Operational asset replacement strategy: A real options approach

European Journal of Operational Research 210 (2011) 318–325 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor Decision Support Operational asset replacement strategy: A real options approach João Zambujal-Oliveira a,⇑, João Duque b,1 a Department of Engineering and Management (DEG), Centre for Management Studies (CEG-IST), Instituto Superior Técnico (IST), Technical University of Lisbon (UTL), Campus Alameda, Av. Rovisco Pais, 1,1049-001 Lisboa, Portugal b Department of Management, Advance (ISEG), Instituto Superior de Economia e Gestão (ISEG), Technical University of Lisbon (UTL), Rua Miguel Lupi, 20, 1249-078 Lisboa, Portugal a r t i c l e i n f o Article history: Received 27 January 2009 Accepted 8 September 2010 Available online 16 September 2010 Keywords: Replacement Real options Uncertainty Equivalent annual cost First passage time a b s t r a c t This paper analyses the problem of replacement by investigating the optimal moment of investment replacement in a given tax environment with a given depreciation policy. An operation and maintenance cost minimization model, based on the definition of equivalent annual cost, is applied to a real options paradigm. The developed methodology allows for an innovative evaluation of the flexibility of replacement process analysis. A new two-factor evaluation function is introduced to quantify decisions on asset replacement under a unique cycle environment. This study improves upon previous findings in the literature as it accounts for autonomous salvage value processes. Based on partial differential equations, this model achieves a general analytical solution and particular numerical solution. The results differ significantly from those observed in one-factor models by showing evidence of over-evaluation in optimal levels of replacement, and by confirming suspicions that different types of uncertainties produce nonmonotonous effects on the optimal replacement level. The scientific contribution of this study lies in new and stronger approaches to equivalent annual cost literature, supplying an algorithm for operation and maintenance cost minimization that is conditioned by autonomous salvage value. This study also contributes to the real options literature by developing a two-factor model with Brownian processes applied to asset replacement. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction The traditional analysis for comparing assets of different life spans and selecting the optimal replacement level makes use of the equivalent annual cost (EAC). This methodology implies the calculation of a cost stream for current and alternative assets. The method may also consider a depreciation tax shelter, a cost of postponing replacement and an asset replacement. Assuming an infinite time horizon, the replacement cycle will correspond to the minimum cost. Among other assumptions, the traditional methodology assumes a similar Operation and Maintenance Costs (OMC) structure for future replacement assets, a known salvage value, and certainty in tax policy. One of the major problems of using the evaluation method to make a replacement decision is the failure to consider uncertainty, implicitly or explicitly. In deterministic analysis, there is no uncertainty about the salvage value and timing to replace the equipment. Thus, there is a crucial insufficiency of the discounted-cash-flow (DCF) approaches to capital budgeting because they cannot properly evaluate management flexibility to adjust or revise investment decisions. The ⇑ Corresponding author. Tel.: +351 218 417 981; fax: +351 218 417 979. E-mail addresses: j.zambujal.oliveira@ist.utl.pt (J. Zambujal-Oliveira), jduque@ iseg.utl.pt (J. Duque). 1 Tel.: +351 213 925 800; fax: +351 213 922 808. 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.09.011 conventional DCF approaches make implicit assumptions regarding an expected structure of cash flows and assume management loyalty to a precise operating strategy. However, real world and competitive interactions originate cash flows structures different from what managers initially estimated. As new information appears and uncertainty about future cash flows is gradually resolved, managers may decide to exchange one asset for another at several stages of the asset life. Disregarding the uncertainty in the OMC can yield very odd results during the budgeting control phase. The contribution of this model is to provide an approximation of reality and therefore an accurate timing to concretize the replacement equipment investment, diminishing the deviations from the budget (Clark and Rousseau, 2002). Thus, the consideration of uncertainty in the model produces benefits over the minimization of the operating and maintenance costs. This paper examines the implications for the replacement process of not taking a deterministic salvage value. This assumption creates additional significant replacement opportunities. This paper has enriched