2- Supply chain finance for small and medium sized enterprises the case of reverse factoring

International Journal of Physical Distribution & Logistics
Management
Supply chain finance for small and medium sized enterprises: the case of reverse
factoring
Spyridon Damianos Lekkakos, Alejandro Serrano,
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
Article information:
To cite this document:
Spyridon Damianos Lekkakos, Alejandro Serrano, (2016) "Supply chain finance for small and medium
sized enterprises: the case of reverse factoring", International Journal of Physical Distribution &
Logistics Management, Vol. 46 Issue: 4, pp.367-392, https://doi.org/10.1108/IJPDLM-07-2014-0165
Permanent link to this document:
https://doi.org/10.1108/IJPDLM-07-2014-0165
Downloaded on: 10 January 2019, At: 02:16 (PT)
References: this document contains references to 35 other documents.
To copy this document: permissions@emeraldinsight.com
The fulltext of this document has been downloaded 3418 times since 2016*
Users who downloaded this article also downloaded:
(2016),"Does finance solve the supply chain financing problem?", Supply Chain Management:
An International Journal, Vol. 21 Iss 5 pp. 534-549 <a href="https://doi.org/10.1108/
SCM-11-2015-0436">https://doi.org/10.1108/SCM-11-2015-0436</a>
(2015),"Market adoption of reverse factoring", International Journal of Physical Distribution
& Logistics Management, Vol. 45 Iss 3 pp. 286-308 <a href="https://doi.org/10.1108/
IJPDLM-10-2013-0258">https://doi.org/10.1108/IJPDLM-10-2013-0258</a>
Access to this document was granted through an Emerald subscription provided by emeraldsrm:549081 []
For Authors
If you would like to write for this, or any other Emerald publication, then please use our Emerald
for Authors service information about how to choose which publication to write for and submission
guidelines are available for all. Please visit www.emeraldinsight.com/authors for more information.
About Emerald www.emeraldinsight.com
Emerald is a global publisher linking research and practice to the benefit of society. The company
manages a portfolio of more than 290 journals and over 2,350 books and book series volumes, as
well as providing an extensive range of online products and additional customer resources and
services.
Emerald is both COUNTER 4 and TRANSFER compliant. The organization is a partner of the
Committee on Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative for
digital archive preservation.
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
*Related content and download information correct at time of download.
The current issue and full text archive of this journal is available on Emerald Insight at:
www.emeraldinsight.com/0960-0035.htm
Supply chain finance for small
and medium sized enterprises:
the case of reverse factoring
Spyridon Damianos Lekkakos and Alejandro Serrano
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
MIT-Zaragoza International Logistics Program, Zaragoza, Spain
Abstract
Purpose – Faced with increasing pressure to meet short-term financing needs, companies are
looking for ways to unlock potential funds from within the supply chain. Recently, reverse
factoring (RF) has emerged as a financing solution that is initiated by the ordering parties to help
their suppliers secure financing of receivables at favorable terms. The purpose of this paper is to
study the impact of RF schemes on small and medium enterprises’ operational decisions and
performance.
Design/methodology/approach – The authors model a supplier’s inventory replenishment problem
as a multi-stage dynamic program and derive the supplier’s optimal inventory policy for two cases:
no access to external financing; access to external financing through RF or traditional factoring.
A number of numerical experiments assesses the supplier’s operational performance.
Findings – A working capital-dependent base-stock policy is optimal. The optimal policy specifies the
sell-up-to-level of accounts receivable with regard to their maturity. RF considerably improves a
supplier’s operational performance while providing the potential to unlock more than 10 percent of the
supplier’s working capital. When RF is associated with credit-term extension and the supplier has
access to alternative sources of financing, the value of RF is then lower than intuitively expected unless
the interest spread is considerably large.
Originality/value – This is the first attempt to analytically study the impact of RF in a stochastic
multi-period setting.
Keywords Inventory management, Dynamic programming, Working capital management,
Supply chain finance, Reverse factoring
Paper type Research paper
Supply chain
finance
367
Received 30 July 2014
Revised 26 May 2015
31 July 2015
17 January 2016
Accepted 19 January 2016
Nomenclature
p
c
h
w'
m
n
xt
qt
Unit revenue
Unit production cost
Unit inventory holding cost
yt
Unit revenue that is retained
for financing operations, 0 ⩽ w'⩽ p
z0t
Integer number of periods
in the buyer-supplier trade
zt
credit agreement
Integer number of periods in the
buyer-supplier trade credit agreement R0tj
under RF, n ⩾ m
On-hand inventory at the beginning
of period t
Inventory replenishment decision in
period t
Available inventory to service
demand in period t
Available cash at the beginning of
period t
Inventory equivalent of the cash
plus on-hand inventory at the
beginning of period t
Size of the A/R that corresponds
to period’s t−j sales and is pending
at the beginning of period t,
j ¼ 1, …, m(n)
The authors would like to thank two Guest Editors and two anonymous referees for their useful
suggestions.
International Journal of Physical
Distribution & Logistics
Management
Vol. 46 No. 4, 2016
pp. 367-392
© Emerald Group Publishing Limited
0960-0035
DOI 10.1108/IJPDLM-07-2014-0165
IJPDLM
46,4
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
368
R 0t
Rt
Lji
Vector of the pending A/R at the
Lt
beginning of period t
Inventory equivalent of the vector
of pending A/R at the beginning
rr
of period t
Portion of the A/R generated in period rt
i that is sold in period j, Lttj p R0tj for
β
j ∈[i+1, i+n]
Vector of the factoring decision
in period t, where Lttj p R0tj for
all t and j
Interest rate per period for invoice
discounting under RF
Interest rate per period for invoice
discounting under TF
Single-period’s discount rate
Introduction
The current economic conditions, as shaped after the 2008 global financial crisis, along
with the ensuing liquidity constraints and raised sensitivity toward risk in the financial
markets, have created significant issues for companies trying to finance operations and
efficiently manage their working capital. In this environment of relatively low liquidity,
the cost of financing has increased and suppliers, especially small and medium
enterprises (SMEs), are finding it more difficult to obtain the credit they need. The
empirical findings in Campello et al. (2010) suggest that, in the aftermath of the 2008
financial crisis, the deterioration of the SME borrowing capacity has often caused
underinvestment problems. The scarcity of cheap external financing has driven many
firms to look across their financial supply chain for opportunities to improve the
management of working capital, optimize their cash flows, and unlock trapped cash.
Supply chain finance involves the use of financial instruments, processes, and
technologies that facilitate interventions in the financial supply chain by tracking
events in the physical supply chain (e.g. placement of purchase order, inventory
replenishment, order shipment, invoice approval, etc.).
Reverse factoring (RF), the most popular instrument among the different supply
chain finance schemes, has been initiated by large firms with high-quality credit rating
as a mechanism for soothing their suppliers’ financing problems. It involves a threeparty arrangement between a buyer (hereafter, “she”), a factor (usually a bank), and a
supplier (hereafter, “he”). In this arrangement, the buyer promises she will promptly
pay the invoices from her trade transactions with the supplier to the factor, in order for
the factor to provide an approved-invoice-based financing solution to the supplier. That
is, if the supplier wishes to get payment for an approved invoice earlier than its due
date, he can sell the relevant invoice to the factor at a discount that is based on the
buyer’s credit rating. This is possible because the factor in an RF scheme becomes an
essential partner in the supply chain and is able to transfer the financial risk from the
supplier to the buyer. Since our research is relevant for both SMEs and capitalconstrained suppliers, hereafter we use “SME” and “supplier” interchangeably.
In principle, the reason why RF is gaining popularity is because a well-designed
program is supposed to provide advantages to all three parties involved. By expediting the
cash flows from his accounts receivable (A/R) at favorable terms, the supplier can
efficiently manage his working capital and achieve a higher operational performance at a
lower cost. The buyer can achieve direct financial returns through payment-term extension,
a return-oriented strategy, and/or operational benefits through service-level improvements,
a risk-oriented strategy (van der Vliet et al., 2013). Finally, RF enables the factor to make a
profit through service-related fees and cross-selling opportunities. In addition, financing
against the buyer’s credit rating results in decreased portfolio risk which means banks
need less capital reserves in order to meet central bank solvency requirements.
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
Reports in trade journals refer to different strategic orientations in the
implementation of RF programs. A recent example of a “return-oriented approach” is
Procter & Gamble’s decision in April 2013 to extend its payment terms for all suppliers
by 30 days. The firm’s RF program was initiated to help suppliers finance their
increased working capital requirements due to that extension. Using a similar
approach, Unilever has been able to achieve a $2 billion working capital reduction in a
three-year time span (Seifert and Seifert, 2011). Other companies, such as Volvo, Scania,
and Caterpillar, have followed a more “risk-oriented approach” targeted toward helping
their suppliers support their own growth with the expectation of increasing demand for
their end products. Similarly, Philips uses RF to obtain preferred-buyer status with its
suppliers and reduce the risk of disruption in times of shortage. Finally, RF programs
have occasionally been initiated in response to disruptions in the financial markets. For
example, WalMart’s “Supplier Alliance Program,” was offered to more than a thousand
of its apparel suppliers, many of which SMEs, in the aftermath of the 2009 Chapter 11
bankruptcy filing by CIT Group Inc., an established commercial lender.
The study of RF naturally lies on the interface of supply chain management and
finance. While the availability of an alternative form of low-cost financing makes RF
attractive to SMEs, the assessment of the tradeoff between lower cost of financing and
payment-term extension requires an integrated finance/operations approach. From our
discussions with SMEs that participate in RF programs, we realized that these
companies are actively seeking to coordinate their financial (i.e. how much to factor)
and operational (i.e. how much to produce) decisions, which fall under the responsibility
of different functions within the organization, in order to optimize their overall returns.
Moreover, since these firms rely heavily on their internal capital for financing small
investment programs, it is equally important to assess how much cash can be freed up
from their working capital without, though, jeopardizing their service levels.
