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Risk Sensitive Inventory Management withFinancial Hedging1
Süleyman ÖzekiciKoç University
Department of Industrial EngineeringSar¬yer, ·Istanbul
Games and Decisions in Reliability and Risk WorkshopStatistical and Applied Mathematical Sciences Institute
16-20 May 2016, Durham, NC
1Joint work with F. Karaesmen and C. CanyakmazS. Özekici Financial Hedging in Inventory Management GDRR 1 / 24
Outline
Introduction and Motivation
Literature
Static Financial Hedging Model
Dynamic Financial Hedging Model
Numerical Illustration
Future Research
S. Özekici Financial Hedging in Inventory Management GDRR 2 / 24
Introduction and Motivation
Demand and supply uncertainties are primary sources of riskfor inventory managers
Price volatility is also major source of risk
Commodity based raw material prices, exchange rate�uctuations are typical examples
Exposure to these risks have signi�cant impacts on inputcosts, sales prices and volume
Successful inventory management can create even more valuein �uctuating price environments
One can use �nancial markets to cope with price andexchange rate �uctuations
S. Özekici Financial Hedging in Inventory Management GDRR 3 / 24
Risk-Sensitive Inventory Management
Typical models are based on adjusting the replenishmentpolicy to optimize a certain measure such as
the mean of the cash �owthe variance of the cash �owutility of the decision makerprobability of achieving a certain target pro�t
Expected Utility Models: Boukazis and Sobel [OR�92],Agrawal and Seshandi [MSOM�00], Chen et al. [OR�07]
Mean-Variance Models: Lau [JORS�80], Berman andSchnabel [IJSS�86], Chen and Federgruen [WP�00], Wu et al.[OMEGA�09]
Value-at-Risk Models: Luciano et al. [IJPE�03], Tapiero[EJOR�05], Özler et al. [IJPE�09]
Minimum-Variance Models: Okyay et al. [ORS�14], Tekinand Özekici [IIETr�15]
S. Özekici Financial Hedging in Inventory Management GDRR 4 / 24
Financial Hedging
The use of �nancial markets to reduce the price and inventoryrisk has gained importance
A �nancial hedge is an investment position to reduce thee¤ect of potential losses/gains incurred from anotherinvestment
Hedges can be constructed using stocks, indices, futures,options, swaps, etc.
Future contracts are most widely used to hedge the risks incommodity prices, energy prices, foreign currencies, interestrates, etc.
Gaur and Seshadri [MSOM�05], Caldentey and Haugh[MOR�06], Chod et al. [MS�10], Okyay et al. [ORS�14], Tekinand Özekici [IIETr�15]
S. Özekici Financial Hedging in Inventory Management GDRR 5 / 24
The Model
Single-period inventory model with �uctuating sales prices andmodulated demandSales period is [0; T ]Stochastic price process P = fPt; t � 0g (compounded to time T )P0 : Purchase price at time 0Sales price at time t is f (Pt) (for example, f (p) = �p; where � > 1 isthe markup)The customer arrival process is N = fNt : t � 0g with intensity process� = f�t = � (Pt) ; t � 0g (Doubly stochastic Poisson process)h(p): holding cost, b(p): backordering costCash �ow at time T under order-up-to decision y is
CF (y;N ;P) = �P0y+NTXj=1
f�PTj���b (PT ) (NT � y)+ + h (PT ) (y �NT )+
�
S. Özekici Financial Hedging in Inventory Management GDRR 6 / 24
Price and Arrival Processes
S. Özekici Financial Hedging in Inventory Management GDRR 7 / 24
Security Prices and Trading Times
Price related risks: sales price and demand (also purchase price inmultiperiod inventory model)We assume that there are M �nancial securities which arecorrelated with the price process PS(i) =
