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Most literatures prefer loan-to-value ratios (LTV) decisions in supply chain finance (SCF) on the way of profit maximization. This paper attempts to discuss the relationship between LTV and market risk of the loan in inventory financing of SCF from the perspective of value at risk (VaR) for the critical value of LTV corresponding to extreme value of loan VaR to prevent the bank from the risks caused by LTV decisions under the extreme position of price-decline in commodity market. Different from the traditional method of VaR only considering the asset value, we incorporate the borrower’s financial and procurement positions into VaR model. We demonstrate the critical value of LTV corresponding to extrema of the value-at-risk of loan in nonlinear analysis, as well as the critical order quantity that can monotonically affect the relationship between LTV and loan VaR in linear analysis, followed by the conclusion that higher investment may not mean higher risk from the perspective of VaR in inventory financing of SCF. Furthermore, the impact of parameters involving financial and procurement positions of the borrower is discussed to explore the affections to the bank from the borrower’s procurement decisions.

In recent years, supply chain finance (SCF) has been increasingly looked at by European and global enterprises and financial institutes. A survey of “How has the importance of supply chain finance to your organization changed over the past 12 months?” from the Treasury Today’s European Corporate Treasury Benchmarking Study 2010 in association with J.P. Morgan shows that 42.2% of the respondents have a view of “Increased in importance”; proportions of “Remained the same” and “Not on our agenda” are respectively 31.6% and 25.3%; only 0.9% of respondents think that “decreased in importance”. In fact, the modern concept of SCF stems from the world-class enterprises’ global business outsourcing under the trend of cost minimization in 1980s [

Comparing with the traditional credits mainly providing letter of credit business, the SCF concentrates on providing account prepayments financing, inventory pledged financing and factoring [

The objective of this paper is to show the nonlinear relationship between loan-to-value ratio (LTV) and loan value-at-risk (VaR) in inventory financing of SCF. Commercial banks, suppliers (such as manufacturers), buyers (such as retailers) and logistic enterprises participate in this financial behavior. Relying on suppliers’ credits on which their cooperating with the banks based, buyers generally have budget constraints or financing with strategies, finance their debtors to commercial banks with the purchase pledging and regulating by the enterprises closely cooperating with banks who offer financial supports for buyers to place orders being used to pay off the loans after selling in a commodity market. In this financing business, banks play a key role in mitigating the capital pressures in supply chain, however, undertaking a level of risks, which may be mainly caused by the marketability and market price of pledged inventory with the character of self-liquidation [

Given the self-liquidity feature of inventory financing in SCF, we mainly analyzed the banks’ loan-to-value ratio (LTV) decision, which is effective in controlling banking stability through decreasing the sensitivity of mortgage default risk to fluctuation of assets price [

In uncertainty environment, risks and losses are inevitable, but the worst consequence may be predicted, and the measures according to extreme situations are also useful to common, that is, the extreme case may be better to reflect the real world. Just like VaR, the research method used in this paper, which can summarize the worst dollar loss over a target horizon that will not exceed with a given level of confidence and be applied to most financial prices, stock prices, bond prices, exchange rates and commodities. For instance, the Basel Committee on Banking Supervision declared that the banks market risks could be measured by the combination of VaR and internal model. Furthermore, as a standard method for measuring and reporting market risk, VaR not only reforms the traditional financial risk management but also can easily be used to measure and report market risks in a single number with unified unit and to communicate with the top management, shareholders as well as help financial institutions to confront their exposure to financial risks [

By considering the first order and second order conditions of loan VaR model with the general distribution and log-normal distribution of the buyer’s demand under extreme situations of dramatic price-decline, the analytic formulas of the critical order quantity and critical LTV were calculated. The former determines the monotonic property of the linear relationship between LTV and loan VaR when the order quantity is not limited; while the later has an influence on the LTV corresponding to extreme values of loan VaR, and the recessive analytic formula is provided from which the critical values of LTV corresponding to the local maximums and minimums of loan VaR can be calculated, which prevent the bank from the extreme potential loss deriving from LTV decisions. Furthermore, the impacts of parameters relating to borrower’s financial position, procurement and the loan itself on the relationship between LTV and loan VaR were analyzed in numerical examples. However, the problems of setting loan margin, setting the proportion of inventory pledged to total purchase amount, choosing semi-finished product as inventory pledged and LTV decisions of the bank with an attitude of risk- neutral in inventory financing of SCF were not analyzed in this paper.

