Finally, () ( ) 2 * 11 1 1 11 2(1 ) f DD w R πππ −+− ⎛⎞ =++− ⎜⎟ ⎝⎠ . (7.59) More generally, calculating the implied risk neutral distribution in terms of derivatives will be more precise, albeit numerically based. When the demand is a random function of random prices as well, a similar approach can be used by considering a stochastic constraint, summarizing the potential demand realizations in terms of an order and a portfolio of deri- vatives contracts that replicate (that is can meet) the demand at all prices and in all situations. The implication of our approach is that a retailer acting on a risk neutral probability (i.e., in complete markets) is subject to the same laws of finance, presuming that “without assuming risks”, there are no profits. In other words, a retailer, acting as an intermediary between, say, the supplier and end market customers, “must have some informational advantage” or some other advan- tages that will allow him to make (arbitrage) profits. Of course in a practical setting, the retailer and the supplier base their analyses on both observa- tions of market behavior and their instructed beliefs in regard to the future states of potential prices. Such beliefs recur both with respect to the demand and price uncertainty, which are combined in retailers risk attitudes in deter- mining an optimal order policy. The risk neutral probability imbeds these beliefs (assuming their expression in the price of derivatives) while the relationship between demand and price, has made it possible to remain within the simple complete markets framework that has allowed our calcu- lations. A generalization to more complex situations can be considered as well. From the analysis in the previous section, we clearly saw that our results depend on knowing the risk neutral probability. In practice however, retailers and suppliers possess private information which can lead them to believe that they have in fact an informational advantage. In this case, while market prices are what they are, a retailer for example, may think that the market errs in the specification of the risk neutral distribution. In other words, say that the retailer has a private information regarding the future demand and therefore an information regarding prices (assuming that the demand is indeed a function of prices), or a direct information regarding the future prices. Let (.) P f be a private probability estimate of the future prices and let the retailer utility function (expressing his risk attitudes) be (.)u . In this case, the retailer private utility and information would lead him to maximize the expected next period profit subject to the current prices. An explicit relationship, between these (see also Jackwerth 408 7 RISK AND SUPPLY CHAINS 7.5 SELECTED CASES AND PROBLEMS 409 1999, 2000, Ait-Sahalia .and Lo 1998, 2000) is given by (as seen and proved earlier): (.) (.) ln (.) (.) P r RN f d A df ⎧⎫ ⎛⎞ ⎪⎪ = ⎨⎬ ⎜⎟ ⎪⎪ ⎝⎠ ⎩⎭ , ( ) (.) exp ( ) (.) P r RN f Azdz f = ∫ . (7.60) This relationship states that a decision maker’s index of risk aversion (.) r A , expressing his personal risk attitude, combined with his subjective assessment of future states (prices for example) determines the risk neutral distribution of these future states. Thus, given any two of these terms, the third can be calculated. Therefore, introducing in our optimality equation the implied risk neutral distribution we have: () * 11 11 ** 11 11111 (1 ) exp ( ) ( ) ; frP wR Azdzf dDD ππ ππ π ππ π −− ⎛⎞ ⎜⎟ += − = ⎜⎟ ⎝⎠ ∫∫ . (7.61) For example, if the retailer has an exponential utility function whose index of absolute risk aversion is α , then integration of (7.61)) yields: ( ) () ln P RN f f π ξαπ π += or ( ) ( ) RN P fef ξαπ π π −− = % . (7.62) where ξ is an integration constant defined by the risk neutral distribution: () () 00 1() RN P fdeefdeL ξαπ ξ π πππα ∞∞ −− − == = ∫∫ . (7.63) In equation (7.63), α is the Laplace Transform of the retailer subjective estimate of the future prices (i.e. his private information). As a result, ()eL ξ α = and finally, () ( ) () P RN ef f L απ π π α − = % . (7.64) As a result, we have: () () * 1 1 1 1 ** 11 1111 (1 ) ; () P f ef wR dDD L π απ π π π ππ α − − += = ∫ . (7.65) Again, let the private information of the retailer indicate a price distri- bution which is uniform, that is: () * 1 1 1 ** 111111 11 1 (1 ) ; () f wR edDD L π απ π π ππ απ π − − +− += = ⎡⎤ − ⎣⎦ ∫ , (7.66) or * 11 * 11 1 22 11 111 (1 ) () f wR e e L απ απ ππ αα αα απ π − − −− +− ⎧ ⎫ ⎛⎞⎛⎞ ⎪ ⎪ += −− − ⎨ ⎬ ⎜⎟⎜⎟ ⎡⎤ − ⎪ ⎪ ⎝⎠⎝⎠ ⎩⎭ ⎣⎦ .(7.67) Therefore, it is a solution of: () * 11 * ** 11 111111 22 11 (1 ) ( ) ; f eewRLDD απ απ ππ α ππ π αα αα − − −− +− ⎛⎞⎛⎞ ⎡⎤ += +−+ − = ⎜⎟⎜⎟ ⎣⎦ ⎝⎠⎝⎠ ,(7.68) where: 1 11 1 11 11 1 () x ee Ledx π α παπ α π α ππ ππ + − + − −− − +− +− − == ⎡⎤ ⎡⎤ −− ⎣⎦ ⎣⎦ ∫ . (7.69) Consequently, () () * 11 11 * ** 11 1111 22 11 (1 ) ; f ee wReeDD απ απ απ απ ππ π αα α α −−+ − −− −− ⎛⎞ ⎛ ⎞ =+−+− = ⎜⎟ ⎜ ⎟ ⎝⎠ ⎝ ⎠ .