nS(i)t ; t � 0
o: The price process for security i
(compounded to time T )S =
�S(1); S(2); :::; S(M)
�: The vector of security price processes
T = (t0; t1; t2; :::; tn�1) : Prespeci�ed trading times (t0 = 0,tn = T )
�k =��(1)k ; �
(2)k ; :::; �
(M)k
�: Portfolio decision at time tk
� = (�0; �1; :::; �n�1) : Financial hedging strategy or portfolio
S. Özekici Financial Hedging in Inventory Management GDRR 8 / 24
Financial Cash Flow
The �nal payo¤ of the �nancial portfolio at time T as
G (�;S) =MXi=1
n�1Xk=0
�(i)k
�S(i)tk+1
� S(i)tk�=
n�1Xk=0
�Tk 4 Sk = �T 4 S
4Sk = Stk+1 � Stk : Vector of net payo¤s for holding one unitof each security during (tk; tk+1)
�k;4Sk are M � 1 column vectors�;4S are Mn� 1 column vectors
S. Özekici Financial Hedging in Inventory Management GDRR 9 / 24
The Hedged Cash Flow
Total hedged cash �ow at time T is
HCF (�; y;N ;P; S) = CF (y;N ;P) +G (�; S)
The objective of the inventory manager is to solve
maxy�0
E [HCF (� (y) ; y;N ;P; S)]
subject to� (y) = argmin
�V ar (HCF (�; y;N ;P; S))
S. Özekici Financial Hedging in Inventory Management GDRR 10 / 24
Static Model: Minimum-Variance Portfolio
Portfolio chosen once at the beginning only (n = 1; t0 = 0)Covariance matrix
Cij = Cov�S(i)T ; S
(j)T
�Covariance vector
�i (y) = Cov�CF (y;N ;P) ; S(i)T
�= Cov
0@NTXj=1
f�PTj�; S
(i)T
1A� Cov �h (PT ) (y �NT )+ ; S(i)T ��Cov
�b (PT ) (NT � y)+ ; S(i)T
�Theorem
V ar (�; y;N ;P; S) is convex in � and minimum-variance portfolio for order quantityy is given by
�� (y) = �C�1� (y)
S. Özekici Financial Hedging in Inventory Management GDRR 11 / 24
Static Model: Optimal Base-Stock Level
Assumption
The function
E�(h (PT ) + b (PT )) 1fNT�yg
��Cov
�(h (PT ) + b (PT )) 1fNT�yg; ST
�TC�1E [4S]
is increasing in y:
TheoremThe optimal order quantity that maximizes the expected cash �ow using theminimum-variance portfolio �� (y) = �C�1� (y) is
y� = inf�y � 0;E
�(h (PT ) + b (PT )) 1fNT�yg
��Cov
�(h (PT ) + b (PT )) 1fNT�yg; ST
�TC�1E [4S]
o� E [b (PT )]� P0 � Cov (b (PT ) ; ST )T C�1E [4S]
o
S. Özekici Financial Hedging in Inventory Management GDRR 12 / 24
Static Model: Zero Expected Financial Gain
If E [4S] = 0 (or security prices are martingales), thenAssumption 1 is always satis�edIn this case, optimal order-up-to level is
y�= inf�y � 0;E
�(h (PT ) + b (PT )) 1fNT�yg
�� E [b (PT )]� P0
This is not the newsvendor solution since N and P are dependentWe obtain the newsvendor solution
y�= inf
�y � 0;PfNT � yg �
b� ch+ b
�if P0 = c; h (p) = h and b (p) = b
S. Özekici Financial Hedging in Inventory Management GDRR 13 / 24
Static Model: Demand is Independent of Prices
If N is a Poisson process with rate � and independent of P , then Assumption 1 is always satis�ed.In this case
� (y) = �
TZ0
Cov (f (Pt) ; ST ) dt� E�(y �NT )+
�Cov (h (PT ) ; ST )
�E�(NT � y)+
�Cov (b (PT ) ; ST )
and the optimal order-up-to level is
y� = inf
(y � 0;P fNT � yg �
E [b (PT )]� P0 � Cov (b (PT ) ; ST )T C�1E [4S]E [h (PT )] + E [b (PT )]� Cov (h (PT ) + b (PT ) ; ST )T C�1E [4S]
)
If E [4S] = 0; then
y� = inf
�y � 0 : P fNT � yg �
E [b (PT )]� P0E [h (PT )] + E [b (PT )]
�
S. Özekici Financial Hedging in Inventory Management GDRR 14 / 24
Static Model: Single Financial Security (Future)
If S is a future written on PT , then S0 = P0 and ST = PTLet us assume that b (PT ) = b+ PT ; h (PT ) = h� PTIn this case, optimal order-up-to level is
y� = inf
�y � 0 : P fNT � yg �
b+ (E [PT ]� P0)b+ h+ (1� )P0
�and the minimum-variance portfolio is
�� = E�(NT � y)+
�� E
�(y �NT )+
�� �
TZ0