This paper is organized as follows. Section 2 made several basic assumptions being followed by establishing the model. Section 3 analyzed the model from the perspectives of linear and nonlinear relationships of LTV and loan VaR. In Section 4, numerical examples were used to explore the linear and nonlinear relationships between LTV and loan VaR with considering the affection parameters. The conclusion was made in Section 6.

Inventory financing of SCF is different from the traditional financing with the following properties: 1) The third party, frequently the core enterprise in supply chain, secures for the borrower, such as the retailer in supply chain, with the credit itself instead of her property; 2) Self-liquidity exists in the financing with pledging the borrower’s purchase, which is used to repay the loan through the commodity market; 3) Borrowers without real properties may finance from the bank in a shorter loan period. Thus, basic assumptions are needed before modeling.

The nonlinear relationship between LTV and loan VaR of the commercial bank in supply chain inventory financing is analyzed. It refers to banking decisions that contain loan interest rate, loan period and LTV; the borrower’s initial wealth, purchase or demand that will act as the collateral, which can be sold in the commodity market for paying off the loan. Thus, we set up model based on the following assumptions.

1) Loan interest rate remains stable during the loan period within one year. The sales cycle of pledged inventory (liquidity) will be considered when the bank makes decisions of loan periods, which are negatively related to the liquidity of the collateral, and loan Interest rates are normally expressed for a period of one year.

2) Initial wealth is the only factor classifying retailers by the bank and not only retailers being lack of cash but also the ones owning enough initial wealth may participate in supply chain inventory financing.

3) The retailer orders from her suppliers without idea of actual demand, only the probability distribution of demand [

We assume that the retailer owning initial wealth h orders the size q at a wholesale price p from her suppliers with no idea of the actual demand, only the probability distribution of demand

The loan that is represented as l can be differ from the retailer’s initial wealth h when (1) h is little even cannot afford the m (

where

Follows the Equation (1), the return of the bank

The bank who participates in supply chain inventory financing is risk averse and requires finished products or raw materials as collaterals. Banks prefer finished goods or raw materials to semi-finished products with a high specificity and a low liquidity in the commodity market.

One of the significant characters of supply chain inventory financing is self-liquidating, which the payment of a loan derives from sales of a trade financially supporting by the bank. In this paper, a retailer purchases products depending on a financial support of a bank who requires the borrower pledging the whole or part of the products as collaterals, which can be paid off after being sold in the market, that is, market price and interest rate may become factors of leading to market risk. In model assumption, we assume the interest rate remains stable during the loan period and the market price of the pledged inventory is the only factor affecting the market risk, that is, the bank’s lowest return

VaR can be defined as the dollar loss relative to what was expected for an asset over a target horizon that will not be exceed with a given level of confidence, which implies the identity of the asset during a given horizon, However, both return and loss of a loan in supply chain inventory financing derive from the market value of the pledged inventory. This means there exist a contradiction when we analysis the market risk of bank using the VaR method. For dealing with this problem, we define

where only consider the lowest market value of the “discounted” (corresponding to LTV) collateral but not all. Thus, according to the definition of VaR, the loan LTV of the bank in supply chain inventory financing over a target horizon T at a confidence level of

Followed by first-order and second-order conditions, which were given by

where

In model assumption, we assumed that suppliers only know the probability distribution of the retailer’s demand

Either borrowers or lenders, the order quantity q can be one of the key factors in inventory financing. Although merely being able to directly affect by commercial banks who may prudentially consider the order quantity to make banking decisions and prevent themselves from the risk of over-order. For this reason, Buzacott and Zhang (2004) analyze the maximum order quantity, which can affect the retailer’s bankruptcy risk and bank’s return. In their model, the maximum order quantity is determined by the retailer’s initial wealth, unit purchase cost and a proportion similar to LTV. In this paper, the linear relationship between LTV and loan VaR is analyzed, followed by the condition of the linear relationship, and critical order quantity

Lemma 1. There is linear relationships between LTV and loan VaR following

where

Specifically,

and

Proof. Obviously,

If

Let

Also consider the first order condition, let

Then

In the above analysis, it’s mainly to analyze the monotonically affection of the critical order quantity

Lemma 2. A nonlinear relationship between LTV and loan VaR exists when

Proof. Let

Since

Follows the Equation (4), then

That is, the local maximum of loan VaR exists if

Lemma 2 indicates that the nonlinear relationship between LTV and loan VaR exists under certain conditions, that is, there exists corresponding values of loan LTV leading to extreme VaR of loan.

