REAL OPTION MODELING FOR VALUING WORKER FLEXIBILITY

Transcription

1 REAL OPTION MODELING FOR VALUING WORKER FLEXIBILITY Harriet Black Nembhar Davi A. Nembhar Ayse P. Gurses Department of Inustrial Engineering University of Wisconsin-Maison 53 University Avenue Maison, WI , U.S.A. Abstract We propose using a real options framework to quantify the financial value of cross training. We moel the investment ecision in cross training as a series of European call options with the same exercise price but with ifferent maturity ates. We use the Black-Scholes formula an the binomial tree approach to fin the value of having the option to cross train. A case stuy shows the application of our propose moel to value the option of cross training in an environment where the price of the prouct is stochastic. We also emonstrate the avantage of using the option valuation metho as oppose to the traitional iscounte cash flow analysis to quantify the value of cross training. Keywors Real Options, Cross Training, Worker Flexibility. Introuction In orer to be able to survive in toay s competitive an volatile environment, many organizations are trying to increase the flexibility of their resources. Several authors have consiere option valuation methos to quantify the flexibility provie by flexible manufacturing systems, flexible prouction technology or other machinery having multiple uses [, 9, 0,, 8]. To aress the nee of organizations to measure the value of flexibility, Nembhar, Shi an Park [7] propose using real option moels for managing system changes. Nembhar, Shi an Aktan [6] evaluate the ecision to introuce quality control charts by using option valuation methos. A real options moel to assess the value of prouct outsourcing given a set of uncertain market conitions has also been evelope [5]. The objective of this paper is to evelop a financial moel that will quantify the value of increase worker flexibility, particularly through cross training which is unertaken to create a multi-skille workforce. We propose using a real options framework, as oppose to a traitional iscounte cash flow analysis, to capture the uncertainty in the market. There are several benefits of cross training programs. In a system where employees are cross-traine, ealines are more likely to be met. Since multi-skille employees can work on some aitional jobs other than their primary job uring their regular working hours, cross training can significantly reuce overtime costs ue to absenteeism. Cross training programs may also have significant costs. The company must pay not only the stanar labor wage uring cross training, but also there may be a cost associate with a loss of prouctive labor. Instructor costs are also involve for many cross training programs [5]. 2. Using Option Moels to Evaluate Investment Decisions in Cross Training Programs An option is the right, but not the obligation to take an action in the future. The holer of the option oes not have to exercise this right [7]. When options are associate with investment opportunities that are not financial instruments, then these options are calle real options. Uner the real option pricing approach, risk-neutral pricing is use as

2 oppose to risk-ajuste iscounting [2]. The two basic types of options are call options an put options. A call option is the right to buy an asset by a certain ate for a preetermine price. A put option is the right to sell an asset by a certain ate for a preetermine price. This preetermine price is calle the exercise price or the strike price [7]. Options can also be categorize as American or European. A European option gives the holer the right to exercise the option only on the expiration ate. An American option gives the holer the right to exercise the option on or before the expiration ate. In this paper, we propose moeling the investment ecision in cross training as a series of European options that expire at ifferent time points. Consier a system with only one prouct type. The price of the prouct is a stochastic variable whereas the output is eterministic. Let S (t) be the sales price for the prouct uring the time interval beginning at time t. Let Q be the number of units prouce uring a specific time interval. Then, total revenue R(t) of the prouct per time interval that begins at time t is R ( t ) S ( t )Q At this point, we make two assumptions. First, no inventory is allowe in the system. Secon, we assume that there is sufficient eman for the prouct so that the company can sell everything it prouces uring a specific time perio. Then the total profit P(t) per time interval that begins at time t can be efine as P( t ) R( t ) WL VQ F where W is the number of workers in the system, L is the wage per labor per time perio, V enotes the variable costs per unit of prouct other than the irect labor cost, an F enotes the fixe prouction cost per time interval. This moel assumes that the number of workers in the system is constant. In other wors, hiring aitional employees on a temporary basis is not an option. In this paper, we attempt to quantify the value of increasing the multifunctionality of the existing workers in an organization by cross training instea of hiring temporary workers. Let g be the percentage increase in the output level ue to investing in cross-training programs an N be the cost of cross training per time interval. Then the profit moels without an with cross training can be efine respectively as P( t ) S( t )Q WL VQ F Without cross training (a) an P( t) S( t)( + g) Q WL VQ( + g) F N. With cross training If we get the ifference between the two profit values in Equations (a) an (b), we obtain : P( t ) gs( t )Q N VQg. We have the right but not the obligation to cross train workers at a specific time perio in the future. Therefore, the minimum value of P(t) is zero. Since the only stochastic variable in the term gs (t)q is S (t), we can relabel the whole term as S(t). The term N+VQg has no stochastic component an therefore, we can think of this term as K, the exercise price of the call option. Then, at the en of a specific time perio (at maturity), the moel will reuce to the payoff structure of a call option, as follows: (2) P( t ) max( S( t ) K, 0 ) We assume that we sell the prouct at the en of each time perio for the price etermine one perio before. We also assume that the cost of cross training is incurre at the beginning of each time perio. Therefore, we obtain our profit at iscrete points in time. The investment ecision in cross training may be thought of as mae at iscrete points in time. In other wors, we ecie on whether to cross train or not only at the beginning of each time perio, not within the time perio. Therefore, we can think of the option to cross train as a European call option, which we can exercise only at the maturity ate T. We nee five variables to etermine the value of a European option. We can make a one-to-one corresponence between these variables an the option to cross train. The stock price, S(t) is represente by the profit ifference. The exercise price is represente by the investment in cross training plus aitional variable cost. The time to maturity, T, is represente by the en of the training horizon. The risk-free interest rate, r, remains the same. The volatility of the stock price, σ, is represente by the volatility of the price of the prouct. (b)