�tdt
where
�t =Cov (f (Pt) ; PT )
V ar (PT )
S. Özekici Financial Hedging in Inventory Management GDRR 15 / 24
Dynamic Model
Trading times are t0 = 0; t1; t2; :::; tn�1Assumption: Security prices are martingales
In this case, the minimum-variance problem for every ydecision is reduced to
min�Eh�CF (y;N;P) + �T 4 S
�2iThis objective is separable in terms of dynamic programming
S. Özekici Financial Hedging in Inventory Management GDRR 16 / 24
Dynamic Programming: State Transitions
We use four states X;W;P; S
Inventory level transitions
Xk+1 = Xk �N[tk;tk+1]X0 = y
Wealth level transitions
Wk+1 = Wk +R[tk;tk+1] + �Tk 4 Sk
W0 = 0
Operational revenue during [tk; tk+1]
R[tk;tk+1] =
N[tk;tk+1]Xj=1
f�PTj+tk
�S. Özekici Financial Hedging in Inventory Management GDRR 17 / 24
Dynamic Programming Formulation
Objective function is
E
24 CF (y;N ;P) + n�1Xk=0
�Tk 4 Sk
!235 = E h�Wn ��b (Ptn) (�Xn)+ + h (Ptn)X+
n
��2iDP formulation is
Vk (x;w; p; s) = min�kE�Vk+1
�Xtk �N[tk;tk+1];Wtk +R[tk;tk+1] + �k 4 Sk; Ptk+1 ; Stk+1
�jXtk = x;Wtk = w;Ptk = p; Stk = s]
with boundary condition
Vn (x;w; p; s) =�w � b (p) (�x)+ � h (p)x+
�2Here Vk is the value function at trading time tk
S. Özekici Financial Hedging in Inventory Management GDRR 18 / 24
Notation
Ck (s)ij = Cov�S(i)tk+1
; S(j)tk+1
j S(i)tk = s(i); S
(j)tk= s(j)
�R[tk;tn] =
n�1Xj=k
R[tj ;tj+1]
�k (x; p; s)j = Cov�R[tk;tn] � b (Ptn)
�N[tk;tn] � x
�+ � h (Ptn) �x�N[tk;tn]�+ ; S(j)tk+1 jPtk = p; S(j)tk = s(j)�
gk (x;w; p) = E
��w +R[tk;tn] � b (Ptn)
�N[tk;tn] � x
�+ � h (Ptn) �x�N[tk;tn]�+�2 jPtk = p�
hk (x; p; s) = ��k (x; p; s)T Ck (s)�1 �k (x; p; s)+E�hk+1
�x�N[tk;tk+1]; Ptk+1 ; Stk+1
�j Ptk = p; Stk = s
�
S. Özekici Financial Hedging in Inventory Management GDRR 19 / 24
The Optimal Solution
TheoremValue function for period k is
Vk (x;w; p; s) = gk (x;w; p) + hk (x; p; s)
and the minimum-variance portfolio is
��k (x; p; s) = �Ck (s)�1 �k (x; p; s)
TheoremOptimal order-up-to level that maximizes the expected hedgedcash �ow is
y� = inf�y � 0;E
�(h (PT ) + b (PT )) 1fNT�yg
�� E [b (PT )]� P0
Risk-neutral solutionMartingale security price processes
S. Özekici Financial Hedging in Inventory Management GDRR 20 / 24
Numerical Illustration
Schwartz and Smith [MS�00] uses a price model that describeslong and short term behaviours of commodities
Pt = e�t+�t
whered�t = ���tdt+ ��dW
(�)t
is an Ornstein-Uhlenbeck process that models the short-termdeviations (mean reverting to zero) and
d�t = ��dt+ ��dW(�)t
is a geometric Brownian motion that models the long-termequilibrium (dW (�)
t dW(�)t = �dt)
We use the risk-neutral version where
dPt = (�� + ���)PtdW1t + ��
p1� �2PtdW 2
t
S. Özekici Financial Hedging in Inventory Management GDRR 21 / 24
Financial Securities and Parameters
T = 1; f (p) = 2p, b (p) = 4 + p, h (p) = 1,�t = (90� 1:4Pt)+
P0 = 20; �� = 0:25; �� = 0:15, � = 0:3
S(1)t = Pt : Future
S(2)t = E
�(PT � 20)+ j Pt
�: Call option
S. Özekici Financial Hedging in Inventory Management GDRR 22 / 24
E¤ect of Financial Hedging on Risk Reduction
Mean � 2000
S. Özekici Financial Hedging in Inventory Management GDRR 23 / 24
Conclusion and Future Work
Financial hedging of an inventory system where a stochasticprice process modulates the demand and sales prices
We characterize the static and dynamic �nancial hedgingpolicies
We also analyzed the multi-period inventory version
Di¤erent objective functions (mean-variance, utility functions)
Continuous-time versions
Budget constraint
S. Özekici Financial Hedging in Inventory Management GDRR 24 / 24