the real options literature with a relatively simple technique to derive a set of solutions in a problem with two variables. This modeling may serve to deal with any similar problems. Moreover, markets are increasingly becoming susceptible to the phenomenon of reusing durable real assets. There are several examples of markets for second-hand assets, such as vehicles, buildings, production equipment, aircraft and ships. The most J. Zambujal-Oliveira, J. Duque / European Journal of Operational Research 210 (2011) 318–325 studied aspects in these markets are the residual resale value, liquidity, the level of information and market specificity. Studies from Shleifer and Vishny (1992), Pulvino (1998) and Banerji (2009) constitute illustrations of research on the salvage value and its relationship with market progress. Disregarding the uncertainty in the costs can lead to very biased results in the process of budget development, producing related deviations in the execution of planned activities (Ierapetritou and Zukuim, 2009). To address these problems, Rust (1985) suggests that higher asset deterioration indicates a greater OMC value. Arboleda and Abraham (2006) confirm that higher deterioration rates may increase the maintenance costs due to the incremental frequency of the preservation routines. Rust described the OMC evolution as an arithmetic Brownian motion with constant drift and constant volatility. However, there is evidence that shows that this motion is not the most appropriate to model OMC. Ye (1990) continues this analysis of the replacement problem by considering the OMC as an Itô process (the main difference between a simple Brownian motion and an Itô process is that the drift and variance coefficients of the process became functions of the current state and time). This approach implies that OMC returns to an initial state, each time a replacement occurs. So, Ye assumes that all replacement assets are stochastically comparable in that they all have the same initial OMC and goes one step further in assuming a new concept of physical deterioration that increases stochastically and influences decidedly the model of Mauer and Ott (1995), through the introduction of a geometric Brownian motion (GBM) to modulate OMC. This action bypasses a shortage of the arithmetic Brownian motion, because it may give negatives values, which is impossible for OMC. Yilmaz (2001) confirms that increasing uncertainty and decreasing life expectancy can increase the optimal stopping barrier and Lai et al. (2000) present a case study about practices of replacement of engines. Martzoukos and Trigeorgis (2002) also observe the conditions required to increase both the tax revenue and incentives by using depreciation policy. Arkin and Slastnikov (2007) enhance this last work, studying the effect of depreciation allowances on the investment timing and government tax. Myers and Majd (1990) and McDonald and Siegel (1986) study the option to abandon a project in exchange for its salvage value or best alternative use. The abandonment option problem is similar to the timing option one, which can be evaluated with an adequate interpretation analysis of variables. Other real options, such as to switch inputs or outputs in production, can be evaluated in the same way. The main obstacle to such real option applications emerges from the potential incapacity to determine the asset values. Sometimes, these values can be recovered from models that use observable prices as inputs, such as the model of Brennan and Schwartz (1985) that prices a mine with information provided by the gold futures markets. The next section describes in detail the two-factor replacement model. Section 3 shows a numerical case study, which permits validation of the theoretical model. Section 4 checks the model’s robustness through a sensitivity analysis of some model parameters, and Section 5 concludes the paper, presenting some conclusions about the outcomes and explaining the contributions. 2. A two-factor replacement model The theoretical framework of the paper considers a method of asset replacement by extending previous models, described in the previous section, to an environment with stochastic salvage value. Therefore, this paper considers an uncertain salvage value (estimated value of an asset at the end of its useful life, also called abandonment value) as its OMC. This means that when an asset replacement occurs, the salvage value can be different from 319 previous estimates. The modulation of the salvage value as a geometric Brownian motion (GBM) will produce a different relationship between OMC (C) and salvage value (S), from which a new optimal replacement level results. Our model considers a firm operating at a fixed level of output with two geometric Brownian motions, one for C and S another for: dc ¼ aC Cdt þ rC CdzC ; ð1Þ ds ¼ aS Sdt þ rS SdzS ; ð2Þ with instantaneous drifts (average increase per unit of time) aC P 0, aS 6 0 and instantaneous volatilities (standard deviation) rC