Supply Contracts with Financial Hedging René Caldentey Martin Haugh Stern School of Business NYU Integrated Risk Management in Operations and Global Supply Chain Management: Risk, Contracts and Insurance June 006.
Motivation Financial Market Procurement Suppliers' Management Inventory/Production Capacity/Processes Product Design Corporation Distribution Pricing Policies Demand Forecast Problem: Select an operational strategy and a financial strategy that optimize the value of the corporation. Supply Contracts with Financial Hedging
Motivation A Methodology that effectively integrates Operational and Financial activities by Maximizing the Economic Value of the corporation. Modeling the risk preferences of decision makers. Using financial markets for hedging purposes. Recognizing the incompleteness of the problem. Taking advantage of all available information. max {Economic Value} subject to and subject to problem dynamics risk management constraint(s). Operational risk in customers taste, machine breakdowns, unreliable suppliers, etc... Financial risk in exchange rates, commodity prices, interest rates, etc... Supply Contracts with Financial Hedging 3
Outline of Talk The Modelling Framework - Motivation for maximizing economic value of operating profits. - Financial market & informational structure Supply Contracts with Financial Hedging - The Wholesale Price Contract with Budget Constraint Conclusions & Further Research Supply Contracts with Financial Hedging 4
Modelling Framework Financial Market & Information Supply Contracts with Financial Hedging 5
Motivating the Modelling Framework Consider two series cash-flows arising from two operating policies I 1 and I : C (1) (I 1 ) = (c (1) 1 (I 1), c (1) (I 1),..., c (1) n (I 1)) C () (I ) = (c () 1 (I ), c () (I ),..., c () n (I )) If probability distribution of C (1) and C () identical then most operations models are indifferent between the two. However, if C (1) positively correlated with financial market for example, and C () negatively correlated with financial market then economic value of C () greater than economic value of C (1). models should take this into account as they get ever more complex and include additional sources of uncertainty Do so by using an equivalent martingale measure, Q, or stochastic discount factor consistent with only a hedging motivation for financial trading Supply Contracts with Financial Hedging 6
The Modelling Framework H(I) the time T cumulative payoff from some operating policy, I The operating policy may be static or dynamic. G(θ) the time T gain from a self-financing trading strategy, θ Operational and financial hedging problem is min I,G(θ) EQ [H(I)] subject to and subject to problem dynamics risk management constraint(s) 8 < : R(H(I), G(θ)) 0 a.s. E P [R(H(I), G(θ))] 0. consistent with only a hedging motivation for financial trading G(θ) a martingale under Q so E Q [G(θ)] = 0 assuming initial capital assigned to hedging strategy is zero Supply Contracts with Financial Hedging 7
Financial Market Model Probability space (Ω, F, P). (W 1t, W t ) a standard Brownian motion. A single risky stock with price dynamics dx t = µx t dt + σx t dw 1t. Cash account available with r 0. F t is the filtration generated by (W 1t, W t ). F X t is the filtration generated by X t. Operating profits, H(I) F T, a function of (W 1t, W t ). The gain process, G(θ) t of a self-financing trading strategy θ t is G(θ) t = Z t 0 θ s dx s. An equivalent martingale measure, Q, under which X t and G(θ) t are martingales. Supply Contracts with Financial Hedging 8