Motivated by our interaction with RF-financed SMEs, our research intends to
answer the following questions: How is an SME’s inventory replenishment decision
affected by the availability of RF financing? What is the value proposition for an
SME and how is this affected by his operational and financial characteristics? While
the focus of this paper is on the implications of RF financing for SME firms, our
research also has interest for the buyers. By gaining insight on the operational impact
and value potential of RF, the buyers can better select which suppliers to take onboard,
decide whether and by how much to extend their payment terms, communicate the
potential benefits to gain suppliers’ participation, and negotiate their service-level
contractual terms.
To address the first question, we study a multi-period model of a self-financed SME
which replenishes his inventory to satisfy stochastic demand from a single
downstream buyer in a lost-sales operating environment. The study of RF in a
multi-period model is more suitable for capturing the supplier’s tradeoff between the
benefit from the relaxation of his financial constraints at low cost and the higher
financial needs from trade credit extension. We formulate our problem as a Markov
decision process and derive the optimal inventory policy when the supplier: has no
access to external financing; and is able to sell his A/R through RF or traditional
factoring (TF). For both cases, we show that a working capital-dependent base-stock
policy is optimal. In the second case, the optimal policy specifies the sell-up-to level of
the A/R with regard to their time-to-maturity.
To address the second question, we conduct a number of numerical experiments to
assess the impact on the SME’s performance of some key operational parameters
Supply chain
finance
369
IJPDLM
46,4
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
370
involved in the problem, such as the SME’s working capital policy, demand variability,
profit margin, and access to external financing. In line with the anecdotal evidence from
trade journals, our results suggest that RF allows the supplier to unlock a considerable
portion of his working capital, which can be used in other productive investments.
Also, consistent with our intuition, the value of RF increases with the length of existing
credit terms and with profit margin. However, the value of RF is not as high as
intuitively expected when RF is associated with credit-term extension and the supplier
has guaranteed access to TF, unless the credit spread between RF and TF is
substantially large.
The remainder of the paper is organized as follows. Next section provides a review
of the related literature. Then, we analyze and derive the optimal policy for the basecase model of no access to financing and we extend our discussion to the RF and TF
cases. Then, we present the results of our numerical analyses and discuss the
managerial implications. Finally, we conclude with our summary and possible
extensions. The technical results and proofs are presented in the Appendix.
Literature review
To put our work into a broader context, we provide a brief review of the streams of
literature on the interface of supply chain and finance that are related to our paper:
trade credit, sourcing from risky suppliers, inventory models with financial constraints,
and receivables financing.
Trade credit is an important source of financing for a capital-constrained firm.
There are several theories in the finance literature that attempt to explain the purpose
of trade credit such as price discrimination and financing advantages (see Petersen and
Rajan, 1997 and Seifert et al., 2013 for reviews of this literature). The operations
management stream of research treats trade credit as a demand risk-sharing
mechanism. In this literature, the supplier of goods is usually a financially
unconstrained firm, which, acting as the leader in a transactional game with a
financially constrained retailer, decides on the credit terms to maximize his profit.
In this transaction, the supplier trades off the higher returns from increased sales
associated with favorable credit terms with the bankruptcy cost associated with
downstream demand risk. Representative work in this field includes but is not limited
to Kouvelis and Zhao (2012), Caldentey and Chen (2010), Yang and Birge (2010), and Cai
et al. (2014) for single-period models; and Haley and Higgins (1973), Maddah et al. (2004),
and Gupta and Wang (2009) for multi-period models. Our work is aligned with the
market power rational of trade credit (Klapper, 2006), by which the credit terms are
largely determined by the buyer’s bargaining power. In this context, our work studies
the impact of financing solutions initiated by the buyers to help their financially
constrained suppliers.
Some papers more explicitly consider the issue of a buyer sourcing from risky
suppliers. Babich et al. (2007) study the impact of suppliers’ default correlation on a
buyer’s ordering diversification strategy. Babich (2010) studies the optimal joint
capacity ordering and financial subsidy policy for a manufacturer sourcing from a
capital-constrained supplier facing the risk of bankruptcy. Swinney and Netessine
(2009) study the impact of long-term contracts, as a supply chain coordination
mechanism, when a buyer sources from suppliers with uncertain production costs that
are prone to default. While RF is intended to address similar concerns, our model does
not treat buyers as strategic players because our focus is on supplier inventory policies
in light of RF financing.
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
There are a few multi-period models in the existing literature that consider a firm’s
optimal inventory policy in the presence of financial constraints with or without
permissible delays in payments. Chao et al. (2008) study the dynamic inventory control
problem with lost sales of a self-financed retailer and they show that a capitaldependent base-stock inventory policy is optimal. Shi et al. (2013) extend this model to
consider access to riskless bank financing and derive similar results. Li et al. (2013)
consider a firm that decides upon short-term debt, production quantity, and dividends
to maximize shareholder value. Protopappa-Sieke and Seifert (2010) study the
inventory decision of a self-financed firm that is subject to replenishment and payment
delays and working capital restrictions; they use numerical analyses to show the
sensitivity of the firm’s operational and financial performance on the problem’s
parameters. Zeballos et al. (2013) extend this work to consider access to short-term debt.
Luo and Shang (2013) study analytically a self-financed entrepreneurial firm that
periodically orders inventory to satisfy non-decreasing demand with upstream and
downstream payment delays. By utilizing a working capital approach and penalizing
late payments to upstream suppliers, they prove that a myopic, working capitaldependent base-stock policy is optimal. As in Luo and Shang (2013), our paper studies
the inventory replenishment decision of a self-financed firm with downstream payment
delays. Our contribution to this research is twofold. First, we show that desirable
properties of the focal firm’s profit function are preserved when we relax the firm’s
financial constraint to incorporate provision of receivables financing. Second, we
analytically show that a myopic base-stock policy is optimal for the relaxed problem
and characterize it.
Within the literature on receivables financing, Klapper’s (2006) empirical study suggests
that TF and RF are growing financing sources for both large corporations and SMEs in
countries with greater economic development and growth. Randall and Farris (2009)
and Hofmann and Kotzab (2010) use a conceptual approach to demonstrate how value can
be created when supply chain participants collaborate on their cash management and
leverage on the cost of financing of the most creditworthy party in a supply chain. An early
analytical model in the finance literature that focusses on a firm’s A/R factoring decision is
the work of Sopranzetti (1999). The model solves for the breakeven point of a firm’s A/R
credit quality above which factoring can help mitigate the firm’s underinvestment problem.
Our work also relates to some recent research on asset-based lending, which can be
considered a generalization of TF. Buzacott and Zhang (2004) study the impact of assetbased lending on the production decision of a capital-constrained manufacturing firm. In a
manufacturer-bank game, they show that the availability of asset-based financing
enhances the manufacturer’s ability to grow while any information asymmetry can be
resolved when the bank decides both the credit limit and the interest rate. Our model is
similar to the multi-period case in Buzacott and Zhang (2004); however, we consider
stochastic instead of deterministic demand.
The analytical study of RF is relatively limited. Tanrisever et al. (2012) study the
operational and financial decisions of an SME, operating over a single period with
stochastic demand arrival, when RF financing is available. They characterize the
participation constraints for the supply chain members and show that the value of RF
is greater when the credit spread is large, the payment extension period is short, the
demand volatility is high, and the SME’s working capital requirements are relatively
high. van der Vliet et al. (2015) study the same problem in a periodic-review infinitehorizon model with stochastic demand, in which the supplier decides both on his
inventory base stock and his cash management policies. By means of simulation
Supply chain
finance
371
IJPDLM
46,4
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
372
optimization their findings on the supplier’s tradeoff between lower cost of financing
and payment-term extension are similar to those of Tanrisever et al. (2012). Our work
has similarities with the aforementioned research in the sense that we also study
the supplier’s problem. A key difference is on the availability of unsecured bank
financing which is excluded in our model. Also, with the cash retention policy being an
external parameter in our model, we are able to analytically show the optimality of a
base-stock policy.
The base-case model
Consider a periodic-review inventory control problem where a self-financed SME sells a
single product to a single downstream buyer on credit. The SME is risk neutral and
makes inventory replenishment decisions to meet stochastic demand over a T-period
planning horizon. Let Dt be the demand realized during each period t, t ¼ 1,…,T. We
assume that D1, …, DT are independent and identically distributed (i.i.d.) nonnegative
random variables. Let f(⋅) and F(⋅) be, respectively, their probability density and
cumulative distribution functions. Any unsold inventory in periods t ¼ 1, …, T−1 is
carried over to be used in subsequent periods while unmet demand in each period is
lost. This can be the case in a competitive business environment where the buyer can
efficiently use several suppliers for satisfying her sourcing needs. Let p, c, and h be,
respectively, the unit revenue, production cost, and inventory holding cost. For
analytical tractability, we assume that at the end of the planning horizon, the unit
salvage value of any leftover inventory is equal to the unit production cost c.
We assume that the SME’s production costs are incurred at the beginning of each
period, while his sales to the buyer take place under a net-term agreement. That is, trade
credit is characterized by a single term that specifies the number of days after goods
delivery – and invoice approval – within which the outstanding invoice is expected to be
paid in full. We assume that the credit term is exogenously determined within the
industry in which the SME operates (Klapper et al., 2010). For example, in the Procter &
Gamble’s case discussed in the introduction, the firm justified her decision to extend her
payment terms as a necessary alignment with the established terms in her industry.
Let m denote the integer number of periods in the trade credit agreement. That is, payment
for sales made in period t is due at the beginning of the period t + m + 1. Without loss of
generality, the firm’s inventory replenishment and shipping lead times are zero.
The SME has no access to external financing and his production decision is
constrained by available cash at the beginning of the period (this assumption is relaxed in
the next section where the RF case is studied). Our assumption of a self-financed SME is
motivated by some recent empirical findings from the finance literature that suggest that,
in the aftermath of the 2008 financial crisis, firms of all sizes (and particularly SMEs) have
seen their lines of credit shrink, face higher financing costs, and have difficulties in
financing valuable investments (Ivashina and Scharfstein, 2010; Campello et al., 2010).