P 0, rS P 0. This model also assumes two stochastic equivalent assets for which the initial OMC CN P 0 and the initial salvage value2 SN > 0 change according to Eqs. (1) and (2). The values established CN for SN and should be collected through econometric analysis of historical data and should prevent an infinite extension or a value very close to zero for the exchange period. This section examines how the change of the salvage value framework affects the optimal replacement level. Other assumptions of this model are: there is a single asset at a given time (the decision to replace is equivalent to deciding when to exercise an exchange option3 and in that case, the replacement decision can be seen as problem of option valuation), and production does not expand or contract (in order to guarantee that OMC are independent from production updates). Therefore, when the OMC reach a certain level, the current asset sale occurs and another stochastically equivalent asset replaces it. Thus, we have a cost minimization problem to determine the optimal replacement level using a two-factor model. With l being the risk-adjusted discount rate, V(Ct, t) is the function that determines the value of the replacement trigger variable, the expected present value of a stream of current and future operating costs, corresponding to the expected discounted OMC: 1 Z VðC t ; tÞ ¼ min E Ct 0  ðC t ð1  sÞ  sda #ðtÞÞelt dt : ð3Þ From the previous expression, the cost flow results from subtracting after-tax OMC Ct(1  s) from the tax shield4 sda#(t). By hypothesis, Ct corresponds to OMC as described in (3), s is the tax rate, da corresponds to the depreciation rate and #(t) indicates the book value given by: a #ðtÞ ¼ Pð1  uÞed t ; ð4Þ where P is the acquisition price and u is the investment tax credit rate. We can see from (3) that there is a functional dependence of V() on both Ct and t. As we have two variables (Ct,t), problem simplification justifies the adoption of an infinite horizon time framework, relaxing V() from the dependence of the calendar date t. In this setting, the function V() is common to all periods, although it will be evaluated at different points. In the two-factor model, the behaviour of C* with changes in the depreciation rate depends on two effects. Considering an initial cost CN, the effect of tax savings, the depreciation costs can be defined as: C d ¼ Pda ð1  uÞ C CN  dZa 5 ð5Þ and the effect of converting this cost into a perpetuity through the opportunity cost is: 2 Initial depreciation level. An exchange option corresponds to the right to swap one asset for another under defined conditions. 4 A tax shield is the reduction in income taxes that results from taking an allowable deduction from taxable income. 5 Risk-adjusted drift rate:Z = aC  1/2rC. 3 J. Zambujal-Oliveira, J. Duque / European Journal of Operational Research 210 (2011) 318–325 320  a  a   a  d 1 d d r d ¼ r f  aC  r2C :   1  2 Z Z Z ð6Þ Let Exc(C, S) denote the value of an exchange option as a function of the current OMC (C) and the current salvage value S. Formally, the exchange option is given by the following expression, ExcðC; SÞ ¼ VðCÞ  XðC; SÞ; where function V(C) represents the present value of the OMC and function X(C, S) symbolizes the present value of the exchange costs. These costs include the present value of the OMC and the acquisition price of the new asset, and net salvage value of the current asset. The right to acquire an asset at the strike price of selling the other asset describes a long position on an exchange option. The same exchange option can be seen as the right to sell an asset at the strike price of buying the other asset. Assuming the distribution of risks associated with OMC by financial assets and using the contingent claims approach, the exchange option Exc(C, S) must satisfy the following equation (Merton, 1973):  1 ExcCC r2C C 2 þ 2ExcCS rC rC rS CSqCS þ ExcSS r2S S2 2 þ ðr f  dC ÞExcC C þ ðr f  dS ÞExcS S ¼ rf Exc ExcðC; SÞ ¼ C rS rC !ka2a S ½kb C x1 þ kc C x2 ; ð7Þ where ka, kb, and kc are constants, and x1 and x2 represent the roots of a quadratic equation derived in the Appendix and given in (29). To determine the solution to the replacement problem, we must calculate three constants ki, i 2 [a, b, c] and a replacement critical level (C*, S*). In order to achieve this, Eq. (8) must satisfy five boundary conditions. In order to determine (C*, S*), the following discriminatory boundary conditions should be applied in planning the trigger levels for OMC and depreciation. The first one implies its satisfaction by Exc(C*, S*) upon the replacement level. ð9Þ where VðC  Þ ¼ !  