Risk Sensitive Inventory Management with Financial Hedging 1

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1 Risk Sensitive Inventory Management with Financial Hedging 1 Süleyman Özekici Koç University Department of Industrial Engineering Sar yer, Istanbul Games and Decisions in Reliability and Risk Workshop Statistical and Applied Mathematical Sciences Institute May 2016, Durham, NC 1 Joint work with F. Karaesmen and C. Canyakmaz S. Özekici Financial Hedging in Inventory Management GDRR 1 / 24

2 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

3 Introduction and Motivation Demand and supply uncertainties are primary sources of risk for 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 input costs, sales prices and volume Successful inventory management can create even more value in uctuating price environments One can use nancial markets to cope with price and exchange rate uctuations S. Özekici Financial Hedging in Inventory Management GDRR 3 / 24

4 Risk-Sensitive Inventory Management Typical models are based on adjusting the replenishment policy to optimize a certain measure such as the mean of the cash ow the variance of the cash ow utility of the decision maker probability 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 and Schnabel [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], Tekin and Özekici [IIETr 15] S. Özekici Financial Hedging in Inventory Management GDRR 4 / 24

5 Financial Hedging The use of nancial markets to reduce the price and inventory risk has gained importance A nancial hedge is an investment position to reduce the e ect of potential losses/gains incurred from another investment Hedges can be constructed using stocks, indices, futures, options, swaps, etc. Future contracts are most widely used to hedge the risks in commodity prices, energy prices, foreign currencies, interest rates, etc. Gaur and Seshadri [MSOM 05], Caldentey and Haugh [MOR 06], Chod et al. [MS 10], Okyay et al. [ORS 14], Tekin and Özekici [IIETr 15] S. Özekici Financial Hedging in Inventory Management GDRR 5 / 24

6 The Model Single-period inventory model with uctuating sales prices and modulated demand Sales period is [0; T ] Stochastic price process P = fp t ; t 0g (compounded to time T ) P 0 : Purchase price at time 0 Sales price at time t is f (P t ) (for example, f (p) = p; where > 1 is the markup) The customer arrival process is N = fn t : t 0g with intensity process = f t = (P t ) ; t 0g (Doubly stochastic Poisson process) h(p): holding cost, b(p): backordering cost Cash ow at time T under order-up-to decision y is XN T CF (y; N ; P) = P 0 y+ f P Tj b (PT ) (N T y) + + h (P T ) (y N T ) + j=1 S. Özekici Financial Hedging in Inventory Management GDRR 6 / 24

7 Price and Arrival Processes S. Özekici Financial Hedging in Inventory Management GDRR 7 / 24

8 Security Prices and Trading Times Price related risks: sales price and demand (also purchase price in multiperiod inventory model) We assume that there are M nancial securities which are correlated n with theoprice process P S (i) = S (i) t ; t 0 : The price process for security i (compounded to time T ) S = S (1) ; S (2) ; :::; S (M) : The vector of security price processes T = (t 0 ; t 1 ; t 2 ; :::; t n 1 ) : Prespeci ed trading times (t 0 = 0, t n = T ) k = (1) k ; (2) k ; :::; (M) k : Portfolio decision at time t k = ( 0 ; 1 ; :::; n 1 ) : Financial hedging strategy or portfolio S. Özekici Financial Hedging in Inventory Management GDRR 8 / 24

9 Financial Cash Flow The nal payo of the nancial portfolio at time T as G (; S) = MX nx 1 (i) k i=1 k=0 S (i) t k+1 S (i) nx 1 t k = T k 4 S k = T 4 S k=0 4S k = S tk+1 S tk : Vector of net payo s for holding one unit of each security during (t k ; t k+1 ) k ; 4S k are M 1 column vectors ; 4S are Mn 1 column vectors S. Özekici Financial Hedging in Inventory Management GDRR 9 / 24

10 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 max y0 subject to E [HCF ( (y) ; y; N ; P; S)] (y) = arg minv ar (HCF (; y; N ; P; S)) S. Özekici Financial Hedging in Inventory Management GDRR 10 / 24

11 Static Model: Minimum-Variance Portfolio Portfolio chosen once at the beginning only (n = 1; t 0 = 0) Covariance matrix C ij = Cov S (i) T ; S(j) T Covariance vector i (y) = Cov CF (y; N ; P) ; S (i) T 0 1 XN T = (i) f P Tj ; S A j=1 T Cov b (P T ) (N T Cov h (P T ) (y y) + ; S (i) T N T ) + ; S (i) T Theorem V ar (; y; N ; P; S) is convex in and minimum-variance portfolio for order quantity y is given by (y) = C 1 (y) S. Özekici Financial Hedging in Inventory Management GDRR 11 / 24

12 Static Model: Optimal Base-Stock Level Assumption The function E (h (P T ) + b (P T )) 1 fnt yg Cov (h (P T ) + b (P T )) 1 fnt yg; S T T C 1 E [4S] is increasing in y: Theorem The optimal order quantity that maximizes the expected cash ow using the minimum-variance portfolio (y) = C 1 (y) is y = inf y 0; E (h (P T ) + b (P T )) 1 fnt yg o T Cov (h (P T ) + b (P T )) 1 fnt yg; S T C 1 E [4S] o E [b (P T )] P 0 Cov (b (P T ) ; S T ) T C 1 E [4S] S. Özekici Financial Hedging in Inventory Management GDRR 12 / 24

13 Static Model: Zero Expected Financial Gain If E [4S] = 0 (or security prices are martingales), then Assumption 1 is always satis ed In this case, optimal order-up-to level is y = inf y 0; E (h (P T ) + b (P T )) 1 fnt yg E [b (PT )] P 0 This is not the newsvendor solution since N and P are dependent We obtain the newsvendor solution y = inf y 0; P fn T yg b c h + b if P 0 = c; h (p) = h and b (p) = b S. Özekici Financial Hedging in Inventory Management GDRR 13 / 24