Let w0 be the unit revenue that is retained for financing operations according to the firm’s
working capital policy, 0 ⩽ w0 ⩽ p. Essentially, w0 is determined by the financial manager
and summarizes how much of the supplier’s internally generated cash is tied up in
financing his working capital. Since the growth of SMEs is usually constrained by their
internal financing (Carpenter and Petersen, 2002), w0 also captures the SME’s cash
allocation policy between short- and long-term investments.
state
by a vector
The
of the system at the beginning of each period is summarized
the amount of on-hand inventory, z0t the available cash,
xt ; z0t ; R 0t , where xt denotes
and R 0t ¼ R0tm ; . . .; R0t1 the m-dimensional vector of the firm’s pending A/R. In our
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
notation, R0tm is the most mature A/R and corresponds to period’s t − m sales, the
payment of which will materialize in period t+1. The sequence of events in each period
is as follows: first, at the beginning of each period, after observing the available onhand inventory, xt, and cash, z0t , the firm makes his inventory replenishment decision,
qt. Since the firm is self-financed and all production costs are incurred at the beginning
of the period, the replenishment decision satisfies the cash constraint cqt p z0t . Second,
during the period, demand Dt is realized and serviced by on-hand inventory, xt+qt.
Under the lost-sales assumption, the satisfied demand in each period is equal to min
(xt+qt, Dt). Finally, at the end of the period any unsold inventory, (xt+qt−Dt)+, with
(a)+ ¼ max{a, 0}, is carried over to subsequent periods incurring the corresponding
holding cost. The notation used in the paper is summarized in Nomenclature.
Before proceeding with the mathematical
of the model, it is convenient
formulation
to define yt ¼ xt + qt, zt ¼ xt þ z0t =c; R t ¼ 1=c R 0t , and w ¼ w0 /c. We refer to yt as
the inventory level after the firm’s production decision, which will be the new decision
variable instead of qt; zt represents the inventory equivalent of the cash level plus
available inventory; and Rt represents the inventory equivalent (measured in units) of
the A/R still pending.
Let β be the single-period’s discount rate. Then, the expected profit function in each
period, t ¼ 1, …, T, is given by:
(1)
Gt ðxt ; yt Þ ¼ E bm p min ðyt ; Dt Þhð yt Dt Þ þ cðyt xt Þ
for xt ⩽ yt ⩽ zt, and the dynamics of the system are:
xt þ 1 ¼ ðyt Dt Þ þ
(2)
zt þ 1 ¼ xt þ 1 þ Rtm þ zt yt
(3)
R t þ 1 ¼ ðRtm þ 1 ; . . .; Rt1 ; w min ðyt ; Dt ÞÞ
(4)
The first term in (1) is the expected present value of period t’s sales minus holding cost;
the last term is the inventory replenishment cost. Notice that while the inventory holding
cost is incorporated in the expected profit function, it is not considered in the cash flow
dynamics in (3). We justify this by assuming that the firm uses accrual accounting for cost
recognition; i.e., costs are recognized when they occur rather than when the payment is
made (Luo and Shang, 2013). For example, h may represent insurance or obsolescence
reserve costs which are settled at the end of the planning horizon. Also, notice that the
A/R vector in (4) captures the portion of receivables that is retained for operations (as this
is determined by w). We will maintain this convention throughout the paper.
The decision problem for a risk-neutral firm is to decide upon an ordering policy to
maximize the expected profit over the planning horizon, given an initial state (x1, z1, R1).
Let Vt(xt, zt, Rt) be the maximum expected terminal wealth over all feasible solutions,
given a state (xt, zt, Rt). This results in the following dynamic programming formulation:
(5)
V t ðxt ; zt ; R t Þ ¼ max Gt ðxt ; yt Þþ bE ½V t þ 1 ðxt þ 1 ; zt þ 1 ; R t þ 1 Þ
V T þ 1 ðxT þ 1 ; zT þ 1 ; R T þ 1 Þ ¼ cxT þ 1
(6)
The terminal wealth function, VT+1(⋅,⋅,⋅), does not contain any elements of the A/R
vector since their expected present values have been incorporated into the profit
functions, Gt(⋅,⋅), of the periods that they were realized. To this end, the A/R vector in
the base-case model captures additional information that only affects the cash flow
Supply chain
finance
373
IJPDLM
46,4
constraint in subsequent periods. Consequently, the firm’s fundamental tradeoff is
between investing in inventory to avoid lost sales vs incurring the inventory holding
cost associated with this decision.
374
Optimal policy
Next, we will show that a myopic policy is optimal for the dynamic problem presented
above. The myopic maximization problem at period t can be written as:
(7)
maxxt p yt p zt J t ðyt Þ
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
where:
J t ðyt Þ ¼ E bm p min ðyt ; Dt ÞðhbcÞðyt Dt Þ þ cyt
(8)
here βcE[(yt−Dt)+] represents the salvage value of unsold inventory in the myopic system.
Observe that the myopic problem in (7) is a newsvendor type of problem. Let S be
the optimal base-stock level for the unconstrained problem. Given that function (8) is
concave in y, S is the classical newsvendor solution:
bm pc
(9)
S ¼ F 1 m
b p þ hbc
Next, we establish the optimal policy for the myopic problem in (7):
Lemma 1. For any given initial state (xt, zt, Rt), a state dependent base-stock policy
y^ n ðx; zÞ is optimal for the myopic problem in (7), where:
8
>
< z; if z p S;
n
y^ ðx; zÞ ¼ S; if x pS p z;
>
: x; otherwise:
and S is given by (9).
The proof is omitted.
Lemma 1 states that the optimal base-stock level for the myopic problem, S, may not
be achievable due to the working capital constraint represented by z. When x ⩽ S, the
optimal achieved production quantity is min(z, S); whereas, if x W S, not producing is
optimal. Therefore, the capital constraint in our model serves a similar role to the
capacity constraint in inventory control problems (e.g. Angelus and Porteus, 2002). The
optimal policies in both cases are capital/capacity-dependent although the capital
constraint in our model is endogenous.
Let ynt ðzt Þ ¼ minðzt ; S Þ and Bt ¼ fðxt ; zt Þ A R2 : xt p ynt ðzt Þg be the region where the
initial inventory does not exceed the optimal base-stock level ynt ðzt Þ. We next state our
main result in this section which is an adaptation of Proposition 4 in Luo and Shang
(2013) to our model:
P1. For the model (5)-(6), given an initial state (xt, zt, Rt) and (xt, zt)∈Bt, in each period
t ¼ 1, …, T:
(1) the objective function can be decomposed as Vt(xt, zt, Rt) ¼ cxt+Wt(zt, Rt), where
Wt is jointly concave; and
(2) a working capital-dependent base-stock policy is optimal, under which the
optimal production up-to-level is ynt ðzt Þ ¼ minðzt ; S Þ, where S is given by (9).
Supply chain
finance
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
In words, if the on-hand inventory at the beginning of period 1, x1, is less than or equal
to the unconstrained optimal base-stock level, S, a myopic policy that in each period
raises inventory up to min(zt, S) is optimal for the multi-period model.
The RF and TF models
Having addressed the base-case model where the SME is self-financed, in this
section we consider the case of an SME that can factor his receivables to obtain
additional funds. In the RF model, the buyer initiates an RF program to facilitate
the supplier’s financing of operations. The supplier may choose to factor the
invoices from his trade transactions with the buyer after the invoices are approved,
i.e., at any time after the end of the period when the corresponding sales are made.
The buyer’s RF program is characterized by two terms, namely, rr and n. The first
term, rr, is the period’s interest rate at which the relevant invoices are discounted
when factored. We assume that rr ¼ rb+b, where rb is the interest rate applicable
to the buyer and b is the bank’s fee for facilitating the program. The second term, n,
is the new credit term required by the buyer. In order to participate in the RF
program the supplier must agree to extending the credit period from m to n periods,
where n⩾m.
Following the same notation as in the base-case model, the state of the system
at the
0
0
;
z
;
R
beginning
of
each
period,
t
¼
1,
…,
T,
is
summarized
by
a
vector
x
t
t
t , where
R 0t ¼ R0tn ; . . .; R0t1 is now the n-dimensional vector of the firm’s retained A/Rs, the
payment for which is still pending. In the RF model, the firm decides not only on the
production quantity, qt, but also on whether to factor any of the pending A/Rs. We
assume that the supplier can discount any portion of the outstanding A/R. From our
discussions with SME suppliers to corporations in the consumer goods industry, we
saw that invoice approvals take place on fixed dates in each month; therefore, each
period’s A/R often contains a large number of distinct invoices, which justify this
assumption. Let Lji denote the portion of the A/Rs generated in period i that is sold in
period j, where j∈[i+1,
i+n]. Then,
the factoring decision in period t can be captured by
the vector Lt ¼ Lttn ; . . .; Ltt1 , where Lttj p R0tj for all t ¼ 1, …, T and j ¼ 1, …, n.
The expected profit function in each period, t ¼ 1, …, T, is then:
Gt ðxt ; qt ; L t Þ ¼ E bn p min ðxt þ qt ; Dt Þhðxt þ qt Dt Þ þ
cqt n h
i
X
1ð1r r Þj Lttn þ j1
(10)
j¼1
P
for 0 ⩽ cqt ⩽ z0t þ nj¼1 ð1 − r r Þ j Ltt−nþj−1 and Ltt−j ⩽ R0t−j , for all t=1, …, T and j=1, …, n.
The last term in (10) corresponds to the interest charged by the bank when the
supplier sells some of his pending A/Rs. In the next section, we study the properties of
the expected profit function and derive the optimal policy for the single-period problem
as an intermediate step before addressing the multi-period problem.