3 3. Option Valuation Moels We use two ifferent methos to value the European call option given in Equation 2: The Black-Scholes formula an the binomial tree approach. The avantage of the Black-Scholes formula is that it gives a precise analytic solution for European options [2]. The binomial tree approach gives an approximate solution that converges to the correct option value as the number of steps in the tree increases [2]. However, it has the avantage of giving a visual layout of the values of the stochastic variable for all jumps at each time interval. 3. Black-Scholes Formula The value of a European call option at time zero (c) on a non-ivien paying stock can be calculate by where c( S,t ) SN( ) Ke r(t t ) N(2 ) ln(s / K ) + (r + σ 2 / 2)(T t ) σ T t 2 σ T t (3a) (3b) an N(x) is the cumulative probability istribution function for a variable that is normally istribute with a mean of zero an a stanar eviation of.0. S is the stock price at time zero, K is the exercise price, r is the constant continuously compoune risk-free interest rate, σ is the volatility of the unerlying asset an T is the time to maturity of an option. In orer to value the option of cross training by the Black-Scholes formula, we nee to replace the variable S in Equations (3a) an (3b) by gs (0)Q. As will be explaine later in etail, we moel the investment ecision in cross training as a package of European options. Each of these options has the same exercise price (K) but a ifferent maturity ate (T). Workers are cross-traine if the aitional profit gaine by cross training is larger than the cost of cross training. 3.2 Binomial Tree Approach Suppose that we have a European call option. If we enote the initial value of the unerlying asset by S, an up move for this variable by u an a own move by, then at the en of one time perio, the value of the unerlying asset will be either us or S, where u>>0. In this case, we can calculate the value of this European call option at time zero by c [ pc er t + ( p ) c ] e r t p u where the payoff from the option is c u if the value of the unerlying asset moves up to Su an the payoff from the option is c if the value of the unerlying asset moves own to S at the en of one time perio. p is known as the t risk-neutral probability of an up movement. We can calculate the u an values by u e σ an /u. It is important to note that p is not the actual probability of an up movement. We can enote the actual probability of an up movement with q. As can be seen from Equations (4a) an (4b), we o not nee to know the value of q in orer to fin the value of an option at time zero. This is because we are using a risk-neutral valuation technique to fin the value of the option at time zero. As can be seen from Equation (4b), uner the risk-neutral pricing approach, the option value is equal to its expecte payoff in a risk-neutral worl iscounte at the risk-free interest rate. t is the length of the time at each step of the binomial tree. In fact, the binomial tree metho is a iscrete approximation to the Black-Scholes continuous time moel. As the length of the time at each step of the tree ( t) ecreases, the value calculate by the binomial metho gets closer to the value obtaine by the Black-Scholes formula. We can value multi-perio options by extening the one-step binomial tree to n steps an working backwar through the tree. In our moel, the initial value of the unerlying stochastic variable is S gs (0)Q. (4a) (4b)