(7.70) In this situation as well, the price estimates of the derivatives by the retailer are: () 1 1 0 1 1() P f ef d RL π απ π π π ππ α + − − = + ∫ . (7.71) () () 1 1 00 1 ;(,0) 1() P f ef Ck Maxk d RL π απ π π π ππ α + − − =− + ∫ , (7.72) () () 1 1 00 1 ;() 1() P f ef P hMaxh d RL π απ π π π ππ α + − − =− + ∫ . (7.73) In this illustration, we shall provide an application of Value at Risk as out- lined in the previous section. Assume that in a supply chain an individual firm target costs Q a part for a product to be assembled by a supply chain and let a random variable, z , be the realized cost with a known probability distribution function. The actual development cost is a function of the firm’s operational strategy and investments. To finance the production cost, an amount equal to the target cost is borrowed at the bank at the rate r . The following objective is then defined, consisting of the costs of over or under meeting the target cost. Explicitly, if z Q> , then the firm (a supplier) is penalized at a rate of r α > while if the supplier cost is below the target, z Q< , then the resulting cost of such a deviation is penalized at a rate of β α < . As a result, the following (asymmetric) objective is defined ; , Q M in rQ E z Q z Q E z Q z Q r αβ βαα Φ= + ⎡− ≥ ⎤+ ⎡− ≤ ⎤ < > ⎣⎦⎣⎦ .(7.74) Let the cumulative probability distribution of the cost be (.) C F , then the expected cost is: 410 7 RISK AND SUPPLY CHAINS Value at Risk, Safety is First and Target Costing 7.5 SELECTED CASES AND PROBLEMS 411 () () 0 () (); Q CC Q Q M in rQ z Q dF z z Q dF z r αββ ∞ =+ − + − > Φ ∫∫ . (7.75) The least objective target cost is thus found by setting to zero the first derivative of the objective function above, or: [ ] 1() ()0 CC rFQFQ αβ −− − = . Thus, an optimal target cost is (), , , C r FQ r α ξ αβα αβ − == << − (7.76) or, in other words, the target cost is given by *1 1 () , , , CC r QF F r α ξ αβα αβ −− ⎛⎞ − == << ⎜⎟ − ⎝⎠ (7.77) where Q * is the optimal value. Of course, if development costs are a function of their effort (or some other variable of interest) and the costs are charged when performing above or below the target cost, then a firm’s objective consists in minimizeing the following: 0 () ( ) ( ) Q CC u Q M in h u zdF z u zdF z u αβ ∞ =+ + Φ ∫∫ . (7.78) For example, let the cost probability distribution have a Weibull proba- bility distribution defined by 1() () () , 0, 0 a abuz fz abuz e a τ −− = ≥> , then the cumulative probability distribution is () () 1 a buz Fz e − =− with mean and variance 22 121 () () 1, var() () 1 1, aa Ez bu z bu aaa ⎡ ⎤ ⎛⎞ ⎛⎞⎛⎞ =Γ+ = Γ+−Γ+ ⎜⎟ ⎜⎟⎜⎟ ⎢ ⎥ ⎝⎠ ⎝⎠⎝⎠ ⎣ ⎦ .(7.79) Thus, the optimal target cost is in this case is: 1 1 () *1 ln ln , , , a bu C rr QF r r αβ α βα αβ α − ⎧⎫ ⎛⎞ −− ⎪⎪ ⎛⎞ == << ⎨⎬ ⎜⎟ ⎜⎟ −− ⎝⎠ ⎝⎠ ⎪⎪ ⎩⎭ . (7.80) The Target Costing problem defined above can be generalized further to the firm outsourcing parts to multiple suppliers. In this case, a simple formulation of the problem faced by the “central firm” is given by () () ,, 11 1 0 () (); k k Q nn n kkCk kCk Q kk k Q M in r Q z Q dF z z Q dF z r αββ ∞ == = =+ − + − > ∑∑ ∑ Φ ∫∫ .(7.81) while each supplying firm, seeks to minimize the following ,, 0 () () () k k Q kCkkCkk u Q M in h u zdF z u zdF z u αβ ∞ =+ + Φ ∫∫ . (7.82) These equations define a game between the supply chain manager and the individual supplying firms. In this game, the “central firm” determines the target cost for each firm, while the individual firms optimize with respect to the efforts furnished. Evidently, for each firm k, we have the following: ( ) ,, 1() ()0; Ck k Ck k rFQFQ r αββ −− − = > . (7.83) Or , () = Ck k r FQ α β α − − , (7.84) while optimization of the effort by the individual firm is defined as stated earlier. 7.6 COLLABORATION, RISKS AND SUPPLY CHAINS Collaboration in supply chains assumes a growing importance due to the profit that results from economies of scale, in technology, in production, in market power, in introducing entry barriers and thereby reducing some of the associated risks for firms. At the same time however, internal risks such as lock-in contracts, risk sharing, risk transfer, size risk etc, have to be dealt with. The risks sustained are of course a function of the contractual, behavioral and collaboration attitudes in use. For example, vertical integra- tion or hierarchical control; subcontractors-contractual relationships; franchi- ses; joint ventures and partnerships; strategic alliances; reciprocity agreements etc. all have benefits and risks. These risks derive mostly form the supply chain leadership rules and incentives (inducing power asymmetries) and by information asymmetry. For example, when two parties engage in a con- tractual relationship which is costly to break apart, or lock-in contracts, there may be risks for one or the other party or both. For this reason, the profit of collaboration by reducing the number of suppliers and building trustworthy relationships, engaging in long term supplies and exchange, locking oneself in dependence of any kind (such as joint technology, Intranets, joint plann- ing, technology sharing etc.) is also a “two edge sword”. In this sense collaboration in supply chains is not a “free lunch”. Celebrated cases are of course outsourcing and franchises. 