The single-period RF case
To reduce the dimension of the problem, we argue that whenever the supplier resorts to
factoring of his A/Rs to finance his operations, it is optimal to do so in decreasing A/R
375
IJPDLM
46,4
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
376
maturity order. In other words, the supplier will be better off if at time t he sells a more
mature A/R, say R0tn , before selling any part of the less mature A/R, say R0tn þ 1 , since
the cost associated with the latter is higher. To see this, notice that, in the expected
profit function (10), the penalty from selling any layer of A/R increases with the A/R’s
time-to-maturity. Therefore, when adapted to the single-period myopic problem, the
expected profit function in (10) can be written as:
J t qt ; xt ; z0t ; R 0t ¼ E½bn p min ðxt þ qt ; Dt ÞðhbcÞðxt þ qt Dt Þ þ cqt
1
1 cqt z0t 1fz0 o cqt p z0 þ ð1rr ÞR0tn g
t
t
1r r
"
j
n1 X
X
1
1
0
1 Rtn þ i1 þ
1
i
ð1r r Þ
ð1r r Þj þ 1
j¼1
i¼1
!#
j
X
i 0
0
cqt zt ð1r r Þ Rtn þ i1
i¼1
1
fz0t þ
P
j
i¼1
ð1r r Þi R0tn þ i1 o cqt p z0t þ
jþ1
P
ð1r r Þi R0tn þ i1 g
(11)
i¼1
where 1{⋅} is the indicator function (i.e. 1{x W 0} ¼ 1 if x W 0, and 0 otherwise). The
number of terms of the expected profit function in (11) is driven by the credit term, n.
Consequently, the last n terms in (11) correspond to the interest charged by the bank
when the supplier sells his pending A/Rs in increasing maturity order. For example,
consider the case that the SME’s production decision, q t , is such that
z0t o cqt p z0t þ ð1r r ÞR0tn , i.e., the SME factors only his most mature A/R. Then, the
first indicator function in (11) will be equal to 1 and the corresponding term will
represent the interest paid to the bank for that decision, while the remaining n−1
indicators will be equal to 0. Notice that through the indicator functions, it is now the
inventory replenishment decision, qt, that drives the depletion of A/R. Consequently,
the problem is reduced to one with a single decision variable. Lemma 2 provides the
properties of the myopic single-period’s profit function:
Lemma 2. The function J t ðqt ; xt ; z0t ; R 0t Þ is continuous in qt, xt,z0t , and R 0t , and has the
following properties:
h
i
P
(1) it is concave in qt, for qt A xt ; xt þ 1c z0t þ nj¼1 ð1r r Þj R0tn þ j1 ; and
(2) it is increasing and concave in xt, z0t , and R 0t (component-wise).
Lemma 2 suggests that there is an optimal inventory replenishment policy that
maximizes the single period expected profit function of (11). To characterize the optimal
solution, it is convenient to introduce new system variables
and definitions as in the
previous section. Define y t ¼ x t+qt , R t ¼ 1=c R 0t , and the vector
t1
0
tn
¼ zt þ ð1r r ÞRtn , and so on up
z t ¼ ðzt ; ztn
t ; . . .; zt Þ, where zt ¼ xt þ zt =c, zt
n
t1
t2
to zt ¼ zt þ ð1r r Þ Rt1 . Here, the first element of the vector zt represents the
inventory equivalent of the cash level at the beginning of the period plus the available
inventory, while the subsequent elements correspond to the same quantity
supplemented with the inventory equivalent from selling each layer of the A/R
vector in decreasing maturity order. Also, define:
!
bn pð1rc Þk
bn pc
1
1
r
and stn þ k1 ¼ F
;
S¼F
bn p þ hbc
bn p þ hbc
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
for k ¼ 1; 2; . . .; n:
(12)
The quantities S, st−n, …, and st−1 represent the optimal inventory replenishment
decisions (order-up-to levels), as they are derived from the first-order condition of (11),
and correspond to dives in the A/R layers in decreasing maturity order. For example,
st−n is the optimal order-up-to level when the firm only sells part of the most mature
A/R, R0tn . Notice that the optimal production decisions, S, st−n, …, and st−1, are
t1
p zt1
achievable only if S ⩽ zt, stn p ztn
t , …, and s
t , respectively. Next, we
formalize the optimal policy for the single-period problem:
Lemma 3. For any given initial state (xt, zt), a capital-dependent base-stock policy
ynt ðxt ; z t Þ is optimal for the myopic problem in (11), where:
ynt ðxt ; z t Þ ¼
8
max fxt ; S g;
>
>
>
>
>
zt ;
>
>
>
>
>
stn ;
>
>
>
> ztn ;
>
>
>
> t
>
<...
stn þ k ;
>
>
>
>
þk
>
ztn
;
>
t
>
>
>
>
>
...
>
>
>
>
>
st1 ;
>
>
>
: zt1 ;
t
if S p zt ;
if stn p zt o S;
if zt o stn p ztn
t ;
if stn þ 1 p ztn
o stn ;
t
...
þ k1
þk
if ztn
o stn þ k p ztn
; k ¼ 1; . . .; n2;
t
t
(13)
þk
if stn þ k þ 1 p ztn
o stn þ k ; k ¼ 1; . . .; n2;
t
...
if zt2
o st1 p zt1
t
t ;
t1
if zt o st1 ;
and S, st−n, …, and st−1 are given by (12).
The proof is omitted.
The cases considered in (13) are mutually exclusive and collectively exhaustive;
thus, there can only be one optimal decision for any initial state realization.
Figure 1 depicts the construction of the optimal policy in Lemma 3 when n ¼ 3. The
different curves correspond to the first-order derivatives of the expected profit function
for different utilizations of A/R layers in financing production. For example, the solid
line that crosses the horizontal axis at y ¼ S (the unconstrained optimal) corresponds to
the first-order derivative of (11) for yt ⩽ zt. It follows that if zt ⩾ S, the optimal
replenishment decision is given by max{xt, S}. Similarly, if st−3 ⩽ zt o S, the
optimal replenishment decision is equal to zt since the marginal profit from selling the
most mature A/R (as given by the corresponding dotted line) is negative in this region.
In general, the optimal policy is automatically determined if we position the working
capital vector, zt, in parallel to the horizontal axis and examine the position of the
breaking points (of zt) relative to the first-order condition crossing points.
Supply chain
finance
377
IJPDLM
46,4
G (x ,y ,z )
y t t t t
yz t
z t yz t
t– 3
t– 2
z tt– 3yz t
z tt – 2yz tt–1
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
378
Figure 1.
Construction of the
single-period optimal
policy (Lemma 3)
for n ¼ 3
0
s t –1
s t –2 s t–3 S
y
ztt–3
zt
ztt–2
ztt–1
0
In Figure 2, we provide an example for n ¼ 3, given an arbitrary initial state (xt, zt).
First, notice that the profit function is not differentiable at the breaking points. Also,
t2
are not large enough to reach their corresponding optimal
notice that zt, zt3
t , and zt
inventory replenishment decision, S, st−3, and st−2, respectively. So, the firm must
decide whether or not to sell the most recent A/R. However, by doing so, the marginal
profit for the firm would be negative. Therefore, the optimal decision would be to raise
inventory up to zt2
t ; that is, to sell the two most mature A/Rs, but not the latest one,
o st2 .
which is what Lemma 3 suggests for st1 p zt2
t
The multi-period RF case
Once the single-period case has been studied, in this section we derive the optimal policy
for the multi-period RF model. With the variable transformations introduced in the singleperiod RF case, the dynamic programming model of the problem under study can be
written as:
V t ðxt ; z t Þ ¼ max fGt ðxt ; yt ; z t Þþ bE½V t þ 1 ðxt þ 1 ; z t þ 1 Þg
(14)
V T þ 1 ðxT þ 1 ; z T þ 1 Þ ¼ cxT þ 1
(15)
G (x ,y ,z )
y t t t t
Optimal solution
y* = z tt–2
Figure 2.
Example of the
application of
Lemma 3 for n ¼ 3
and for arbitrary
initial state (xt, zt)
0
s t– 1
zt
z tt– 3
z tt–2
s t– 2
s t– 3
S
zt
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
for xt pyt p zt1
t , where:
Gt ðxt ; yt ; z t Þ ¼ E bn pminðyt ; Dt Þhðyt Dt Þ þ cðyt xt Þ
1
1 cðyt zt Þ 1fzt o yt p ztn g
t
1r r
"
j1 n1
X
X
1
þi
þ i1
c
1 ztn
ztn
t
t
iþ1
ð1r r Þ
i¼0
j¼1
1
þ j1
þ
1 yt ztn
1ztn þ j1 o y p ztn þ j t
jþ1
t
t
t
ð1r r Þ
Supply chain
finance
379
(16)
where in (16) we use the convention that ztn1
zt . The dynamics of the states are
t
given by:
xt þ 1 ¼ ð yt Dt Þ þ
(17)
þ1
t
z t þ 1 ¼ zt þ 1 ; ztn
t þ 1 ; . . .; zt þ 1
(18)
where:
zt þ 1 ¼ xt þ 1 þ ðzt yt Þ þ þ
þ1
ztn
¼ zt þ 1 þ
t þ1
þ
1
Þ
ðztn min yt ; ztn
t
1r r t
1
þ1
þ1 þ
ðztn
min yt ; ztn
Þ
t
t
1r r
þk
þ k1
ztn
¼ ztn
þ
t þ1
t þ1
1 tn þ k
ðz
minð yt ; zttn þ k ÞÞ þ
1r r t
for k ¼ 2, …, n−1, and:
n
ztt þ 1 ¼ zt1
t þ 1 þ ð1r r Þ w minð yt ; D t Þ
(19)
We next define the set Bt ¼ {(xt, zt)∈R : xt ⩽ S}, which establishes the region where
the initial inventory does not exceed the unconstrained optimal base-stock level S.
Consequently, for (xt, zt)∈Bt, the single-period optimal inventory replenishment
decision from Lemma 3 is only a function of zt; i.e., ynt ðxt ; z t Þ ¼ ynt ðz t Þ.
Having presented the problem, we show next that a base-stock policy is optimal:
n+2
P2. For the model (14)-(16), given an initial state (xt, zt)∈Bt, in each period
t ¼ 1, …, T:
(1) the objective function can be decomposed as Vt(xt, zt) ¼ cxt+Wt(zt), where Wt is
jointly concave; and
(2) a working capital-dependent base-stock policy is optimal where the optimal
production up-to-level is ynt ðz t Þ, as given by (13).