4 4. Using Real Options to Value Worker Flexibility: A Case Stuy We have a manufacturing system proucing only one type of prouct. Suppose that no inventory can be carrie an we can sell every unit we can prouce. Currently we cannot meet the eman completely because we o not have enough capacity. However, we know that if we coul prouce more, our customers woul buy from us instea of our competitors. We are preicting that we can sell approximately 20% more if we have enough capacity. We are currently using a new machine that we bought a few years ago. However, the capacity of that machine is not enough to meet the eman. Before buying this new machine, we were using an ol machine that we still have but not using at all. This ol machine requires ifferent worker skills than the skills our workers currently have. We have the option to cross train our workers so that they can use both the ol machine an the new machine. However, in orer to cross train our workers we nee to make an investment. We want to evaluate this investment alternative using the real options framework. We know that the price of our prouct is stochastic. The price ata for the last 0 months is given in Table 2. Currently, the price of our prouct is $5.68. The thir column shows the logarithm of the increase rate of price for two consecutive months. If we enote ln [S (t+)/s (t)] by m t, then we can calculate the yearly volatility of the price of the prouct as [7] σ 2 ( ) 8 m t m 9 t (5) Table 3 inclues aitional ata for the manufacturing system. Currently, there are 800 workers in the system. The number of units prouce in the system can be increase by 20% if we cross train our workers. One worker can prouce 80 units of the prouct without cross training an 96 units with cross training in one month. The increase in the number of units prouce per worker is mainly ue to the fact that workers have some ile time if only one machine is working. When we put the ol machine into use as well, workers can work on that machine instea of sitting ile. The total numbers of units prouce in the system per month with an without cross training are 64,000 an 76,800, respectively. Variable costs other than the irect labor cost is $2 per unit. The monthly wage of one worker is $300. The cost of cross training is $40,000 an is consiere fixe for the next 2 months. The risk free interest rate is 8% per month. Table 2. Price ata for one prouct t S (t) m t t S (t) m t Average Volatility Table 4. The twelve option values as calculate by the Black-Scholes formula Month Option Value ($) Month Option Value ($) 7,54 7 0, , , , , ,804 0,30 5 9,229,78 6 9, ,23 Total 8,30 Table 3. Data for the manufacturing system Number of workers in the system (W) 800 % increase in the output level ue to cross-training (g) 20% Number of units prouce per worker in one month Without cross training 80 With cross training 96 Total number of units prouce per month Without cross training (Q) 64,000 With cross training (+g)q 76,800 Other variable costs per unit of prouct (V) 2 Aitional variable cost ue to cross training (VQg) $25,600 Wage per labor per month (L) $300 Cost of cross training per time perio (N) $40,000 Exercise price (K N+ VQg) $65,600 Risk-free interest rate (r) 8% The initial value of the unerlying stochastic variable (S) $72,678

5 We can moel this investment problem by using a series of European options. We can assume that the company has a package of twelve European options with ifferent maturity ates. The first option matures in one month, the secon option in two months an so on. In other wors, the company has the option to cross train workers uring a specific month for the next twelve months. Workers can use the ol machine uring a specific month only if they are cross-traine uring that month. Management can ecie on whether to cross train or not epening on the price of the prouct for a specific month. In other wors, management has the flexibility to make a ecision after the uncertainty about the price of the prouct is clarifie. 4. Black-Scholes Formula We value the option of cross training for a one-year time perio. Since management can make a ecision to cross train or not uring each month, the company has twelve ifferent European options with ifferent maturities. We can use the Black-Scholes formula to calculate the value of each option. The value of the option with twelve months of maturity can be calculate as ln($72678 / $65600) + ( / 2 )( 0 ) Note that the initial value of the unerlying stochastic variable S$72,678. The values of the other eleven options can be calculate in a similar manner (Table 4). The total value of having the option to cross train for twelve months is $8,30. This value can be foun by summing each of the iniviual option values in the package. 4.2 Binomial Tree c( S, t) $72,678 N( ) $65,600 e 0.08*( 0) N(3.03) $2,23. At each step of the binomial tree, the value of the unerlying stochastic variable can either increase by u or ecrease by. The u an values can be calculate respectively by u e σ t e / an u. The risk-neutral probability of an up move (p) can be calculate using Equation (4b): er t p u e0.08/ (6) We can now use Equation (4a) to calculate the value of the option by going backwar on the tree one step at a time until we reach time zero. For example, the value of the option at time zero can be calculate by a one-step backwar calculation from the binomial lattice as follows: The option to cross-train shoul be exercise only if the payoff from the option at month twelve is positive. The value of the European call option with a maturity of twelve months is equal to $2,22. The values of the remaining options with ifferent maturities can be foun by constructing a ifferent binomial tree for each option an working backwars through each tree in a similar fashion. The twelve option values [ 0.69* * 0478] $2, 22 e 0.08/ 2 Table 5. The twelve option values calculate by the binomial tree metho Month Option Value ($) Month Option Value ($) 7,54 7 0, , , , , ,804 0, ,229,77 6 9, ,22 Total 8,28 calculate by using the binomial tree metho are given in Table 5. The total value of having the option to cross train for twelve months is foun to be $8,28. When the values in Table 5 are compare with the values in Table 4, we can see that the values obtaine from binomial lattice approach are very close to the values obtaine from the Black- Scholes formula.