14 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 TZ (y) = Cov (f (P t ) ; S T ) dt E (y N T ) + Cov (h (P T ) ; S T ) and the optimal order-up-to level is 0 E (N T y) + Cov (b (P T ) ; S T ) ( ) y E [b (P T )] P 0 Cov (b (P T ) ; S T ) T C 1 E [4S] = inf y 0; P fn T yg E [h (P T )] + E [b (P T )] Cov (h (P T ) + b (P T ) ; S T ) T C 1 E [4S] If E [4S] = 0; then y = inf E [b (P T )] P 0 y 0 : P fn T yg E [h (P T )] + E [b (P T )] S. Özekici Financial Hedging in Inventory Management GDRR 14 / 24

15 Static Model: Single Financial Security (Future) If S is a future written on P T, then S 0 = P 0 and S T = P T Let us assume that b (P T ) = b + P T ; h (P T ) = h P T In this case, optimal order-up-to level is y = inf y 0 : P fn T yg b + (E [P T ] P 0 ) b + h + (1 )P 0 and the minimum-variance portfolio is = E (N T y) + E (y N T ) + TZ 0 t dt where t = Cov (f (P t) ; P T ) V ar (P T ) S. Özekici Financial Hedging in Inventory Management GDRR 15 / 24

16 Dynamic Model Trading times are t 0 = 0; t 1 ; t 2 ; :::; t n 1 Assumption: Security prices are martingales In this case, the minimum-variance problem for every y decision is reduced to h mine CF (y; N; P) + T 4 S i 2 This objective is separable in terms of dynamic programming S. Özekici Financial Hedging in Inventory Management GDRR 16 / 24

17 Dynamic Programming: State Transitions We use four states X; W; P; S Inventory level transitions Wealth level transitions X k+1 = X k N [tk ;t k+1 ] X 0 = y W k+1 = W k + R [tk ;t k+1 ] + T k 4 S k W 0 = 0 Operational revenue during [t k ; t k+1 ] R [tk ;t k+1 ] = N [tk ;t k+1 ] X j=1 f P Tj +t k S. Özekici Financial Hedging in Inventory Management GDRR 17 / 24

18 Dynamic Programming Formulation Objective function is 2! n X h E 4 CF (y; N ; P) + T k 4 S 5 k = E DP formulation is k=0 W n b (Ptn ) ( X n ) + + h (P tn ) X n + 2 i V k (x; w; p; s) = min k E V k+1 X tk N [tk ;t k+1 ]; W tk + R [tk ;t k+1 ] + k 4 S k ; P tk+1 ; S tk+1 with boundary condition jx tk = x; W tk = w; P tk = p; S tk = s] V n (x; w; p; s) = w b (p) ( x) + h (p) x + 2 Here V k is the value function at trading time t k S. Özekici Financial Hedging in Inventory Management GDRR 18 / 24

19 Notation C k (s) ij = Cov S (i) ; tk+1 S(j) j tk+1 S(i) tk nx 1 R [tk;tn] = j=k R [tj;tj+1] = s(i) ; S (j) = s(j) tk k (x; p; s) j = Cov R [tk;tn] b (P tn ) N [tk;tn] x + h (P tn ) x + (j) N [tk;tn] ; S jp tk+1 tk = p; S(j) = s(j) tk g k (x; w; p) = E w + R [tk;tn] b (P tn) N [tk;tn] x h (P tn ) x N [tk;tn] jptk = p h k (x; p; s) = k (x; p; s) T C k (s) 1 k (x; p; s)+e h k+1 x N [tk;tk+1]; P tk+1 ; S tk+1 j Ptk = p; S tk = s S. Özekici Financial Hedging in Inventory Management GDRR 19 / 24

20 The Optimal Solution Theorem Value function for period k is V k (x; w; p; s) = g k (x; w; p) + h k (x; p; s) and the minimum-variance portfolio is k (x; p; s) = C k (s) 1 k (x; p; s) Theorem Optimal order-up-to level that maximizes the expected hedged cash ow is y = inf y 0; E (h (P T ) + b (P T )) 1 fnt yg E [b (PT )] P 0 Risk-neutral solution Martingale security price processes S. Özekici Financial Hedging in Inventory Management GDRR 20 / 24

21 Numerical Illustration Schwartz and Smith [MS 00] uses a price model that describes long and short term behaviours of commodities where d t = P t = e t + t t dt + dw () t is an Ornstein-Uhlenbeck process that models the short-term deviations (mean reverting to zero) and d t = dt + dw () t is a geometric Brownian motion that models the long-term equilibrium (dw () t dw () t = dt) We use the risk-neutral version where dp t = ( + ) P t dw 1 t + p 1 2 P t dw 2 t S. Özekici Financial Hedging in Inventory Management GDRR 21 / 24

22 Financial Securities and Parameters T = 1; f (p) = 2p, b (p) = 4 + p, h (p) = 1, t = (90 1:4P t ) + P 0 = 20; = 0:25; = 0:15, = 0:3 S (1) t = P t : Future S (2) t = E (P T 20) + j P t : Call option S. Özekici Financial Hedging in Inventory Management GDRR 22 / 24

23 E ect of Financial Hedging on Risk Reduction Mean 2000 S. Özekici Financial Hedging in Inventory Management GDRR 23 / 24

24 Conclusion and Future Work Financial hedging of an inventory system where a stochastic price process modulates the demand and sales prices We characterize the static and dynamic nancial hedging policies 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