P2 states that if the on-hand inventory at the beginning of period 1, x1, is less than
or equal to the unconstrained optimal base-stock level, S, then, in each period
IJPDLM
46,4
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
380
t ¼ 1, …, T, the capital-dependent base-stock policy specified by the myopic
problem is optimal. The optimality of a myopic solution, when the supplier operates
in an environment characterized by stationary demand and cost parameters, is
useful since it simplifies the decisions of the firm’s operations and financial
managers. In particular, the firm’s managers, instead of dealing with the complicate
longer term problem (given by the dynamic programming model), will only have to
consider each period’s starting conditions to make their planning decisions for
the period.
We do not explicitly model the TF problem because its only difference from the
presented RF model is on the applicable terms. In particular, the interest rate for TF, let
rt, is expected to be strictly greater than rr. This is due to the deadweight costs related
with bankruptcy risk, information asymmetry, and other agency costs a bank is subject
to in its transactions with an SME (Dietsch and Petey, 2002), which are eliminated with
the buyer’s intermediation in an RF arrangement. Therefore, our results in this section
also hold for the TF model upon simply replacing (rr, n) with (rt, m) since TF does not
involve any credit extensions.
Numerical study
Having analytically derived the SME’s optimal policy for each of the cases studied in
the previous two sections, in this part we conduct a series of numerical experiments to
assess the impact of the operational parameters involved in the problem on the SME’s
performance. We use Monte Carlo simulation to test the firm’s performance in each
case, since an analytical comparison is not possible due to the endogenous nature of the
financial state of the system. Our parameter selection was carefully made to reflect the
operating environment of an SME. However, our analysis is not exhaustive but is
concentrated on the impact of some key parameter values on the firm’s performance.
As such, our numerical simulations serve exposition purposes and, being subject to
limitations, need to be verified in practice.
We consider a supplier that operates over a 12-month planning horizon and we test
three scenarios, denoted by S(m, n), for (m, n) values of (1,2), (2,3), and (2,2). Thus, RF is
associated with a credit extension of one period in the first two scenarios, whereas there
is no credit extension in the third scenario. The demand in each period is normally
distributed (truncated to avoid negative realizations) with μ ¼ 100 and σ ¼ 30. In each
simulation 1,000 replications are generated.
The nominal parameter values are fixed to p ¼ 1.4, c ¼ 1, w0 ¼ 1, h ¼ 0.2, β ¼ 0.9975,
and rr ¼ 0.005. The selection for w0 being equal to the production cost is reasonable
for an SME that faces i.i.d. demand and that relies on internally generated cash to
finance his capital investments. The selection for rr corresponds to an annual rate of
about 6 percent (a buyer interest rate of 5 percent plus an 1 percent bank fee), which is
in line with the figures provided in Seifert and Seifert’s (2009) study of 23 RF
programs. The initial inventory, x1, is fixed
to zero under all of the scenarios, while
the initial cash and A/R vector, z01 ; R 01 is fixed to (60,60,60), (60,60,60,60), and
(120,60,60) for S(1, 2), S(2, 3), and S(2, 2), respectively. For example, the three values of 60
in the S(1, 2) scenario correspond to the initial cash level, the most mature A/R, and the
most recent A/R. Accordingly, when the supplier does
not participate in RF (and
consequently, does not concede any credit extension), z01 ; R 01 is equal to (120,60)
and
(120,60,60) for scenarios S(1,2) and S(2,3), respectively. Our selection for z01 ; R 01 is
consistent with our focus on the operational decisions and performance of a
moderately cash-constrained SME.
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
Table I shows the results of our simulations with the nominal parameter values for
some key performance variables. The profit calculation corresponds to the present
value of the earnings from operations at the end of the planning horizon and
incorporates the inventory holding cost. Total inventory refers to the sum of achieved
base-stock level over the 12 periods.
On average, the RF case results outperform the ones without RF, in terms of both
service level and profitability, under all scenarios. To test the significance of the profit
differences shown in Table I we conducted, per scenario, a paired difference test for
means (t-test) at 99.9 percent confidence level. The t-statistics ( p-values) are equal to
23.5 (8.4E−98), 40.7 (1.8E−214), and 51.9 (3.4E−286) for scenarios (1,2), (2,3), and (2,2),
respectively. These results suggest that the profit under RF is significantly greater
than that without.
The demand coverage performance with RF is always greater than that without due
to the flexibility RF provides (through the relaxation of the cash constraint) in
supporting a replenishment decision as close as possible to the unconstrained optimal.
Total inventory with RF is no less than 0.5 percent of the unconstrained
optimal (as shown in the left panels of Figure 3 for w ¼ 1) under all scenarios.
Consequently, the impact of RF on demand coverage is very close to the expected
performance of the unconstrained supplier.
While the service-level volatility (as reflected on the corresponding ranges) with RF
is low, that is not the case with profit. This is due to the fact that performance on profit
is more sensitive to demand realizations. While in our simulations the profit with RF is,
on average, no less than 1.25 percent of the achieved profit of an unconstrained SME
under all scenarios, there is no guarantee that RF will always outperform the “no RF”
operations. Our simulations outcomes show that the profit without RF is greater than
the profit with 25.2, 8.1, and 3.9 percent of the times in scenarios S(1, 2), S(2, 3), and S(2, 2),
respectively. The results in Table I also show that the value of RF increases with the
credit term, as the average profit differential under scenarios S(2, 3) and S(2, 2) is
considerably greater than that in scenario S(1, 2). This may explain why RF is usually
adopted in supply chains that already operate with long upstream payment delays.
Finally, we tested the sensitivity of the supplier’s performance
on
the initial
conditions, as determined by the size of the elements of the z01 ; R 01 vector. We
considered the cases of 10 percent richer and 10 percent poorer supplier. Our results
show that the SME’s performance with RF, in terms of both profitability and service
level, is quite robust. In the “no RF” case, though, the performance is sensitive to the
initial conditions with the adverse impact from a 10 percent decrease to the initial
conditions be greater than the favorable impact from a 10 percent increase. While it is
intuitive that the more financially constrained an SME is the greater the value
proposition of RF, this result may also have implications for an SME operating in an
environment where the demand demonstrates some seasonality. With seasonal
demand, the SME may find himself with less (more) cash at the end of the low (high)
demand cycle. Then, the robustness of the RF results implies that the SME’s
performance may be less sensitive to the cash fluctuations inherent in a seasonal
demand environment.
Impact of working capital policy
One key argument in trade journals on the benefits of RF is the potential for a
substantial reduction in the net working capital for both parties. Consequently, the
working capital that is freed-up due to RF financing may be used in other investments.
Supply chain
finance
381
324.9 (216.0, 345.4)
71.2 (57.5, 90.1)
10% poorer
Profit ($)
355.3 (249.6, 391.9) 391.1 (251.5, 510.0)
Demand coverage (%)
78.2 (63.1, 96.7)
93.2 (80.2, 100.0)
Note: The minimum and maximum are shown in parentheses
RF
386.0 (213.5, 491.0)
92.8 (81.2, 100.0)
389.5 (249.6, 503.1)
93.0 (79.8, 100.0)
386.8 (241.8, 496.9)
92.9 (81.7, 100.0)
10.15 (−12.2, 31.0)
28.20 (11.0, 38.7)
S(2, 3)
350.6 (253.9, 383.7)
78.1 (63.1, 93.9)
No RF
373.5 (241.4, 421.2)
84.1 (68.8, 98.4)
393.5 (250.0, 503.7)
92.8 (81.1, 100.0)
3.63 (−11.4, 16.4)
16.63 (7.2, 23.4)
RF
394.4 (253.0, 508.5)
93.1 (80.8, 100.0)
395.2 (232.8, 478.0)
89.4 (74.3, 100.0)
10% richer
Profit ($)
Demand coverage (%)
Table I.
Supplier’s average
operational
performance
379.2 (258.7, 435.4)
84.2 (70.4, 100.0)
S(1, 2)
324.1 (218.8, 345.4)
71.5 (56.2, 95.2)
373.8 (248.2, 422.0)
83.9 (65.5, 99.6)
RF
393.7 (236.0, 505.6)
93.1 (81.7, 100.0)
399.4 (233.3, 517.3)
93.2 (81.6, 100.0)
395.7 (219.9, 498.4)
93.1 (82.6, 100.0)
12.72 (−12.2, 33.1)
29.25 (9.5, 39.7)
S(2, 2)
350.5 (244.4, 383.7)
78.1 (62.9, 95.9)
No RF
382
Nominal
Profit ($)
Demand coverage (%)
ΔProfit (% over no RF)
ΔTotal inventory no RF)
No RF
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
IJPDLM
46,4
Total inventory (no. of items)
Total inventory (no. of items)
1.1
m =2, n =3
w( )
1.1
1.2
1.2
1.4
w( )
1.3
RF
No RF
1.4
Unconstrained
1.3
300
350
400
450
200
200
1
1
500
0.9
0.9
250
300
350
250
0.8
0.8
RF
No RF
Unconstrained
400
450
700
900
1,100
1,300
1,500
500
700
900
1,100
1,300
m =1, n=2
Total profit ( )
1,500
Total profit ( )
0.8
0.8
0.9
0.9
1
1
w( )
1.1
m =2, n=3
w( )
1.1
m=1, n=2
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
1.2
1.2
1.3
1.3
No RF
1.4
RF
No RF
1.4
RF
Supply chain
finance
383
Figure 3.
Impact of working
capital policy on
supplier’s operational
performance
IJPDLM
46,4
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
384
This potential is obvious for a buyer that links her RF program to credit-term extension
due to the resulting increase in her accounts payable. To explore the impact of RF on
the supplier’s working capital, we test the SME’s operational performance relative to
his working capital policy, as captured by the retained unit revenue. The rationale is
that if the supplier’s performance with RF is satisfactory under a more aggressive
working capital policy (i.e. smaller w), then he may decide to retain less for current
investment and more for long-term investment (e.g. fixed assets). For a self-financed
SME, a smaller w translates to a direct reduction in working capital. Figure 3 shows the
impact of w on the SME’s performance. The dotted line corresponds to the optimal
replenishment decision of an unconstrained supplier.
The SME’s performance with RF is comparable to that achieved with more
conservative working capital policies (i.e. larger w) and “no RF.” Table II summarizes the
simulation results for a case where w is equal to 0.9 and 1, respectively, for RF and “no RF.”