Information In general, operating profits are function of (X t, W t ), that is, H(I) F T. Incomplete Information: Demand, product quality, customer tastes not always observable. Only information related to X t is available. Trading strategies must satisfy: θ t F X t, t [0, T ]. Complete Information: The evolution of X t and B t are both available. Trading strategies satisfy: θ t F t, t [0, T ]. Supply Contracts with Financial Hedging 9
Supply Contracts with Financial Hedging The Wholesale Price Contract with Budget Constaint Supply Contracts with Financial Hedging 10
The Wholesale Price Contract Manufacturer Production Cost: C(q) (1): w (): q Retailer Revenue: P(q) q Clearance Price (3): P(q) Non-Cooperative Operation: Solution Concept: Stackelberg game. 1. At time t = 0, Manufacturer offers a wholesale price w to the Retailer.. At time t = 0, Retailer orders a quantity q. 3. At time t = T, a random clearance price P (q) is realized: P (q) = A q. The random price A is correlated to the financial market X t. Retailer operates under a budget constraint: w q B. Supply Contracts with Financial Hedging 11
Types of Contracts (w,q) contract is decided Clearance Price P(q) is realized Payoffs are determined Production takes place t = 0 Simple Contract t = T time (w τ, q τ ) contract is decided Market Signal X τ is observed Clearance Price P(q) is realized Payoffs are determined Production takes place t = 0 t = τ t = T time Flexible Contract (w τ, q τ ) contract is decided Market Signal X τ is observed Trading Gain G τ is observed Clearance Price P(q) is realized Payoffs are determined Financial hedging takes place Production takes place t = 0 t = τ t = T time Flexible Contract with Financial Hedging Remark: The time τ can be a decision variable: a deterministic time or a F X τ -stopping time. Supply Contracts with Financial Hedging 1
Some Notation ) (Ω, F, Q) probability space. ) X t : Time t value of a Q-martingale tradable security (t [0, T ]). ) F t = σ(x s ; 0 s t) filtration generated by X t with F T F. ) E Q τ [E] := EQ [E F τ ] for all E F. ) P (q) = A q: retail price at time T with A F. ) Āτ = E Q [A F τ ] for any F t -stopping time τ T. ) c τ : per unit manufacturing cost at time τ [0, T ]. ) w τ F τ : manufacturer s wholesale price menu. ) q τ F τ : retailer s ordering quantity menu. ) G τ F τ : retailer s trading gains (or losses) with E Q [G τ ] = 0. ) Budget constraint: w τ q τ B τ := B + G τ. Assumption. For all τ [0, T ], Ā τ c τ. Supply Contracts with Financial Hedging 13
Flexible Contract Consider a fixed τ [0, T ]. Step 1: At t = 0, the manufacturer offers a wholesale menu w τ F τ. Step : A retailer, with no access to the financial market, decides the ordering level q τ F τ solving n o Π F R (w τ) = E Q 0 max E Q τ [(A q τ w τ ) q τ ] qτ 0 subject to w τ q τ B, for all ω Ω. The optimal solution (retailer s reaction) is q(w τ ) = min(āτ w τ +, ) B. w τ Step 3: The manufacturer problem is (Stackelberg leader) Π F M = EQ 0 max wτ cτ {(w τ c τ ) q τ (w τ )}. Supply Contracts with Financial Hedging 14
Flexible Contract Proposition. (Flexible Contract Solution) Under Assumption, the equilibrium solution for the flexible contract is where w F τ = Āτ + δ F τ δ F τ := max c τ, and q F τ = Āτ δ F τ 4 q(āτ 8 B)+., The equilibrium expected payoffs of the players are then given by Π F M τ = (Āτ + δ F τ c τ) (Āτ δ F τ ) 8 and Π F R τ = (Āτ δ F τ ). 16 Remarks: -) δ F τ is a modified production cost ( c τ) that is induced by a limited budget B. -) w F τ decreases in B and q F τ, Π F M τ and Π F R τ increase in B. Supply Contracts with Financial Hedging 15