412 7 RISK AND SUPPLY CHAINS 7.6 COLLABORATION, RISKS AND SUPPLY CHAINS 413 Outsourcing (as discussed in Chapters 2 and 4) is essentially defined as the transfer of previously in-house activities to a third party (see also, Gattorna 1988; La Londe and Cooper 1989 for additional review of this problem). In such a transfer, economies of scale may be reached while fixed cost invest- ments can be reduced rendering the outso urcing more agile-flexible. At the same time, there may be opportunity risks based on the search for self interest such as information asymmetries as stated above. The questions firms struggle with prior to outsourcing are therefore both complex and numerous. Should a firm strive to maintain its capacity or turn to an external (and therefore hardly controllable) supplier? Will a firm’s technological positioning (and therefore its knowledge base in the future) be reduced? What are the firm’s strategic options and contingent plans? These and other questions are important risk problems to contend with. For example, an essential motivation when outsourcing inventory, arises from economies of scale, risk and focus. These motivations presume that economic advantages arise from collaboration and exchange between firms, leading to a firm restructuring its organization to deal with its external supply chain. A typical example would in practice be to focus on a JIT (Just in Time) manufacturing strategy while outsource the management of inventories to a carefully selected supplier (although outsourcing and JIT might not be correlated). Such a practice can lead to numerous problems however. Spe- cifically, when several firms act on the same market and outsource to a common supplier, they may augment significantly the demand volatility faced by the supplier (and thereby augment costs). Such risk considera- tions are therefore essential and to be accounted for when reaching the decision to outsource inventories. Thus, inventory outsourcing involves not only reduced costs and the potential to focus on core competencies, but also risks. The two main risks tory activities, namely, risks assumed at the inventory and order stage and risks assumed ex-post once uncertainty in demands is revealed and supplies received. These risks are of course dependent on different factors such as supply delays and the preferential supplier-firm relationship. As a result, inventory outsourcing can be conceived in numerous ways, based on model relationships, which involve wholly or partly, arm’s length contractual and conflicting partnerships. From a supplier’s point of view the concern to maintain firms-clients that have outsourced as well as minimizing the costs of managing inventories are the prime objective. See also Baghana and Cohen 1998, Janssen and Kok 1999, Ritchken and Tapiero 1986, Outsourcing and Risks (ex-ante and ex-post) in this case include the outsourcing of critical inven- Tapiero and Grando 2006, Van Donk and van der Vaart 2005, Tsay et al. 1998. To manage outsourcing risks, a number of approaches is suggested in the literature. For example, essential factors to reckon with in reaching the decision to outsource require that we understand the specific competitive advantage the firm has, and recognize the firm’s resource heterogeneity, the effects of imperfect mobility and its internal alignment. In this context, a firm to manage risks and seek one or several suppliers ought to: (i) Retain the resources responsible for competitive advantage; (ii) Avoid monopolistic or oligopolistic supply markets and (iii) Manage the risk of post-contractual dependency. In implementing the decision to outsource, negotiations relating to supply prices, supply security and assurances, back up and alternative supplies are the issues a firm will be confronted with. Should the firm have one or more suppliers? To what extend can a firm depend on its suppliers? Can contracts negotiated between two firms be reciprocal, in a manner that one will depend reciprocally on the other! What are the penalties for non conformance to contract terms? These are a sample of the many questions one may raise that can have risk implica- tions. For this reason, in car manufacturing supply chains in Japan, several suppliers are used, emphasizing the independent development of parts, integrated into a whole at the Car manufacturer. As a result, outsourcing and external supply relationships are