1s ðC N Þn Pda sð1  uÞ   ðC  Þn C r f  aC r f  aC  12 ðn  1Þnr2C  ð10Þ and _ XðC  ; S Þ ¼ VðC N Þ þ Pð1  uÞ  ½S  sðS  #ðC  ÞÞ: ES ðC  ; S Þ ¼ V S ðC  Þ  XS ðC  ; S Þ; ð11Þ At the critical level (C*, S*), the exchange option value Exc(C*, S*) must equal the difference between the expected discounted value of after-tax OMC and the total alternative cost value function X(C*, S*). The value of X(C*, S*) reflects the sum of the expected discounted value of after-tax OMC, in the instant after replacement, with the net acquisition price of an alternative asset P(1  u) minus the after-tax salvage value (salvage value S* minus capital _ gains tax sðS  # ðC  ÞÞ). Eqs. (12) and (13) must ensure that the smooth past condition is satisfied (Dixit and Pindyck, 1994). In conjunction with other conditions, these boundary conditions permit the determination of ð12Þ ð13Þ Condition (14) describes the behaviour of function Exc(C, S) when the OMC approach the minimal allowed value CN. Thus, when C assumes values next to CN, the probability that C grows until C* is very low, significantly diminishing the probability of an asset replacement occurring. As such, the value of the exchange option Exc(C, S) will tend towards zero: lim ExcðC; SÞ ¼ 0: C>C N ð14Þ When OMC become very high relative to the salvage value S, the increase in value of the exchange option should equal the savings gained between the OMC of the current asset and the OMC of the alternative asset: V C ðCÞ  V C ðC N Þ; resulting in the following condition: lim ExcC ðC; SÞ ¼ V C ðCÞ  V C ðC N Þ: ð8Þ ExcðC  ; S Þ ¼ VðC  Þ  XðC  ; S Þ; EC ðC  ; S Þ ¼ V C ðC  Þ  XC ðC  ; S Þ; C>1 with the risk-adjusted drift rate of OMC aC ¼ rf  dC , the risk-adjusted drift rate of salvage value aS ¼ rf  dS , and the risk-free rate of interest rf. The convenience yields of each stochastic variable are represented by dC and dS. From Eq. (7) and according to the Appendix, the general solution outcomes are: S three constants, which exist in (8). Therefore, the function in Eq. (8) must satisfy the following equations: ð15Þ When C* goes up, the salvage value must go down to make the replacement process economically viable. When salvage value is a function of the OMC and the salvage value adjustment is immediate, the replacement process just needs to examine the level of OMC. Otherwise, the replacement process needs to wait for an appropriate salvage value to consummate the replacement. So, as time goes by, OMC must go up in order to justify the capital cost originating from the asset replacement. As a base to our solution, we revised, corrected and prepared for this model the numerical case belonging to Mauer and Ott (1995). 3. Description of the numerical case Before continuing with our solution through modeling, the numerical case, designed to test critical asset replacement solutions, should be described. As the results of this numerical case are going to serve as a comparison for the new model, all previous parameter values, with the exception of aC, have been accepted from Mauer and Ott (1995). Relative to aC, we initially consider a growth rate aC = 0.15 and a volatility rC = 0.10. As the value of aC is greater than the discount rate rf = 0.07, by Gordon’s Model, we consider aC = 0.06 (see Table 1). Mun (2003) presents a case with aC = 0.10 and rC = 0.35, where rC/aC = 3.5. For aC = 0.06, we obtain rC/aC = 1.66, which is a satisfactory of Mun’s previous ratio. Real options valuation (contingent claims approach) assumes complete markets and substitutes the real drift by a risk-neutral drift. This process is equivalent to deduct the risk premium from the real drift. For this purpose, we use the Sharpe ratio as a proxy to the price of risk for making the risk adjustment. Bernstein and Damodaran (1998) and Hull (1993) describe this concept as the premium demanded by the market to compensate for each unit of risk. Taking the total risk premium equal to gmrCqCm, the adjusted growth rate value aC will be: aC ¼ aC  gm rC qCm : ð16Þ We estimate the market risk price using a market index return rate as an evaluation pattern. Therefore, the market risk price comes from the ratio between the market risk premium lm  rf and the market standard deviation rm: gm ¼ lm  r f : rm ð17Þ J. Zambujal-Oliveira, J. Duque / European Journal of Operational Research 210 (2011) 318–325 Table 1 Set of parameters and values of the numerical case. Parameter Symbol Value Risk-free interest rate Cost drift Volatility of cost Salvage value drift Volatility of salvage value Market risk price Minimal cost Acquisition price Investment tax credit rate Tax rate Depreciation rate rf 0.07 0.06 0.10 0.06 0.10 0.4 1 10 0 0.30 0.50 aC rC aS rS gm CN P u s da Table 2 Description of the depreciation rate for annual periods. Period 1 2 3 4 5 Depreciation rate 39.35% 23.87% 14.47% 8.78% 5.33% According to Ibbotson Associates (2006), the market risk premium for this case is (lm  rf) = 0.08 with a volatility of rm = 0.02. These values result in a market risk price of gm = 0.40. A lack of correlation between the OMC and the systematic factor of evaluation, which produces an adjusted growth rate, aC with an annual value of 0.06, is assumed. Concerning the new asset characteristics, an acquisition price P = 10 and an OMC initial value CN = 1 are also specified. Thus, V(CN) corresponds to the after-tax value of the cost to replace a stochastic asset. In respect to tax parameters, the numerical case includes a credit investment rate u, whose value represents the possibility of reinvestment of the amount resulting from an asset sale. The numerical case also defines an initial value u = 0, a tax rate s = 0.30, and a depreciation rate da = 0.50. The depreciation method follows a negative exponential function. In discrete terms, exponential depreciation corresponds to a regimen similar to the one described in Table 2. 