Flexible Contract vs. Simple Contract Define w F := E Q 0 [w F τ ], q F := E Q 0 [q F τ ], Π F M := EQ 0 [Π F M τ ] and Π F R := EQ 0 [Π F R τ ]. Proposition. Suppose that B Ā τ c τ 8 almost surely. Then w F w S, q F q S, Π F M Π S M and Π F R Π S R. However, if B max Ā τ c τ 8, Ā c 0 8 almost surely then w F = w S + c τ c 0 and q F = q S c τ c 0 4 and Π F M Π S M and Π F R Π S R if and only if E Q 0 [(Āτ c τ ) ] (Ā c 0). Supply Contracts with Financial Hedging 16
Flexible Contract vs. Simple Contract Wholesale Price Ordering Level 0.5 Flexible vs. Simple 1.8 1.6 1.4 Flexible 0.4 0.3 0. Simple Flexible 1.5 1 Π R F /ΠR S Π P F /ΠP S Case 1 1. Simple 1 0 0.6 1. 0.1 0 0 0.6 1. 0.5 0 0.6 1. 1.8 1.6 1.4 1. Simple Flexible 0.5 0.4 0.3 0. 0.1 Simple Flexible 1.6 1.4 1. 1 0.8 0.6 0.4 Π R F /ΠR S Π P F /ΠP S Case 1 0 0.6 1. 0 0 0.6 1. Budget (B) 0. 0 0.6 1. Ā τ Uniform[1, 3], c 0 = 0.3, c τ = 0.35 (case 1) and c τ = 0.7 (case ). Supply Contracts with Financial Hedging 17
Flexible Contract: Efficiency 0.55 Q F τ 3.5 W F τ 0.7 P F τ 0.5 0.6 0.45 3 0.5 A τ 3 c τ 0.4.5 0.4 0.35 0.3 0.3 0 0.5 1 0 0.5 1 0. 0 0.5 1 0.8 1.8 0.4 0.75 0.7 1.7 0.35 0.65 1.6 0.3 0.6 1.5 0.5 A τ 3 c τ 0.55 0.5 1.4 0. 0.45 0 0.5 1 1.3 0 0.5 1 Budget (B) 0.15 0 0.5 1 Ā τ =, c τ = 0.6 (top) and c τ = 1. (bottom). Supply Contracts with Financial Hedging 18
Flexible Contract with Financial Hedging (w τ, q τ ) contract is decided Market Signal X τ is observed Trading Gain G τ is observed Clearance Price P(q) is realized Payoffs are determined Financial hedging takes place Production takes place t = 0 t = τ t = T time Flexible Contract with Financial Hedging Step 1: At t = 0, and for a fixed τ T, the manufacturer offers a price menu w τ F X τ. Step : In response, at t = 0, the retailer selects an optimal ordering menu q τ (w τ) F X τ solving Π H R (w τ) = max qτ 0, Gτ E Q [(A q τ ) q τ w τ q τ ] subject to w τ q τ B + G τ, for all ω Ω E Q [G τ ] = 0. Step 3: The manufacturer selects the optimal wholesale price menu w τ solving Π H M (w τ) = max wτ E Q wτ q τ (w τ) c τ q τ (w τ). Supply Contracts with Financial Hedging 19
Flexible Contract with Financial Hedging Proposition. (Retailer s Optimal Strategy) Let Q τ, X and X c be defined as follows Q τ Āτ w τ +, X {ω Ω : B Q τ w τ }, and X c Ω X. Case 1: Suppose that E Q [Q τ w τ ] B. Then q τ (w τ) = Q τ and there are infinitely many choices of the optimal claim, G τ. One natural choice is to take G τ = [Q τ w τ B] δ if ω X 1 if ω X c δ R R X c [Q τ w τ B] dq X [B Q τ w τ ] dq. Remark: In this case, it is possible to completely eliminate the budget constraint by trading in the financial market. Supply Contracts with Financial Hedging 0
Flexible Contract with Financial Hedging Proposition. (Continuation) Case : Suppose that B < E Q [Q τ w τ ]. Then Āτ w τ + (1 + λ) Āτ q τ (w τ ) = where λ 0 solves E "w Q w τ (1 + λ) τ + # = B. Proposition. Let φ inf (Producer s Optimal Strategy and the Stackelberg Solution) (φ 1 such that E Q " Ā τ (φ c τ ) 8 + # B ). Then, w τ = Āτ +φ cτ and q Āτ τ = φ + cτ 4 and the players expected payoffs satisfy Π H M τ = (Āτ + φ c τ c τ ) (Āτ φ c τ ) + 8 and Π H R τ = ((Āτ φ c τ ) + ). 16 Remark: When q τ = 0, the manufacturer decides to overcharge the retailer making the supply chain non-operative. This is never the case if the retailer does not have not access to the financial market. Supply Contracts with Financial Hedging 1