Even with a more aggressive working capital policy, with RF the service level is
always higher, whereas the profit is, on average, slightly higher in scenario S(1, 2) and
significantly improved in scenarios S(2, 3) and S(2, 2). These results agree with the
findings in Seifert and Seifert (2011) that the average working capital reduction from
RF for suppliers is 14 percent. Consequently, in evaluating his participation in an RF
program, an SME should also consider the flexibility that RF provides in productive
usages of his freed-up working capital. For example, consider an SME that evaluates a
capital investment on some productivity improving equipment, but the cost of external
financing (if available) makes this investment unattractive. With RF, the SME could
temporarily use a more aggressive working capital policy and use the freed-up working
capital to finance the cash outflows associated with the investment, without
jeopardizing his service level with the buyer.
Impact of demand variability and profit margin
As one would expect, the value of RF when compared to “no RF” decreases with
demand variability and increases with profit margin (Figure 4). This is a direct
consequence of the inventory replenishment decision with RF being closer to the
unconstrained optimal. Thus, the SME is able to realize a higher profit when the
demand uncertainty is low and the profit margin is higher.
Impact of access to external financing
Next, we test how RF performs in relation to TF as a function of rt. As expected, since
both TF and RF enable an inventory replenishment decision that is close to the
unconstrained optimal, in our simulations the total inventory in both cases is almost
S(1, 2)
No RF (w ¼ 1)
Table II.
Supplier’s average
operational
performance with an
aggressive working
capital policy
under RF
Profit
Demand
coverage (%)
ΔProfit (% over
no RF)
ΔTotal inventory
(% over no RF)
S(2, 3)
RF (w ¼ 0.9)
No RF (w ¼ 1)
S(2, 2)
RF (w ¼ 0.9)
No RF (w ¼ 1)
RF (w ¼ 0.9)
379.1 (272.1, 434.4) 382.7 (247.5, 456.3) 350.8 (247.5, 383.7) 383.5 (219.4, 500.3) 350.7 (238.2, 383.7) 392.9 (231.8, 500.3)
84.5 (71.4, 98.7)
88.5 (75.3, 100.0)
78.4 (62.3, 94.3)
93.0 (78.0, 100.0)
78.4 (62.4, 99.7)
93.2 (74.9, 100.0)
0.99 (−10.6, 18.3)
9.16 (−12.8, 30.4)
11.86 (−12.2, 31.3)
7.44 (1.6, 15.6)
28.31 (9.2, 38.6)
28.70 (12.7, 39.2)
Note: The minimum and maximum are shown in parentheses
Total inventory (no. of items)
Total inventory (no. of items)
600
800
1,000
1,200
1,400
1,600
500
700
900
1,100
1,300
1.2
10
1.4
20
p( )
1.6
m=2, n=3
30
m=2, n=3
1.8
40
RF
No RF
2
50
Unconstrained
RF
No RF
Unconstrained
Total profit ( )
1,500
Total profit ( )
0
200
400
600
800
1,000
1.2
10
1,200
300
350
400
450
500
1.4
20
p( )
1.6
m=2, n=3
30
m=2, n=3
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
1.8
40
50
RF
2
No RF
RF
No RF
Supply chain
finance
385
Figure 4.
Impact of demand
uncertainty and
profit margin on
supplier’s operational
performance
IJPDLM
46,4
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
386
identical. Also, one would expect RF to outperform TF, due to the lower cost of
financing, under all circumstances. To see this, notice that under TF the cost of selling
$1 of an A/R with time-to-maturity equal to one period is equal to rt. Under RF, the cost
of selling $1 of an A/R with time-to-maturity equal to two or three periods is $0.009975
and $0.014925, respectively. Thus, given that the inventory replenishment decision is
identical in both RF and TF, RF is expected to consistently outperform TF for all values
of rt ⩾ 0.015 under all scenarios. However, the results shown in Figure 5 contradict our
intuition, even when we eliminate the impact of the discount factor, β, on the profit
function for each period (bottom panels in Figure 5).
The explanation for this counterintuitive result lies in the impact of the longer
payment term (associated with RF) on the supplier’s capability to finance his inventory
replenishment decision without resorting to A/R liquidation. Our results show that
under TF the supplier does not need to sell any A/Rs to finance his production decision
20.1 and 13.2 percent of the times in the S(1, 2) and S(2, 3) scenarios, respectively.
The corresponding numbers for the RF case are 0.13 and 0.02 percent.
Finally, Figure 6 shows that the benefit for a supplier increases almost linearly with
the spread between RF and TF, as expressed by the ratio rt/rr, when there is no
credit-term extension involved with RF.
In our discussions with SMEs we found that the cost differential between RF and
other types of bank financing (such as TF or asset-based financing) can be quite high.
However, if the suppliers have access to external financing with relatively competitive
terms, they may be reluctant to participate in an RF program involving a credit-term
extension. In these cases, and particularly if sourcing from alternative suppliers is
expensive, buyers may benefit if they associate their RF program with a service-level
clause instead of a credit-term extension.
Conclusions and future work
This paper studies the implications of RF financing on the operational decisions and
performance of a cash-constrained SME. We model the SME’s problem as a multi-stage
dynamic program and derive his optimal policy for the case of no access to external
financing and the cases of receivables financing through RF and TF. Under mild
assumptions, we find that a working capital-dependent base-stock policy is optimal. For
the RF and TF cases, the optimal policy specifies the A/R maturity level-up-to at which
selling the corresponding invoice is profitable. Our numerical experiments suggest that
RF considerably improves the SME’s operational performance; its value is higher in
industries that operate with long credit periods; it increases the robustness of the SME’s
performance to cash fluctuations; and it provides the potential to unlock more than
10 percent of SME’s working capital. However, when RF is associated with credit-term
extension and the SME has access to alternative sources of financing (such as TF), the
value of RF is not as high as intuitively expected unless the credit spread is quite large.
Our results have clear implications for the supply chain and financial managers of
both SMEs and buyers in understanding the potential and trade-offs associated with
RF. A key takeaway is that the SMEs, when evaluating their participation in RF
programs, they should consider not only the direct benefit from increased service level
and profitability, but also the potential for profitable usages of the freed-up working
capital. The buyers, on the other hand, should consider the financial flexibility of their
suppliers when deciding the terms of their RF programs, since arbitrary selections of
credit extension may fail to induce the participation of relatively strong suppliers with
adverse effects on the existing service levels.
0.0125
rt
0.015
0.0175
0.01
0.0125
rt
0.015
0.0175
0.015
375
370
0.0075
370
0.0075
385
390
395
400
rt
0.015
0.0175
RF
0.0125
0.0175
TF
0.01
rt
m=2, n=3, =1
0.0125
RF
TF
405
380
0.02
0.01
m=2, n=3, =0.9975
410
415
420
375
RF
385
0.02
370
0.0075
375
380
385
390
395
400
405
410
415
420
380
TF
390
395
400
405
410
415
420
370
0.0075
m=1, n=2, =1
RF
380
375
TF
0.01
m=1, n=2, =0.9975
385
390
395
400
405
410
415
420
Total profit ( )
Total profit ( )
Total profit ( )
Total profit ( )
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
0.02
0.02
Supply chain
finance
387
Figure 5.
Comparison of TF
and RF on supplier’s
operational
performance
IJPDLM
46,4
4.00
m=2, n=2
3.50
% profit increase
3.00
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
388
Figure 6.
Impact of TF-RF
credit differential on
supplier’s profit
2.50
2.00
1.50
1.00
0.50
0.00
1.5
2
2.5
3
3.5
4
rt /rr
There is fertile area for future research on supply chain finance for SMEs. First, our work
could possibly be extended to consider other issues that are important in evaluating an
RF program, such as non-stationary demand characteristics, different levels for supplier’s
financial flexibility, and uncertainty in the buyer’s creditworthiness. Empirical research,
based on SME case studies or analysis of supplier-portfolios for a specific buyer/industry,
may further test our findings on the pecking order of A/R factoring, service-level
improvement, and freed-up working capital, and enhance our understanding of the RF
value proposition. Finally, there is high potential for the study of other supply chain
finance solutions that gradually appear in the industry, such as pre-shipment, higher-tier,
or third-party logistics financing, in order to evaluate the main drivers and their
suitability for different supply chain characteristics.
References
Angelus, A. and Porteus, E.L. (2002), “Simultaneous capacity and production management of
short-life-cycle, produce-to-stock goods under stochastic demand”, Management Science,
Vol. 48 No. 3, pp. 399-413.
Babich, V. (2010), “Independence of capacity ordering and financial subsidies to risky suppliers”,
Manufacturing & Service Operations Management, Vol. 12 No. 4, pp. 583-607.
Babich, V., Burnetas, A.N. and Ritchken, P.H. (2007), “Competition and diversification effects in
supply chains with supplier default risk”, Manufacturing & Service Operations Management,
Vol. 9 No. 2, pp. 123-146.
Buzacott, J.A. and Zhang, R.Q. (2004), “Inventory management with asset-based financing”,
Management Science, Vol. 50 No. 9, pp. 1274-1292.
Cai, G., Chen, X. and Xiao, Z. (2014), “The roles of bank and trade credit: theoretical analysis
and empirical evidence”, Production and Operations Management, Vol. 23 No. 4,
pp. 583-598.
Caldentey, R. and Chen, X. (2010), “The role of financial services in procurement contracts”,
in Kouvelis, P., Boyabatli, O., Dong, L. and Li, R. (Eds), Handbook of Integrated Risk
Management in Global Supply Chain, John Wiley & Sons, Inc., Hoboken, NJ, pp. 289-326.
Campello, M., Graham, J.R. and Harvey, C.R. (2010), “The real effects of financial constraints:
evidence from a financial crisis”, Journal of Financial Economics, Vol. 97 No. 3, pp. 470-487.
Carpenter, E. and Petersen, B.C. (2002), “Is the growth of small firms constrained by internal
finance?”, The Review of Economics and Statistics, Vol. 84 No. 2, pp. 298-309.