4. Characteristics of the solution and sensitivity analysis The OMC critical level is endogenous and results from Eq. (8) in conjunction with the applied boundary conditions defined in the previous section. The numerical simulation obtained from the actual model produced the following percentage changes. From Table 3, we observe that a substantial critical level adjustment is possible (243.0%). This could be due to the introduction of 321 decreasing dynamics for salvage value S could motivate an anticipation of asset replacement. The percentage change in C* and S* indicate that in the early stages of the replacement process, the level of salvage value may have a higher relative importance than OMC. A more detailed observation highlights an even larger variation in the replacement period, which results from the application of the critical level to a first passage time distribution. These results seem to confirm the intuition that the introduction of a two-factor function would induce strong variations in the cost replacement critical level, confirming some weaknesses in the previous model. These indications lead us to conduct a comparative analysis based on behavioural standards and to conduct an analysis of the impact of variations of each parameter in the determination of the optimal replacement policy. Henceforth, we examine the impact of changes in parameter values on the replacement model by analyzing replacement boundary values and optimal replacement periods for different states of nature. In this way, we can define a set of panels to isolate the effect of varying each parameter, and verify the critical level sensitivity, associated with each parameter value variation. Our analysis begins with the observation of the impact of varying parameters which comprise the salvage value S, described in Eq. (2). Table 4 shows the critical level and critical period updates resulting from the variation of drift rate aS. Elevating the average rate of decrease aS produces two positive effects: a significant increase of the critical OMC level and a reduction in the exercise period. The intuitive explanation for these effects resides in the more distant intersection point associated with a flatter slope. The effect of changing the volatility of salvage value rS (shown in Table 5 at 0.05 intervals) will depend on its position relative to the volatility of cost rC. If rC > rS, the critical level should go down and if rC 6 rS the critical level should go up. From Table 5, we verify an ascent of optimal cost C* coincident with a rise in replacement timing T*. As expected, the introduction of rS does not significantly modify the function of the replacement model but induces a lower replacement critical level. Table 5 also shows that the simple consideration rS of results in a 1.1% decrease in a new critical C*, compared to the revised numerical case. The reason for this behaviour seems to reside in the evidence that less volatile markets create fewer investment opportunities, originating from economic savings from asset replacement processes. As our horizon of analysis is only the next replacement event, the variation in volatility produces rS small effects on the initial replacement value VN. As Dobbs (2002) and Dixit (1989) suggest, variations in volatility intervene with the value of the asset exchange option. In this case, the exchange option becomes influenced not only by rC but Table 3 Percentage change of the critical values to the model of Mauer and Ott (1995). Mod. C* DC* (%) S* DS* (%) T* D T* (%) V* DV* (%) Vn DVn (%) R(M&O) Actual 2.26 1.14 17.3 58.4 3.53 6.00 20.9 105.2 13.20 1.08 97.0 83.8 23.032 82.184 3.9 243.0 15.507 78.325 32.1 243.0 Mod: R(M&O) – revised Mauer and Ott (1995); C*: critical level of costs; S*: critical level of salvage value; T*: expected time; V*: project value at critical level; Vn: project value at initial level. Table 4 Effect of increasing the salvage value growth rate (as). aS DaS (%) C* DC* (%) S* DS* (%) T* DT* (%) V* DV* (%) Vn DVn (%) 0.09 0.06 0.03 50.0 0.0 50.0 1.079 1.137 1.157 5.1 0.0 1.7 6.501 5.999 5.218 8.4 0.0 13.0 0.940 1.085 1.328 13.3 0.0 22.4 90.290 82.184 83.189 9.9 0.0 1.2 83.254 78.325 78.325 6.3 0.0 0.0 aS: salvage value drift; C*: critical level of costs; S*: critical level of salvage value; T*: expected time; V*: project value at critical level; Vn: project value at initial level. J. Zambujal-Oliveira, J. Duque / European Journal of Operational Research 210 (2011) 318–325 322 Table 5 Effect of changing the standard