Lemma 2 identifies the properties of convexity and concavity of LTV-loan VaR curve, which denotes the existing of the extreme value of loan VaR, then the Theorem 1 is is immediately followed by the analytic formula of

Theorem 1. Assume the retailer’s demand

where

sents the proportion of the lending margin to loan amount, p represents unit wholesale price, h represents the retailer’s initial wealth,

Proof. Since there exists the maximum and minimum of loan VaR,

then

then

where

Assume

then

Then

Then the differential equation is solved as follows,

Since

Since

Then,

Since,

Then,

Then,

To some extent,

Based on the above analysis, we specifically assume the retailer’s demand

positive real numbers (Wiki) with parameters

Lemma 3. If

where

The proof follows the Lemma 1,

Lemma 4. If

Proof. Since

and

Follows Lemma 2,

Substitute

Followed by

Substitute u from

where

Specifically, if

Lemma 4 makes the manager of the bank clearly analyze the LTV decisions only if the initial wealth of the retailer h, the margin proportion of the loan

Theorem 2. If

where

Proof. Since

Follows Theorem 1, the value of LTV corresponding to

where

Follows 3.3, log-normal distribution of the retailer’s demand

The mode is the point of global maximum of the probability density function (Wikipedia)

In Lemma 3, the loan VaR has a monotonic linear-increase with LTV as q was in

viously,

The value of loan VaR can either be positive or negative, higher value of abstract of

Based on our analysis in Lemma 2 that higher LTV may corresponding to relatively lower loan VaR, whereas, relatively lower LTV may lead to higher risk level of potential loss of bank loan. According to Lemma 4, the local maximum and local minimum of loan VaR may respectively exist on the left and right sides of the critical LTV that can be given by

According to the analytic formula of LTV^{*}, it’s not difficult to see that LTV^{*} is positive with

retailer’s demand

nonlinear relationship between LTV and loan VaR will be analyzed. Specifically, the proportion

0 | 0.0000 | 1.2 | 0.8750 | 19.6549 | 10 | 11.5858 |

0 | 0.0000 | 1.05 | 1.0000 | 22.4627 | 25 | −2.6642 |

0 | 0.0000 | 0.95 | 1.1053 | 24.8280 | 25 | −0.1642 |

3.5 | 0.1111 | 1.05 | 1.0000 | 22.4627 | 30 | −7.9142 |

8.4 | 1.0000 | 1.05 | 1.0000 | 22.4627 | 8 | 15.1858 |

11 | 1.7460 | 1.05 | 1.0000 | 22.4627 | 6 | 17.2858 |

h and

lowed by

From

Instead of limiting the order quantity q, LTV was set as an limit to prevent the bank from loan risk. As a matter

h | v_{min} | beta | gamma | LTV^{*} | VaR_{l}_{min} | LTV_{l}_{min} | VaR_{l}_{max} | LTV_{l}_{max} |
---|---|---|---|---|---|---|---|---|

15 | 1.50 | 1.4286 | 0.7000 | 1.3083 | - | - | 14.7109 | 1.00 |

15 | 0.85 | 1.4286 | 1.2353 | 1.3083 | - | - | 21.2109 | 1.00 |

5 | 1.50 | 0.4762 | 0.7000 | 0.4361 | 1.9192 | 0.44 | 5.8652 | 0.43 |

5 | 0.85 | 0.4762 | 1.2353 | 0.4361 | 4.7792 | 0.44 | 8.6602 | 0.43 |

of fact, the bank generally has no ability to control the borrower’s order quantity, which may have an influence on the loan VaR. Let

Furthermore,

T and

As an example, let

q | v_{min} | beta | gamma | LTV^{*} | VaR_{l}_{min} | LTV_{l}_{min} | VaR_{l}_{max} | LTV_{l}_{max} |
---|---|---|---|---|---|---|---|---|