Flexible Contract with Financial Hedging Proposition. The manufacturer always prefers the H-contract to the F-Contract. On the other hand, the retailer s preferences are Undetermined H-Contract H-Contract H-Contract = F-Contract H-Contract or F-Contract Small Budget Retailer's Preferences Large Budget.4. 1.8 1.6 1.4 1. 0.35 0.3 0.5 0. 0.15 0.1 0.05 Wholesale Price H Contract F Contract 0 0.5 1 1.5 Budget (B) Producer s Payoff H Contract F Contract 0 0 0.5 1 1.5 Budget (B) 0.4 0.35 0.3 0.5 0. 0.15 0.1 0.05 Ordering Level 0 0 0.5 1 1.5 Budget (B) 0.18 0.16 0.14 0.1 0.1 0.08 0.06 0.04 0.0 H Contract F Contract Retailer s Payoff H Contract F Contract 0 0 0.5 1 1.5 Budget (B) Supply Contracts with Financial Hedging
Flexible Contract with Financial Hedging Efficiency On path-by-path basis, the Centralized system is not necessarily more efficient than the Decentralized Supply Chain! ω Ω such that q H C τ = 0 and q H τ > 0. Remark: This is never the case under a Flexible Contract without Hedging. On average, the Centralized solution is more efficient than the Decentralized solution. E Q 0 [q H C τ] E Q 0 [q H τ ]. Supply Contracts with Financial Hedging 3
Optimal Production Postponement The manufacturer can choose the optimal time to execute the contract solving subject to Π H P = max τ E Q 0 φ = inf [ ( Ā τ + φ c τ c τ ) (Āτ φ c τ ) + ] { ψ 1 : E Q 0 8 ξ [ (Ā τ ψ c τ 8 ξ ) + ] B }. Two possibilities: - Open Loop: τ is a deterministic time selected at time t = 0. - Closed Loop: τ is an F t -stopping time. Modeling Assumptions: Financial Market X t is a diffusion process dx t = σ(x t ) dw t. Operations Additive: A = F (X T ) + ε, E Q [ε] = 0, or Multiplicative: A = ε F (X T ), ε 0 and E Q [ε] = 1 Production: c τ = c 0 + α τ κ, for all τ [0, 1]. Supply Contracts with Financial Hedging 4
Optimal Open-Loop Production Postponement 8.5 8 κ = 8 7.5 7 Profits 6.5 κ =1 κ = 4 6 5.5 5 κ = 0.5 0 0. 0.4 0.6 0.8 1 Time (τ) Optimal open-loop production postponement for four different production cost functions. X 0 = 1, σ(x) = X, F (X) = + X and c τ = 0.3 + 0.7 τ κ. Supply Contracts with Financial Hedging 5
Optimal Close-Loop Production Postponement 14 κ = 8 7 κ = 4 1 6 X(τ) 10 8 6 ρ(t) X(τ) 5 4 3 ρ(t) 4 Optimal 0 0 0. 0.4 0.6 0.8 1 Time (τ) 5 κ = 1 1 0 0 0. 0.4 0.6 0.8 1 Time (τ) 10 κ = 0.5 Optimal X(τ) 4 3 ρ(t) Optimal X(τ) 8 6 4 ρ(t) Optimal 1 0 0. 0.4 0.6 0.8 1 Time (τ) 0 0 0. 0.4 0.6 0.8 1 Time (τ) Optimal continuation region for four different manufacturing cost functions. X 0 = 1, σ(x) = X, F (X) = + X and c τ = 0.3 + 0.7 τ κ. Supply Contracts with Financial Hedging 6
Optimal Close-Loop vs. Optimal Open-Loop Production Postponement κ Open-Loop Payoff Closed-Loop Payoff % Increase 0.5 7.9 7.9 0.0% 1 7.9 7.305 0.% 4 7.71 7.99 3.7% 8 8.09 8.33 3.8% Optimal continuation region for four different manufacturing cost functions. X 0 = 1, σ(x) = X, F (X) = + X and c τ = 0.3 + 0.7 τ κ. Supply Contracts with Financial Hedging 7
Summary Simple extension to the traditional wholesale contract. The proposed procurement contracts uses the Financial Market as: A source of public information upon which contracts can be written. A means for financial hedging to mitigate the impact of the budget constraint. Consistent with the notions of production postponement and demand forecast. Managerial Insights: Manufacturer and Retailer incentives are not always aligned as a function of B. Manufacturer prefers retailers that have access to the financial market. With hedging, the supply chain might not operate in some states ω Ω. In some cases, financial hedging eliminates the budget constraint. Optimal time τ of the contract balances Var(Āτ) and c τ. Extensions: Other types of contracts: quantity discount, buy-back, etc. Include other sources of uncertainty: exchange rates, interest rates, credit risk. Supply Contracts with Financial Hedging 8