deviation of salvage value (rs). rS DrS (%) C* DC* (%) S* DS* (%) T* DT* (%) V* DV* (%) Vn DVn (%) 0.05 0.10 0.15 50.0 0.0 50.0 1.125 1.137 1.192 1.1 0.0 4.8 6.568 5.999 5.623 9.5 0.0 6.3 0.940 1.085 1.192 13.4 0.0 9.8 81.599 82.184 85.104 0.7 0.0 3.6 78.325 78.325 78.325 0.0 0.0 0.0 rS: salvage value volatility; C*: critical level of costs; S*: critical level of salvage value; T*: expected time; V*: project value at critical level; Vn: project value at initial level. Table 6 Effect of changing the acquisition price (P). P DP (%) C* DC* (%) S* DS* (%) T* DT* (%) V* DV* (%) Vn DVn (%) 5 10 15 50.0 0.0 50.0 1.014 1.137 1.328 10.8 0.0 16.8 5.077 5.999 5.166 15.4 0.0 13.9 0.355 1.085 3.578 67.3 0.0 229.8 78.466 82.184 93.908 4.5 0.0 14.3 76.660 78.325 82.487 2.1 0.0 5.3 P: acquisition price; C*: critical level of costs; S*: critical level of salvage value; T*: expected time; V*: project value at critical level; Vn: project value at initial level. also rS by, whose increase provokes a delay in the moment that the asset exchange option is chosen. Uncertainty in S produces a new optimal boundary where rS and rC work against each other. An increase rS in can induce the decision to replace by increasing the possibility of a future price decline, while an increase in rC induces a choice to keep the asset because future OMC are expected to descend. Thus, delay or advancement of the optimal replacement moment will depend on the combined effect of these two volatilities (Brach, 2002). The next table registers a two-factor function panel with the effect of varying the acquisition price P. In the previous tests, P varying upwards led to an increase in C*(16.8%), establishing a higher level for exercising the replacement option. This panel sets up positive and negative variations (50%) in the acquisition price, using a standard level of P = 10, which results in the following table: Table 6 shows the effect of varying the acquisition price in terms of the critical replacement level and an increase in the discount OMC from growth in the acquisition price P. This critical level behaviour results from the decline in the attractiveness of the alternative asset, resulting from an increase in the cost of the new asset. In the Adkins (2005) replacement model, where the critical revenue is the basis for model functionality, incremental increases in the investment cost have the effect of making the asset less attractive for purposes of exchange. Consequently, as the critical revenue value is a decreasing function of investment cost, the decision to exercise asset replacement will be delayed for lower levels of the exchange option. This analysis seems to contradict Keles and Hartman (2004), who relate the impact of variation in acquisition price to the critical decision of asset replacement. A possible explanation for the conclusions drawn by Keles and Hartman could be a feature of the budgetary restriction, existing in their replacement model. Another parameter that influences appreciably the alternative cost X(C, S) is the tax credit rate u, whose increase produces a reduction of C* because of the reduction of P(1  u) and the increased attractiveness of a new and improved asset cost. Table 7 suggests that an increase in tax credits acts as an incentive for asset exchange, which further suggests two other effects. The first effect is the reduction of the net acquisition price. The second effect is the corresponding decrease in the asset salvage value (from the change in the depreciation base). In functional terms, the increase in u corresponds to a negative variation in the acquisition price P, which is a similar effect to the one previously discussed in the analysis of the acquisition price. It should be noted that a constant proportionality in all parameters might be found, except in salvage value, where the tax credit reduction has an effect approximately four times greater than its rise. This finding brings opportunities for the definition of tax policies for markets of used assets. The tax credit rate u, the tax rate s, and the depreciation rate da constitute the tax vector. While variation in u affects the level of the acquisition price and the depreciation base, the change in s has an impact not only on the tax savings value given by C, but also on the resulting taxation. The growth of tax ratessuggests an increase in the critical level for asset exchange, and consequently, an increase in the critical period. This results from the fact that incremental increases in the tax rate also increase the taxes charged to capital gains received from reduction of the net salvage value. In this situation, the new asset becomes less attractive, and maintenance of the current asset is favoured by the reduction C(1  s) and by the increased contribution of depreciation cost Cd to the reduction in total costs. Table 8 shows an increase in C* resulting from tax rate s growth. This scenario is an outcome of lower cost flows and lower current values and results from the