Chao, X., Chen, J. and Wang, S. (2008), “Dynamic inventory management with cash flow
constraints”, Naval Research Logistics, Vol. 55 No. 8, pp. 758-768.
Dietsch, M. and Petey, J. (2002), “The credit risk is SME loans portfolios: modeling issues, pricing,
and capital requirements”, Journal of Banking and Finance, Vol. 26 Nos 2-3, pp. 303-322.
Gupta, D. and Wang, L. (2009), “A stochastic inventory model with trade credit”, Manufacturing
& Service Operations Management, Vol. 11 No. 1, pp. 4-18.
Haley, W. and Higgins, R. (1973), “Inventory policy and trade credit financing”, Management
Science, Vol. 20 No. 4, pp. 464-471.
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
Heyman, D.P. and Sobel, M.J. (1984), Stochastic Models in Operations Research, Vol. II,
McGraw-Hill, New York, NY.
Hofmann, E. and Kotzab, H. (2010), “A supply chain-oriented approach of working capital
management”, Journal of Business Logistics, Vol. 31 No. 2, pp. 305-330.
Ivashina, V. and Scharfstein, D. (2010), “Bank lending during the financial crisis of 2008”, Journal
of Financial Economics, Vol. 97 No. 3, pp. 319-338.
Klapper, L. (2006), “The role of factoring for financing small and medium enterprises”, Journal of
Banking and Finance, Vol. 30 No. 11, pp. 3111-3130.
Klapper, L., Laeven, L. and Rajan, R. (2010), “Trade credit contracts”, Working Paper No. 5328,
The World Bank, Washington, DC.
Kouvelis, P. and Zhao, W. (2012), “Financing the newsvendor: supplier vs bank, and the structure
of optimal trade credit contracts”, Operations Research, Vol. 60 No. 3, pp. 566-580.
Li, L., Shubik, M. and Sobel, M.J. (2013), “Control of dividends, capital subscriptions, and physical
inventories”, Management Science, Vol. 59 No. 5, pp. 1107-1124.
Luo, W. and Shang, K. (2013), “Managing inventory for entrepreneurial firms with trade credit
and payment defaults”, working paper, IESE Business School, University of Navarra,
Barcelona.
Maddah, B., Jaber, M. and Abboud, N. (2004), “Periodic review (s, S) inventory model with
permissible delay in payments”, Journal of the Operational Research Society, Vol. 55 No. 2,
pp. 147-159.
Petersen, M.A. and Rajan, R.G. (1997), “Trade credit: theory and evidence”, The Review of
Financial Studies, Vol. 10 No. 3, pp. 661-691.
Protopappa-Sieke, M. and Seifert, R.W. (2010), “Interrelating operational and financial performance
measurements”, European Journal of Operational Research, Vol. 204 No. 3, pp. 439-448.
Randall, W.S. and Farris, M.T. (2009), “Supply chain financing: using cash-to-cash variables to
strengthen the supply chain”, International Journal of Physical Distribution & Logistics
Management, Vol. 39 No. 8, pp. 669-689.
Seifert, D., Seifert, R.W. and Protopappa-Sieke, M. (2013), “A review of trade credit literature:
opportunities for research in operations”, European Journal of Operational Research,
Vol. 231 No. 2, pp. 245-256.
Seifert, R.W. and Seifert, D. (2009), “Supply chain finance – what’s it worth?”, IMD Perspectives
for Managers, Vol. 178, pp. 1-4.
Seifert, R.W. and Seifert, D. (2011), “Financing the chain”, International Commerce Review, Vol. 10
No. 1, pp. 32-44.
Shi, J., Katehakis, M. and Melamed, B. (2013), “Cash-flow based dynamic inventory management”,
working paper, Robinson College of Business, Georgia State University, Atlanta, GA.
Sopranzetti, B.J. (1999), “Selling accounts receivable and the underinvestment problem”,
Quarterly Review of Economics and Finance, Vol. 39 No. 2, pp. 291-301.
Supply chain
finance
389
IJPDLM
46,4
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
390
Swinney, R. and Netessine, S. (2009), “Long-term contracts under the threat of supplier default”,
Manufacturing & Service Operations Management, Vol. 11 No. 1, pp. 109-127.
Tanrisever, F., Cetinay, H., Reindorp, M.J. and Fransoo, J.C. (2012), “Value of reverse factoring in
multi-stage supply chains”, working paper, Eindhoven University of Technology,
Eindhoven.
van der Vliet, K., Reindorp, M.J. and Fransoo, J.C. (2013), “Maximizing the value of supply chain
finance”, Working Paper No. 405, Eindhoven University of Technology, Eindhoven.
van der Vliet, K., Reindorp, M.J. and Fransoo, J.C. (2015), “The price of reverse factoring:
financing rates vs payment delays”, European Journal of Operational Research, Vol. 242
No. 3, pp. 842-853.
Yang, S.A. and Birge, J.R. (2010), “How inventory is (should be) financed: trade credit in supply
chains with demand uncertainty and costs of financial distress”, working paper, London
Business School, London.
Zeballos, A.C., Seifert, R.W. and Protopappa-Sieke, M. (2013), “Single product, periodic review
inventory model with working capital requirements and short-term debt”, Computers &
Operations Research, Vol. 40 No. 12, pp. 2940-2949.
Appendix
Proof of P1. We prove both (1) and (2) by induction.
For t ¼ T:
V T ðxT ; zT ; R T Þ ¼ maxxT p yT p zT GT ðxT ; yT Þþ bE cð yT DT Þ þ
¼ maxxT p yT p zT E bm p minð yT ; DT ÞðhbcÞð yT DT Þ þ cð yT xT Þ
This is the single-period problem, where given Lemma
1 and when (xT, zT)∈BT,
ynT ðzT Þ ¼ minðzT ; S Þ. Moreover, V T ðxT ; zT ; R T Þ ¼ cxT þ J T ynT ðzT Þ is achievable.
Assume that for (xt+1, zt+1)∈Bt+1, Vt+1(xt+1, zt+1, Rt+1) ¼ cxt+1+Wt+1(zt+1, Rt+1), where
Wt+1(zt+1, Rt+1) is jointly concave, and ynt þ 1 ðzt þ 1 Þ ¼ minðzt þ 1 ; S Þ is optimal. Then, for (xt, zt)∈Bt
the value function in period t, Vt(xt, zt, Rt), can be written as:
V t ðxt ; zt ; R t Þ ¼ maxxt p yt p zt Gt ðxt ; yt Þ þbE ½V t þ 1 ðxt þ 1 ; zt þ 1 ; R t þ 1 Þ
¼ maxxt p yt p zt E bm pminðyt ; Dt Þhðyt Dt Þ þ cðyt xt Þ
þbE V t þ 1 ð yt Dt Þ þ ; ð yt Dt Þ þ þRtm þzt yt ; R t þ 1
We will consider two cases for (xt, zt)∈Bt:
Case 1. zt ⩽ S. In this case we claim that yt ¼ zt is optimal. To see this, note that
xt+1 ¼ (zt−Dt)+ ⩽ S and xt+1 ¼ (zt − Dt)+ ⩽ xt+1+Rt−n+1 ¼ zt+1; thus, xt+1 ⩽ min(zt+1, S), i.e., (xt+1,
zt+1)∈Bt+1. Define the (m−1)-dimension vector resulting after removal of the first element from Rt
as R 1
t . Then, from induction and Lemma 1:
V t ðxt ; zt ; R t Þ ¼ maxxt p yt p zt E bm p minð yt ; Dt Þhð yt Dt Þ þ cðyt xt Þ
þ bE ½xt þ 1 þbE ½W t þ 1 ðzt þ 1 ; R t þ 1 Þ
n ¼ cxt þmaxxt p yt p zt E bm p minð yt ; Dt Þhð yt Dt Þ þ cyt þbE ð yt Dt Þ þ
h
io
;
w
min
ð
z
;
D
Þ
þbE W t þ 1 ðzt Dt Þ þ þ Rtm þzt yt ; R 1
t
t
t
¼ cxt þW t ðzt ; R t Þ
where W t ðzt ; R t Þ ¼ J t ðzt Þþ bE½W t þ 1 ððzt Dt Þ þ þRtm ; ðR 1
t ; w minðzt ; D t ÞÞÞ, which for (xt, zt)
∈ Bt is achievable:
Case 2. zt W S. In this case we claim that yt ¼ S is optimal. To see this, note that xt+1 ¼
(S−Dt)+ ⩽ S and xt+1 ¼ (S − Dt)+ ⩽ xt+1 + Rt−n+1 + zt − S ¼ zt+1; thus, xt+1⩽min(zt+1, S ), i.e.,
(xt+1, zt+1) ∈ Bt+1. Then, from induction and Lemma 1:
V t ðxt ; zt ; R t Þ ¼ maxxt p yt p zt E½bm p min ð yt ; Dt Þhð yt Dt Þ þ cð yt xt Þ
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
þbE ½xt þ 1 þ bE ½W t þ 1 ðzt þ 1 ; R t þ 1 Þ
n ¼ cxt þ maxxt p yt p zt E bm p minð yt ; Dt Þhð yt Dt Þ þ cyt þ bE ð yt Dt Þ þ
h
io
þbE W t þ 1 ð yt Dt Þ þ þ Rtm þzt yt ; R 1
t ; w minð yt ; D t Þ
¼ cxt þW t ðzt ; R t Þ
where W t ðzt ; R t Þ ¼ J t ðS Þ þbE½W t þ 1 ððSDt Þ þ þRtm þzt S; ðR 1
t ; w minðS; Dt ÞÞÞ, which for
(xt, zt)∈Bt is achievable. Summarizing the above two cases, we prove the optimality of the y*(z)
base-stock policy and the decomposition of Vt(xt, zt, Rt). Moreover, since Wt+1(⋅,⋅) is jointly
concave from induction, Wt(⋅,⋅) is also jointly concave. Therefore, the sufficient conditions for a
myopic optimal (Heyman and Sobel, 1984, Section 3.3) are satisfied.