6 5. Conclusions In this paper, we propose using a real options framework to quantify the value of increase workforce flexibility through investing in cross training programs. We use a real options framework, as oppose to a traitional iscounte cash flow analysis because of two main reasons. First, in our problem, we have the right but not the obligation to make an investment in cross training program uring each perio. Since our problem has an asymmetric payoff structure, using the traitional iscounte cash flow analysis may lea us to erroneous ecisions. The secon reason is that investments in the cross training programs shoul be contingent upon the ynamics of the market conitions. The real options framework helps us to consier the market ynamics in our ecisions by capturing the uncertainty associate with variables such as price an eman. We moele the investment in cross training as a package of European call options, with each option having the same exercise price (K) but a ifferent maturity ate (T). The aitional profit gaine ue to cross training uring a specific perio correspons to the stock price in a financial option. The only variable in our option moel is the price of the prouct. Since our moel inclues a series of European call options with only one variable, we use the Black-Scholes formula an the binomial tree metho to calculate the value of each option. The Black-Scholes formula gives us a close form solution whereas the binomial metho has the avantage of visually laying out possible future profit gains ue to cross training. If the option value for a specific perio is foun to be positive, then management shoul invest in cross training for that perio. References. Aggarwal, R., 99, Justifying Investments in Flexible Manufacturing Technology, Managerial Finance Black, F., an Scholes, M., 973, The Pricing of Options an Corporate Liabilities, Journal of Political Economy, 8, Campbell, G. M., 999, Cross-Utilization of Workers Whose Capabilities Differ, Management Science, 45(5), Dijkstra, M. C., Krron L. G., Van Nunen, J. A. E. E. an Salomon, M., 99, A DSS for Capacity Planning of Aircraft Maintenance Personnel. appears in Prouction Economics: Issues an Challenges of the 90s, Grubbstom, R. W., Hinterhuber, H. H. an Lunquist J. (es), (Amsteram: Elsevier Science). 5. Ebeling, A. C., an Lee C. Y., 994, Cross-training Effectiveness an Profitability, International Journal of Prouction Research, 32(2), Gerwin, D., 993, Manufacturing Flexibility: A Strategic Perspective, Management Science, 39(4), Hull, J., 2000, Options, Futures, an Other Derivatives. Fourth Eition, Prentice Hall, Upper Sale River, N.J. 8. Hoer, J., Mello, A. S., an Sick, G., 200, Valuing Real Options: Can Risk-Ajuste Discounting Be Mae to Work? Journal of Applie Corporate Finance, 492, Kamra, B. an Ernst, R., 995, Multi-prouct Manufacturing with Stochastic Input Process an Output Yiel Uncertainty, appears in Real Options in Capital Investment: New Contributions, Trigeorgis, L. (e.), New York, Praeger. 0. Kulatilaka, N., 988, Valuing the Flexibility of Flexible Manufacturing Systems, IEEE Transactions in Engineering Management, Kulatilaka, N., an Trigeorgis, L., The General Flexibility to Switch: Real Options Revisite, International Journal of Finance, Luenberger, D. G., 998, Investment Science. Oxfor, Oxfor University Press. 3. Molleman, E., an Slomp, J., 999, Functional Flexibility an Team Performance, International Journal of Prouction Research, 37(8), Molleman, E., an Van Knippenberg, A., 995, Work Reesign an the Balance of Control Within a Nursing Context, Human Relations, 48, Nembhar, H. B., Shi, L., an Aktan, M., 200, A Real Options Design for Prouct Outsourcing Proceeings of the 200 Winter Simulation Conference. 6. Nembhar, H. B., Shi, L., an Aktan, M., 200, A Real Options Design for Quality Control Charts, The Engineering Economist. 7. Nembhar, H. B., Shi, L., an Park, C., 2000, Real Option Moels for Managing Manufacturing System Changes in the New Economy, The Engineering Economist, 45(3),

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