OMC, and net revenue increase. The handling of the tax rate has effects on the proportional level of replacement investment. Ranging by 50%, the investment level will change only by 15%. However, this proportionality does not hold for the optimal time of replacement. This feature allows tax policies, which substantially delay the replacement process, without influencing the salvage value. As Table 9 illustrates, when the depreciation rate da increases (50%), the critical level oscillates around a reference value. Thus, while increases in the depreciation rate up to 0.5 provoke critical level growth, increases in the depreciation rate above 0.5 cause a reduction in the critical level. In the two-factor model, there are various effects. As OMC move away from their initial value, tax savings mitigate them and this modifies the opportunity cost used to discount the net tax depreciation cost. When da < 0.50 there is an Table 7 Effect of changing the tax credit rate (u). u Du (%) C* DC* (%) S* DS* (%) T* DT* (%) V* DV* (%) Vn DVn (%) 0.00 0.05 0.10 100.0 0.0 100.0 1.137 1.112 1.089 2.3 0.0 2.0 5.999 6.302 6.402 4.8 0.0 1.6 1.085 0.791 0.548 37.1 0.0 30.8 82.184 80.852 79.695 1.6 0.0 1.4 78.325 77.908 77.492 0.5 0.0 0.5 @: tax credit rate; C*: critical level of costs; S*: critical level of salvage value; T*: expected time; V*: project value at critical level; Vn: project value at initial level. J. Zambujal-Oliveira, J. Duque / European Journal of Operational Research 210 (2011) 318–325 323 Table 8 Effect of changing the tax rate s. s Ds (%) C* DC* (%) S* DS* (%) T* DT* (%) V* DV* (%) Vn DVn (%) 0.10 0.30 0.50 66.7 0.0 66.7 1.030 1.137 1.386 9.4 0.0 21.9 6.454 5.999 5.998 7.6 0.0 0.0 0.078 1.085 4.333 92.8 0.0 299.4 94.807 82.184 70.033 15.4 0.0 14.8 92.775 78.325 63.874 18.4 0.0 18.4 s tax rate; C*: critical level of costs; S*: critical level of salvage value; T*: expected time; V*: project value at critical level; Vn: project value at initial level. Table 9 Effect of depreciation rate (da) variation. da Dda (%) C* DC* (%) S* DS* (%) T* DT* (%) V* DV* (%) Vn DVn (%) 0.25 0.50 0.75 50.0 0.0 50.0 1.011 1.137 1.078 11.1 0.0 5.2 6.175 5.999 6.225 2.9 0.0 3.8 0.011 1.085 0.434 98.9 0.0 60.0 50.734 82.184 76.531 38.3 0.0 6.9 48.949 78.325 73.014 37.5 0.0 6.8 da: depreciation rate; C*: critical level of costs; S*: critical level of salvage value; T*: expected time; V*: project value at critical level; Vn: project value at initial level. incentive to delay replacement because tax savings produce a reduction in the next replacement cost and a potential increase in capital gains. Otherwise, if da P 0.50, the depreciation rate growth contributes to an erosion of the asset’s taxable base, which accelerates the replacement process. According to Dixit and Pindyck (1994), including the role of depreciation diminishes the investment opportunities of the project. The analogy with the replacement problem is the reduction in the incentive to replace the asset. As changes in depreciation rate influence, relevantly, the investment levels V* but have little impact on the parameters value, tax policies (depreciation- rate-based) permit changes to the investment levels of the markets with slight changes in the equilibrium values. 5. Conclusions Conventional replacement models assume that the salvage value of an asset at any moment will be equal to the present value of the residual cash flow stream. For numerous reasons, the salvage value could be different from the present value of its future cash flow. This paper presented a new methodology for approaching the optimal asset replacement problem, considering a fixed tax regimen applied to a one-cycle problem. It demonstrates how it is possible to evaluate OMC, using a two-model factor. This model incorporates the flexibility of choosing the appropriate salvage value to make an optimal replacement decision. Thus, we analyse the replacement decision in a one-cycle environment where salvage value S is decreasing and follows a GBM. Thus, a new formula for identifying the optimum level of replacement with two uncertainties is available. This formula is even more interesting because it assumes the non-existence of first-degree homogeneity between uncertainties. The numerical case provides some outcomes and demonstrates the ease of use to real options practitioners. Besides, with these different dynamics for salvage value, we collect evidence concerning the anticipation of the asset replacement decision. This evidence confirms the significant influence of salvage value in the evaluation of OMC. The analysis made with the tax vector (tax rate, depreciation rate and tax credit rate) shows interesting properties, providing different tools for policy-makers who need to regulate used assets markets. Each component of the tax vector permits a different intervention in the market (constant investment level with different values of OMC and salvage value or similar uncertainties values with different replacement level). However, the model