Proof of Lemma 2. The continuity in qt is straightforward. To show the concavity of the profit
function in qt, notice that despite the fact that the function is not differentiable at the breaking
points, we can still consider the first-order and second-order derivatives. Differentiating (11) with
respect to qt yields:
8 n
b pcðbn p þhbcÞF ðxt þqt Þ;
>
>
> n
c
>
b p1r
ðbn pþ hbcÞF ðxt þqt Þ;
>
>
r
>
>
>
>
< n
n
@ c
0
0
J q ; xt ; zt ; R t ¼ b pð1rr Þ j ðb p þ hbcÞF ðxt þ qt Þ;
>
@qt t t
>
>
>
>
>
>
> for j ¼ 2; . . .; n
>
>
:
if cqt pz0t ;
if z0t o cqt p z0t þ ð1r r ÞR0tn ;
if z0t þ
j1
X
ð1rr Þi R0tn þ i1 ocqt
i¼1
p z0t þ
j
X
ð1rr Þi R0tn þ i1 :
i¼1
and ð@2 =@q2t Þ J t qt ; xt ; z0t ; R 0t ¼ ðbn p þhbcÞf ðxt þ qt Þ.
P
Thus, the concavity for all qt A ½xt ; xt þð1=cÞ ðz0t þ nj¼1 ð1rr Þ j R0tn þ j1 Þ follows directly
0
2
from ð@ =@q2t Þ J t ðqt ; xt ; z0t ; R t Þo 0.
To show the increasing property and joint concavity in xt, z0t , and R 0t , we first examine the
first-order derivatives:
@ J q ; xt ; z0t ; R 0t ¼ bn pðbn p þhbcÞF ðxt þ qt Þ
@xt t t
@ J q ; xt ; z0t ; R 0t ¼
@z0t t t
n1
X
1
1
1 1fz0t o cqt o z0t þ ð1rr ÞR0tn g þ
1
jþ1
1rr
ð
1r
rÞ
j¼1
1
z0t þ
j
P
k¼1
ð1rr Þk R0tn þ k1 o cqt o z0t þ
jþ1
P
k¼1
ð1rr Þk R0tn þ k1
Supply chain
finance
391
IJPDLM
46,4
∂
J
∂R0t−nþk t
n−2
1
1
qt ; xt ; z0t ; R 0t ¼ ∑ −
−1
þ
−1 ð1−rr Þkþ1
kþ1
ð1−rr Þ
ð1−rr Þ jþ2
j¼k
1
jþ1
jþ2
i¼1
i¼1
z0t þ∑ ð1−rr Þi R0t−nþi−1 <cqt <z0t þ∑ ð1−rr Þi R0t−nþi−1
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
392
for k ¼ 0, ..., n−2
The increasing property follows from the fact that all first-order derivatives are positive
(look at the remarks
below). Taking the second-order derivative we get
ð@2 =@x2t Þ J t qt ; xt ; z0t ; R 0t ¼ ðbn p þhbcÞf ðxt þ qt Þo 0, while the second-order derivative for
z0t and the A/R elements is zero. Then, the joint concavity in xt, z0t , and R 0t follows from the fact
that the corresponding Hessian is negative semi-definite.
Some remarks follow:
(1) Notice that the first-order derivative of the profit function with regard to an A/R element
may not be positive for some rr. By solving for rr, we can show that the profit function
increases in A/Rs with maturity of two, three, four, five, and six periods as long as rr is
less than or equal to 38, 15, 7, 4, and 3 percent, respectively. For short planning period
intervals (say one to three months), the corresponding RF discount factor is expected to
be well below these values.
(2) We do not consider the derivative over R0t1 since this quantity appears in the last term of
the profit function at the indicator function. However, we argue that the profit function
also weakly increases in R0t1 by considering two cases. If the optimal qt is not a
boundary solution, the derivative over R0t1 will be zero. If the optimal qt is a boundary
solution, an increase in R0t1 will relax the budget constraint resulting in a positive partial
derivative.
Proof of P2. The proof is similar to the proof of P1 and is omitted. The last induction step in the
RF case considers all 2n+2 possible cases for the elements of zt. In each case, we have shown that
the optimal replenishment decision in Lemma 3, ynt , for (xt, zt)∈Bt, is always a function of zt; and
is, thus, achievable.
About the authors
Spyridon Damianos Lekkakos is a Postdoctoral Research Fellow at the Zaragoza Logistics
Center. His research interests lie in the intersection of supply chain management and finance.
Spyridon Damianos Lekkakkos is the corresponding author and can be contacted at:
slekkakos@zlc.edu.es
Alejandro Serrano is a Professor of Supply Chain Management at the MIT-Zaragoza
International Logistics Program, and a Research Associate at the MIT Center for Transportation
and Logistics. He teaches regularly at other masters and executive programs both in Europe and
Latin America.
For instructions on how to order reprints of this article, please visit our website:
www.emeraldgrouppublishing.com/licensing/reprints.htm
Or contact us for further details: permissions@emeraldinsight.com
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
This article has been cited by:
1. Zericho Marak, Deepa Pillai. 2019. Factors, Outcome, and the Solutions of Supply Chain Finance:
Review and the Future Directions. Journal of Risk and Financial Management 12:1, 3. [Crossref]
2. Mengyue Wang, Hongxuan Huang. 2018. The design of a flexible capital-constrained global
supply chain by integrating operational and financial strategies. Omega . [Crossref]
3. YousefiSara, Sara Yousefi, Farzipoor SaenReza, Reza Farzipoor Saen, Seyedi HosseininiaSeyed
Shahrooz, Seyed Shahrooz Seyedi Hosseininia. Developing an inverse range directional measure
model to deal with positive and negative values. Management Decision, ahead of print. [Abstract]
[Full Text] [PDF]
4. AliZulqurnain, Zulqurnain Ali, GongbingBi, Bi Gongbing, MehreenAqsa, Aqsa Mehreen.
Predicting supply chain effectiveness through supply chain finance. The International Journal of
Logistics Management, ahead of print. [Abstract] [Full Text] [PDF]
5. AliZulqurnain, Zulqurnain Ali, GongbingBi, Bi Gongbing, MehreenAqsa, Aqsa Mehreen. Does
supply chain finance improve SMEs performance? The moderating role of trade digitization.
Business Process Management Journal, ahead of print. [Abstract] [Full Text] [PDF]
6. Lima Zhao, Arnd Huchzermeier. 2018. Managing supplier financial distress with advance payment
discount and purchase order financing. Omega . [Crossref]
7. Judith Martin, Erik Hofmann. 2018. Towards a framework for supply chain finance for the supply
side. Journal of Purchasing and Supply Management . [Crossref]
8. Matthijs van Bergen, Michiel Steeman, Matthew Reindorp, Luca Gelsomino. 2018. Supply chain
finance schemes in the procurement of agricultural products. Journal of Purchasing and Supply
Management . [Crossref]
9. SongHua, Hua Song, LuQiang, Qiang Lu, YuKangkang, Kangkang Yu, QianCheng, Cheng Qian.
How do knowledge spillover and access in supply chain network enhance SMEs’ credit quality?.
Industrial Management & Data Systems, ahead of print. [Abstract] [Full Text] [PDF]
10. Cristof Bals. 2018. Toward a supply chain finance (SCF) ecosystem – Proposing a framework and
agenda for future research. Journal of Purchasing and Supply Management . [Crossref]
11. Luca Mattia Gelsomino, Ronald de Boer, Michiel Steeman, Alessandro Perego. 2018. An
optimisation strategy for concurrent Supply Chain Finance schemes. Journal of Purchasing and
Supply Management . [Crossref]
12. SongHua, Hua Song, YuKangkang, Kangkang Yu, LuQiang, Qiang Lu. 2018. Financial service
providers and banks’ role in helping SMEs to access finance. International Journal of Physical
Distribution & Logistics Management 48:1, 69-92. [Abstract] [Full Text] [PDF]
13. Volodymyr Babich, Panos Kouvelis. 2018. Introduction to the Special Issue on Research at the
Interface of Finance, Operations, and Risk Management (iFORM): Recent Contributions and
Future Directions. Manufacturing & Service Operations Management 20:1, 1-18. [Crossref]
14. Zhuang Xiao, Yixiang Tian, Zheng Yuan. 2018. The Impacts of Regulations and Financial
Development on the Operations of Supply Chains with Greenhouse Gas Emissions. International
Journal of Environmental Research and Public Health 15:2, 378. [Crossref]
15. Lima Zhao, Arnd Huchzermeier. Supply Chain Finance 105-119. [Crossref]
16. Viktor Elliot, Ted Lindblom. The Impact of Recent Regulatory Reform on the Use of Supply
Chain Finance: The Case of Reverse Factoring 11-30. [Crossref]
Downloaded by ISTANBUL KULTUR UNIVERSITY At 02:16 10 January 2019 (PT)
17. Stephen Gong. Supply Chain Finance and Risk Management 11-33. [Crossref]
18. Abhinayan Basu Bal, Viktor Elliot, Ted Lindblom, Lars-Göran Malmberg, Trisha Rajput, Johan
Woxenius. Different Perspectives on Supply Chain Finance—In Search of a Holistic Approach
35-54. [Crossref]
19. HobergKai, Kai Hoberg, Protopappa-SiekeMargarita, Margarita Protopappa-Sieke,
SteinkerSebastian, Sebastian Steinker. 2017. How do financial constraints and financing costs affect
inventories? An empirical supply chain perspective. International Journal of Physical Distribution &
Logistics Management 47:6, 516-535. [Abstract] [Full Text] [PDF]
20. LarsenSamuel Brüning, Samuel Brüning Larsen, JacobsenPeter, Peter Jacobsen. 2016. Revenue in
reverse? An examination of reverse supply chain enabled revenue streams. International Journal of
Physical Distribution & Logistics Management 46:8, 783-804. [Abstract] [Full Text] [PDF]
21. CaniatoFederico, Federico Caniato, GelsominoLuca Mattia, Luca Mattia Gelsomino,
PeregoAlessandro, Alessandro Perego, RonchiStefano, Stefano Ronchi. 2016. Does finance solve
the supply chain financing problem?. Supply Chain Management: An International Journal 21:5,
534-549. [Abstract] [Full Text] [PDF]