 9, the probability of moral hazard of consumer credit Pm changes in the same direction with the probability of default Pd. Especially, If the consumer credit moral hazard has not occur(Pm=0). The default rate of consumer credit equals to the probability of failure of Project B, that is& If the consumer credit moral hazard occurs (Pm=0). The default rate of consumer credit equals to the probability of failure of Project A that is 1-Pa.

Conclusion 3: (1)The default rate of consumer credit Pd e[& 1-Pa], if 0and 1-Pa are all unchanged, the probability of moral hazard of consumer credit Pm and the default rate Pd changes in the same direction. (2) When the default rate of consumer credit Pd is unchanged, the probability of failure of Project A that is 1 -Pa is increasing; or 1-Pa is unchanged, while d is decreasing, then the probability of moral hazard of consumer credit Pm will decrease.

Definition 2: The interval [d, 1-Pa] is called the constraint interval of default rate of consumer credit.

Apparently, on one hand, the bigger the interval is, the smaller the probability of the moral hazard of consumer credit is, on the other hand, when the constraint interval of the default rate is determined, for the

probability of the moral hazard of consumer credit changes in the same direction with the default rate, the individual that have greater credit risk(the default rate Pd fall into the right half of interval [d, 1-Pa]), once the asset further deteriorate, the probability of moral hazard of it will further increase.

6. Conclusions

This paper discusses the inner mechanism of moral hazard of personal credit and credit risk. From the theoretical level, this paper gets some important conclusions as follows: (1) When moral hazard exists in consumer credit, bank lending rates r and personal credit probability of occurrence of moral hazard change in the same direction. (2) Consumer credit default probability and the probability of occurrence of moral hazard are in non-linear relationship. (3) There is an exogenous lending rate, making the probability of default of consumer credit reached extremes.(4) There is a constraint interval for the probability of default, when the interval is determined, the probability of individual credit risk of moral hazard Pm and the probability of default Pd changes in the same direction and the greater range is the smaller the probability of personal credit of moral hazard is.

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