described in this paper has some minor limitations. The first is that it applies only to the next replacement and not to an infinite chain of replacements. The second is that the tax parameters do not incorporate uncertainty. It is predictable that expanding the model in this manner will drastically influence the critical level of replacement. This is a consequence of the inclusion of a Poisson process and a multi-cycle environment, which involves a consideration of a OMC perpetuity. Appendix A This section uses the Method of Characteristics to find a new system of coordinates and reduce the differential equation to its canonical form. This reduction allows the application of the Method of Separation of Variables (Weinberger, 1995). This application will result in a closed solution on which boundary conditions are applied. Following Polyanin (2001), we begin our analysis with a general form of a second order partial differential equation: aExcxx þ 2bExcxy þ cExcyy þ dExcx þ eExcy þ fExc ¼ g; ð18Þ where a, b, c, d, e, f, g are coefficients of the equation classified as parabolic in the cases where b2  ac = 0. Thus, it is possible to reduce Eq. (7) to its canonical form through the introduction of a new system of coordinates (h, g): Exchh ¼ /ðh; g; Exc; Exch ; Excg Þ; ð19Þ using this equation, we obtain the values for the coefficients of expression (7) a¼ 1 2 2 rC; 2 C b¼ 1 rC rS CSqCS ; 2 and c ¼ 1 2 2 rS; 2 s ð20Þ from which we obtain the following determinant: 2 b  ac ¼ 1 2 2 2 2 2 r r C S qCS  1 : 4 C S Admitting that Eq. (18) is classified as parabolic, we need to change the system coordinates, (C, S) ? (h, g), and it will be necessary to solve the following equation: gC þ rS S g ¼ 0; rC C S @g where gC ¼ @C and gS ¼ @@Sg. The solution is: dS b rS S ¼ ¼ dC a rC C and, rearranging this, we get: J. Zambujal-Oliveira, J. Duque / European Journal of Operational Research 210 (2011) 318–325 324 ds rS dC ¼ ; S rC C q2 Q 00 R rS lnðCÞ þ S0 ; lnðSÞ ¼ rC with from which results: S gðC; SÞ ¼ S0 ¼ rS ð21Þ : C rc For h, we choose a function that intercepts the lines of constants, such as: hðC; SÞ ¼ C: ð22Þ Differentiating the expressions (21) and (22): gC ðC; SÞ ¼  rS S rS rC C rC þ1 hC ðC; SÞ ¼ 1; hS ðC; SÞ ¼ 0; @h for hC ¼ @C and hS ¼ @h . Assuming Exc(C, S) = v(h, g), we calculate: @S ExcS ¼ C rS S ExcSS ¼ ExcCS ¼ 2rS C rC 1 rS rC g Y ¼ aS  n Y ¼ aC ; rC C rC þ1 v hg þ r2S S2 2r S r2C C rC þ2 v gg ; rf QR ¼ 0; ð25Þ aC rS ; rC nn Q þqQ 0 Q n Q r f Q 2 ¼ ka ; where ka is a constant. Thus, the previous expression allows us to obtain the following differential equations: q2 Q 00 v gg ; v hg  g Y where rf corresponds to the risk-free rate of interest. Splitting Eq. (25) into two separate equations, one a function of R and another a function of Q, and 2rS S rS þrQR0 1 2 r; 2 C 2 nn Y g Y ¼0 n Y þqQ 0 ð26Þ   2 Q r f  ka ¼ 0: ð27Þ To find the expression for R, we manipulate (26): rS S rC 2r S þ1 C rC 2 v gg : Just before making the substitution in Eq. (7), we simplify the following expression: 1 1 ExcCC r2C C 2 þ ExcCS rC rS CS þ ExcSS r2S S2 2 2 1 2 ¼ ðExcC rC C þ ExcS rS SÞ : 2   1 2 2 ar rC h v hh ¼ rf v  aS  C S gv g  aC hv h : 2 rC dR ka dr ¼ g ; Q r R 2 lnðRÞ ¼ ka g lnðrÞ þ k1 ; Q k2 a ð23Þ Substituting the new coordinate and in the last equation: ð24Þ To find a general solution, Abell and Braselton (1997) suggest the transformation q = 0 and r = g, producing the function v(h, g) = v(q, r). The solution will result from the product of two functions, each one depending only on one independent variable. This process is known as the Method of Separation of Variables (Weinberger, 1995) and serves to convert a partial differential equation into an ordinary differential equation. Thus, considering: dQ Q0 ¼ dq ¼ ka R  r f R0 v g; 1 C ; rS rC C rC þ1v g ExcCC ¼ v hh  nn Y n Y g Y r f R0 q2 Q 00  ¼ R gS ðC; SÞ ¼ C ; rS rC þqQ 0 R ; rS rC ExcC ¼ v h  nn Y dR and R0 ¼ dr g Q RðrÞ ¼ kr ðrÞ ; where kr is a constant. Proceeding in similar way for Q, we verify the presence of a Cauchi–Euler equation for which the following general solution exists: QðqÞ ¼ k1 q-1 þ k2 q-2 ; ð28Þ where -1;2 ffi pffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffipffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 8a 8ða Þ2 2 r2 ðk2a rf ÞaC 2 ðk2a rf Þr2C 2 ka  rf  12 16  k2 rC þ k2 rC r2 þ k2 rC  2ðk2a rf ÞrC ð a fÞ C ð a fÞ f a pffiffiffi ¼ : 2rC ð29Þ Consequently, the expression v(q, r) = Q(q)R(r) takes the following form: k2 a and v ðq; rÞ ¼ QðqÞRðrÞ; differentiating v(), we obtain: v h ¼ v q ¼ Q 0 R; v g ¼ v r ¼ QR0 ; v hh ¼ v qq ¼ Q 00 R: Applying these expressions to (24), one transformation produces: v ðq; rÞ ¼ Q ðqÞRðrÞ ¼ ðk1 q -1 -2 þ k2 q Þkr r   k2 k2 a a v ðq; rÞ ¼ k1 kr q-1 raS þ k2 kr q-2 raS ; v ðq; rÞ ¼ k1 kr qv ðq; rÞ ¼ kA q- 1 1 k2 a g Q ; k2 a r aS þ k2 kr q-2 r aS ; k2 a k2 a r aS þ kB q-2 r aS : ð30Þ Replacing q and r by the equivalent terms in C and S, we achieve the general solution (8). J. 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