extremely varied with different types of supplier relationship; with different costs and rewards associated in each relationship. They are also varied with relationships designed to meet the supply chain specific needs and spanning “arm’s length”- contractual, con- flictual, limited or full partnership that may be fixed or varying over time. As seen earlier, each relationship entails its own risks of supplying faulty material and products, information asymmetry and power risks (of moral hazard, adverse selection). In such an environment, the risk management of suppliers and outsourcing depends far more on organizational and pro- perly conceived contracts than just technical analysis, albeit such an ana- lysis is important as we shall see below through examples and in the next chapter as well. For example, single sourcing versus multiple sourcing can compound the supplies variation of firms, long term and locked in con- tracts can lead one firm to be totally dependent on the other (although long term contracts are considered important for sustaining a supply chain). Of course, a mutual commitment, a shift form a conflictual to a collaborative based on trade-offs and sharing, maximizing mutual understanding and an exchange of information leading to trustworthy and credible commit- ments are basic ingredients in outsourcing, supplier and supply chain relationships. These problems are of the utmost importance requiring a 414 7 RISK AND SUPPLY CHAINS 7.6 COLLABORATION, RISKS AND SUPPLY CHAINS 415 strategic approach to risk. For simplicity, we often reduce these problems to a treatable format as will be shown in a specific case below. Franchises are an old and broadly practiced economic arrangement, origi- nating in the Middle Ages (X and XII the Centuries) where landed lords granted territorial rights to cultivate land by some in their local population. It expanded dramatically at the beginning of the century in both the US and Europe. The French Cotton firm (Lainiere de Roubaix, Laine Penguin) seeking to sell its textile expanded into 350 franchisees in less than ten years while in the US, Antitrust Laws of 1929 led US firms to the creation of distribution franchises by US car manufacturers. The expansion of franchises, mostly in services, has been since then spectacular, accounting for a substantial percentage of service and logistics activity. In France for example, there were 34 Franchisers in 1970 compared to 600 in 1990 and, of course, this number has expanded since the European Union integration. Franchises are an approach to collaboration between a franchiser—the firm, and franchisees, contracted for the purpose of exploiting a particular concept or advantage provided by the franchiser. It is mostly an economic agreement based on an exchange between parties made for profit, with each of the parties expecting to draw some advantage from the agreement. This general principle underlies franchise contracts, outsourcing agree- ments, joint partnerships etc. Franchises in particular, are essentially a contract between two legally independent firms establishing a long-term relationship where the franchiser grants to the franchisee the right to use the franchiser’s trademark, the use of a specific (potentially patented) tech- nology etc., In exchange, the franchisee pays a lump sum fee and annual royalties at an agreed percentage of sales. A franchise may involve several other provisions as well as options that each of the parties may grant to the other. For example, risk sharing, exclu- sive territories with optional agreement appended to these agreements, promotional efforts sharing, buy-back provisions (Marvel 1982, Rey 1992, Rey and Tirole 1986, Tirole 1988, Mathewson and Winter 1986, Klein and Saft 1885). These contractual relationships are broadly used. Over one third of all retail sales in the US occur through a franchise system. For example, in many cases, production may be centralized while distribution may be franchised (e.g. car selling, some food and department stores, fast food, clothing trademarks etc.). In some cases as well, image and advertis- ing is centralized but production is decentralized, franchised to companies focused in manufacturing (as it is increasingly the case). Franchises The economic rationale for franchises arises due to the very high set up costs in selling as well as to problem of managing complex and diffused distribution systems. Thus, a franchisor may construct a franchising system where franchisees would invest parts, if not all, of the required local investment. Typically, such an agreement is made for definite or indefinite periods of time, which the owner of a protected trademark grants to fran- chisees, for some consideration, the right to operate under this trademark for the purpose of producing or distributing a product or service (Caves and Murphy 1976). Because