5 | 1.50 | 1.9048 | 0.7000 | 0.8722 | 10.4435 | 0.88 | 18.4997 | 0.87 |

5 | 0.85 | 1.9048 | 1.2353 | 0.8722 | 13.3035 | 0.88 | 21.3272 | 0.87 |

15 | 1.50 | 0.6349 | 0.7000 | 0.8722 | -2.7565 | 0.88 | 5.4497 | 0.87 |

15 | 0.85 | 0.6349 | 1.2353 | 0.8722 | 5.8235 | 0.88 | 13.9322 | 0.87 |

q | v_{min} | beta | gamma | LTV^{*} | VaR_{l}_{max} | LTV_{l}_{max} |
---|---|---|---|---|---|---|

5 | 1.50 | 2.8571 | 0.7000 | 1.3083 | 22.2109 | 1.00 |

5 | 0.85 | 2.8571 | 1.2353 | 1.3083 | 25.4609 | 1.00 |

15 | 1.50 | 0.9524 | 0.7000 | 1.3083 | 7.2109 | 1.00 |

15 | 0.85 | 0.9524 | 1.2353 | 1.3083 | 16.9609 | 1.00 |

3 | 3.00 | 0.8298 | 4.1276 | 0.83 | 6.8954 | 0.82 |

3 | 3.50 | 0.5033 | 1.9879 | 0.51 | 4.7557 | 0.50 |

9 | 3.00 | 0.8298 | 4.8630 | 0.83 | 7.5300 | 0.82 |

9 | 3.50 | 0.5033 | 2.3124 | 0.51 | 5.1594 | 0.50 |

T | mu | LTV^{*} | VaR_{l}_{max} | LTV_{l}_{max} |
---|---|---|---|---|

3 | 3.00 | 2.3708 | 8.8333 | 1.00 |

3 | 3.50 | 1.4380 | 10.1796 | 1.00 |

9 | 3.00 | 2.3708 | 9.6200 | 1.00 |

9 | 3.50 | 1.4380 | 11.0073 | 1.00 |

51.84% ((4.1276 − 1.9879)/4.1276 * 100%) and 31.03% ((6.8954 − 4.7557)/6.8954 * 100%), companion with the increase of

Furthermore, as in

In this paper, the problem of the relationship between LTV and loan VaR was dealt with to explore the critical LTV that could affect the extreme values of loan VaR, which was the worst potential loss of the loan causing by LTV decisions and price-decline of the inventory pledged in commodity market. Although several literatures concentrate on the issue of LTV decisions in inventory financing of SCF or the applications of VaR method, there are few studies focusing on the relationship between LTV and loan VaR, meanwhile, considering the borrower’s positions of financial and procurement.

Firstly, we assume that the borrower’s demand follows generally distribution, followed by the general conditions of linear and nonlinear relationships between LTV and loan VaR, as well as the critical order quantity in linear analysis and the critical values of LTV corresponding to extreme values of loan VaR in nonlinear analysis, meanwhile, the log-normal distribution of the borrower’s demand was assumed based on the general model, with the specific results and conclusions. Moreover, the critical order quantity follows the established linear relationship and has an influence on the monotonic property of loan VaR to LTV. In particular, the loan VaR is positive with LTV as the real order quantity is less than the critical value, whereas, with negative value and is negative with LTV, that is, the higher quantity the borrower orders, the lower loan VaR the bank will suffer under the precondition of no order which limits to the borrower. In addition, the initial wealth of the retailer merely influences the loan risk level relative to LTV, as a matter of fact, both

However, the problems of setting loan margin, setting the proportion of inventory pledged to total purchase amount, choosing semi-finished product as inventory pledged and LTV decisions of the bank with an attitude of risk-neutral in inventory financing of SCF were not analyzed in this paper, and the following problems would be fatherly considered, including 1) Consider the first order and second order conditions with the proportion determine the loan margin; 2) Multiply the proportion of inventory pledged to total purchase amount as calculating the loan amount; 3) Consider the buy-back decisions to the collateral with semi-product, which has a high level

of specificity and weaken liquidity in commodity market; 4) Further considering the bank with risk-neutral attitude with an objective of profit-maximization.

The authors would like to thank the support by Project of Outstanding Young Teachers’ Training in Higher Education Institutions of Guangxi and a grant of Guangxi Philosophy and Social Science Fund (13BGL009).

Zhigao Liao,Xin Yu,Jiuping Xu, (2015) Criticality Analysis on Value-at-Risk Model of Loan-to-Value Ratios Decision in Inventory Financing of Supply Chain Finance. Open Access Library Journal,02,1-17. doi: 10.4236/oalib.1102224