the value of such assets is defined by their use, these contracts involve difficult contractual relations. Franchisee fees assume then many variations such as royalties, or commission, resale price maintenance, exclusive territories, exclusive dealing as well exclusivity relationships of various sorts with reciprocal agreements for the conduct of mutual services. The study of franchises involves as a result many issues such as resource constraints (thus the franchise will grant access to finan- cial capital, market expertise and managerial talent of franchisee); incentive issues where the franchise system provides strong incentive for both parties to perform well; and of course an economy of scale where the franchiser assumes responsibility for economic activities where economies of scale can be realized. Traditionally, an expected utility framework based on the parties’ utilities for money is used to value franchise contracts (see, for example, Blair and Kaserman 1982, Caves and Murphy 1976, Mathewson and Winter 1986, Rubin 1978, Rubin and Carter, 1990). Such an approach is subjective how- ever expressing the value that each of the parties draws from the agree- ment based on valuations that are no easily revealed. For example, each of the parties may calculate the discounted utility of gains and losses, summa- rized in a “flow of funds”, over a relevant planning horizon. And on the basis of appropriate assumptions regarding the policies and managerial procedures adopted, a pricing “objective” is determined (see Kaufman and Dant 2001, Lafontaine 1992, Kaufman and Lafontaine 1994, Sen 1993). This price is not the market price for the franchise agreement and does not convey the true discount rate (which is both time and risk sensitive). In addition franchising risk is imbedded in both the franchise contract and the ex-post controls applied to manage the franchisee-franchisor rela- tionships once the contract is signed. For example, a typical franchise con- tract consists of a lump sum payment which may or may not be refundable and involves optional choices just as the relationship maintained over at least a certain length of time, at which the franchise can be renegotiated (as a way to commit the franchisee to entrepreneurial activity and safeguard from misuse of the franchise). Similarly, an advantage (or disadvantage) can be gotten through a tax on current inputs, such as selling current input 416 7 RISK AND SUPPLY CHAINS 7.6 COLLABORATION, RISKS AND SUPPLY CHAINS 417 at prices larger than the franchisor’s marginal cost. For some contracts the franchisor supply parts of the fixed operating costs (when he leases land that he owns) combined (or not) with provisions to recapture the franchise (which alters the franchisee utility). Thus, even with the most stringent contract, franchises are subject to many risks. Risks of “milking” the fran- chise; asymmetry risks (in power and in information) and other risks resulting in sub-performing franchisees can harm the franchise brand as a whole. We will next consider a number of simple problems to highlight only some of these issues. Below we shall consider two problems demonstrating alternative app- roaches to dealing with risk in both outsourcing and in franchises. The former is a straightforward expected minimization problem, while the latter provides an approach to pricing the franchise. We consider first a problem of “inventory outsourcing” (Tapiero and Grando 2006) with the supplier a leader, having full information of the outsourcing firm’s demand distributions and parameters. This leads as we saw earlier to a Stackelberg game meaning that one of the parties in the game is a leader, aware of the other party—the follower, his motivations and his decisions. When dealing with an independent demand of the parties, the supplier bene- fits from (statistical) risk aggregation. On the other hand, if parties demands are dependent, this may lead to an unwieldy situation which requires that a risk management policy be adopted by the supplier (such as building an aggregate inventory as well as buying call options for further supplies, as shown in Ritchken and Tapiero 1986 and highlighted in the revised Example 2.1 in this chapter). 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Baghana and Cohen 199 8, Janssen and Kok 199 9, Ritchken and Tapiero 198 6, Outsourcing and Risks (ex-ante and ex-post) in this case include the outsourcing of critical inven- Tapiero and Grando. these (see also Jackwerth 408 7 RISK AND SUPPLY CHAINS 7.5 SELECTED CASES AND PROBLEMS 4 09 199 9, 2000, Ait-Sahalia .and Lo 199 8, 2000) is given by (as seen and proved earlier): (.) (.) ln (.). collaboration in supply chains is not a “free lunch”. Celebrated cases are of course outsourcing and franchises. 412 7 RISK AND SUPPLY CHAINS 7.6 COLLABORATION, RISKS AND SUPPLY CHAINS 413 Outsourcing